3.550 \(\int \frac{x^4}{\sqrt [3]{a+b x^3}} \, dx\)

Optimal. Leaf size=38 \[ \frac{x^5 \left (a+b x^3\right )^{2/3} \, _2F_1\left (1,\frac{7}{3};\frac{8}{3};-\frac{b x^3}{a}\right )}{5 a} \]

[Out]

(x^5*(a + b*x^3)^(2/3)*Hypergeometric2F1[1, 7/3, 8/3, -((b*x^3)/a)])/(5*a)

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Rubi [A]  time = 0.0555906, antiderivative size = 51, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{x^5 \sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{5}{3};\frac{8}{3};-\frac{b x^3}{a}\right )}{5 \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^3)^(1/3),x]

[Out]

(x^5*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/3, 5/3, 8/3, -((b*x^3)/a)])/(5*(a
+ b*x^3)^(1/3))

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Rubi in Sympy [A]  time = 6.40882, size = 42, normalized size = 1.11 \[ \frac{x^{5} \left (a + b x^{3}\right )^{\frac{2}{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{5 a \left (1 + \frac{b x^{3}}{a}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**3+a)**(1/3),x)

[Out]

x**5*(a + b*x**3)**(2/3)*hyper((1/3, 5/3), (8/3,), -b*x**3/a)/(5*a*(1 + b*x**3/a
)**(2/3))

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Mathematica [A]  time = 0.0475584, size = 64, normalized size = 1.68 \[ \frac{x^2 \left (-a \sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )+a+b x^3\right )}{4 b \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x^3)^(1/3),x]

[Out]

(x^2*(a + b*x^3 - a*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/3, 2/3, 5/3, -((b*
x^3)/a)]))/(4*b*(a + b*x^3)^(1/3))

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Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{{x}^{4}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^3+a)^(1/3),x)

[Out]

int(x^4/(b*x^3+a)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^3 + a)^(1/3),x, algorithm="maxima")

[Out]

integrate(x^4/(b*x^3 + a)^(1/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^3 + a)^(1/3),x, algorithm="fricas")

[Out]

integral(x^4/(b*x^3 + a)^(1/3), x)

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Sympy [A]  time = 2.40587, size = 37, normalized size = 0.97 \[ \frac{x^{5} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac{8}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**3+a)**(1/3),x)

[Out]

x**5*gamma(5/3)*hyper((1/3, 5/3), (8/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(1/3)*
gamma(8/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^3 + a)^(1/3),x, algorithm="giac")

[Out]

integrate(x^4/(b*x^3 + a)^(1/3), x)