Optimal. Leaf size=305 \[ -\frac{\left (\sqrt{3} \sqrt{c} d-\sqrt{a} e\right ) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{a} e+\sqrt{3} \sqrt{c} d\right ) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}-\frac{\left (\sqrt{3} \sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{c} d-\sqrt{3} \sqrt{a} e\right ) \tan ^{-1}\left (\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{6 a^{5/6} c^{2/3}}+\frac{d \tan ^{-1}\left (\frac{\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}-\frac{e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}} \]
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Rubi [A] time = 0.248568, antiderivative size = 305, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.471, Rules used = {1416, 635, 203, 260, 634, 617, 204, 628} \[ -\frac{\left (\sqrt{3} \sqrt{c} d-\sqrt{a} e\right ) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{a} e+\sqrt{3} \sqrt{c} d\right ) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}-\frac{\left (\sqrt{3} \sqrt{a} e+\sqrt{c} d\right ) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{c} d-\sqrt{3} \sqrt{a} e\right ) \tan ^{-1}\left (\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}+\sqrt{3}\right )}{6 a^{5/6} c^{2/3}}+\frac{d \tan ^{-1}\left (\frac{\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}-\frac{e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}} \]
Antiderivative was successfully verified.
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Rule 1416
Rule 635
Rule 203
Rule 260
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{d+e x^3}{a+c x^6} \, dx &=\frac{\int \frac{\frac{2 \sqrt [3]{c} d}{\sqrt [3]{a}}-\left (\frac{\sqrt{3} \sqrt{c} d}{\sqrt{a}}-e\right ) x}{1-\frac{\sqrt{3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{6 a^{2/3} \sqrt [3]{c}}+\frac{\int \frac{\frac{2 \sqrt [3]{c} d}{\sqrt [3]{a}}+\left (\frac{\sqrt{3} \sqrt{c} d}{\sqrt{a}}+e\right ) x}{1+\frac{\sqrt{3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{6 a^{2/3} \sqrt [3]{c}}+\frac{\int \frac{\frac{\sqrt [3]{c} d}{\sqrt [3]{a}}-e x}{1+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{3 a^{2/3} \sqrt [3]{c}}\\ &=\frac{d \int \frac{1}{1+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{3 a}-\frac{e \int \frac{x}{1+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{3 a^{2/3} \sqrt [3]{c}}-\frac{\left (\sqrt{3} \sqrt{c} d-\sqrt{a} e\right ) \int \frac{-\frac{\sqrt{3} \sqrt [6]{c}}{\sqrt [6]{a}}+\frac{2 \sqrt [3]{c} x}{\sqrt [3]{a}}}{1-\frac{\sqrt{3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{12 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{3} \sqrt{c} d+\sqrt{a} e\right ) \int \frac{\frac{\sqrt{3} \sqrt [6]{c}}{\sqrt [6]{a}}+\frac{2 \sqrt [3]{c} x}{\sqrt [3]{a}}}{1+\frac{\sqrt{3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{12 a^{5/6} c^{2/3}}+\frac{\left (d-\frac{\sqrt{3} \sqrt{a} e}{\sqrt{c}}\right ) \int \frac{1}{1+\frac{\sqrt{3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{12 a}+\frac{\left (d+\frac{\sqrt{3} \sqrt{a} e}{\sqrt{c}}\right ) \int \frac{1}{1-\frac{\sqrt{3} \sqrt [6]{c} x}{\sqrt [6]{a}}+\frac{\sqrt [3]{c} x^2}{\sqrt [3]{a}}} \, dx}{12 a}\\ &=\frac{d \tan ^{-1}\left (\frac{\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}-\frac{e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}-\frac{\left (\sqrt{3} \sqrt{c} d-\sqrt{a} e\right ) \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{3} \sqrt{c} d+\sqrt{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}-\frac{\left (\sqrt{3} \sqrt{c} d-3 \sqrt{a} e\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1+\frac{2 \sqrt [6]{c} x}{\sqrt{3} \sqrt [6]{a}}\right )}{18 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{3} \sqrt{c} d+3 \sqrt{a} e\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{1}{3}-x^2} \, dx,x,1-\frac{2 \sqrt [6]{c} x}{\sqrt{3} \sqrt [6]{a}}\right )}{18 a^{5/6} c^{2/3}}\\ &=\frac{d \tan ^{-1}\left (\frac{\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}-\frac{\left (\sqrt{c} d+\sqrt{3} \sqrt{a} e\right ) \tan ^{-1}\left (\sqrt{3}-\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{c} d-\sqrt{3} \sqrt{a} e\right ) \tan ^{-1}\left (\sqrt{3}+\frac{2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{5/6} c^{2/3}}-\frac{e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}}-\frac{\left (\sqrt{3} \sqrt{c} d-\sqrt{a} e\right ) \log \left (\sqrt [3]{a}-\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}+\frac{\left (\sqrt{3} \sqrt{c} d+\sqrt{a} e\right ) \log \left (\sqrt [3]{a}+\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{12 a^{5/6} c^{2/3}}\\ \end{align*}
