Optimal. Leaf size=72 \[ -\frac{\log (x)}{2 a^2}+\frac{3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 a^2}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a^3+b^3 x}+a}{\sqrt{3} a}\right )}{a^2} \]
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Rubi [A] time = 0.0621119, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ -\frac{\log (x)}{2 a^2}+\frac{3 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )}{2 a^2}-\frac{\sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a^3+b^3 x}+a}{\sqrt{3} a}\right )}{a^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x*(a^3 + b^3*x)^(2/3)),x]
[Out]
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Rubi in Sympy [A] time = 5.37111, size = 65, normalized size = 0.9 \[ - \frac{\log{\left (x \right )}}{2 a^{2}} + \frac{3 \log{\left (a - \sqrt [3]{a^{3} + b^{3} x} \right )}}{2 a^{2}} - \frac{\sqrt{3} \operatorname{atan}{\left (\frac{\sqrt{3} \left (\frac{a}{3} + \frac{2 \sqrt [3]{a^{3} + b^{3} x}}{3}\right )}{a} \right )}}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x/(b**3*x+a**3)**(2/3),x)
[Out]
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Mathematica [A] time = 0.0285316, size = 95, normalized size = 1.32 \[ -\frac{-2 \log \left (a-\sqrt [3]{a^3+b^3 x}\right )+2 \sqrt{3} \tan ^{-1}\left (\frac{2 \sqrt [3]{a^3+b^3 x}+a}{\sqrt{3} a}\right )+\log \left (a \sqrt [3]{a^3+b^3 x}+\left (a^3+b^3 x\right )^{2/3}+a^2\right )}{2 a^2} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x*(a^3 + b^3*x)^(2/3)),x]
[Out]
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Maple [A] time = 0.013, size = 88, normalized size = 1.2 \[{\frac{1}{{a}^{2}}\ln \left ( \sqrt [3]{{b}^{3}x+{a}^{3}}-a \right ) }-{\frac{1}{2\,{a}^{2}}\ln \left ( \left ({b}^{3}x+{a}^{3} \right ) ^{{\frac{2}{3}}}+\sqrt [3]{{b}^{3}x+{a}^{3}}a+{a}^{2} \right ) }-{\frac{\sqrt{3}}{{a}^{2}}\arctan \left ({\frac{\sqrt{3}}{3\,a} \left ( a+2\,\sqrt [3]{{b}^{3}x+{a}^{3}} \right ) } \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x/(b^3*x+a^3)^(2/3),x)
[Out]
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Maxima [A] time = 1.47607, size = 117, normalized size = 1.62 \[ -\frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (a + 2 \,{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right )}{a^{2}} - \frac{\log \left (a^{2} +{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}} a +{\left (b^{3} x + a^{3}\right )}^{\frac{2}{3}}\right )}{2 \, a^{2}} + \frac{\log \left (-a +{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^3*x + a^3)^(2/3)*x),x, algorithm="maxima")
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Fricas [A] time = 0.22109, size = 111, normalized size = 1.54 \[ -\frac{2 \, \sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (a + 2 \,{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right ) + \log \left (a^{2} +{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}} a +{\left (b^{3} x + a^{3}\right )}^{\frac{2}{3}}\right ) - 2 \, \log \left (-a +{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}}\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^3*x + a^3)^(2/3)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.37894, size = 134, normalized size = 1.86 \[ \frac{\log{\left (1 - \frac{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x}}{a} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{2} \Gamma \left (\frac{4}{3}\right )} + \frac{e^{\frac{4 i \pi }{3}} \log{\left (1 - \frac{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x} e^{\frac{2 i \pi }{3}}}{a} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{2} \Gamma \left (\frac{4}{3}\right )} + \frac{e^{\frac{2 i \pi }{3}} \log{\left (1 - \frac{b \sqrt [3]{\frac{a^{3}}{b^{3}} + x} e^{\frac{4 i \pi }{3}}}{a} \right )} \Gamma \left (\frac{1}{3}\right )}{3 a^{2} \Gamma \left (\frac{4}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x/(b**3*x+a**3)**(2/3),x)
[Out]
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GIAC/XCAS [A] time = 0.230379, size = 119, normalized size = 1.65 \[ -\frac{\sqrt{3} \arctan \left (\frac{\sqrt{3}{\left (a + 2 \,{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}}\right )}}{3 \, a}\right )}{a^{2}} - \frac{{\rm ln}\left (a^{2} +{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}} a +{\left (b^{3} x + a^{3}\right )}^{\frac{2}{3}}\right )}{2 \, a^{2}} + \frac{{\rm ln}\left ({\left | -a +{\left (b^{3} x + a^{3}\right )}^{\frac{1}{3}} \right |}\right )}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b^3*x + a^3)^(2/3)*x),x, algorithm="giac")
[Out]