Optimal. Leaf size=145 \[ -\frac{(b c-a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{4/3}}+\frac{(b c-a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{4/3}}+\frac{d x}{b} \]
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Rubi [A] time = 0.0780682, antiderivative size = 145, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.412, Rules used = {388, 200, 31, 634, 617, 204, 628} \[ -\frac{(b c-a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{4/3}}+\frac{(b c-a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{4/3}}+\frac{d x}{b} \]
Antiderivative was successfully verified.
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Rule 388
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{c+d x^3}{a+b x^3} \, dx &=\frac{d x}{b}-\frac{(-b c+a d) \int \frac{1}{a+b x^3} \, dx}{b}\\ &=\frac{d x}{b}+\frac{(b c-a d) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{2/3} b}+\frac{(b c-a d) \int \frac{2 \sqrt [3]{a}-\sqrt [3]{b} x}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{3 a^{2/3} b}\\ &=\frac{d x}{b}+\frac{(b c-a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\frac{(b c-a d) \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 a^{2/3} b^{4/3}}+\frac{(b c-a d) \int \frac{1}{a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2} \, dx}{2 \sqrt [3]{a} b}\\ &=\frac{d x}{b}+\frac{(b c-a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\frac{(b c-a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{4/3}}+\frac{(b c-a d) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{a^{2/3} b^{4/3}}\\ &=\frac{d x}{b}-\frac{(b c-a d) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{4/3}}+\frac{(b c-a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\frac{(b c-a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{4/3}}\\ \end{align*}
Mathematica [A] time = 0.0647341, size = 129, normalized size = 0.89 \[ \frac{-(b c-a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+6 a^{2/3} \sqrt [3]{b} d x+2 (b c-a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} (b c-a d) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{6 a^{2/3} b^{4/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 195, normalized size = 1.3 \begin{align*}{\frac{dx}{b}}-{\frac{ad}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{c}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\frac{ad}{6\,{b}^{2}}\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{c}{6\,b}\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{\sqrt{3}ad}{3\,{b}^{2}}\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{\sqrt{3}c}{3\,b}\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}}}} \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 = 1.71623, size = 926, normalized size = 6.39 \begin{align*} \left [\frac{6 \, a^{2} b d x - 3 \, \sqrt{\frac{1}{3}}{\left (a b^{2} c - a^{2} b d\right )} \sqrt{\frac{\left (-a^{2} b\right )^{\frac{1}{3}}}{b}} \log \left (\frac{2 \, a b x^{3} + 3 \, \left (-a^{2} b\right )^{\frac{1}{3}} a x - a^{2} - 3 \, \sqrt{\frac{1}{3}}{\left (2 \, a b x^{2} + \left (-a^{2} b\right )^{\frac{2}{3}} x + \left (-a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{\frac{\left (-a^{2} b\right )^{\frac{1}{3}}}{b}}}{b x^{3} + a}\right ) - \left (-a^{2} b\right )^{\frac{2}{3}}{\left (b c - a d\right )} \log \left (a b x^{2} - \left (-a^{2} b\right )^{\frac{2}{3}} x - \left (-a^{2} b\right )^{\frac{1}{3}} a\right ) + 2 \, \left (-a^{2} b\right )^{\frac{2}{3}}{\left (b c - a d\right )} \log \left (a b x + \left (-a^{2} b\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b^{2}}, \frac{6 \, a^{2} b d x + 6 \, \sqrt{\frac{1}{3}}{\left (a b^{2} c - a^{2} b d\right )} \sqrt{-\frac{\left (-a^{2} b\right )^{\frac{1}{3}}}{b}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (-a^{2} b\right )^{\frac{2}{3}} x + \left (-a^{2} b\right )^{\frac{1}{3}} a\right )} \sqrt{-\frac{\left (-a^{2} b\right )^{\frac{1}{3}}}{b}}}{a^{2}}\right ) - \left (-a^{2} b\right )^{\frac{2}{3}}{\left (b c - a d\right )} \log \left (a b x^{2} - \left (-a^{2} b\right )^{\frac{2}{3}} x - \left (-a^{2} b\right )^{\frac{1}{3}} a\right ) + 2 \, \left (-a^{2} b\right )^{\frac{2}{3}}{\left (b c - a d\right )} \log \left (a b x + \left (-a^{2} b\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.780621, size = 71, normalized size = 0.49 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{4} + a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}, \left ( t \mapsto t \log{\left (- \frac{3 t a b}{a d - b c} + x \right )} \right )\right )} + \frac{d x}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09841, size = 217, normalized size = 1.5 \begin{align*} \frac{d x}{b} - \frac{{\left (b c - a d\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} + \frac{\sqrt{3}{\left (\left (-a b^{2}\right )^{\frac{1}{3}} b c - \left (-a b^{2}\right )^{\frac{1}{3}} a d\right )} \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 (\left (-a b^{2}\right )^{\frac{1}{3}} b c - \left (-a b^{2}\right )^{\frac{1}{3}} a d\right )} \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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