Optimal. Leaf size=213 \[ \frac{x \left (a e^2-b d e+c d^2\right )}{3 d e^2 \left (d+e x^3\right )}+\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (4 c d^2-e (2 a e+b d)\right )}{18 d^{5/3} e^{7/3}}-\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (4 c d^2-e (2 a e+b d)\right )}{9 d^{5/3} e^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (4 c d^2-e (2 a e+b d)\right )}{3 \sqrt{3} d^{5/3} e^{7/3}}+\frac{c x}{e^2} \]
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Rubi [A] time = 0.225653, antiderivative size = 213, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {1409, 388, 200, 31, 634, 617, 204, 628} \[ \frac{x \left (a e^2-b d e+c d^2\right )}{3 d e^2 \left (d+e x^3\right )}+\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (4 c d^2-e (2 a e+b d)\right )}{18 d^{5/3} e^{7/3}}-\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (4 c d^2-e (2 a e+b d)\right )}{9 d^{5/3} e^{7/3}}+\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (4 c d^2-e (2 a e+b d)\right )}{3 \sqrt{3} d^{5/3} e^{7/3}}+\frac{c x}{e^2} \]
Antiderivative was successfully verified.
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Rule 1409
Rule 388
Rule 200
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{a+b x^3+c x^6}{\left (d+e x^3\right )^2} \, dx &=\frac{\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^3\right )}-\frac{\int \frac{c d^2-e (b d+2 a e)-3 c d e x^3}{d+e x^3} \, dx}{3 d e^2}\\ &=\frac{c x}{e^2}+\frac{\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^3\right )}-\frac{\left (4 c d^2-e (b d+2 a e)\right ) \int \frac{1}{d+e x^3} \, dx}{3 d e^2}\\ &=\frac{c x}{e^2}+\frac{\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^3\right )}-\frac{\left (4 c d^2-e (b d+2 a e)\right ) \int \frac{1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{9 d^{5/3} e^2}-\frac{\left (4 c d^2-e (b d+2 a e)\right ) \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}{9 d^{5/3} e^2}\\ &=\frac{c x}{e^2}+\frac{\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^3\right )}-\frac{\left (4 c d^2-e (b d+2 a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} e^{7/3}}+\frac{\left (4 c d^2-e (b d+2 a e)\right ) \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}{18 d^{5/3} e^{7/3}}-\frac{\left (4 c d^2-e (b d+2 a e)\right ) \int \frac{1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{6 d^{4/3} e^2}\\ &=\frac{c x}{e^2}+\frac{\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^3\right )}-\frac{\left (4 c d^2-e (b d+2 a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} e^{7/3}}+\frac{\left (4 c d^2-e (b d+2 a e)\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{18 d^{5/3} e^{7/3}}-\frac{\left (4 c d^2-e (b d+2 a e)\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}\right )}{3 d^{5/3} e^{7/3}}\\ &=\frac{c x}{e^2}+\frac{\left (c d^2-b d e+a e^2\right ) x}{3 d e^2 \left (d+e x^3\right )}+\frac{\left (4 c d^2-e (b d+2 a e)\right ) \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{3 \sqrt{3} d^{5/3} e^{7/3}}-\frac{\left (4 c d^2-e (b d+2 a e)\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{5/3} e^{7/3}}+\frac{\left (4 c d^2-e (b d+2 a e)\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{18 d^{5/3} e^{7/3}}\\ \end{align*}
Mathematica [A] time = 0.19949, size = 199, normalized size = 0.93 \[ \frac{\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (4 c d^2-e (2 a e+b d)\right )}{d^{5/3}}+\frac{6 \sqrt [3]{e} x \left (e (a e-b d)+c d^2\right )}{d \left (d+e x^3\right )}-\frac{2 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (4 c d^2-e (2 a e+b d)\right )}{d^{5/3}}+\frac{2 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt{3}}\right ) \left (4 c d^2-e (2 a e+b d)\right )}{d^{5/3}}+18 c \sqrt [3]{e} x}{18 e^{7/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 345, normalized size = 1.6 \begin{align*}{\frac{cx}{{e}^{2}}}+{\frac{ax}{3\,d \left ( e{x}^{3}+d \right ) }}-{\frac{bx}{3\,e \left ( e{x}^{3}+d \right ) }}+{\frac{dxc}{3\,{e}^{2} \left ( e{x}^{3}+d \right ) }}+{\frac{2\,a}{9\,de}\ln \left ( x+\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{b}{9\,{e}^{2}}\ln \left ( x+\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{4\,cd}{9\,{e}^{3}}\ln \left ( x+\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{a}{9\,de}\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{b}{18\,{e}^{2}}\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{2\,cd}{9\,{e}^{3}}\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{2\,\sqrt{3}a}{9\,de}\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{\sqrt{3}b}{9\,{e}^{2}}\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{4\,d\sqrt{3}c}{9\,{e}^{3}}\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}}}} \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 = 1.38885, size = 1548, normalized size = 7.27 \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 = 1.98364, size = 206, normalized size = 0.97 \begin{align*} \frac{c x}{e^{2}} + \frac{x \left (a e^{2} - b d e + c d^{2}\right )}{3 d^{2} e^{2} + 3 d e^{3} x^{3}} + \operatorname{RootSum}{\left (729 t^{3} d^{5} e^{7} - 8 a^{3} e^{6} - 12 a^{2} b d e^{5} + 48 a^{2} c d^{2} e^{4} - 6 a b^{2} d^{2} e^{4} + 48 a b c d^{3} e^{3} - 96 a c^{2} d^{4} e^{2} - b^{3} d^{3} e^{3} + 12 b^{2} c d^{4} e^{2} - 48 b c^{2} d^{5} e + 64 c^{3} d^{6}, \left ( t \mapsto t \log{\left (\frac{9 t d^{2} e^{2}}{2 a e^{2} + b d e - 4 c d^{2}} + x \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12166, size = 306, normalized size = 1.44 \begin{align*} c x e^{\left (-2\right )} - \frac{\sqrt{3}{\left (4 \, \left (-d e^{2}\right )^{\frac{1}{3}} c d^{2} - \left (-d e^{2}\right )^{\frac{1}{3}} b d e - 2 \, \left (-d e^{2}\right )^{\frac{1}{3}} a e^{2}\right )} \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 (-3\right )}}{9 \, d^{2}} + \frac{{\left (4 \, c d^{2} - b d e - 2 \, a e^{2}\right )} \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} e^{\left (-2\right )} \log \left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} \right |}\right )}{9 \, d^{2}} - \frac{{\left (4 \, \left (-d e^{2}\right )^{\frac{1}{3}} c d^{2} - \left (-d e^{2}\right )^{\frac{1}{3}} b d e - 2 \, \left (-d e^{2}\right )^{\frac{1}{3}} a e^{2}\right )} e^{\left (-3\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 )}{18 \, d^{2}} + \frac{{\left (c d^{2} x - b d x e + a x e^{2}\right )} e^{\left (-2\right )}}{3 \,{\left (x^{3} e + d\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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