Optimal. Leaf size=244 \[ -\frac{-a b e-a (c d-a f)+b^2 d}{2 a^3 x^2}+\frac{\log \left (a+b x^2+c x^4\right ) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{4 a^4}-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)-a b^3 e+b^4 d\right )}{2 a^4 \sqrt{b^2-4 a c}}-\frac{\log (x) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{a^4}+\frac{b d-a e}{4 a^2 x^4}-\frac{d}{6 a x^6} \]
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Rubi [A] time = 0.572902, antiderivative size = 244, 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} \[ -\frac{-a b e-a (c d-a f)+b^2 d}{2 a^3 x^2}+\frac{\log \left (a+b x^2+c x^4\right ) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{4 a^4}-\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)-a b^3 e+b^4 d\right )}{2 a^4 \sqrt{b^2-4 a c}}-\frac{\log (x) \left (a^2 c e-a b^2 e-a b (2 c d-a f)+b^3 d\right )}{a^4}+\frac{b d-a e}{4 a^2 x^4}-\frac{d}{6 a x^6} \]
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{d+e x^2+f x^4}{x^7 \left (a+b x^2+c x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{d+e x+f x^2}{x^4 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{d}{a x^4}+\frac{-b d+a e}{a^2 x^3}+\frac{b^2 d-a b e-a (c d-a f)}{a^3 x^2}+\frac{-b^3 d+a b^2 e-a^2 c e+a b (2 c d-a f)}{a^4 x}+\frac{b^4 d-a b^3 e+2 a^2 b c e+a^2 c (c d-a f)-a b^2 (3 c d-a f)+c \left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) x}{a^4 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{d}{6 a x^6}+\frac{b d-a e}{4 a^2 x^4}-\frac{b^2 d-a b e-a (c d-a f)}{2 a^3 x^2}-\frac{\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log (x)}{a^4}+\frac{\operatorname{Subst}\left (\int \frac{b^4 d-a b^3 e+2 a^2 b c e+a^2 c (c d-a f)-a b^2 (3 c d-a f)+c \left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^4}\\ &=-\frac{d}{6 a x^6}+\frac{b d-a e}{4 a^2 x^4}-\frac{b^2 d-a b e-a (c d-a f)}{2 a^3 x^2}-\frac{\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log (x)}{a^4}+\frac{\left (b^3 d-a b^2 e+a^2 c e-a b (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 a^4}+\frac{\left (b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^4}\\ &=-\frac{d}{6 a x^6}+\frac{b d-a e}{4 a^2 x^4}-\frac{b^2 d-a b e-a (c d-a f)}{2 a^3 x^2}-\frac{\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log (x)}{a^4}+\frac{\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}-\frac{\left (b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 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 a^4}\\ &=-\frac{d}{6 a x^6}+\frac{b d-a e}{4 a^2 x^4}-\frac{b^2 d-a b e-a (c d-a f)}{2 a^3 x^2}-\frac{\left (b^4 d-a b^3 e+3 a^2 b c e+2 a^2 c (c d-a f)-a b^2 (4 c d-a f)\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 a^4 \sqrt{b^2-4 a c}}-\frac{\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log (x)}{a^4}+\frac{\left (b^3 d-a b^2 e+a^2 c e-a b (2 c d-a f)\right ) \log \left (a+b x^2+c x^4\right )}{4 a^4}\\ \end{align*}
Mathematica [A] time = 0.351751, size = 416, normalized size = 1.7 \[ \frac{\frac{3 \log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (a^2 c \left (e \sqrt{b^2-4 a c}-2 a f+2 c d\right )+a b^2 \left (-e \sqrt{b^2-4 a c}+a f-4 c d\right )+a b \left (-2 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}+3 a c e\right )+b^3 \left (d \sqrt{b^2-4 a c}-a e\right )+b^4 d\right )}{\sqrt{b^2-4 a c}}+\frac{3 \log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (a^2 c \left (e \sqrt{b^2-4 a c}+2 a f-2 c d\right )-a b^2 \left (e \sqrt{b^2-4 a c}+a f-4 c d\right )+a b \left (-2 c d \sqrt{b^2-4 a c}+a f \sqrt{b^2-4 a c}-3 a c e\right )+b^3 \left (d \sqrt{b^2-4 a c}+a e\right )+b^4 (-d)\right )}{\sqrt{b^2-4 a c}}-12 \log (x) \left (a^2 c e-a b^2 e+a b (a f-2 c d)+b^3 d\right )+\frac{3 a^2 (b d-a e)}{x^4}-\frac{2 a^3 d}{x^6}+\frac{6 a \left (a b e+a (c d-a f)+b^2 (-d)\right )}{x^2}}{12 a^4} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.014, size = 523, normalized size = 2.1 \begin{align*} -{\frac{d}{6\,a{x}^{6}}}-{\frac{e}{4\,a{x}^{4}}}+{\frac{bd}{4\,{a}^{2}{x}^{4}}}-{\frac{f}{2\,a{x}^{2}}}+{\frac{be}{2\,{a}^{2}{x}^{2}}}+{\frac{cd}{2\,{a}^{2}{x}^{2}}}-{\frac{{b}^{2}d}{2\,{a}^{3}{x}^{2}}}-{\frac{\ln \left ( x \right ) bf}{{a}^{2}}}-{\frac{\ln \left ( x \right ) ce}{{a}^{2}}}+{\frac{\ln \left ( x \right ){b}^{2}e}{{a}^{3}}}+2\,{\frac{\ln \left ( x \right ) bcd}{{a}^{3}}}-{\frac{\ln \left ( x \right ){b}^{3}d}{{a}^{4}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bf}{4\,{a}^{2}}}+{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) e}{4\,{a}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}e}{4\,{a}^{3}}}-{\frac{c\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bd}{2\,{a}^{3}}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{3}d}{4\,{a}^{4}}}-{\frac{fc}{a}\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}f}{2\,{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{3\,bce}{2\,{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{c}^{2}d}{{a}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{e{b}^{3}}{2\,{a}^{3}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-2\,{\frac{{b}^{2}cd}{{a}^{3}\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{4}d}{2\,{a}^{4}}\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 = 13.7013, size = 1747, normalized size = 7.16 \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 [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 = 1.14577, size = 423, normalized size = 1.73 \begin{align*} \frac{{\left (b^{3} d - 2 \, a b c d + a^{2} b f - a b^{2} e + a^{2} c e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} - \frac{{\left (b^{3} d - 2 \, a b c d + a^{2} b f - a b^{2} e + a^{2} c e\right )} \log \left (x^{2}\right )}{2 \, a^{4}} + \frac{{\left (b^{4} d - 4 \, a b^{2} c d + 2 \, a^{2} c^{2} d + a^{2} b^{2} f - 2 \, a^{3} c f - a b^{3} e + 3 \, a^{2} b c e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} a^{4}} + \frac{11 \, b^{3} d x^{6} - 22 \, a b c d x^{6} + 11 \, a^{2} b f x^{6} - 11 \, a b^{2} x^{6} e + 11 \, a^{2} c x^{6} e - 6 \, a b^{2} d x^{4} + 6 \, a^{2} c d x^{4} - 6 \, a^{3} f x^{4} + 6 \, a^{2} b x^{4} e + 3 \, a^{2} b d x^{2} - 3 \, a^{3} x^{2} e - 2 \, a^{3} d}{12 \, a^{4} x^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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