Optimal. Leaf size=188 \[ -\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (a e^2-b d e+c d^2\right )}{6 d^{2/3} e^{7/3}}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (a e^2-b d e+c d^2\right )}{3 d^{2/3} e^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (a e^2-b d e+c d^2\right )}{\sqrt{3} d^{2/3} e^{7/3}}-\frac{x (c d-b e)}{e^2}+\frac{c x^4}{4 e} \]
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Rubi [A] time = 0.211115, antiderivative size = 188, 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 = {1411, 388, 200, 31, 634, 617, 204, 628} \[ -\frac{\log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (a e^2-b d e+c d^2\right )}{6 d^{2/3} e^{7/3}}+\frac{\log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (a e^2-b d e+c d^2\right )}{3 d^{2/3} e^{7/3}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right ) \left (a e^2-b d e+c d^2\right )}{\sqrt{3} d^{2/3} e^{7/3}}-\frac{x (c d-b e)}{e^2}+\frac{c x^4}{4 e} \]
Antiderivative was successfully verified.
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Rule 1411
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}{d+e x^3} \, dx &=\frac{c x^4}{4 e}+\frac{\int \frac{4 a e-(4 c d-4 b e) x^3}{d+e x^3} \, dx}{4 e}\\ &=-\frac{(c d-b e) x}{e^2}+\frac{c x^4}{4 e}-\left (-a-\frac{d (c d-b e)}{e^2}\right ) \int \frac{1}{d+e x^3} \, dx\\ &=-\frac{(c d-b e) x}{e^2}+\frac{c x^4}{4 e}+\frac{\left (a+\frac{d (c d-b e)}{e^2}\right ) \int \frac{1}{\sqrt [3]{d}+\sqrt [3]{e} x} \, dx}{3 d^{2/3}}+\frac{\left (a+\frac{d (c d-b e)}{e^2}\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}{3 d^{2/3}}\\ &=-\frac{(c d-b e) x}{e^2}+\frac{c x^4}{4 e}+\frac{\left (c d^2-b d e+a e^2\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac{\left (c d^2-b d e+a e^2\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}{6 d^{2/3} e^{7/3}}+\frac{\left (a+\frac{d (c d-b e)}{e^2}\right ) \int \frac{1}{d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2} \, dx}{2 \sqrt [3]{d}}\\ &=-\frac{(c d-b e) x}{e^2}+\frac{c x^4}{4 e}+\frac{\left (c d^2-b d e+a e^2\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac{\left (c d^2-b d e+a e^2\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{7/3}}+\frac{\left (c d^2-b d e+a e^2\right ) \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} e^{7/3}}\\ &=-\frac{(c d-b e) x}{e^2}+\frac{c x^4}{4 e}-\frac{\left (c d^2-b d e+a e^2\right ) \tan ^{-1}\left (\frac{\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt{3} \sqrt [3]{d}}\right )}{\sqrt{3} d^{2/3} e^{7/3}}+\frac{\left (c d^2-b d e+a e^2\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{3 d^{2/3} e^{7/3}}-\frac{\left (c d^2-b d e+a e^2\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{6 d^{2/3} e^{7/3}}\\ \end{align*}
Mathematica [A] time = 0.15456, size = 176, normalized size = 0.94 \[ \frac{-\frac{2 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right ) \left (e (a e-b d)+c d^2\right )}{d^{2/3}}+\frac{4 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right ) \left (e (a e-b d)+c d^2\right )}{d^{2/3}}-\frac{4 \sqrt{3} \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt{3}}\right ) \left (e (a e-b d)+c d^2\right )}{d^{2/3}}+12 \sqrt [3]{e} x (b e-c d)+3 c e^{4/3} x^4}{12 e^{7/3}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.004, size = 313, normalized size = 1.7 \begin{align*}{\frac{c{x}^{4}}{4\,e}}+{\frac{bx}{e}}-{\frac{cdx}{{e}^{2}}}+{\frac{a}{3\,e}\ln \left ( x+\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}-{\frac{bd}{3\,{e}^{2}}\ln \left ( x+\sqrt [3]{{\frac{d}{e}}} \right ) \left ({\frac{d}{e}} \right ) ^{-{\frac{2}{3}}}}+{\frac{c{d}^{2}}{3\,{e}^{3}}\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{bd}{6\,{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{c{d}^{2}}{6\,{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{\sqrt{3}a}{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{\sqrt{3}bd}{3\,{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{\sqrt{3}c{d}^{2}}{3\,{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 = 1.43311, size = 1096, normalized size = 5.83 \begin{align*} \left [\frac{3 \, c d^{2} e^{2} x^{4} + 6 \, \sqrt{\frac{1}{3}}{\left (c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} \sqrt{-\frac{\left (d^{2} e\right )^{\frac{1}{3}}}{e}} \log \left (\frac{2 \, d e x^{3} - 3 \, \left (d^{2} e\right )^{\frac{1}{3}} d x - d^{2} + 3 \, \sqrt{\frac{1}{3}}{\left (2 \, d e x^{2} + \left (d^{2} e\right )^{\frac{2}{3}} x - \left (d^{2} e\right )^{\frac{1}{3}} d\right )} \sqrt{-\frac{\left (d^{2} e\right )^{\frac{1}{3}}}{e}}}{e x^{3} + d}\right ) - 2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \left (d^{2} e\right )^{\frac{2}{3}} \log \left (d e x^{2} - \left (d^{2} e\right )^{\frac{2}{3}} x + \left (d^{2} e\right )^{\frac{1}{3}} d\right ) + 4 \,{\left (c d^{2} - b d e + a e^{2}\right )} \left (d^{2} e\right )^{\frac{2}{3}} \log \left (d e x + \left (d^{2} e\right )^{\frac{2}{3}}\right ) - 12 \,{\left (c d^{3} e - b d^{2} e^{2}\right )} x}{12 \, d^{2} e^{3}}, \frac{3 \, c d^{2} e^{2} x^{4} + 12 \, \sqrt{\frac{1}{3}}{\left (c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )} \sqrt{\frac{\left (d^{2} e\right )^{\frac{1}{3}}}{e}} \arctan \left (\frac{\sqrt{\frac{1}{3}}{\left (2 \, \left (d^{2} e\right )^{\frac{2}{3}} x - \left (d^{2} e\right )^{\frac{1}{3}} d\right )} \sqrt{\frac{\left (d^{2} e\right )^{\frac{1}{3}}}{e}}}{d^{2}}\right ) - 2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \left (d^{2} e\right )^{\frac{2}{3}} \log \left (d e x^{2} - \left (d^{2} e\right )^{\frac{2}{3}} x + \left (d^{2} e\right )^{\frac{1}{3}} d\right ) + 4 \,{\left (c d^{2} - b d e + a e^{2}\right )} \left (d^{2} e\right )^{\frac{2}{3}} \log \left (d e x + \left (d^{2} e\right )^{\frac{2}{3}}\right ) - 12 \,{\left (c d^{3} e - b d^{2} e^{2}\right )} x}{12 \, d^{2} e^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.07036, size = 175, normalized size = 0.93 \begin{align*} \frac{c x^{4}}{4 e} + \operatorname{RootSum}{\left (27 t^{3} d^{2} e^{7} - a^{3} e^{6} + 3 a^{2} b d e^{5} - 3 a^{2} c d^{2} e^{4} - 3 a b^{2} d^{2} e^{4} + 6 a b c d^{3} e^{3} - 3 a c^{2} d^{4} e^{2} + b^{3} d^{3} e^{3} - 3 b^{2} c d^{4} e^{2} + 3 b c^{2} d^{5} e - c^{3} d^{6}, \left ( t \mapsto t \log{\left (\frac{3 t d e^{2}}{a e^{2} - b d e + c d^{2}} + x \right )} \right )\right )} + \frac{x \left (b e - c d\right )}{e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10266, size = 279, normalized size = 1.48 \begin{align*} \frac{\sqrt{3}{\left (\left (-d e^{2}\right )^{\frac{1}{3}} c d^{2} - \left (-d e^{2}\right )^{\frac{1}{3}} b d e + \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 )}}{3 \, d} - \frac{{\left (c d^{2} e^{2} - b d e^{3} + a e^{4}\right )} \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} e^{\left (-4\right )} \log \left ({\left | x - \left (-d e^{\left (-1\right )}\right )^{\frac{1}{3}} \right |}\right )}{3 \, d} + \frac{1}{4} \,{\left (c x^{4} e^{3} - 4 \, c d x e^{2} + 4 \, b x e^{3}\right )} e^{\left (-4\right )} + \frac{{\left (\left (-d e^{2}\right )^{\frac{1}{3}} c d^{2} - \left (-d e^{2}\right )^{\frac{1}{3}} b d e + \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 )}{6 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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