3.553 \(\int \frac{1}{x^5 \sqrt [3]{a+b x^3}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0509573, 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 \[ -\frac{\sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (-\frac{4}{3},\frac{1}{3};-\frac{1}{3};-\frac{b x^3}{a}\right )}{4 x^4 \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^5*(a + b*x^3)^(1/3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 6.14683, size = 48, normalized size = 1.26 \[ - \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, - \frac{4}{3} \\ - \frac{1}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{4 a x^{4} \left (1 + \frac{b x^{3}}{a}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**5/(b*x**3+a)**(1/3),x)

[Out]

-(a + b*x**3)**(2/3)*hyper((1/3, -4/3), (-1/3,), -b*x**3/a)/(4*a*x**4*(1 + b*x**
3/a)**(2/3))

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Mathematica [B]  time = 0.0554079, size = 82, normalized size = 2.16 \[ \frac{-a^2-b^2 x^6 \sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )+a b x^3+2 b^2 x^6}{4 a^2 x^4 \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^5*(a + b*x^3)^(1/3)),x]

[Out]

(-a^2 + a*b*x^3 + 2*b^2*x^6 - b^2*x^6*(1 + (b*x^3)/a)^(1/3)*Hypergeometric2F1[1/
3, 2/3, 5/3, -((b*x^3)/a)])/(4*a^2*x^4*(a + b*x^3)^(1/3))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5}}{\frac{1}{\sqrt [3]{b{x}^{3}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^5/(b*x^3+a)^(1/3),x)

[Out]

int(1/x^5/(b*x^3+a)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(1/3)*x^5),x, algorithm="maxima")

[Out]

integrate(1/((b*x^3 + a)^(1/3)*x^5), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}} x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(1/3)*x^5),x, algorithm="fricas")

[Out]

integral(1/((b*x^3 + a)^(1/3)*x^5), x)

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Sympy [A]  time = 3.07627, size = 44, normalized size = 1.16 \[ \frac{\Gamma \left (- \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{1}{3} \\ - \frac{1}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 \sqrt [3]{a} x^{4} \Gamma \left (- \frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**5/(b*x**3+a)**(1/3),x)

[Out]

gamma(-4/3)*hyper((-4/3, 1/3), (-1/3,), b*x**3*exp_polar(I*pi)/a)/(3*a**(1/3)*x*
*4*gamma(-1/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{1}{3}} x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^3 + a)^(1/3)*x^5),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(1/3)*x^5), x)