Optimal. Leaf size=134 \[ \frac{b \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 \sqrt [3]{d} e^{2/3}}-\frac{b \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 \sqrt [3]{d} e^{2/3}}-\frac{b \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} \sqrt [3]{d} e^{2/3}}+\frac{c \log \left (d+e x^3\right )}{3 e} \]
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Rubi [A] time = 0.113145, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 10, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.526, Rules used = {1593, 1871, 12, 292, 31, 634, 617, 204, 628, 260} \[ \frac{b \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 \sqrt [3]{d} e^{2/3}}-\frac{b \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 \sqrt [3]{d} e^{2/3}}-\frac{b \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} \sqrt [3]{d} e^{2/3}}+\frac{c \log \left (d+e x^3\right )}{3 e} \]
Antiderivative was successfully verified.
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Rule 1593
Rule 1871
Rule 12
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rule 260
Rubi steps
\begin{align*} \int \frac{b x+c x^2}{d+e x^3} \, dx &=\int \frac{x (b+c x)}{d+e x^3} \, dx\\ &=c \int \frac{x^2}{d+e x^3} \, dx+\int \frac{b x}{d+e x^3} \, dx\\ &=\frac{c \log \left (d+e x^3\right )}{3 e}+b \int \frac{x}{d+e x^3} \, dx\\ &=\frac{c \log \left (d+e x^3\right )}{3 e}-\frac{b \int \frac{1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{3 \sqrt [3]{d} \sqrt [3]{e}}+\frac{b \int \frac{\sqrt [3]{d}+\sqrt [3]{e} x}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{3 \sqrt [3]{d} \sqrt [3]{e}}\\ &=-\frac{b \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 \sqrt [3]{d} e^{2/3}}+\frac{c \log \left (d+e x^3\right )}{3 e}+\frac{b \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 \sqrt [3]{d} e^{2/3}}+\frac{b \int \frac{1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{2 \sqrt [3]{e}}\\ &=-\frac{b \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 \sqrt [3]{d} e^{2/3}}+\frac{b \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 \sqrt [3]{d} e^{2/3}}+\frac{c \log \left (d+e x^3\right )}{3 e}+\frac{b \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{\sqrt [3]{d} e^{2/3}}\\ &=-\frac{b \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} \sqrt [3]{d} e^{2/3}}-\frac{b \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 \sqrt [3]{d} e^{2/3}}+\frac{b \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 \sqrt [3]{d} e^{2/3}}+\frac{c \log \left (d+e x^3\right )}{3 e}\\ \end{align*}
Mathematica [A] time = 0.0230979, size = 122, normalized size = 0.91 \[ \frac{b \sqrt [3]{e} \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )-2 b \sqrt [3]{e} \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )-2 \sqrt{3} b \sqrt [3]{e} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt{3}}\right )+2 c \sqrt [3]{d} \log \left (d+e x^3\right )}{6 \sqrt [3]{d} e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 108, normalized size = 0.8 \begin{align*} -{\frac{b}{3\,e}\ln \left ( x+\sqrt [3]{{\frac{d}{e}}} \right ){\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+{\frac{b}{6\,e}\ln \left ({x}^{2}-\sqrt [3]{{\frac{d}{e}}}x+ \left ({\frac{d}{e}} \right ) ^{{\frac{2}{3}}} \right ){\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+{\frac{b\sqrt{3}}{3\,e}\arctan \left ({\frac{\sqrt{3}}{3} \left ( 2\,{x{\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}-1 \right ) } \right ){\frac{1}{\sqrt [3]{{\frac{d}{e}}}}}}+{\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 = 6.38466, size = 2425, normalized size = 18.1 \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 = 0.299123, size = 75, normalized size = 0.56 \begin{align*} \operatorname{RootSum}{\left (27 t^{3} d e^{3} - 27 t^{2} c d e^{2} + 9 t c^{2} d e + b^{3} e - c^{3} d, \left ( t \mapsto t \log{\left (x + \frac{9 t^{2} d e^{2} - 6 t c d e + c^{2} d}{b^{2} e} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10152, size = 162, normalized size = 1.21 \begin{align*} -\frac{\sqrt{3} \left (-d e^{2}\right )^{\frac{2}{3}} b \arctan \left (\frac{\sqrt{3}{\left (2 \, x + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}\right )}}{3 \, \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}}}\right ) e^{\left (-2\right )}}{3 \, d} + \frac{1}{3} \, c e^{\left (-1\right )} \log \left ({\left | x^{3} e + d \right |}\right ) + \frac{\left (-d e^{2}\right )^{\frac{2}{3}} b e^{\left (-2\right )} \log \left (x^{2} + \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} x + \left (-d e^{\left (-1\right )}\right )^{\frac{2}{3}}\right )}{6 \, d} - \frac{\left (-d e^{\left (-1\right )}\right )^{\frac{2}{3}} b \log \left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} \right |}\right )}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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