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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]