Optimal. Leaf size=134 \[ \frac{a \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac{a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac{a \tan ^{-1}\left (\frac{\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} \sqrt [3]{e}}-\frac{c \log \left (d-e x^3\right )}{3 e} \]
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Rubi [A] time = 0.0891273, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {1871, 12, 200, 31, 634, 617, 204, 628, 260} \[ \frac{a \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac{a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac{a \tan ^{-1}\left (\frac{\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} \sqrt [3]{e}}-\frac{c \log \left (d-e x^3\right )}{3 e} \]
Antiderivative was successfully verified.
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Rule 1871
Rule 12
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 260
Rubi steps
\begin{align*} \int \frac{a+c x^2}{d-e x^3} \, dx &=c \int \frac{x^2}{d-e x^3} \, dx+\int \frac{a}{d-e x^3} \, dx\\ &=-\frac{c \log \left (d-e x^3\right )}{3 e}+a \int \frac{1}{d-e x^3} \, dx\\ &=-\frac{c \log \left (d-e x^3\right )}{3 e}+\frac{a \int \frac{1}{\sqrt [3]{d}-\sqrt [3]{e} x} \, dx}{3 d^{2/3}}+\frac{a \int \frac{2 \sqrt [3]{d}+\sqrt [3]{e} x}{d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{3 d^{2/3}}\\ &=-\frac{a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}-\frac{c \log \left (d-e x^3\right )}{3 e}+\frac{a \int \frac{1}{d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{2 \sqrt [3]{d}}+\frac{a \int \frac{\sqrt [3]{d} \sqrt [3]{e}+2 e^{2/3} x}{d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{6 d^{2/3} \sqrt [3]{e}}\\ &=-\frac{a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac{a \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac{c \log \left (d-e x^3\right )}{3 e}-\frac{a \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1+\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{d^{2/3} \sqrt [3]{e}}\\ &=\frac{a \tan ^{-1}\left (\frac{\sqrt [3]{d}+2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} \sqrt [3]{e}}-\frac{a \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )}{3 d^{2/3} \sqrt [3]{e}}+\frac{a \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} \sqrt [3]{e}}-\frac{c \log \left (d-e x^3\right )}{3 e}\\ \end{align*}
Mathematica [A] time = 0.0299887, size = 123, normalized size = 0.92 \[ \frac{a e^{2/3} \log \left (d^{2/3}+\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )-2 a e^{2/3} \log \left (\sqrt [3]{d}-\sqrt [3]{e} x\right )+2 \sqrt{3} a e^{2/3} \tan ^{-1}\left (\frac{\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}+1}{\sqrt{3}}\right )-2 c d^{2/3} \log \left (d-e x^3\right )}{6 d^{2/3} e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.041, size = 111, normalized size = 0.8 \begin{align*} -{\frac{a}{3\,e}\ln \left ( x-\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a}{6\,e}\ln \left ({x}^{2}+\sqrt [3]{{\frac{d}{e}}}x+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{a\sqrt{3}}{3\,e}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+1 \right ) } \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{c\ln \left ( e{x}^{3}-d \right ) }{3\,e}} \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 [C] time = 6.29991, size = 2820, normalized size = 21.04 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.413526, size = 70, normalized size = 0.52 \begin{align*} - \operatorname{RootSum}{\left (27 t^{3} d^{2} e^{3} - 27 t^{2} c d^{2} e^{2} + 9 t c^{2} d^{2} e - a^{3} e^{2} - c^{3} d^{2}, \left ( t \mapsto t \log{\left (x + \frac{- 3 t d e + c d}{a e} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09519, size = 128, normalized size = 0.96 \begin{align*} -\frac{1}{3} \, c e^{\left (-1\right )} \log \left ({\left | x^{3} e - d \right |}\right ) + \frac{\sqrt{3} a \arctan \left (\frac{\sqrt{3}{\left (d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + 2 \, x\right )} e^{\frac{1}{3}}}{3 \, d^{\frac{1}{3}}}\right ) e^{\left (-\frac{1}{3}\right )}}{3 \, d^{\frac{2}{3}}} + \frac{a e^{\left (-\frac{1}{3}\right )} \log \left (d^{\frac{1}{3}} x e^{\left (-\frac{1}{3}\right )} + x^{2} + d^{\frac{2}{3}} e^{\left (-\frac{2}{3}\right )}\right )}{6 \, d^{\frac{2}{3}}} - \frac{a e^{\left (-\frac{1}{3}\right )} \log \left ({\left | -d^{\frac{1}{3}} e^{\left (-\frac{1}{3}\right )} + x \right |}\right )}{3 \, d^{\frac{2}{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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