Optimal. Leaf size=97 \[ \frac{\left (2 a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 c^2 \sqrt{b^2-4 a c}}+\frac{(c d-b e) \log \left (a+b x^3+c x^6\right )}{6 c^2}+\frac{e x^3}{3 c} \]
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Rubi [A] time = 0.119684, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {1474, 773, 634, 618, 206, 628} \[ \frac{\left (2 a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 c^2 \sqrt{b^2-4 a c}}+\frac{(c d-b e) \log \left (a+b x^3+c x^6\right )}{6 c^2}+\frac{e x^3}{3 c} \]
Antiderivative was successfully verified.
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Rule 1474
Rule 773
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^5 \left (d+e x^3\right )}{a+b x^3+c x^6} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{x (d+e x)}{a+b x+c x^2} \, dx,x,x^3\right )\\ &=\frac{e x^3}{3 c}+\frac{\operatorname{Subst}\left (\int \frac{-a e+(c d-b e) x}{a+b x+c x^2} \, dx,x,x^3\right )}{3 c}\\ &=\frac{e x^3}{3 c}+\frac{(c d-b e) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^3\right )}{6 c^2}-\frac{\left (b c d-b^2 e+2 a c e\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^3\right )}{6 c^2}\\ &=\frac{e x^3}{3 c}+\frac{(c d-b e) \log \left (a+b x^3+c x^6\right )}{6 c^2}+\frac{\left (b c d-b^2 e+2 a c e\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^3\right )}{3 c^2}\\ &=\frac{e x^3}{3 c}+\frac{\left (b c d-b^2 e+2 a c e\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 c^2 \sqrt{b^2-4 a c}}+\frac{(c d-b e) \log \left (a+b x^3+c x^6\right )}{6 c^2}\\ \end{align*}
Mathematica [A] time = 0.0702844, size = 93, normalized size = 0.96 \[ \frac{\frac{2 \left (-2 a c e+b^2 e-b c d\right ) \tan ^{-1}\left (\frac{b+2 c x^3}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+(c d-b e) \log \left (a+b x^3+c x^6\right )+2 c e x^3}{6 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 175, normalized size = 1.8 \begin{align*}{\frac{e{x}^{3}}{3\,c}}-{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ) be}{6\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ) d}{6\,c}}-{\frac{2\,ae}{3\,c}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}e}{3\,{c}^{2}}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{bd}{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 = 1.92803, size = 664, normalized size = 6.85 \begin{align*} \left [\frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} e x^{3} +{\left (b c d -{\left (b^{2} - 2 \, a c\right )} e\right )} \sqrt{b^{2} - 4 \, a c} \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 ) +{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d -{\left (b^{3} - 4 \, a b c\right )} e\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}, \frac{2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} e x^{3} + 2 \,{\left (b c d -{\left (b^{2} - 2 \, a c\right )} e\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{{\left (2 \, c x^{3} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d -{\left (b^{3} - 4 \, a b c\right )} e\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 9.09558, size = 434, normalized size = 4.47 \begin{align*} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \left (4 a c - b^{2}\right )} - \frac{b e - c d}{6 c^{2}}\right ) \log{\left (x^{3} + \frac{- a b e - 12 a c^{2} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \left (4 a c - b^{2}\right )} - \frac{b e - c d}{6 c^{2}}\right ) + 2 a c d + 3 b^{2} c \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \left (4 a c - b^{2}\right )} - \frac{b e - c d}{6 c^{2}}\right )}{2 a c e - b^{2} e + b c d} \right )} + \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \left (4 a c - b^{2}\right )} - \frac{b e - c d}{6 c^{2}}\right ) \log{\left (x^{3} + \frac{- a b e - 12 a c^{2} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \left (4 a c - b^{2}\right )} - \frac{b e - c d}{6 c^{2}}\right ) + 2 a c d + 3 b^{2} c \left (\frac{\sqrt{- 4 a c + b^{2}} \left (2 a c e - b^{2} e + b c d\right )}{6 c^{2} \left (4 a c - b^{2}\right )} - \frac{b e - c d}{6 c^{2}}\right )}{2 a c e - b^{2} e + b c d} \right )} + \frac{e x^{3}}{3 c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35018, size = 128, normalized size = 1.32 \begin{align*} \frac{x^{3} e}{3 \, c} + \frac{{\left (c d - b e\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \, c^{2}} - \frac{{\left (b c d - b^{2} e + 2 \, a c 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^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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