Optimal. Leaf size=112 \[ -\frac{\left (-a b e-2 a c d+b^2 d\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 a^2 \sqrt{b^2-4 a c}}+\frac{(b d-a e) \log \left (a+b x^3+c x^6\right )}{6 a^2}-\frac{\log (x) (b d-a e)}{a^2}-\frac{d}{3 a x^3} \]
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Rubi [A] time = 0.197486, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24, Rules used = {1474, 800, 634, 618, 206, 628} \[ -\frac{\left (-a b e-2 a c d+b^2 d\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 a^2 \sqrt{b^2-4 a c}}+\frac{(b d-a e) \log \left (a+b x^3+c x^6\right )}{6 a^2}-\frac{\log (x) (b d-a e)}{a^2}-\frac{d}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 1474
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{d+e x^3}{x^4 \left (a+b x^3+c x^6\right )} \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int \frac{d+e x}{x^2 \left (a+b x+c x^2\right )} \, dx,x,x^3\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (\frac{d}{a x^2}+\frac{-b d+a e}{a^2 x}+\frac{b^2 d-a c d-a b e+c (b d-a e) x}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^3\right )\\ &=-\frac{d}{3 a x^3}-\frac{(b d-a e) \log (x)}{a^2}+\frac{\operatorname{Subst}\left (\int \frac{b^2 d-a c d-a b e+c (b d-a e) x}{a+b x+c x^2} \, dx,x,x^3\right )}{3 a^2}\\ &=-\frac{d}{3 a x^3}-\frac{(b d-a e) \log (x)}{a^2}+\frac{(b d-a e) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^3\right )}{6 a^2}+\frac{\left (b^2 d-2 a c d-a b e\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^3\right )}{6 a^2}\\ &=-\frac{d}{3 a x^3}-\frac{(b d-a e) \log (x)}{a^2}+\frac{(b d-a e) \log \left (a+b x^3+c x^6\right )}{6 a^2}-\frac{\left (b^2 d-2 a c d-a b e\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^3\right )}{3 a^2}\\ &=-\frac{d}{3 a x^3}-\frac{\left (b^2 d-2 a c d-a b e\right ) \tanh ^{-1}\left (\frac{b+2 c x^3}{\sqrt{b^2-4 a c}}\right )}{3 a^2 \sqrt{b^2-4 a c}}-\frac{(b d-a e) \log (x)}{a^2}+\frac{(b d-a e) \log \left (a+b x^3+c x^6\right )}{6 a^2}\\ \end{align*}
Mathematica [C] time = 0.0521102, size = 130, normalized size = 1.16 \[ \frac{\text{RootSum}\left [\text{$\#$1}^3 b+\text{$\#$1}^6 c+a\& ,\frac{-\text{$\#$1}^3 a c e \log (x-\text{$\#$1})+\text{$\#$1}^3 b c d \log (x-\text{$\#$1})-a b e \log (x-\text{$\#$1})-a c d \log (x-\text{$\#$1})+b^2 d \log (x-\text{$\#$1})}{2 \text{$\#$1}^3 c+b}\& \right ]}{3 a^2}+\frac{\log (x) (a e-b d)}{a^2}-\frac{d}{3 a x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 191, normalized size = 1.7 \begin{align*} -{\frac{d}{3\,a{x}^{3}}}+{\frac{\ln \left ( x \right ) e}{a}}-{\frac{b\ln \left ( x \right ) d}{{a}^{2}}}-{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ) e}{6\,a}}+{\frac{\ln \left ( c{x}^{6}+b{x}^{3}+a \right ) bd}{6\,{a}^{2}}}-{\frac{be}{3\,a}\arctan \left ({(2\,c{x}^{3}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{2\,cd}{3\,a}\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}d}{3\,{a}^{2}}\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 = 4.11086, size = 845, normalized size = 7.54 \begin{align*} \left [\frac{{\left (a b e -{\left (b^{2} - 2 \, a c\right )} d\right )} \sqrt{b^{2} - 4 \, a c} x^{3} \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^{3} - 4 \, a b c\right )} d -{\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x^{3} \log \left (c x^{6} + b x^{3} + a\right ) - 6 \,{\left ({\left (b^{3} - 4 \, a b c\right )} d -{\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x^{3} \log \left (x\right ) - 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} d}{6 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{3}}, \frac{2 \,{\left (a b e -{\left (b^{2} - 2 \, a c\right )} d\right )} \sqrt{-b^{2} + 4 \, a c} x^{3} \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^{3} - 4 \, a b c\right )} d -{\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x^{3} \log \left (c x^{6} + b x^{3} + a\right ) - 6 \,{\left ({\left (b^{3} - 4 \, a b c\right )} d -{\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x^{3} \log \left (x\right ) - 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} d}{6 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [A] time = 1.36632, size = 173, normalized size = 1.54 \begin{align*} \frac{{\left (b d - a e\right )} \log \left (c x^{6} + b x^{3} + a\right )}{6 \, a^{2}} - \frac{{\left (b d - a e\right )} \log \left ({\left | x \right |}\right )}{a^{2}} + \frac{{\left (b^{2} d - 2 \, a c d - a 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} a^{2}} + \frac{b d x^{3} - a x^{3} e - a d}{3 \, a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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