main.dart

import 'dart:async';

import 'package:flutter/material.dart';

void main() => runApp(new MyApp());

class MyApp extends StatelessWidget {
  @override
  Widget build(BuildContext context) {
    return new MaterialApp(
      title: 'Flutter Demo',
      theme: new ThemeData(
        primarySwatch: Colors.blue,
      ),
      home: new MyHomePage(title: 'Flutter Demo Home Page'),
    );
  }
}

class MyHomePage extends StatefulWidget {
  MyHomePage({Key key, this.title}) : super(key: key);
  final String title;

  @override
  _MyHomePageState createState() => new _MyHomePageState();
}

enum ScoreWidgetStatus {
  HIDDEN,
  BECOMING_VISIBLE,
  VISIBLE,
  BECOMING_INVISIBLE
}

class _MyHomePageState extends State<MyHomePage> with TickerProviderStateMixin {
  int _counter = 0;
  ScoreWidgetStatus _scoreWidgetStatus = ScoreWidgetStatus.HIDDEN;
  final duration = new Duration(milliseconds: 300);
  final oneSecond = new Duration(seconds: 1);
  Timer holdTimer, scoreOutETA;
  AnimationController scoreInAnimationController, scoreOutAnimationController,
      scoreSizeAnimationController;
  Animation scoreOutPositionAnimation;

  initState() {
    super.initState();
    scoreInAnimationController = new AnimationController(duration: new Duration(milliseconds: 150), vsync: this);
    scoreInAnimationController.addListener((){
      setState(() {}); // Calls render function
    });

    scoreOutAnimationController = new AnimationController(vsync: this, duration: duration);
    scoreOutPositionAnimation = new Tween(begin: 100.0, end: 150.0).animate(
      new CurvedAnimation(parent: scoreOutAnimationController, curve: Curves.easeOut)
    );
    scoreOutPositionAnimation.addListener((){
      setState(() {});
    });
    scoreOutAnimationController.addStatusListener((status) {
      if (status == AnimationStatus.completed) {
        _scoreWidgetStatus = ScoreWidgetStatus.HIDDEN;
      }
    });

    scoreSizeAnimationController = new AnimationController(vsync: this, duration: new Duration(milliseconds: 150));
    scoreSizeAnimationController.addStatusListener((status) {
      if(status == AnimationStatus.completed) {
        scoreSizeAnimationController.reverse();
      }
    });
    scoreSizeAnimationController.addListener((){
      setState(() {});
    });
  }

  dispose() {
   super.dispose();
   scoreInAnimationController.dispose();
   scoreOutAnimationController.dispose();
  }

  void increment(Timer t) {
    scoreSizeAnimationController.forward(from: 0.0);
    setState(() {
      _counter++;
    });
  }

  void onTapDown(TapDownDetails tap) {
    // User pressed the button. This can be a tap or a hold.
    if (scoreOutETA != null) {
      scoreOutETA.cancel(); // We do not want the score to vanish!
    }
    if(_scoreWidgetStatus == ScoreWidgetStatus.BECOMING_INVISIBLE) {
      // We tapped down while the widget was flying up. Need to cancel that animation.
      scoreOutAnimationController.stop(canceled: true);
      _scoreWidgetStatus = ScoreWidgetStatus.VISIBLE;
    }
    else if (_scoreWidgetStatus == ScoreWidgetStatus.HIDDEN ) {
        _scoreWidgetStatus = ScoreWidgetStatus.BECOMING_VISIBLE;
        scoreInAnimationController.forward(from: 0.0);
    }
    increment(null); // Take care of tap
    holdTimer = new Timer.periodic(duration, increment); // Takes care of hold
  }

  void onTapUp(TapUpDetails tap) {
    // User removed his finger from button.
    scoreOutETA = new Timer(oneSecond, () {
      scoreOutAnimationController.forward(from: 0.0);
      _scoreWidgetStatus = ScoreWidgetStatus.BECOMING_INVISIBLE;
    });
    holdTimer.cancel();
  }

  Widget getScoreButton() {
    var scorePosition = 0.0;
    var scoreOpacity = 0.0;
    var extraSize = 0.0;
    switch(_scoreWidgetStatus) {
      case ScoreWidgetStatus.HIDDEN:
        break;
      case ScoreWidgetStatus.BECOMING_VISIBLE :
      case ScoreWidgetStatus.VISIBLE:
        scorePosition = scoreInAnimationController.value * 100;
        scoreOpacity = scoreInAnimationController.value;
        extraSize = scoreSizeAnimationController.value * 10;
        break;
      case ScoreWidgetStatus.BECOMING_INVISIBLE:
        scorePosition = scoreOutPositionAnimation.value;
        scoreOpacity = 1.0 - scoreOutAnimationController.value;
    }
    return new Positioned(
        child: new Opacity(opacity: scoreOpacity, child: new Container(
            height: 50.0 + extraSize,
            width: 50.0  + extraSize,
            decoration: new ShapeDecoration(
              shape: new CircleBorder(
                  side: BorderSide.none
              ),
              color: Colors.pink,
            ),
            child: new Center(child:
            new Text("+" + _counter.toString(),
              style: new TextStyle(color: Colors.white,
                  fontWeight: FontWeight.bold,
                  fontSize: 15.0),))
        )),
        bottom: scorePosition
    );
  }

  Widget getClapButton() {
    // Using custom gesture detector because we want to keep increasing the claps
    // when user holds the button.

    var extraSize = 0.0;
    if (_scoreWidgetStatus == ScoreWidgetStatus.VISIBLE || _scoreWidgetStatus == ScoreWidgetStatus.BECOMING_VISIBLE) {
      extraSize = scoreSizeAnimationController.value * 10;
    }
    return new GestureDetector(
        onTapUp: onTapUp,
        onTapDown: onTapDown,
        child: new Container(
          height: 60.0 + extraSize ,
          width: 60.0 + extraSize,
          padding: new EdgeInsets.all(10.0),
          decoration: new BoxDecoration(
              border: new Border.all(color: Colors.pink, width: 1.0),
              borderRadius: new BorderRadius.circular(50.0),
              color: Colors.white,
              boxShadow: [
                new BoxShadow(color: Colors.pink, blurRadius: 8.0)
              ]
          ),
          child: new ImageIcon(
              new AssetImage("images/clap.png"), color: Colors.pink,
              size: 40.0),
        )
    );
  }

  @override
  Widget build(BuildContext context) {
    return new Scaffold(
      appBar: new AppBar(
        title: new Text(widget.title),
      ),
      body: new Center(
        child: new Column(
          mainAxisAlignment: MainAxisAlignment.center,
          children: <Widget>[
            new Text(
              'You have pushed the button this many times:',
            ),
            new Text(
              '$_counter',
              style: Theme
                  .of(context)
                  .textTheme
                  .display1,
            ),
          ],
        ),
      ),
      floatingActionButton: new Padding(
          padding: new EdgeInsets.only(right: 20.0),
          child: new Stack(
            alignment: FractionalOffset.center,
            overflow: Overflow.visible,
            children: <Widget>[
              getScoreButton(),
              getClapButton(),
            ],
          )
      ),
    );
  }
}

Do you feel that flutter's animation system is hard to use?? I can implement the same effect with less code in Android or iOS.
To run a animation in Android, just ObjectAnimator.OfFloat(v, from, to).set..set..play()