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

Optimal. Leaf size=88 \[ -\frac{1}{2} b^{2/3} \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )+\frac{b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\left (a+b x^3\right )^{2/3}}{2 x^2} \]

[Out]

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

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Rubi [A]  time = 0.0555762, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{1}{2} b^{2/3} \log \left (\sqrt [3]{a+b x^3}-\sqrt [3]{b} x\right )+\frac{b^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )}{\sqrt{3}}-\frac{\left (a+b x^3\right )^{2/3}}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^(2/3)/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 17.5087, size = 129, normalized size = 1.47 \[ - \frac{b^{\frac{2}{3}} \log{\left (- \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{3} + \frac{b^{\frac{2}{3}} \log{\left (\frac{b^{\frac{2}{3}} x^{2}}{\left (a + b x^{3}\right )^{\frac{2}{3}}} + \frac{\sqrt [3]{b} x}{\sqrt [3]{a + b x^{3}}} + 1 \right )}}{6} + \frac{\sqrt{3} b^{\frac{2}{3}} \operatorname{atan}{\left (\sqrt{3} \left (\frac{2 \sqrt [3]{b} x}{3 \sqrt [3]{a + b x^{3}}} + \frac{1}{3}\right ) \right )}}{3} - \frac{\left (a + b x^{3}\right )^{\frac{2}{3}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.17757, size = 129, normalized size = 1.47 \[ \frac{1}{6} b^{2/3} \left (\log \left (\frac{b^{2/3} x^2}{\left (a+b x^3\right )^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1\right )-2 \log \left (1-\frac{\sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a+b x^3}}+1}{\sqrt{3}}\right )\right )-\frac{\left (a+b x^3\right )^{2/3}}{2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

-(a + b*x^3)^(2/3)/(2*x^2) + (b^(2/3)*(2*Sqrt[3]*ArcTan[(1 + (2*b^(1/3)*x)/(a +
b*x^3)^(1/3))/Sqrt[3]] - 2*Log[1 - (b^(1/3)*x)/(a + b*x^3)^(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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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ( b{x}^{3}+a \right ) ^{{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^(2/3)/x^3,x)

[Out]

int((b*x^3+a)^(2/3)/x^3,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)^(2/3)/x^3,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)^(2/3)/x^3,x, algorithm="fricas")

[Out]

Timed out

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Sympy [A]  time = 4.26734, size = 42, normalized size = 0.48 \[ \frac{a^{\frac{2}{3}} \Gamma \left (- \frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, - \frac{2}{3} \\ \frac{1}{3} \end{matrix}\middle |{\frac{b x^{3} e^{i \pi }}{a}} \right )}}{3 x^{2} \Gamma \left (\frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(2/3)*gamma(-2/3)*hyper((-2/3, -2/3), (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. \[ \int \frac{{\left (b x^{3} + a\right )}^{\frac{2}{3}}}{x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^3 + a)^(2/3)/x^3, x)