\(\int \frac {x}{(3-2 \sqrt {x})^{3/4}} \, dx\) [297]

Optimal result

Integrand size = 15, antiderivative size = 69 \[ \int \frac {x}{\left (3-2 \sqrt {x}\right )^{3/4}} \, dx=-\frac {27}{2} \sqrt [4]{3-2 \sqrt {x}}+\frac {27}{10} \left (3-2 \sqrt {x}\right )^{5/4}-\frac {1}{2} \left (3-2 \sqrt {x}\right )^{9/4}+\frac {1}{26} \left (3-2 \sqrt {x}\right )^{13/4} \]

[Out]

-27/2*(3-2*x^(1/2))^(1/4)+27/10*(3-2*x^(1/2))^(5/4)-1/2*(3-2*x^(1/2))^(9/4)+1/26*(3-2*x^(1/2))^(13/4)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {x}{\left (3-2 \sqrt {x}\right )^{3/4}} \, dx=\frac {1}{26} \left (3-2 \sqrt {x}\right )^{13/4}-\frac {1}{2} \left (3-2 \sqrt {x}\right )^{9/4}+\frac {27}{10} \left (3-2 \sqrt {x}\right )^{5/4}-\frac {27}{2} \sqrt [4]{3-2 \sqrt {x}} \]

[In]

Int[x/(3 - 2*Sqrt[x])^(3/4),x]

[Out]

(-27*(3 - 2*Sqrt[x])^(1/4))/2 + (27*(3 - 2*Sqrt[x])^(5/4))/10 - (3 - 2*Sqrt[x])^(9/4)/2 + (3 - 2*Sqrt[x])^(13/
4)/26

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 272

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^3}{(3-2 x)^{3/4}} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (\frac {27}{8 (3-2 x)^{3/4}}-\frac {27}{8} \sqrt [4]{3-2 x}+\frac {9}{8} (3-2 x)^{5/4}-\frac {1}{8} (3-2 x)^{9/4}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\frac {27}{2} \sqrt [4]{3-2 \sqrt {x}}+\frac {27}{10} \left (3-2 \sqrt {x}\right )^{5/4}-\frac {1}{2} \left (3-2 \sqrt {x}\right )^{9/4}+\frac {1}{26} \left (3-2 \sqrt {x}\right )^{13/4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.52 \[ \int \frac {x}{\left (3-2 \sqrt {x}\right )^{3/4}} \, dx=-\frac {4}{65} \sqrt [4]{3-2 \sqrt {x}} \left (144+24 \sqrt {x}+10 x+5 x^{3/2}\right ) \]

[In]

Integrate[x/(3 - 2*Sqrt[x])^(3/4),x]

[Out]

(-4*(3 - 2*Sqrt[x])^(1/4)*(144 + 24*Sqrt[x] + 10*x + 5*x^(3/2)))/65

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 2.

Time = 0.29 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.29

method	result	size
meijerg	\(\frac {3^{\frac {1}{4}} x^{2} {}_{2}^{}{\moversetsp {}{\mundersetsp {}{F_{1}^{}}}}\left (\frac {3}{4},4;5;\frac {2 \sqrt {x}}{3}\right )}{6}\)	\(20\)
derivativedivides	\(-\frac {27 \left (3-2 \sqrt {x}\right )^{\frac {1}{4}}}{2}+\frac {27 \left (3-2 \sqrt {x}\right )^{\frac {5}{4}}}{10}-\frac {\left (3-2 \sqrt {x}\right )^{\frac {9}{4}}}{2}+\frac {\left (3-2 \sqrt {x}\right )^{\frac {13}{4}}}{26}\)	\(46\)
default	\(-\frac {27 \left (3-2 \sqrt {x}\right )^{\frac {1}{4}}}{2}+\frac {27 \left (3-2 \sqrt {x}\right )^{\frac {5}{4}}}{10}-\frac {\left (3-2 \sqrt {x}\right )^{\frac {9}{4}}}{2}+\frac {\left (3-2 \sqrt {x}\right )^{\frac {13}{4}}}{26}\)	\(46\)

