Optimal. Leaf size=70 \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac{2 \left (\frac{B}{\sqrt [3]{a}}+\frac{C}{\sqrt [3]{b}}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3}} \]
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Rubi [A] time = 0.0665512, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 49, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.082, Rules used = {1863, 31, 617, 204} \[ \frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}-\frac{2 \left (\frac{B}{\sqrt [3]{a}}+\frac{C}{\sqrt [3]{b}}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
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Rule 1863
Rule 31
Rule 617
Rule 204
Rubi steps
\begin{align*} \int \frac{\sqrt [3]{a} \sqrt [3]{b} B+2 a^{2/3} C+b^{2/3} B x+b^{2/3} C x^2}{a+b x^3} \, dx &=\frac{C \int \frac{1}{\frac{\sqrt [3]{a}}{\sqrt [3]{b}}+x} \, dx}{\sqrt [3]{b}}+\frac{\left (\sqrt [3]{b} B+\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 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\left (2 \left (\frac{B}{\sqrt [3]{a}}+\frac{C}{\sqrt [3]{b}}\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )\\ &=-\frac{2 \left (\frac{B}{\sqrt [3]{a}}+\frac{C}{\sqrt [3]{b}}\right ) \tan ^{-1}\left (\frac{\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt{3} \sqrt [3]{a}}\right )}{\sqrt{3}}+\frac{C \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}\\ \end{align*}
Mathematica [A] time = 0.0464287, size = 122, normalized size = 1.74 \[ \frac{\sqrt [3]{a} C \left (-\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+\log \left (a+b x^3\right )+2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )\right )-2 \sqrt{3} \left (\sqrt [3]{a} C+\sqrt [3]{b} B\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt{3}}\right )}{3 \sqrt [3]{a} \sqrt [3]{b}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 310, normalized size = 4.4 \begin{align*}{\frac{B}{3}\sqrt [3]{a}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){b}^{-{\frac{2}{3}}} \left ({\frac{a}{b}} \right ) ^{-{\frac{2}{3}}}}+{\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{B}{6}\sqrt [3]{a}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){b}^{-{\frac{2}{3}}} \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{B\sqrt{3}}{3}\sqrt [3]{a}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){b}^{-{\frac{2}{3}}} \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{B}{3}\ln \left ( x+\sqrt [3]{{\frac{a}{b}}} \right ){\frac{1}{\sqrt [3]{b}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B}{6}\ln \left ({x}^{2}-\sqrt [3]{{\frac{a}{b}}}x+ \left ({\frac{a}{b}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{b}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\frac{B\sqrt{3}}{3}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{b}}}{\frac{1}{\sqrt [3]{{\frac{a}{b}}}}}}+{\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 [B] time = 11.3042, size = 1058, normalized size = 15.11 \begin{align*} \left [\frac{\sqrt{\frac{1}{3}} b \sqrt{-\frac{C^{2} a b^{\frac{1}{3}} + 2 \, B C a^{\frac{2}{3}} b^{\frac{2}{3}} + B^{2} a^{\frac{1}{3}} b}{a b}} \log \left (-\frac{C^{3} a^{2} + B^{3} a b - 2 \,{\left (C^{3} a b + B^{3} b^{2}\right )} x^{3} + 3 \,{\left (C^{3} a + B^{3} b\right )} a^{\frac{2}{3}} b^{\frac{1}{3}} x - 3 \, \sqrt{\frac{1}{3}}{\left ({\left (2 \, B^{2} b x^{2} + C^{2} a x + B C a\right )} a^{\frac{2}{3}} b^{\frac{2}{3}} +{\left (2 \, C^{2} a b x^{2} - B C a b x - B^{2} a b\right )} a^{\frac{1}{3}} -{\left (2 \, B C a b x^{2} - B^{2} a b x + C^{2} a^{2}\right )} b^{\frac{1}{3}}\right )} \sqrt{-\frac{C^{2} a b^{\frac{1}{3}} + 2 \, B C a^{\frac{2}{3}} b^{\frac{2}{3}} + B^{2} a^{\frac{1}{3}} b}{a b}}}{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}} b \sqrt{\frac{C^{2} a b^{\frac{1}{3}} + 2 \, B C a^{\frac{2}{3}} b^{\frac{2}{3}} + B^{2} a^{\frac{1}{3}} b}{a b}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left ({\left (2 \, C^{2} x + B C\right )} a^{\frac{2}{3}} b^{\frac{2}{3}} -{\left (2 \, B C b x + B^{2} b\right )} a^{\frac{1}{3}} +{\left (2 \, B^{2} b x - C^{2} a\right )} b^{\frac{1}{3}}\right )} \sqrt{\frac{C^{2} a b^{\frac{1}{3}} + 2 \, B C a^{\frac{2}{3}} b^{\frac{2}{3}} + B^{2} a^{\frac{1}{3}} b}{a b}}}{C^{3} a + B^{3} b}\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 [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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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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