3.535 \(\int \frac{\left (a+b x^3\right )^{2/3}}{x^5} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.051555, 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 \[ -\frac{\left (a+b x^3\right )^{2/3} \, _2F_1\left (-\frac{4}{3},-\frac{2}{3};-\frac{1}{3};-\frac{b x^3}{a}\right )}{4 x^4 \left (\frac{b x^3}{a}+1\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(2/3)/x^5,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.87596, size = 48, normalized size = 1.26 \[ - \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, - \frac{4}{3} \\ - \frac{1}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{4 x^{4} \left (1 + \frac{b x^{3}}{a}\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**(2/3)/x**5,x)

[Out]

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

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Mathematica [B]  time = 0.0450994, size = 82, normalized size = 2.16 \[ \frac{-a^2+b^2 x^6 \sqrt [3]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )-3 a b x^3-2 b^2 x^6}{4 a x^4 \sqrt [3]{a+b x^3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^(2/3)/x^5,x]

[Out]

(-a^2 - 3*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*x^4*(a + b*x^3)^(1/3))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{5}} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(2/3)/x^5,x)

[Out]

int((b*x^3+a)^(2/3)/x^5,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)/x^5,x, algorithm="maxima")

[Out]

integrate((b*x^3 + a)^(2/3)/x^5, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{5}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)/x^5,x, algorithm="fricas")

[Out]

integral((b*x^3 + a)^(2/3)/x^5, x)

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Sympy [A]  time = 3.36184, size = 46, normalized size = 1.21 \[ \frac{a^{\frac{2}{3}} \Gamma \left (- \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, - \frac{2}{3} \\ - \frac{1}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{4} \Gamma \left (- \frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**(2/3)/x**5,x)

[Out]

a**(2/3)*gamma(-4/3)*hyper((-4/3, -2/3), (-1/3,), b*x**3*exp_polar(I*pi)/a)/(3*x
**4*gamma(-1/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{5}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(2/3)/x^5,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(2/3)/x^5, x)