Optimal. Leaf size=132 \[ \frac{(b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac{d \log \left (d+e x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}+\frac{d \log \left (a+b x^2+c x^4\right )}{4 \left (a e^2-b d e+c d^2\right )} \]
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Rubi [A] time = 0.157051, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1251, 800, 634, 618, 206, 628} \[ \frac{(b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac{d \log \left (d+e x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}+\frac{d \log \left (a+b x^2+c x^4\right )}{4 \left (a e^2-b d e+c d^2\right )} \]
Antiderivative was successfully verified.
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Rule 1251
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^3}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x}{(d+e x) \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{d e}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac{a e+c d x}{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{d \log \left (d+e x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{a e+c d x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{d \log \left (d+e x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}+\frac{d \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 \left (c d^2-b d e+a e^2\right )}-\frac{(b d-2 a e) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{d \log \left (d+e x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}+\frac{d \log \left (a+b x^2+c x^4\right )}{4 \left (c d^2-b d e+a e^2\right )}+\frac{(b d-2 a e) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{(b d-2 a e) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )}-\frac{d \log \left (d+e x^2\right )}{2 \left (c d^2-b d e+a e^2\right )}+\frac{d \log \left (a+b x^2+c x^4\right )}{4 \left (c d^2-b d e+a e^2\right )}\\ \end{align*}
Mathematica [A] time = 0.07163, size = 114, normalized size = 0.86 \[ \frac{d \sqrt{4 a c-b^2} \left (2 \log \left (d+e x^2\right )-\log \left (a+b x^2+c x^4\right )\right )+2 (b d-2 a e) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{4 \sqrt{4 a c-b^2} \left (e (b d-a e)-c d^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 176, normalized size = 1.3 \begin{align*}{\frac{d\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) }{4\,a{e}^{2}-4\,deb+4\,c{d}^{2}}}+{\frac{ae}{a{e}^{2}-deb+c{d}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{bd}{2\,a{e}^{2}-2\,deb+2\,c{d}^{2}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{d\ln \left ( e{x}^{2}+d \right ) }{2\,a{e}^{2}-2\,deb+2\,c{d}^{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 = 112.082, size = 716, normalized size = 5.42 \begin{align*} \left [\frac{{\left (b^{2} - 4 \, a c\right )} d \log \left (c x^{4} + b x^{2} + a\right ) - 2 \,{\left (b^{2} - 4 \, a c\right )} d \log \left (e x^{2} + d\right ) - \sqrt{b^{2} - 4 \, a c}{\left (b d - 2 \, a e\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 )}{4 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} -{\left (b^{3} - 4 \, a b c\right )} d e +{\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )}}, \frac{{\left (b^{2} - 4 \, a c\right )} d \log \left (c x^{4} + b x^{2} + a\right ) - 2 \,{\left (b^{2} - 4 \, a c\right )} d \log \left (e x^{2} + d\right ) + 2 \, \sqrt{-b^{2} + 4 \, a c}{\left (b d - 2 \, a e\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right )}{4 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d^{2} -{\left (b^{3} - 4 \, a b c\right )} d e +{\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )}}\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 = 1.13567, size = 180, normalized size = 1.36 \begin{align*} -\frac{d e \log \left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{2} e - b d e^{2} + a e^{3}\right )}} + \frac{d \log \left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (c d^{2} - b d e + a e^{2}\right )}} - \frac{{\left (b d - 2 \, a e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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