Optimal. Leaf size=203 \[ \frac{x^2 \left (-c (a f+b e)+b^2 f+c^2 d\right )}{2 c^3}+\frac{\log \left (a+b x^2+c x^4\right ) \left (-b c (c d-2 a f)-a c^2 e+b^2 c e+b^3 (-f)\right )}{4 c^4}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^3 c e+b^4 (-f)\right )}{2 c^4 \sqrt{b^2-4 a c}}+\frac{x^4 (c e-b f)}{4 c^2}+\frac{f x^6}{6 c} \]
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Rubi [A] time = 0.423681, antiderivative size = 203, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {1663, 1628, 634, 618, 206, 628} \[ \frac{x^2 \left (-c (a f+b e)+b^2 f+c^2 d\right )}{2 c^3}+\frac{\log \left (a+b x^2+c x^4\right ) \left (-b c (c d-2 a f)-a c^2 e+b^2 c e+b^3 (-f)\right )}{4 c^4}+\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^3 c e+b^4 (-f)\right )}{2 c^4 \sqrt{b^2-4 a c}}+\frac{x^4 (c e-b f)}{4 c^2}+\frac{f x^6}{6 c} \]
Antiderivative was successfully verified.
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Rule 1663
Rule 1628
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^5 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^2 \left (d+e x+f x^2\right )}{a+b x+c x^2} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{c^2 d+b^2 f-c (b e+a f)}{c^3}+\frac{(c e-b f) x}{c^2}+\frac{f x^2}{c}-\frac{a \left (c^2 d+b^2 f-c (b e+a f)\right )-\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{\left (c^2 d+b^2 f-c (b e+a f)\right ) x^2}{2 c^3}+\frac{(c e-b f) x^4}{4 c^2}+\frac{f x^6}{6 c}-\frac{\operatorname{Subst}\left (\int \frac{a \left (c^2 d+b^2 f-c (b e+a f)\right )-\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^3}\\ &=\frac{\left (c^2 d+b^2 f-c (b e+a f)\right ) x^2}{2 c^3}+\frac{(c e-b f) x^4}{4 c^2}+\frac{f x^6}{6 c}+\frac{\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^4}-\frac{\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^4}\\ &=\frac{\left (c^2 d+b^2 f-c (b e+a f)\right ) x^2}{2 c^3}+\frac{(c e-b f) x^4}{4 c^2}+\frac{f x^6}{6 c}+\frac{\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^4}+\frac{\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^4}\\ &=\frac{\left (c^2 d+b^2 f-c (b e+a f)\right ) x^2}{2 c^3}+\frac{(c e-b f) x^4}{4 c^2}+\frac{f x^6}{6 c}+\frac{\left (b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^4 \sqrt{b^2-4 a c}}+\frac{\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^4}\\ \end{align*}
Mathematica [A] time = 0.143298, size = 193, normalized size = 0.95 \[ \frac{6 c x^2 \left (-c (a f+b e)+b^2 f+c^2 d\right )-3 \log \left (a+b x^2+c x^4\right ) \left (b c (c d-2 a f)+a c^2 e-b^2 c e+b^3 f\right )+\frac{6 \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right ) \left (b^2 c (c d-4 a f)+3 a b c^2 e+2 a c^2 (a f-c d)-b^3 c e+b^4 f\right )}{\sqrt{4 a c-b^2}}+3 c^2 x^4 (c e-b f)+2 c^3 f x^6}{12 c^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.006, size = 474, normalized size = 2.3 \begin{align*}{\frac{f{x}^{6}}{6\,c}}-{\frac{{x}^{4}bf}{4\,{c}^{2}}}+{\frac{{x}^{4}e}{4\,c}}-{\frac{{x}^{2}af}{2\,{c}^{2}}}+{\frac{{b}^{2}f{x}^{2}}{2\,{c}^{3}}}-{\frac{be{x}^{2}}{2\,{c}^{2}}}+{\frac{{x}^{2}d}{2\,c}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) abf}{2\,{c}^{3}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) ae}{4\,{c}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{3}f}{4\,{c}^{4}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}e}{4\,{c}^{3}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bd}{4\,{c}^{2}}}+{\frac{{a}^{2}f}{{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{a{b}^{2}f}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{3\,abe}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{ad}{c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{4}f}{2\,{c}^{4}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{{b}^{3}e}{2\,{c}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}d}{2\,{c}^{2}}\arctan \left ({(2\,c{x}^{2}+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 = 2.82308, size = 1404, normalized size = 6.92 \begin{align*} \left [\frac{2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} f x^{6} + 3 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e -{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} f\right )} x^{4} + 6 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d -{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e +{\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} f\right )} x^{2} + 3 \, \sqrt{b^{2} - 4 \, a c}{\left ({\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d -{\left (b^{3} c - 3 \, a b c^{2}\right )} e +{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} f\right )} \log \left (\frac{2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c -{\left (2 \, c x^{2} + b\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - 3 \,{\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d -{\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e +{\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{12 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}, \frac{2 \,{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} f x^{6} + 3 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} e -{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} f\right )} x^{4} + 6 \,{\left ({\left (b^{2} c^{3} - 4 \, a c^{4}\right )} d -{\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} e +{\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} f\right )} x^{2} - 6 \, \sqrt{-b^{2} + 4 \, a c}{\left ({\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d -{\left (b^{3} c - 3 \, a b c^{2}\right )} e +{\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} f\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - 3 \,{\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d -{\left (b^{4} c - 5 \, a b^{2} c^{2} + 4 \, a^{2} c^{3}\right )} e +{\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{12 \,{\left (b^{2} c^{4} - 4 \, a c^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 35.7045, size = 1044, normalized size = 5.14 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1619, size = 289, normalized size = 1.42 \begin{align*} \frac{2 \, c^{2} f x^{6} - 3 \, b c f x^{4} + 3 \, c^{2} x^{4} e + 6 \, c^{2} d x^{2} + 6 \, b^{2} f x^{2} - 6 \, a c f x^{2} - 6 \, b c x^{2} e}{12 \, c^{3}} - \frac{{\left (b c^{2} d + b^{3} f - 2 \, a b c f - b^{2} c e + a c^{2} e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{4}} + \frac{{\left (b^{2} c^{2} d - 2 \, a c^{3} d + b^{4} f - 4 \, a b^{2} c f + 2 \, a^{2} c^{2} f - b^{3} c e + 3 \, a b c^{2} e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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