3.2269 \(\int \frac{x^3}{\left (a+b x^{3/2}\right )^{2/3}} \, dx\)

Optimal. Leaf size=198 \[ -\frac{10 a^2 \log \left (1-\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{27 b^{8/3}}+\frac{5 a^2 \log \left (\frac{b^{2/3} x}{\left (a+b x^{3/2}\right )^{2/3}}+\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}+1\right )}{27 b^{8/3}}-\frac{10 a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{8/3}}-\frac{5 a x \sqrt [3]{a+b x^{3/2}}}{9 b^2}+\frac{x^{5/2} \sqrt [3]{a+b x^{3/2}}}{3 b} \]

[Out]

(-5*a*x*(a + b*x^(3/2))^(1/3))/(9*b^2) + (x^(5/2)*(a + b*x^(3/2))^(1/3))/(3*b) -
(10*a^2*ArcTan[(1 + (2*b^(1/3)*Sqrt[x])/(a + b*x^(3/2))^(1/3))/Sqrt[3]])/(9*Sqr
t[3]*b^(8/3)) - (10*a^2*Log[1 - (b^(1/3)*Sqrt[x])/(a + b*x^(3/2))^(1/3)])/(27*b^
(8/3)) + (5*a^2*Log[1 + (b^(2/3)*x)/(a + b*x^(3/2))^(2/3) + (b^(1/3)*Sqrt[x])/(a
+ b*x^(3/2))^(1/3)])/(27*b^(8/3))

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Rubi [A]  time = 0.374353, antiderivative size = 198, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.529 \[ -\frac{10 a^2 \log \left (1-\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}\right )}{27 b^{8/3}}+\frac{5 a^2 \log \left (\frac{b^{2/3} x}{\left (a+b x^{3/2}\right )^{2/3}}+\frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}+1\right )}{27 b^{8/3}}-\frac{10 a^2 \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a+b x^{3/2}}}+1}{\sqrt{3}}\right )}{9 \sqrt{3} b^{8/3}}-\frac{5 a x \sqrt [3]{a+b x^{3/2}}}{9 b^2}+\frac{x^{5/2} \sqrt [3]{a+b x^{3/2}}}{3 b} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(a + b*x^(3/2))^(2/3),x]

[Out]

(-5*a*x*(a + b*x^(3/2))^(1/3))/(9*b^2) + (x^(5/2)*(a + b*x^(3/2))^(1/3))/(3*b) -
(10*a^2*ArcTan[(1 + (2*b^(1/3)*Sqrt[x])/(a + b*x^(3/2))^(1/3))/Sqrt[3]])/(9*Sqr
t[3]*b^(8/3)) - (10*a^2*Log[1 - (b^(1/3)*Sqrt[x])/(a + b*x^(3/2))^(1/3)])/(27*b^
(8/3)) + (5*a^2*Log[1 + (b^(2/3)*x)/(a + b*x^(3/2))^(2/3) + (b^(1/3)*Sqrt[x])/(a
+ b*x^(3/2))^(1/3)])/(27*b^(8/3))

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Rubi in Sympy [A]  time = 32.3995, size = 187, normalized size = 0.94 \[ - \frac{10 a^{2} \log{\left (- \frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a + b x^{\frac{3}{2}}}} + 1 \right )}}{27 b^{\frac{8}{3}}} + \frac{5 a^{2} \log{\left (\frac{b^{\frac{2}{3}} x}{\left (a + b x^{\frac{3}{2}}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} \sqrt{x}}{\sqrt [3]{a + b x^{\frac{3}{2}}}} + 1 \right )}}{27 b^{\frac{8}{3}}} - \frac{10 \sqrt{3} a^{2} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} \sqrt{x}}{3 \sqrt [3]{a + b x^{\frac{3}{2}}}} + \frac{1}{3}\right ) \right )}}{27 b^{\frac{8}{3}}} - \frac{5 a x \sqrt [3]{a + b x^{\frac{3}{2}}}}{9 b^{2}} + \frac{x^{\frac{5}{2}} \sqrt [3]{a + b x^{\frac{3}{2}}}}{3 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(a+b*x**(3/2))**(2/3),x)

[Out]

-10*a**2*log(-b**(1/3)*sqrt(x)/(a + b*x**(3/2))**(1/3) + 1)/(27*b**(8/3)) + 5*a*
*2*log(b**(2/3)*x/(a + b*x**(3/2))**(2/3) + b**(1/3)*sqrt(x)/(a + b*x**(3/2))**(
1/3) + 1)/(27*b**(8/3)) - 10*sqrt(3)*a**2*atan(sqrt(3)*(2*b**(1/3)*sqrt(x)/(3*(a
+ b*x**(3/2))**(1/3)) + 1/3))/(27*b**(8/3)) - 5*a*x*(a + b*x**(3/2))**(1/3)/(9*
b**2) + x**(5/2)*(a + b*x**(3/2))**(1/3)/(3*b)

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Mathematica [C]  time = 0.0533722, size = 87, normalized size = 0.44 \[ \frac{5 a^2 x \left (\frac{b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^{3/2}}{a}\right )-5 a^2 x-2 a b x^{5/2}+3 b^2 x^4}{9 b^2 \left (a+b x^{3/2}\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(a + b*x^(3/2))^(2/3),x]

[Out]

(-5*a^2*x - 2*a*b*x^(5/2) + 3*b^2*x^4 + 5*a^2*x*(1 + (b*x^(3/2))/a)^(2/3)*Hyperg
eometric2F1[2/3, 2/3, 5/3, -((b*x^(3/2))/a)])/(9*b^2*(a + b*x^(3/2))^(2/3))

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \[ \int{{x}^{3} \left ( a+b{x}^{{\frac{3}{2}}} \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b*x^(3/2))^(2/3),x)

[Out]

int(x^3/(a+b*x^(3/2))^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x^(3/2) + a)^(2/3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x^(3/2) + a)^(2/3),x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 11.4368, size = 41, normalized size = 0.21 \[ \frac{2 x^{4} \Gamma \left (\frac{8}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{8}{3} \\ \frac{11}{3} \end{matrix}\middle |{\frac{b x^{\frac{3}{2}} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{11}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(a+b*x**(3/2))**(2/3),x)

[Out]

2*x**4*gamma(8/3)*hyper((2/3, 8/3), (11/3,), b*x**(3/2)*exp_polar(I*pi)/a)/(3*a*
*(2/3)*gamma(11/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3}}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(b*x^(3/2) + a)^(2/3),x, algorithm="giac")

[Out]

integrate(x^3/(b*x^(3/2) + a)^(2/3), x)