Optimal. Leaf size=189 \[ \frac{\left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{\left (-a b e-a c d+b^2 d\right ) \log \left (a+b x^2+c x^4\right )}{4 c^2 \left (a e^2-b d e+c d^2\right )}-\frac{d^3 \log \left (d+e x^2\right )}{2 e^2 \left (a e^2-b d e+c d^2\right )}+\frac{x^2}{2 c e} \]
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Rubi [A] time = 0.329083, antiderivative size = 189, 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, 1628, 634, 618, 206, 628} \[ \frac{\left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{\left (-a b e-a c d+b^2 d\right ) \log \left (a+b x^2+c x^4\right )}{4 c^2 \left (a e^2-b d e+c d^2\right )}-\frac{d^3 \log \left (d+e x^2\right )}{2 e^2 \left (a e^2-b d e+c d^2\right )}+\frac{x^2}{2 c e} \]
Antiderivative was successfully verified.
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Rule 1251
Rule 1628
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{x^7}{\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^3}{(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{1}{c e}-\frac{d^3}{e \left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac{a (b d-a e)+\left (b^2 d-a c d-a b e\right ) x}{c \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{x^2}{2 c e}-\frac{d^3 \log \left (d+e x^2\right )}{2 e^2 \left (c d^2-b d e+a e^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{a (b d-a e)+\left (b^2 d-a c d-a b e\right ) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c \left (c d^2-b d e+a e^2\right )}\\ &=\frac{x^2}{2 c e}-\frac{d^3 \log \left (d+e x^2\right )}{2 e^2 \left (c d^2-b d e+a e^2\right )}+\frac{\left (b^2 d-a c d-a b e\right ) \operatorname{Subst}\left (\int \frac{b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2 \left (c d^2-b d e+a e^2\right )}-\frac{\left (b^3 d-3 a b c d-a b^2 e+2 a^2 c e\right ) \operatorname{Subst}\left (\int \frac{1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{x^2}{2 c e}-\frac{d^3 \log \left (d+e x^2\right )}{2 e^2 \left (c d^2-b d e+a e^2\right )}+\frac{\left (b^2 d-a c d-a b e\right ) \log \left (a+b x^2+c x^4\right )}{4 c^2 \left (c d^2-b d e+a e^2\right )}+\frac{\left (b^3 d-3 a b c d-a b^2 e+2 a^2 c e\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^2 \left (c d^2-b d e+a e^2\right )}\\ &=\frac{x^2}{2 c e}+\frac{\left (b^3 d-3 a b c d-a b^2 e+2 a^2 c e\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{2 c^2 \sqrt{b^2-4 a c} \left (c d^2-b d e+a e^2\right )}-\frac{d^3 \log \left (d+e x^2\right )}{2 e^2 \left (c d^2-b d e+a e^2\right )}+\frac{\left (b^2 d-a c d-a b e\right ) \log \left (a+b x^2+c x^4\right )}{4 c^2 \left (c d^2-b d e+a e^2\right )}\\ \end{align*}
Mathematica [A] time = 0.178524, size = 186, normalized size = 0.98 \[ \frac{2 e^2 \left (2 a^2 c e-a b^2 e-3 a b c d+b^3 d\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )+\sqrt{4 a c-b^2} \left (e \left (e \left (a b e+a c d+b^2 (-d)\right ) \log \left (a+b x^2+c x^4\right )-2 c x^2 \left (a e^2-b d e+c d^2\right )\right )+2 c^2 d^3 \log \left (d+e x^2\right )\right )}{4 c^2 e^2 \sqrt{4 a c-b^2} \left (e (b d-a e)-c d^2\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.01, size = 408, normalized size = 2.2 \begin{align*}{\frac{{x}^{2}}{2\,ce}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) abe}{ \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ){c}^{2}}}-{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ) ad}{ \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ) c}}+{\frac{\ln \left ( c{x}^{4}+b{x}^{2}+a \right ){b}^{2}d}{ \left ( 4\,a{e}^{2}-4\,deb+4\,c{d}^{2} \right ){c}^{2}}}-{\frac{{a}^{2}e}{ \left ( a{e}^{2}-deb+c{d}^{2} \right ) c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{3\,abd}{ \left ( 2\,a{e}^{2}-2\,deb+2\,c{d}^{2} \right ) c}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{a{b}^{2}e}{ \left ( 2\,a{e}^{2}-2\,deb+2\,c{d}^{2} \right ){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}^{3}d}{ \left ( 2\,a{e}^{2}-2\,deb+2\,c{d}^{2} \right ){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{{d}^{3}\ln \left ( e{x}^{2}+d \right ) }{2\,{e}^{2} \left ( a{e}^{2}-deb+c{d}^{2} \right ) }} \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 [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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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 = 1.1716, size = 262, normalized size = 1.39 \begin{align*} -\frac{d^{3} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{2} e^{2} - b d e^{3} + a e^{4}\right )}} + \frac{x^{2} e^{\left (-1\right )}}{2 \, c} + \frac{{\left (b^{2} d - a c d - a b e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \,{\left (c^{3} d^{2} - b c^{2} d e + a c^{2} e^{2}\right )}} - \frac{{\left (b^{3} d - 3 \, a b c d - a b^{2} e + 2 \, a^{2} c e\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \,{\left (c^{3} d^{2} - b c^{2} d e + a c^{2} 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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