3.518 \(\int \frac{\sqrt [3]{a+b x^3}}{x^2} \, dx\)

Optimal. Leaf size=138 \[ \frac{1}{6} \sqrt [3]{b} \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )-\frac{\sqrt [3]{a+b x^3}}{x}-\frac{1}{3} \sqrt [3]{b} \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

[Out]

-((a + b*x^3)^(1/3)/x) - (b^(1/3)*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/S
qrt[3]])/Sqrt[3] - (b^(1/3)*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)])/3 + (b^(1/3)
*Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)])/6

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Rubi [A]  time = 0.151522, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.533 \[ \frac{1}{6} \sqrt [3]{b} \log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )-\frac{\sqrt [3]{a+b x^3}}{x}-\frac{1}{3} \sqrt [3]{b} \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )-\frac{\sqrt [3]{b} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(1/3)/x^2,x]

[Out]

-((a + b*x^3)^(1/3)/x) - (b^(1/3)*ArcTan[(1 + (2*b^(1/3)*x)/(a + b*x^3)^(1/3))/S
qrt[3]])/Sqrt[3] - (b^(1/3)*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(1/3)])/3 + (b^(1/3)
*Log[1 + (b^(2/3)*x^2)/(a + b*x^3)^(2/3) + (b^(1/3)*x)/(a + b*x^3)^(1/3)])/6

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Rubi in Sympy [A]  time = 21.4318, size = 126, normalized size = 0.91 \[ - \frac{\sqrt [3]{b} \log{\left (- \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{3} + \frac{\sqrt [3]{b} \log{\left (\frac{b^{\frac{2}{3}} x^{2}}{\left (a + b x^{3}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{6} - \frac{\sqrt{3} \sqrt [3]{b} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} x}{3 \sqrt [3]{a + b x^{3}}} + \frac{1}{3}\right ) \right )}}{3} - \frac{\sqrt [3]{a + b x^{3}}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**(1/3)/x**2,x)

[Out]

-b**(1/3)*log(-b**(1/3)*x/(a + b*x**3)**(1/3) + 1)/3 + b**(1/3)*log(b**(2/3)*x**
2/(a + b*x**3)**(2/3) + b**(1/3)*x/(a + b*x**3)**(1/3) + 1)/6 - sqrt(3)*b**(1/3)
*atan(sqrt(3)*(2*b**(1/3)*x/(3*(a + b*x**3)**(1/3)) + 1/3))/3 - (a + b*x**3)**(1
/3)/x

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Mathematica [C]  time = 0.0383314, size = 66, normalized size = 0.48 \[ \frac{b x^3 \left (\frac{b x^3}{a}+1\right )^{2/3} \, _2F_1\left (\frac{2}{3},\frac{2}{3};\frac{5}{3};-\frac{b x^3}{a}\right )-2 \left (a+b x^3\right )}{2 x \left (a+b x^3\right )^{2/3}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^(1/3)/x^2,x]

[Out]

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

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Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}}\sqrt [3]{b{x}^{3}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(1/3)/x^2,x)

[Out]

int((b*x^3+a)^(1/3)/x^2,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(1/3)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(1/3)/x^2,x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 3.93985, size = 41, normalized size = 0.3 \[ \frac{\sqrt [3]{a} \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, - \frac{1}{3} \\ \frac{2}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 x \Gamma \left (\frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**(1/3)/x**2,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{3} + a\right )}^{\frac{1}{3}}}{x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^(1/3)/x^2,x, algorithm="giac")

[Out]

integrate((b*x^3 + a)^(1/3)/x^2, x)