Optimal. Leaf size=54 \[ \frac{C \log \left (\sqrt [3]{-\frac{a}{b}}-x\right )}{b}-\frac{2 C \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}+1}{\sqrt{3}}\right )}{\sqrt{3} b} \]
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Rubi [A] time = 0.0598418, antiderivative size = 54, 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 = {1869, 31, 617, 204} \[ \frac{C \log \left (\sqrt [3]{-\frac{a}{b}}-x\right )}{b}-\frac{2 C \tan ^{-1}\left (\frac{\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}+1}{\sqrt{3}}\right )}{\sqrt{3} b} \]
Antiderivative was successfully verified.
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Rule 1869
Rule 31
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{2 \left (-\frac{a}{b}\right )^{2/3} C+C x^2}{a+b x^3} \, dx &=-\frac{C \int \frac{1}{\sqrt [3]{-\frac{a}{b}}-x} \, dx}{b}-\frac{\left (\sqrt [3]{-\frac{a}{b}} C\right ) \int \frac{1}{\left (-\frac{a}{b}\right )^{2/3}+\sqrt [3]{-\frac{a}{b}} x+x^2} \, dx}{b}\\ &=\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}-x\right )}{b}+\frac{(2 C) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}\right )}{b}\\ &=-\frac{2 C \tan ^{-1}\left (\frac{1+\frac{2 x}{\sqrt [3]{-\frac{a}{b}}}}{\sqrt{3}}\right )}{\sqrt{3} b}+\frac{C \log \left (\sqrt [3]{-\frac{a}{b}}-x\right )}{b}\\ \end{align*}
Mathematica [B] time = 0.0465669, size = 149, normalized size = 2.76 \[ \frac{C \left (-b^{2/3} \left (-\frac{a}{b}\right )^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+a^{2/3} \log \left (a+b x^3\right )+2 b^{2/3} \left (-\frac{a}{b}\right )^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} b^{2/3} \left (-\frac{a}{b}\right )^{2/3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{3 a^{2/3} b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 132, normalized size = 2.4 \begin{align*}{\frac{2\,C}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\frac{2}{3}}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{C}{3\,b} \left ( -{\frac{a}{b}} \right ) ^{{\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} \left ( -{\frac{a}{b}} \right ) ^{{\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\,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 = 0.982751, size = 140, normalized size = 2.59 \begin{align*} \frac{2 \, \sqrt{3} C \arctan \left (\frac{2 \, \sqrt{3} b x \left (-\frac{a}{b}\right )^{\frac{2}{3}} - \sqrt{3} a}{3 \, a}\right ) + 3 \, C \log \left (x - \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 0.523065, size = 109, normalized size = 2.02 \begin{align*} \frac{C \left (\log{\left (\frac{a}{b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} + x \right )} - \frac{\sqrt{3} i \log{\left (- \frac{a}{2 b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} - \frac{\sqrt{3} i a}{2 b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} + x \right )}}{3} + \frac{\sqrt{3} i \log{\left (- \frac{a}{2 b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} + \frac{\sqrt{3} i a}{2 b \left (- \frac{a}{b}\right )^{\frac{2}{3}}} + x \right )}}{3}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09283, size = 123, normalized size = 2.28 \begin{align*} -\frac{2 \, \sqrt{3} C \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-\frac{a}{b}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-\frac{a}{b}\right )^{\frac{1}{3}}}\right )}{3 \, b} - \frac{{\left (C b^{2} \left (-\frac{a}{b}\right )^{\frac{2}{3}} + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} C\right )} \left (-\frac{a}{b}\right )^{\frac{1}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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