Optimal. Leaf size=118 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-a b e-2 a (c d-a f)+b^2 d\right )}{2 a^2 \sqrt{b^2-4 a c}}+\frac{(b d-a e) \log \left (a+b x^2+c x^4\right )}{4 a^2}-\frac{\log (x) (b d-a e)}{a^2}-\frac{d}{2 a x^2} \]
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Rubi [A] time = 0.285348, antiderivative size = 118, 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} \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right ) \left (-a b e-2 a (c d-a f)+b^2 d\right )}{2 a^2 \sqrt{b^2-4 a c}}+\frac{(b d-a e) \log \left (a+b x^2+c x^4\right )}{4 a^2}-\frac{\log (x) (b d-a e)}{a^2}-\frac{d}{2 a x^2} \]
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^3 \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^2 \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^2}+\frac{-b d+a e}{a^2 x}+\frac{b^2 d-a b e-a (c d-a f)+c (b d-a e) x}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{d}{2 a x^2}-\frac{(b d-a e) \log (x)}{a^2}+\frac{\operatorname{Subst}\left (\int \frac{b^2 d-a b e-a (c d-a f)+c (b d-a e) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^2}\\ &=-\frac{d}{2 a x^2}-\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^2\right )}{4 a^2}+\frac{\left (b^2 d-a b e-2 a (c d-a f)\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2}\\ &=-\frac{d}{2 a x^2}-\frac{(b d-a e) \log (x)}{a^2}+\frac{(b d-a e) \log \left (a+b x^2+c x^4\right )}{4 a^2}-\frac{\left (b^2 d-a b e-2 a (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^2}\\ &=-\frac{d}{2 a x^2}-\frac{\left (b^2 d-a b e-2 a (c d-a f)\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 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^2+c x^4\right )}{4 a^2}\\ \end{align*}
Mathematica [A] time = 0.161855, size = 203, normalized size = 1.72 \[ \frac{\frac{\log \left (-\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (a \left (-e \sqrt{b^2-4 a c}+2 a f-2 c d\right )+b \left (d \sqrt{b^2-4 a c}-a e\right )+b^2 d\right )}{\sqrt{b^2-4 a c}}+\frac{\log \left (\sqrt{b^2-4 a c}+b+2 c x^2\right ) \left (-a \left (e \sqrt{b^2-4 a c}+2 a f-2 c d\right )+b \left (d \sqrt{b^2-4 a c}+a e\right )+b^2 (-d)\right )}{\sqrt{b^2-4 a c}}+4 \log (x) (a e-b d)-\frac{2 a d}{x^2}}{4 a^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.009, size = 227, normalized size = 1.9 \begin{align*} -{\frac{d}{2\,a{x}^{2}}}+{\frac{\ln \left ( x \right ) e}{a}}-{\frac{\ln \left ( x \right ) bd}{{a}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) e}{4\,a}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) bd}{4\,{a}^{2}}}+{f\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{be}{2\,a}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{cd}{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}d}{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}}}}} \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 = 2.99112, size = 873, normalized size = 7.4 \begin{align*} \left [-\frac{{\left (a b e - 2 \, a^{2} f -{\left (b^{2} - 2 \, a c\right )} d\right )} \sqrt{b^{2} - 4 \, a c} x^{2} \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 ) -{\left ({\left (b^{3} - 4 \, a b c\right )} d -{\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x^{2} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \,{\left ({\left (b^{3} - 4 \, a b c\right )} d -{\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x^{2} \log \left (x\right ) + 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} d}{4 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2}}, \frac{2 \,{\left (a b e - 2 \, a^{2} f -{\left (b^{2} - 2 \, a c\right )} d\right )} \sqrt{-b^{2} + 4 \, a c} x^{2} \arctan \left (-\frac{{\left (2 \, c x^{2} + 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^{2} \log \left (c x^{4} + b x^{2} + a\right ) - 4 \,{\left ({\left (b^{3} - 4 \, a b c\right )} d -{\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x^{2} \log \left (x\right ) - 2 \,{\left (a b^{2} - 4 \, a^{2} c\right )} d}{4 \,{\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2}}\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 = 1.12911, size = 182, normalized size = 1.54 \begin{align*} \frac{{\left (b d - a e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{2}} - \frac{{\left (b d - a e\right )} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac{{\left (b^{2} d - 2 \, a c d + 2 \, a^{2} f - a b 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^{2}} + \frac{b d x^{2} - a x^{2} e - a d}{2 \, a^{2} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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