Optimal. Leaf size=61 \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac{2 C \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{b}} \]
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Rubi [A] time = 0.0412275, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {1863, 31, 617, 204} \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac{2 C \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Rule 1863
Rule 31
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{2 a^{2/3} C+b^{2/3} C x^2}{a+b x^3} \, dx &=\frac{\left (\sqrt [3]{a} C\right ) \int \frac{1}{\frac{a^{2/3}}{b^{2/3}}-\frac{\sqrt [3]{a} x}{\sqrt [3]{b}}+x^2} \, dx}{b^{2/3}}+\frac{C \int \frac{1}{\frac{\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx}{\sqrt [3]{b}}\\ &=\frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac{(2 C) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}\\ &=-\frac{2 C \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{b}}+\frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}\\ \end{align*}
Mathematica [A] time = 0.0173002, size = 95, normalized size = 1.56 \[ \frac{C \left (-\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\log \left (a+b x^3\right )+2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{3 \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.005, size = 117, normalized size = 1.9 \begin{align*}{\frac{2\,C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{C}{3\,b}{a}^{{\frac{2}{3}}}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{2\,C\sqrt{3}}{3\,b}{a}^{{\frac{2}{3}}}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{C\ln \left ( b{x}^{3}+a \right ) }{3}{\frac{1}{\sqrt [3]{b}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.05073, size = 463, normalized size = 7.59 \begin{align*} \left [\frac{\sqrt{\frac{1}{3}} C b \sqrt{-\frac{1}{b^{\frac{2}{3}}}} \log \left (\frac{2 \, b x^{3} - 3 \, a^{\frac{2}{3}} b^{\frac{1}{3}} x + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a^{\frac{1}{3}} b x^{2} + a^{\frac{2}{3}} b^{\frac{2}{3}} x - a b^{\frac{1}{3}}\right )} \sqrt{-\frac{1}{b^{\frac{2}{3}}}} - a}{b x^{3} + a}\right ) + C b^{\frac{2}{3}} \log \left (b x + a^{\frac{1}{3}} b^{\frac{2}{3}}\right )}{b}, \frac{2 \, \sqrt{\frac{1}{3}} C b^{\frac{2}{3}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, a^{\frac{2}{3}} b^{\frac{2}{3}} x - a b^{\frac{1}{3}}\right )}}{a b^{\frac{1}{3}}}\right ) + C b^{\frac{2}{3}} \log \left (b x + a^{\frac{1}{3}} b^{\frac{2}{3}}\right )}{b}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.668899, size = 70, normalized size = 1.15 \begin{align*} \operatorname{RootSum}{\left (3 t^{3} b^{\frac{5}{3}} - 3 t^{2} C b^{\frac{4}{3}} + t C^{2} b - C^{3} b^{\frac{2}{3}}, \left ( t \mapsto t \log{\left (x + \frac{3 t \sqrt [3]{a} \sqrt [3]{b} - C \sqrt [3]{a}}{2 C \sqrt [3]{b}} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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