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

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

[Out]

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

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Rubi [A]  time = 0.0520507, 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 \[ -\frac{\sqrt [3]{a+b x^3} \, _2F_1\left (-\frac{2}{3},-\frac{1}{3};\frac{1}{3};-\frac{b x^3}{a}\right )}{2 x^2 \sqrt [3]{\frac{b x^3}{a}+1}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(1/3)/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.99749, size = 46, normalized size = 1.21 \[ - \frac{\sqrt [3]{a + b x^{3}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, - \frac{2}{3} \\ \frac{1}{3} \end{matrix}\middle |{- \frac{b x^{3}}{a}} \right )}}{2 x^{2} \sqrt [3]{1 + \frac{b x^{3}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0489808, size = 66, normalized size = 1.74 \[ \frac{b x^3 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{3},\frac{2}{3};\frac{4}{3};-\frac{b x^3}{a}\right )-a-b x^3}{2 x^2 \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}}\sqrt [3]{b{x}^{3}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(1/3)/x^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(1/3)/x^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^3 + a)^(1/3)/x^3, x)

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Sympy [A]  time = 2.5501, size = 42, normalized size = 1.11 \[ \frac{\sqrt [3]{a} \Gamma \left (- \frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, - \frac{1}{3} \\ \frac{1}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{2} \Gamma \left (\frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(1/3)/x^3, x)