Optimal. Leaf size=273 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (2 a^2 c^3 e-4 a b^2 c^2 e-b^3 c (c d-5 a f)+a b c^2 (3 c d-5 a f)+b^4 c e+b^5 (-f)\right )}{2 c^5 \sqrt{b^2-4 a c}}+\frac{x^4 \left (-c (a f+b e)+b^2 f+c^2 d\right )}{4 c^3}+\frac{x^2 \left (-b c (c d-2 a f)-a c^2 e+b^2 c e+b^3 (-f)\right )}{2 c^4}-\frac{\log \left (a+b x^2+c x^4\right ) \left (-b^2 c (c d-3 a f)-2 a b c^2 e+a c^2 (c d-a f)+b^3 c e+b^4 (-f)\right )}{4 c^5}+\frac{x^6 (c e-b f)}{6 c^2}+\frac{f x^8}{8 c} \]
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Rubi [A] time = 0.8541, antiderivative size = 273, 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} \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (2 a^2 c^3 e-4 a b^2 c^2 e-b^3 c (c d-5 a f)+a b c^2 (3 c d-5 a f)+b^4 c e+b^5 (-f)\right )}{2 c^5 \sqrt{b^2-4 a c}}+\frac{x^4 \left (-c (a f+b e)+b^2 f+c^2 d\right )}{4 c^3}+\frac{x^2 \left (-b c (c d-2 a f)-a c^2 e+b^2 c e+b^3 (-f)\right )}{2 c^4}-\frac{\log \left (a+b x^2+c x^4\right ) \left (-b^2 c (c d-3 a f)-2 a b c^2 e+a c^2 (c d-a f)+b^3 c e+b^4 (-f)\right )}{4 c^5}+\frac{x^6 (c e-b f)}{6 c^2}+\frac{f x^8}{8 c} \]
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^7 \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^3 \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{b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)}{c^4}+\frac{\left (c^2 d+b^2 f-c (b e+a f)\right ) x}{c^3}+\frac{(c e-b f) x^2}{c^2}+\frac{f x^3}{c}+\frac{-a \left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right )-\left (b^3 c e-2 a b c^2 e-b^4 f-b^2 c (c d-3 a f)+a c^2 (c d-a f)\right ) x}{c^4 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x^2}{2 c^4}+\frac{\left (c^2 d+b^2 f-c (b e+a f)\right ) x^4}{4 c^3}+\frac{(c e-b f) x^6}{6 c^2}+\frac{f x^8}{8 c}+\frac{\operatorname{Subst}\left (\int \frac{-a \left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right )-\left (b^3 c e-2 a b c^2 e-b^4 f-b^2 c (c d-3 a f)+a c^2 (c d-a f)\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c^4}\\ &=\frac{\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x^2}{2 c^4}+\frac{\left (c^2 d+b^2 f-c (b e+a f)\right ) x^4}{4 c^3}+\frac{(c e-b f) x^6}{6 c^2}+\frac{f x^8}{8 c}+\frac{\left (b^4 c e-4 a b^2 c^2 e+2 a^2 c^3 e-b^5 f-b^3 c (c d-5 a f)+a b c^2 (3 c d-5 a f)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^5}-\frac{\left (b^3 c e-2 a b c^2 e-b^4 f-b^2 c (c d-3 a f)+a c^2 (c d-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^5}\\ &=\frac{\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x^2}{2 c^4}+\frac{\left (c^2 d+b^2 f-c (b e+a f)\right ) x^4}{4 c^3}+\frac{(c e-b f) x^6}{6 c^2}+\frac{f x^8}{8 c}-\frac{\left (b^3 c e-2 a b c^2 e-b^4 f-b^2 c (c d-3 a f)+a c^2 (c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^5}-\frac{\left (b^4 c e-4 a b^2 c^2 e+2 a^2 c^3 e-b^5 f-b^3 c (c d-5 a f)+a b c^2 (3 c d-5 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^5}\\ &=\frac{\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)\right ) x^2}{2 c^4}+\frac{\left (c^2 d+b^2 f-c (b e+a f)\right ) x^4}{4 c^3}+\frac{(c e-b f) x^6}{6 c^2}+\frac{f x^8}{8 c}-\frac{\left (b^4 c e-4 a b^2 c^2 e+2 a^2 c^3 e-b^5 f-b^3 c (c d-5 a f)+a b c^2 (3 c d-5 a f)\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^5 \sqrt{b^2-4 a c}}-\frac{\left (b^3 c e-2 a b c^2 e-b^4 f-b^2 c (c d-3 a f)+a c^2 (c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 c^5}\\ \end{align*}