[In]

int(x/(3-2*x^(1/2))^(3/4),x,method=_RETURNVERBOSE)

[Out]

1/6*3^(1/4)*x^2*hypergeom([3/4,4],[5],2/3*x^(1/2))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.36 \[ \int \frac {x}{\left (3-2 \sqrt {x}\right )^{3/4}} \, dx=-\frac {4}{65} \, {\left ({\left (5 \, x + 24\right )} \sqrt {x} + 10 \, x + 144\right )} {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {1}{4}} \]

[In]

integrate(x/(3-2*x^(1/2))^(3/4),x, algorithm="fricas")

[Out]

-4/65*((5*x + 24)*sqrt(x) + 10*x + 144)*(-2*sqrt(x) + 3)^(1/4)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.12 (sec) , antiderivative size = 3303, normalized size of antiderivative = 47.87 \[ \int \frac {x}{\left (3-2 \sqrt {x}\right )^{3/4}} \, dx=\text {Too large to display} \]

[In]

integrate(x/(3-2*x**(1/2))**(3/4),x)

[Out]

Piecewise((1280*3**(1/4)*x**(25/2)*(2*sqrt(x) - 3)**(1/4)*exp(-3*I*pi/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3
**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*
x**9 + 47385*3**(1/4)*x**8) + 26304*3**(1/4)*x**(23/2)*(2*sqrt(x) - 3)**(1/4)*exp(-3*I*pi/4)/(-37440*3**(1/4)*
x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**1
0 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 200016*3**(1/4)*x**(21/2)*(2*sqrt(x) - 3)**(1/4)*exp(-3*I*pi
/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 +
140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 331776*sqrt(3)*x**(21/2)/(-37440*3**(1/
4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x
**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 2123820*3**(1/4)*x**(19/2)*(2*sqrt(x) - 3)**(1/4)*exp(-3*
I*pi/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**
11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 2488320*sqrt(3)*x**(19/2)/(-37440*3
**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1
/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 1609632*3**(1/4)*x**(17/2)*(2*sqrt(x) - 3)**(1/4)*ex
p(-3*I*pi/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4
)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 1679616*sqrt(3)*x**(17/2)/(-37
440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*
3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 8960*3**(1/4)*x**12*(2*sqrt(x) - 3)**(1/4)*exp(
-3*I*pi/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*
x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 18432*3**(1/4)*x**11*(2*sqrt(x)
- 3)**(1/4)*exp(-3*I*pi/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2)
+ 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 36864*sqrt(3)*x*
*11/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 +
140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 965520*3**(1/4)*x**10*(2*sqrt(x) - 3)**
(1/4)*exp(-3*I*pi/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160
*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 1244160*sqrt(3)*x**10/
(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140
400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 2548584*3**(1/4)*x**9*(2*sqrt(x) - 3)**(1/4
)*exp(-3*I*pi/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**
(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 2799360*sqrt(3)*x**9/(-374
40*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3
**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 419904*3**(1/4)*x**8*(2*sqrt(x) - 3)**(1/4)*exp(
-3*I*pi/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*
x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 419904*sqrt(3)*x**8/(-37440*3**(
1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)
*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8), Abs(sqrt(x)) > 3/2), (-1280*3**(1/4)*x**(25/2)*(3 - 2*sq
rt(x))**(1/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/
4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 26304*3**(1/4)*x**(23/2)*(3 -
2*sqrt(x))**(1/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3
**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 200016*3**(1/4)*x**(21/2
)*(3 - 2*sqrt(x))**(1/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) +
4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 331776*sqrt(3)*x**
(21/2)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**1
1 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 2123820*3**(1/4)*x**(19/2)*(3 - 2*sq
rt(x))**(1/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/
4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 2488320*sqrt(3)*x**(19/2)/(-3
7440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400
*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 1609632*3**(1/4)*x**(17/2)*(3 - 2*sqrt(x))**(1
/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 +
140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 1679616*sqrt(3)*x**(17/2)/(-37440*3**(1
/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*
x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 8960*3**(1/4)*x**12*(3 - 2*sqrt(x))**(1/4)/(-37440*3**(1
/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*
x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 18432*3**(1/4)*x**11*(3 - 2*sqrt(x))**(1/4)/(-37440*3**(
1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)
*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 36864*sqrt(3)*x**11/(-37440*3**(1/4)*x**(21/2) - 280800
*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4
)*x**9 + 47385*3**(1/4)*x**8) - 965520*3**(1/4)*x**10*(3 - 2*sqrt(x))**(1/4)/(-37440*3**(1/4)*x**(21/2) - 2808
00*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1
/4)*x**9 + 47385*3**(1/4)*x**8) + 1244160*sqrt(3)*x**10/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2)
- 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(
1/4)*x**8) - 2548584*3**(1/4)*x**9*(3 - 2*sqrt(x))**(1/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/
2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3*
*(1/4)*x**8) + 2799360*sqrt(3)*x**9/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x
**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) - 419904*
3**(1/4)*x**8*(3 - 2*sqrt(x))**(1/4)/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*
x**(17/2) + 4160*3**(1/4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8) + 419904
*sqrt(3)*x**8/(-37440*3**(1/4)*x**(21/2) - 280800*3**(1/4)*x**(19/2) - 189540*3**(1/4)*x**(17/2) + 4160*3**(1/
4)*x**11 + 140400*3**(1/4)*x**10 + 315900*3**(1/4)*x**9 + 47385*3**(1/4)*x**8), True))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.65 \[ \int \frac {x}{\left (3-2 \sqrt {x}\right )^{3/4}} \, dx=\frac {1}{26} \, {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {13}{4}} - \frac {1}{2} \, {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {9}{4}} + \frac {27}{10} \, {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {5}{4}} - \frac {27}{2} \, {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {1}{4}} \]