Mathematica [A] time = 0.0981448, size = 334, normalized size = 1.1 \[ -\frac{\left (\sqrt{3} \sqrt [6]{a} \sqrt{c} d-a^{2/3} e\right ) \log \left (-\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a c^{2/3}}-\frac{\left (-a^{2/3} e-\sqrt{3} \sqrt [6]{a} \sqrt{c} d\right ) \log \left (\sqrt{3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{12 a c^{2/3}}+\frac{\left (\sqrt{3} a^{2/3} e+\sqrt [6]{a} \sqrt{c} d\right ) \tan ^{-1}\left (\frac{2 \sqrt [6]{c} x-\sqrt{3} \sqrt [6]{a}}{\sqrt [6]{a}}\right )}{6 a c^{2/3}}+\frac{\left (\sqrt [6]{a} \sqrt{c} d-\sqrt{3} a^{2/3} e\right ) \tan ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a}+2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a c^{2/3}}+\frac{d \tan ^{-1}\left (\frac{\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{5/6} \sqrt [6]{c}}-\frac{e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 \sqrt [3]{a} c^{2/3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.076, size = 329, normalized size = 1.1 \begin{align*}{\frac{c\sqrt{3}d}{12\,{a}^{2}} \left ({\frac{a}{c}} \right ) ^{{\frac{7}{6}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{c}}}x+\sqrt [3]{{\frac{a}{c}}} \right ) }+{\frac{e}{12\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}+\sqrt{3}\sqrt [6]{{\frac{a}{c}}}x+\sqrt [3]{{\frac{a}{c}}} \right ) }+{\frac{d}{6\,a}\sqrt [6]{{\frac{a}{c}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}}+\sqrt{3} \right ) }-{\frac{\sqrt{3}e}{6\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}}+\sqrt{3} \right ) }+{\frac{e}{12\,a}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{c}}}x+\sqrt [3]{{\frac{a}{c}}} \right ) \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}}-{\frac{\sqrt{3}d}{12\,a}\ln \left ({x}^{2}-\sqrt{3}\sqrt [6]{{\frac{a}{c}}}x+\sqrt [3]{{\frac{a}{c}}} \right ) \sqrt [6]{{\frac{a}{c}}}}+{\frac{\sqrt{3}e}{6\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}}-\sqrt{3} \right ) }+{\frac{d}{6\,a}\sqrt [6]{{\frac{a}{c}}}\arctan \left ( 2\,{x{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}}-\sqrt{3} \right ) }-{\frac{e}{6\,a} \left ({\frac{a}{c}} \right ) ^{{\frac{2}{3}}}\ln \left ({x}^{2}+\sqrt [3]{{\frac{a}{c}}} \right ) }+{\frac{d}{3\,a}\sqrt [6]{{\frac{a}{c}}}\arctan \left ({x{\frac{1}{\sqrt [6]{{\frac{a}{c}}}}}} \right ) } \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 = 2.60776, size = 6589, normalized size = 21.6 \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 = 2.13879, size = 165, normalized size = 0.54 \begin{align*} \operatorname{RootSum}{\left (46656 t^{6} a^{5} c^{4} + t^{3} \left (432 a^{4} c^{2} e^{3} - 1296 a^{3} c^{3} d^{2} e\right ) + a^{3} e^{6} + 3 a^{2} c d^{2} e^{4} + 3 a c^{2} d^{4} e^{2} + c^{3} d^{6}, \left ( t \mapsto t \log{\left (x + \frac{- 1296 t^{4} a^{4} c^{2} e - 6 t a^{3} e^{4} + 36 t a^{2} c d^{2} e^{2} - 6 t a c^{2} d^{4}}{3 a^{2} d e^{4} + 2 a c d^{3} e^{2} - c^{2} d^{5}} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12413, size = 397, normalized size = 1.3 \begin{align*} \frac{\left (a c^{5}\right )^{\frac{1}{6}} d \arctan \left (\frac{x}{\left (\frac{a}{c}\right )^{\frac{1}{6}}}\right )}{3 \, a c} - \frac{\left (a c^{5}\right )^{\frac{2}{3}}{\left | c \right |} e \log \left (x^{2} + \left (\frac{a}{c}\right )^{\frac{1}{3}}\right )}{6 \, a c^{5}} + \frac{{\left (\left (a c^{5}\right )^{\frac{1}{6}} c^{3} d - \sqrt{3} \left (a c^{5}\right )^{\frac{2}{3}} e\right )} \arctan \left (\frac{2 \, x + \sqrt{3} \left (\frac{a}{c}\right )^{\frac{1}{6}}}{\left (\frac{a}{c}\right )^{\frac{1}{6}}}\right )}{6 \, a c^{4}} + \frac{{\left (\left (a c^{5}\right )^{\frac{1}{6}} c^{3} d + \sqrt{3} \left (a c^{5}\right )^{\frac{2}{3}} e\right )} \arctan \left (\frac{2 \, x - \sqrt{3} \left (\frac{a}{c}\right )^{\frac{1}{6}}}{\left (\frac{a}{c}\right )^{\frac{1}{6}}}\right )}{6 \, a c^{4}} + \frac{{\left (\sqrt{3} \left (a c^{5}\right )^{\frac{1}{6}} c^{3} d + \left (a c^{5}\right )^{\frac{2}{3}} e\right )} \log \left (x^{2} + \sqrt{3} x \left (\frac{a}{c}\right )^{\frac{1}{6}} + \left (\frac{a}{c}\right )^{\frac{1}{3}}\right )}{12 \, a c^{4}} - \frac{{\left (\sqrt{3} \left (a c^{5}\right )^{\frac{1}{6}} c^{3} d - \left (a c^{5}\right )^{\frac{2}{3}} e\right )} \log \left (x^{2} - \sqrt{3} x \left (\frac{a}{c}\right )^{\frac{1}{6}} + \left (\frac{a}{c}\right )^{\frac{1}{3}}\right )}{12 \, a c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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