Optimal. Leaf size=118 \[ \frac{B \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}-\frac{B \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac{B \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} b^{2/3}} \]
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Rubi [A] time = 0.126368, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {260, 1593, 1871, 12, 292, 31, 634, 617, 204, 628} \[ \frac{B \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}-\frac{B \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}-\frac{B \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} b^{2/3}} \]
Antiderivative was successfully verified.
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Rule 260
Rule 1593
Rule 1871
Rule 12
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \left (-\frac{C x^2}{a+b x^3}+\frac{B x+C x^2}{a+b x^3}\right ) \, dx &=-\left (C \int \frac{x^2}{a+b x^3} \, dx\right )+\int \frac{B x+C x^2}{a+b x^3} \, dx\\ &=-\frac{C \log \left (a+b x^3\right )}{3 b}+\int \frac{x (B+C x)}{a+b x^3} \, dx\\ &=-\frac{C \log \left (a+b x^3\right )}{3 b}+C \int \frac{x^2}{a+b x^3} \, dx+\int \frac{B x}{a+b x^3} \, dx\\ &=B \int \frac{x}{a+b x^3} \, dx\\ &=-\frac{B \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 \sqrt [3]{a} \sqrt [3]{b}}+\frac{B \int \frac{\sqrt [3]{a}+\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\\ &=-\frac{B \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}+\frac{B \int \frac{-\sqrt [3]{a} \sqrt [3]{b}+2 b^{2/3} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{6 \sqrt [3]{a} b^{2/3}}+\frac{B \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \sqrt [3]{b}}\\ &=-\frac{B \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}+\frac{B \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}+\frac{B \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{a} b^{2/3}}\\ &=-\frac{B \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} \sqrt [3]{a} b^{2/3}}-\frac{B \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}+\frac{B \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} b^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.011431, size = 90, normalized size = 0.76 \[ \frac{B \left (\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )\right )}{6 \sqrt [3]{a} b^{2/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 94, normalized size = 0.8 \begin{align*} -{\frac{B}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{6\,b}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B\sqrt{3}}{3\,b}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{a}{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 = 0.99017, size = 760, normalized size = 6.44 \begin{align*} \left [\frac{3 \, \sqrt{\frac{1}{3}} B a b \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \log \left (\frac{2 \, b^{2} x^{3} - a b + 3 \, \sqrt{\frac{1}{3}}{\left (a b x + 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} - 3 \, \left (-a b^{2}\right )^{\frac{2}{3}} x}{b x^{3} + a}\right ) + \left (-a b^{2}\right )^{\frac{2}{3}} B \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} b x + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} B \log \left (b x - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{6 \, a b^{2}}, \frac{6 \, \sqrt{\frac{1}{3}} B a b \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, b x + \left (-a b^{2}\right )^{\frac{1}{3}}\right )} \sqrt{-\frac{\left (-a b^{2}\right )^{\frac{1}{3}}}{a}}}{b}\right ) + \left (-a b^{2}\right )^{\frac{2}{3}} B \log \left (b^{2} x^{2} + \left (-a b^{2}\right )^{\frac{1}{3}} b x + \left (-a b^{2}\right )^{\frac{2}{3}}\right ) - 2 \, \left (-a b^{2}\right )^{\frac{2}{3}} B \log \left (b x - \left (-a b^{2}\right )^{\frac{1}{3}}\right )}{6 \, a b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.134989, size = 26, normalized size = 0.22 \begin{align*} B \operatorname{RootSum}{\left (27 t^{3} a b^{2} + 1, \left ( t \mapsto t \log{\left (9 t^{2} a b + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07905, size = 155, normalized size = 1.31 \begin{align*} -\frac{B \left (-\frac{a}{b}\right )^{\frac{2}{3}} \log \left ({\left | x - \left (-\frac{a}{b}\right )^{\frac{1}{3}} \right |}\right )}{3 \, a} - \frac{\sqrt{3} \left (-a b^{2}\right )^{\frac{2}{3}} B \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 \, a b^{2}} + \frac{\left (-a b^{2}\right )^{\frac{2}{3}} B \log \left (x^{2} + x \left (-\frac{a}{b}\right )^{\frac{1}{3}} + \left (-\frac{a}{b}\right )^{\frac{2}{3}}\right )}{6 \, a b^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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