Optimal. Leaf size=634 \[ \frac{\left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}}-\frac{\left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}}-\frac{\left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}} \]
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Rubi [A] time = 0.727749, antiderivative size = 634, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 7, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.304, Rules used = {1510, 292, 31, 634, 617, 204, 628} \[ \frac{\left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{b-\sqrt{b^2-4 a c}}+\left (b-\sqrt{b^2-4 a c}\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (-\sqrt [3]{2} \sqrt [3]{c} x \sqrt [3]{\sqrt{b^2-4 a c}+b}+\left (\sqrt{b^2-4 a c}+b\right )^{2/3}+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}}-\frac{\left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{\sqrt{b^2-4 a c}+b}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}}-\frac{\left (\frac{2 c d-b e}{\sqrt{b^2-4 a c}}+e\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{\sqrt{b^2-4 a c}+b}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt [3]{\sqrt{b^2-4 a c}+b}} \]
Antiderivative was successfully verified.
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Rule 1510
Rule 292
Rule 31
Rule 634
Rule 617
Rule 204
Rule 628
Rubi steps
\begin{align*} \int \frac{x \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx &=\frac{1}{2} \left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{x}{\frac{b}{2}+\frac{1}{2} \sqrt{b^2-4 a c}+c x^3} \, dx+\frac{1}{2} \left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{x}{\frac{b}{2}-\frac{1}{2} \sqrt{b^2-4 a c}+c x^3} \, dx\\ &=-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3\ 2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}}}+\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{\frac{\sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3\ 2^{2/3} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}}}-\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x} \, dx}{3\ 2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{\frac{\sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+\sqrt [3]{c} x}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{3\ 2^{2/3} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}}}\\ &=-\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}+\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{4 \sqrt [3]{c}}+\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac{\left (b+\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{6\ 2^{2/3} c^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}+\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{1}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{4 \sqrt [3]{c}}+\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \int \frac{-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}}}{\sqrt [3]{2}}+2 c^{2/3} x}{\frac{\left (b-\sqrt{b^2-4 a c}\right )^{2/3}}{2^{2/3}}-\frac{\sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x}{\sqrt [3]{2}}+c^{2/3} x^2} \, dx}{6\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}\\ &=-\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}+\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\left (b-\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\left (b+\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}+\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt{b^2-4 a c}}}\right )}{2^{2/3} c^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}+\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \operatorname{Subst}\left (\int \frac{1}{-3-x^2} \, dx,x,1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}\right )}{2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}\\ &=-\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{1-\frac{2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt{b^2-4 a c}}}}{\sqrt{3}}\right )}{2^{2/3} \sqrt{3} c^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}-\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b-\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}-\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\sqrt [3]{b+\sqrt{b^2-4 a c}}+\sqrt [3]{2} \sqrt [3]{c} x\right )}{3\ 2^{2/3} c^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}+\frac{\left (e+\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\left (b-\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt [3]{b-\sqrt{b^2-4 a c}}}+\frac{\left (e-\frac{2 c d-b e}{\sqrt{b^2-4 a c}}\right ) \log \left (\left (b+\sqrt{b^2-4 a c}\right )^{2/3}-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt{b^2-4 a c}} x+2^{2/3} c^{2/3} x^2\right )}{6\ 2^{2/3} c^{2/3} \sqrt [3]{b+\sqrt{b^2-4 a c}}}\\ \end{align*}
Mathematica [C] time = 0.0312528, size = 59, normalized size = 0.09 \[ \frac{1}{3} \text{RootSum}\left [\text{$\#$1}^3 b+\text{$\#$1}^6 c+a\& ,\frac{\text{$\#$1}^3 e \log (x-\text{$\#$1})+d \log (x-\text{$\#$1})}{2 \text{$\#$1}^4 c+\text{$\#$1} b}\& \right ] \]
Antiderivative was successfully verified.
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Maple [C] time = 0.002, size = 49, normalized size = 0.1 \begin{align*}{\frac{1}{3}\sum _{{\it \_R}={\it RootOf} \left ({{\it \_Z}}^{6}c+{{\it \_Z}}^{3}b+a \right ) }{\frac{ \left ({{\it \_R}}^{4}e+{\it \_R}\,d \right ) \ln \left ( x-{\it \_R} \right ) }{2\,{{\it \_R}}^{5}c+{{\it \_R}}^{2}b}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{3} + d\right )} x}{c x^{6} + b x^{3} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 133.311, size = 920, normalized size = 1.45 \begin{align*} \operatorname{RootSum}{\left (t^{6} \left (46656 a^{4} c^{5} - 34992 a^{3} b^{2} c^{4} + 8748 a^{2} b^{4} c^{3} - 729 a b^{6} c^{2}\right ) + t^{3} \left (- 432 a^{3} b c^{2} e^{3} + 1296 a^{3} c^{3} d e^{2} + 216 a^{2} b^{3} c e^{3} - 648 a^{2} b^{2} c^{2} d e^{2} - 432 a^{2} c^{4} d^{3} - 27 a b^{5} e^{3} + 81 a b^{4} c d e^{2} + 216 a b^{2} c^{3} d^{3} - 27 b^{4} c^{2} d^{3}\right ) + 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 (x + \frac{15552 t^{5} a^{5} c^{5} e^{2} - 11664 t^{5} a^{4} b^{2} c^{4} e^{2} - 15552 t^{5} a^{4} c^{6} d^{2} + 2916 t^{5} a^{3} b^{4} c^{3} e^{2} + 11664 t^{5} a^{3} b^{2} c^{5} d^{2} - 243 t^{5} a^{2} b^{6} c^{2} e^{2} - 2916 t^{5} a^{2} b^{4} c^{4} d^{2} + 243 t^{5} a b^{6} c^{3} d^{2} - 108 t^{2} a^{4} b c^{2} e^{5} + 360 t^{2} a^{4} c^{3} d e^{4} + 63 t^{2} a^{3} b^{3} c e^{5} - 270 t^{2} a^{3} b^{2} c^{2} d e^{4} + 360 t^{2} a^{3} b c^{3} d^{2} e^{3} - 720 t^{2} a^{3} c^{4} d^{3} e^{2} - 9 t^{2} a^{2} b^{5} e^{5} + 45 t^{2} a^{2} b^{4} c d e^{4} - 90 t^{2} a^{2} b^{3} c^{2} d^{2} e^{3} + 180 t^{2} a^{2} b^{2} c^{3} d^{3} e^{2} + 180 t^{2} a^{2} b c^{4} d^{4} e + 72 t^{2} a^{2} c^{5} d^{5} - 45 t^{2} a b^{3} c^{3} d^{4} e - 54 t^{2} a b^{2} c^{4} d^{5} + 9 t^{2} b^{4} c^{3} d^{5}}{2 a^{4} c e^{7} - a^{3} b^{2} e^{7} - a^{3} b c d e^{6} - 2 a^{3} c^{2} d^{2} e^{5} + 2 a^{2} b^{3} d e^{6} - 6 a^{2} b^{2} c d^{2} e^{5} + 15 a^{2} b c^{2} d^{3} e^{4} - 10 a^{2} c^{3} d^{4} e^{3} - a b^{4} d^{2} e^{5} + 5 a b^{3} c d^{3} e^{4} - 15 a b^{2} c^{2} d^{4} e^{3} + 17 a b c^{3} d^{5} e^{2} - 6 a c^{4} d^{6} e + b^{3} c^{2} d^{5} e^{2} - 2 b^{2} c^{3} d^{6} e + b c^{4} d^{7}} \right )} \right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x^{3} + d\right )} x}{c x^{6} + b x^{3} + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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