[In]

integrate(x/(3-2*x^(1/2))^(3/4),x, algorithm="maxima")

[Out]

1/26*(-2*sqrt(x) + 3)^(13/4) - 1/2*(-2*sqrt(x) + 3)^(9/4) + 27/10*(-2*sqrt(x) + 3)^(5/4) - 27/2*(-2*sqrt(x) +
3)^(1/4)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.91 \[ \int \frac {x}{\left (3-2 \sqrt {x}\right )^{3/4}} \, dx=-\frac {1}{26} \, {\left (2 \, \sqrt {x} - 3\right )}^{3} {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {1}{4}} - \frac {1}{2} \, {\left (2 \, \sqrt {x} - 3\right )}^{2} {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {1}{4}} + \frac {27}{10} \, {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {5}{4}} - \frac {27}{2} \, {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {1}{4}} \]

[In]

integrate(x/(3-2*x^(1/2))^(3/4),x, algorithm="giac")

[Out]

-1/26*(2*sqrt(x) - 3)^3*(-2*sqrt(x) + 3)^(1/4) - 1/2*(2*sqrt(x) - 3)^2*(-2*sqrt(x) + 3)^(1/4) + 27/10*(-2*sqrt
(x) + 3)^(5/4) - 27/2*(-2*sqrt(x) + 3)^(1/4)

Mupad [B] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.65 \[ \int \frac {x}{\left (3-2 \sqrt {x}\right )^{3/4}} \, dx=\frac {27\,{\left (3-2\,\sqrt {x}\right )}^{5/4}}{10}-\frac {27\,{\left (3-2\,\sqrt {x}\right )}^{1/4}}{2}-\frac {{\left (3-2\,\sqrt {x}\right )}^{9/4}}{2}+\frac {{\left (3-2\,\sqrt {x}\right )}^{13/4}}{26} \]

[In]

int(x/(3 - 2*x^(1/2))^(3/4),x)

[Out]

(27*(3 - 2*x^(1/2))^(5/4))/10 - (27*(3 - 2*x^(1/2))^(1/4))/2 - (3 - 2*x^(1/2))^(9/4)/2 + (3 - 2*x^(1/2))^(13/4
)/26