Optimal. Leaf size=72 \[ \frac{e \log \left (a+b x^3+c x^6\right )}{6 c}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 c \sqrt{b^2-4 a c}} \]
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Rubi [A] time = 0.0731691, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1468, 634, 618, 206, 628} \[ \frac{e \log \left (a+b x^3+c x^6\right )}{6 c}-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 c \sqrt{b^2-4 a c}} \]
Antiderivative was successfully verified.
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Rule 1468
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^2 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{d+e x}{a+b x+c x^2} \, dx,x,x^3\right )\\ &=\frac{e \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^3\right )}{6 c}+\frac{(2 c d-b e) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^3\right )}{6 c}\\ &=\frac{e \log \left (a+b x^3+c x^6\right )}{6 c}-\frac{(2 c d-b e) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^3\right )}{3 c}\\ &=-\frac{(2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 c \sqrt{b^2-4 a c}}+\frac{e \log \left (a+b x^3+c x^6\right )}{6 c}\\ \end{align*}
Mathematica [A] time = 0.0500176, size = 71, normalized size = 0.99 \[ \frac{e \log \left (a+b x^3+c x^6\right )-\frac{2 (b e-2 c d) \tan ^{-1}\left (\frac{b+2 c x^3}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}}{6 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.002, size = 99, normalized size = 1.4 \begin{align*}{\frac{e\ln \left ( c{x}^{6}+b{x}^{3}+a \right ) }{6\,c}}+{\frac{2\,d}{3}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{be}{3\,c}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \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 = 1.44533, size = 481, normalized size = 6.68 \begin{align*} \left [\frac{{\left (b^{2} - 4 \, a c\right )} e \log \left (c x^{6} + b x^{3} + a\right ) - \sqrt{b^{2} - 4 \, a c}{\left (2 \, c d - b e\right )} \log \left (\frac{2 \, c^{2} x^{6} + 2 \, b c x^{3} + b^{2} - 2 \, a c +{\left (2 \, c x^{3} + b\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{6} + b x^{3} + a}\right )}{6 \,{\left (b^{2} c - 4 \, a c^{2}\right )}}, \frac{{\left (b^{2} - 4 \, a c\right )} e \log \left (c x^{6} + b x^{3} + a\right ) - 2 \, \sqrt{-b^{2} + 4 \, a c}{\left (2 \, c d - b e\right )} \arctan \left (-\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{6 \,{\left (b^{2} c - 4 \, a c^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.38941, size = 287, normalized size = 3.99 \begin{align*} \left (\frac{e}{6 c} - \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{6 c \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{3} + \frac{- 12 a c \left (\frac{e}{6 c} - \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{6 c \left (4 a c - b^{2}\right )}\right ) + 2 a e + 3 b^{2} \left (\frac{e}{6 c} - \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{6 c \left (4 a c - b^{2}\right )}\right ) - b d}{b e - 2 c d} \right )} + \left (\frac{e}{6 c} + \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{6 c \left (4 a c - b^{2}\right )}\right ) \log{\left (x^{3} + \frac{- 12 a c \left (\frac{e}{6 c} + \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{6 c \left (4 a c - b^{2}\right )}\right ) + 2 a e + 3 b^{2} \left (\frac{e}{6 c} + \frac{\sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{6 c \left (4 a c - b^{2}\right )}\right ) - b d}{b e - 2 c d} \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37539, size = 95, normalized size = 1.32 \begin{align*} \frac{e \log \left (c x^{6} + b x^{3} + a\right )}{6 \, c} + \frac{{\left (2 \, c d - b e\right )} \arctan \left (\frac{2 \, c x^{3} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{3 \, \sqrt{-b^{2} + 4 \, a c} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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