3.2268 \(\int \frac{x^4}{\left (a+b x^{3/2}\right )^{2/3}} \, dx\)

Optimal. Leaf size=42 \[ \frac{x^5 \sqrt [3]{a+b x^{3/2}} \, _2F_1\left (1,\frac{11}{3};\frac{13}{3};-\frac{b x^{3/2}}{a}\right )}{5 a} \]

[Out]

(x^5*(a + b*x^(3/2))^(1/3)*Hypergeometric2F1[1, 11/3, 13/3, -((b*x^(3/2))/a)])/(
5*a)

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Rubi [A]  time = 0.098764, antiderivative size = 57, normalized size of antiderivative = 1.36, number of steps used = 3, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176 \[ \frac{x^5 \left (\frac{b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{10}{3};\frac{13}{3};-\frac{b x^{3/2}}{a}\right )}{5 \left (a+b x^{3/2}\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^(3/2))^(2/3),x]

[Out]

(x^5*(1 + (b*x^(3/2))/a)^(2/3)*Hypergeometric2F1[2/3, 10/3, 13/3, -((b*x^(3/2))/
a)])/(5*(a + b*x^(3/2))^(2/3))

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Rubi in Sympy [A]  time = 9.0365, size = 48, normalized size = 1.14 \[ \frac{x^{5} \sqrt [3]{a + b x^{\frac{3}{2}}}{{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{10}{3} \\ \frac{13}{3} \end{matrix}\middle |{- \frac{b x^{\frac{3}{2}}}{a}} \right )}}{5 a \sqrt [3]{1 + \frac{b x^{\frac{3}{2}}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(a+b*x**(3/2))**(2/3),x)

[Out]

x**5*(a + b*x**(3/2))**(1/3)*hyper((2/3, 10/3), (13/3,), -b*x**(3/2)/a)/(5*a*(1
+ b*x**(3/2)/a)**(1/3))

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Mathematica [B]  time = 0.0824257, size = 103, normalized size = 2.45 \[ \frac{\sqrt{x} \left (-14 a^3 \left (\frac{b x^{3/2}}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^{3/2}}{a}\right )+14 a^3+7 a^2 b x^{3/2}-2 a b^2 x^3+5 b^3 x^{9/2}\right )}{20 b^3 \left (a+b x^{3/2}\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x^(3/2))^(2/3),x]

[Out]

(Sqrt[x]*(14*a^3 + 7*a^2*b*x^(3/2) - 2*a*b^2*x^3 + 5*b^3*x^(9/2) - 14*a^3*(1 + (
b*x^(3/2))/a)^(2/3)*Hypergeometric2F1[1/3, 2/3, 4/3, -((b*x^(3/2))/a)]))/(20*b^3
*(a + b*x^(3/2))^(2/3))

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Maple [F]  time = 0.023, size = 0, normalized size = 0. \[ \int{{x}^{4} \left ( a+b{x}^{{\frac{3}{2}}} \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(a+b*x^(3/2))^(2/3),x)

[Out]

int(x^4/(a+b*x^(3/2))^(2/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^(3/2) + a)^(2/3),x, algorithm="maxima")

[Out]

integrate(x^4/(b*x^(3/2) + a)^(2/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^(3/2) + a)^(2/3),x, algorithm="fricas")

[Out]

integral(x^4/(b*x^(3/2) + a)^(2/3), x)

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Sympy [A]  time = 22.9656, size = 41, normalized size = 0.98 \[ \frac{2 x^{5} \Gamma \left (\frac{10}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{10}{3} \\ \frac{13}{3} \end{matrix}\middle |{\frac{b x^{\frac{3}{2}} e^{i \pi }}{a}} \right )}}{3 a^{\frac{2}{3}} \Gamma \left (\frac{13}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(a+b*x**(3/2))**(2/3),x)

[Out]

2*x**5*gamma(10/3)*hyper((2/3, 10/3), (13/3,), b*x**(3/2)*exp_polar(I*pi)/a)/(3*
a**(2/3)*gamma(13/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{\frac{3}{2}} + a\right )}^{\frac{2}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^(3/2) + a)^(2/3),x, algorithm="giac")

[Out]

integrate(x^4/(b*x^(3/2) + a)^(2/3), x)