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

Optimal. Leaf size=38 \[ \frac{x^2 \left (a+b x^3\right )^{2/3} \, _2F_1\left (1,\frac{4}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 a} \]

[Out]

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

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Rubi [A]  time = 0.043048, antiderivative size = 51, normalized size of antiderivative = 1.34, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{x^2 \sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + b*x^3)^(1/3),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.70039, size = 42, normalized size = 1.11 \[ \frac{x^{2} \left (a + b x^{3}\right )^{\frac{2}{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{2 a \left (1 + \frac{b x^{3}}{a}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0280993, size = 52, normalized size = 1.37 \[ \frac{x^2 \sqrt [3]{\frac{a+b x^3}{a}} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )}{2 \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.026, size = 0, normalized size = 0. \[ \int{x{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^3+a)^(1/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*x^3 + a)^(1/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x/(b*x^3 + a)^(1/3), x)

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Sympy [A]  time = 2.21175, size = 37, normalized size = 0.97 \[ \frac{x^{2} \Gamma \left (\frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{2}{3} \\ \frac{5}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} \Gamma \left (\frac{5}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*gamma(2/3)*hyper((1/3, 2/3), (5/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(1/3)*
gamma(5/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/(b*x^3 + a)^(1/3), x)