∫ x_____ (3−2√_x )3∕4 dx [297]

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

Optimal. Leaf size=69 \[ -\frac {27}{2} \sqrt [4]{3-2 \sqrt {x}}+\frac {27}{10} \left (3-2 \sqrt {x}\right )^{5/4}-\frac {1}{2} \left (3-2 \sqrt {x}\right )^{9/4}+\frac {1}{26} \left (3-2 \sqrt {x}\right )^{13/4} \]

[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)

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Rubi [A]
time = 0.01, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \begin {gather*} \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}} \end {gather*}

Antiderivative was successfully veriﬁed.

[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*} \int \frac {x}{\left (3-2 \sqrt {x}\right )^{3/4}} \, dx &=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*}

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Mathematica [A]
time = 0.02, size = 36, normalized size = 0.52 \begin {gather*} -\frac {4}{65} \sqrt [4]{3-2 \sqrt {x}} \left (144+24 \sqrt {x}+10 x+5 x^{3/2}\right ) \end {gather*}

Antiderivative was successfully veriﬁed.

[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

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Maple [A]
time = 0.06, size = 46, normalized size = 0.67


method	result	size

meijerg	\(\frac {3^{\frac {1}{4}} x^{2} \hypergeom \left (\left [\frac {3}{4}, 4\right ], \left [5\right ], \frac {2 \sqrt {x}}{3}\right )}{6}\)	\(20\)
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	\(-\frac {27 \left (3-2 \sqrt {x}\right )^{\frac {1}{4}}}{2}+\frac {27 \left (3-2 \sqrt {x}\right )^{\frac {5}{4}}}{10}-\frac {\left (3-2 \sqrt {x}\right )^{\frac {9}{4}}}{2}+\frac {\left (3-2 \sqrt {x}\right )^{\frac {13}{4}}}{26}\)	\(46\)

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Maxima [A]
time = 2.65, size = 45, normalized size = 0.65 \begin {gather*} \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}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [A]
time = 0.85, size = 25, normalized size = 0.36 \begin {gather*} -\frac {4}{65} \, {\left ({\left (5 \, x + 24\right )} \sqrt {x} + 10 \, x + 144\right )} {\left (-2 \, \sqrt {x} + 3\right )}^{\frac {1}{4}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [C] Result contains complex when optimal does not.
time = 1.08, size = 3303, normalized size = 47.87 \begin {gather*} \text {Too large to display} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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**(1...

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Giac [A]
time = 0.76, size = 63, normalized size = 0.91 \begin {gather*} -\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}} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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)

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Mupad [B]
time = 0.30, size = 45, normalized size = 0.65 \begin {gather*} \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} \end {gather*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[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

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