Optimal. Leaf size=145 \[ -\frac{(A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{4/3}}+\frac{(A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\frac{(A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{4/3}}+\frac{B x}{b} \]
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Rubi [A] time = 0.0768879, 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} \[ -\frac{(A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{4/3}}+\frac{(A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\frac{(A b-a B) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3} a^{2/3} b^{4/3}}+\frac{B x}{b} \]
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{A+B x^3}{a+b x^3} \, dx &=\frac{B x}{b}-\frac{(-A b+a B) \int \frac{1}{a+b x^3} \, dx}{b}\\ &=\frac{B x}{b}+\frac{(A b-a B) \int \frac{1}{\sqrt [3]{a}+\sqrt [3]{b} x} \, dx}{3 a^{2/3} b}+\frac{(A b-a B) \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{B x}{b}+\frac{(A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\frac{(A b-a 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 a^{2/3} b^{4/3}}+\frac{(A b-a B) \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{B x}{b}+\frac{(A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\frac{(A b-a B) \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{(A b-a B) \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{B x}{b}-\frac{(A b-a B) \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{(A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{4/3}}-\frac{(A b-a B) \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 = 0.0608954, size = 129, normalized size = 0.89 \[ \frac{-(A b-a B) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+6 a^{2/3} \sqrt [3]{b} B x+2 (A b-a B) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 \sqrt{3} (A b-a B) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{6 a^{2/3} b^{4/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 195, normalized size = 1.3 \begin{align*}{\frac{Bx}{b}}+{\frac{A}{3\,b}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{Ba}{3\,{b}^{2}}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ) \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}-{\frac{A}{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{Ba}{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{\sqrt{3}A}{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}}}}-{\frac{\sqrt{3}Ba}{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}}}} \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 = 1.51415, size = 900, normalized size = 6.21 \begin{align*} \left [\frac{6 \, B a^{2} b x - 3 \, \sqrt{\frac{1}{3}}{\left (B a^{2} b - A a b^{2}\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 a - A b\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 a - A b\right )} \log \left (a b x + \left (a^{2} b\right )^{\frac{2}{3}}\right )}{6 \, a^{2} b^{2}}, \frac{6 \, B a^{2} b x - 6 \, \sqrt{\frac{1}{3}}{\left (B a^{2} b - A a b^{2}\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 a - A b\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 a - A b\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 = 1.15771, size = 71, normalized size = 0.49 \begin{align*} \frac{B x}{b} + \operatorname{RootSum}{\left (27 t^{3} a^{2} b^{4} - A^{3} b^{3} + 3 A^{2} B a b^{2} - 3 A B^{2} a^{2} b + B^{3} a^{3}, \left ( t \mapsto t \log{\left (- \frac{3 t a b}{- A b + B a} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11964, size = 217, normalized size = 1.5 \begin{align*} \frac{B x}{b} + \frac{{\left (B a - A b\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 a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\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 a - \left (-a b^{2}\right )^{\frac{1}{3}} A b\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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