Optimal. Leaf size=51 \[ -\frac{3 a^2 \log \left (a \sqrt [3]{x}+b\right )}{b^3}+\frac{a^2 \log (x)}{b^3}+\frac{3 a}{b^2 \sqrt [3]{x}}-\frac{3}{2 b x^{2/3}} \]
[Out]
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Rubi [A] time = 0.0822004, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{3 a^2 \log \left (a \sqrt [3]{x}+b\right )}{b^3}+\frac{a^2 \log (x)}{b^3}+\frac{3 a}{b^2 \sqrt [3]{x}}-\frac{3}{2 b x^{2/3}} \]
Antiderivative was successfully verified.
[In] Int[1/((a + b/x^(1/3))*x^2),x]
[Out]
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Rubi in Sympy [A] time = 12.2909, size = 54, normalized size = 1.06 \[ \frac{3 a^{2} \log{\left (\sqrt [3]{x} \right )}}{b^{3}} - \frac{3 a^{2} \log{\left (a \sqrt [3]{x} + b \right )}}{b^{3}} + \frac{3 a}{b^{2} \sqrt [3]{x}} - \frac{3}{2 b x^{\frac{2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(a+b/x**(1/3))/x**2,x)
[Out]
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Mathematica [A] time = 0.0545129, size = 48, normalized size = 0.94 \[ \frac{-6 a^2 \log \left (a \sqrt [3]{x}+b\right )+2 a^2 \log (x)-\frac{3 b \left (b-2 a \sqrt [3]{x}\right )}{x^{2/3}}}{2 b^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/((a + b/x^(1/3))*x^2),x]
[Out]
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Maple [A] time = 0.013, size = 44, normalized size = 0.9 \[ -{\frac{3}{2\,b}{x}^{-{\frac{2}{3}}}}+3\,{\frac{a}{{b}^{2}\sqrt [3]{x}}}-3\,{\frac{{a}^{2}\ln \left ( b+a\sqrt [3]{x} \right ) }{{b}^{3}}}+{\frac{{a}^{2}\ln \left ( x \right ) }{{b}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(a+b/x^(1/3))/x^2,x)
[Out]
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Maxima [A] time = 1.44236, size = 59, normalized size = 1.16 \[ -\frac{3 \, a^{2} \log \left (a + \frac{b}{x^{\frac{1}{3}}}\right )}{b^{3}} - \frac{3 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )}^{2}}{2 \, b^{3}} + \frac{6 \,{\left (a + \frac{b}{x^{\frac{1}{3}}}\right )} a}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^(1/3))*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.235754, size = 63, normalized size = 1.24 \[ -\frac{3 \,{\left (2 \, a^{2} x^{\frac{2}{3}} \log \left (a x^{\frac{1}{3}} + b\right ) - 2 \, a^{2} x^{\frac{2}{3}} \log \left (x^{\frac{1}{3}}\right ) - 2 \, a b x^{\frac{1}{3}} + b^{2}\right )}}{2 \, b^{3} x^{\frac{2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^(1/3))*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.86471, size = 73, normalized size = 1.43 \[ \begin{cases} \frac{\tilde{\infty }}{x^{\frac{2}{3}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{a x} & \text{for}\: b = 0 \\- \frac{3}{2 b x^{\frac{2}{3}}} & \text{for}\: a = 0 \\\frac{a^{2} \log{\left (x \right )}}{b^{3}} - \frac{3 a^{2} \log{\left (\sqrt [3]{x} + \frac{b}{a} \right )}}{b^{3}} + \frac{3 a}{b^{2} \sqrt [3]{x}} - \frac{3}{2 b x^{\frac{2}{3}}} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(a+b/x**(1/3))/x**2,x)
[Out]
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GIAC/XCAS [A] time = 0.215069, size = 66, normalized size = 1.29 \[ -\frac{3 \, a^{2}{\rm ln}\left ({\left | a x^{\frac{1}{3}} + b \right |}\right )}{b^{3}} + \frac{a^{2}{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} + \frac{3 \,{\left (2 \, a b x^{\frac{1}{3}} - b^{2}\right )}}{2 \, b^{3} x^{\frac{2}{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((a + b/x^(1/3))*x^2),x, algorithm="giac")
[Out]