Mathematica [A] time = 0.208072, size = 260, normalized size = 0.95 \[ \frac{-\frac{12 \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right ) \left (-2 a^2 c^3 e+4 a b^2 c^2 e+b^3 c (c d-5 a f)+a b c^2 (5 a f-3 c d)-b^4 c e+b^5 f\right )}{\sqrt{4 a c-b^2}}+6 c^2 x^4 \left (-c (a f+b e)+b^2 f+c^2 d\right )-12 c x^2 \left (b c (c d-2 a f)+a c^2 e-b^2 c e+b^3 f\right )+6 \log \left (a+b x^2+c x^4\right ) \left (b^2 c (c d-3 a f)+2 a b c^2 e+a c^2 (a f-c d)-b^3 c e+b^4 f\right )+4 c^3 x^6 (c e-b f)+3 c^4 f x^8}{24 c^5} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 622, normalized size = 2.3 \begin{align*} -{\frac{5\,{a}^{2}bf}{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{5\,a{b}^{3}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}}}}}-2\,{\frac{a{b}^{2}e}{{c}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{3\,bad}{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{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}d}{4\,{c}^{3}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){a}^{2}f}{4\,{c}^{3}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) ad}{4\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{4}f}{4\,{c}^{5}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{3}e}{4\,{c}^{4}}}+{\frac{{b}^{2}e{x}^{2}}{2\,{c}^{3}}}-{\frac{bd{x}^{2}}{2\,{c}^{2}}}-{\frac{{x}^{6}bf}{6\,{c}^{2}}}-{\frac{{x}^{4}af}{4\,{c}^{2}}}+{\frac{{x}^{4}{b}^{2}f}{4\,{c}^{3}}}-{\frac{be{x}^{4}}{4\,{c}^{2}}}-{\frac{ae{x}^{2}}{2\,{c}^{2}}}-{\frac{{b}^{3}f{x}^{2}}{2\,{c}^{4}}}-{\frac{3\,\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) a{b}^{2}f}{4\,{c}^{4}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) abe}{2\,{c}^{3}}}+{\frac{{a}^{2}e}{{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{{b}^{5}f}{2\,{c}^{5}}\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}e}{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}d}{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{abf{x}^{2}}{{c}^{3}}}+{\frac{{x}^{4}d}{4\,c}}+{\frac{{x}^{6}e}{6\,c}}+{\frac{f{x}^{8}}{8\,c}} \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 = 4.16819, size = 1854, normalized size = 6.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 49.7414, size = 1392, normalized size = 5.1 \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 = 1.15015, size = 413, normalized size = 1.51 \begin{align*} \frac{3 \, c^{3} f x^{8} - 4 \, b c^{2} f x^{6} + 4 \, c^{3} x^{6} e + 6 \, c^{3} d x^{4} + 6 \, b^{2} c f x^{4} - 6 \, a c^{2} f x^{4} - 6 \, b c^{2} x^{4} e - 12 \, b c^{2} d x^{2} - 12 \, b^{3} f x^{2} + 24 \, a b c f x^{2} + 12 \, b^{2} c x^{2} e - 12 \, a c^{2} x^{2} e}{24 \, c^{4}} + \frac{{\left (b^{2} c^{2} d - a c^{3} d + b^{4} f - 3 \, a b^{2} c f + a^{2} c^{2} f - b^{3} c e + 2 \, a b c^{2} e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{5}} - \frac{{\left (b^{3} c^{2} d - 3 \, a b c^{3} d + b^{5} f - 5 \, a b^{3} c f + 5 \, a^{2} b c^{2} f - b^{4} c e + 4 \, a b^{2} c^{2} e - 2 \, a^{2} c^{3} 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^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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