Optimal. Leaf size=150 \[ \frac{\sqrt{a} e \left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a \left (d-e x^2\right )}{4 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac{d^3 \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac{d^3 \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2} \]
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Rubi [A] time = 0.246675, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {1252, 1647, 801, 635, 205, 260} \[ \frac{\sqrt{a} e \left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a \left (d-e x^2\right )}{4 c \left (a+c x^4\right ) \left (a e^2+c d^2\right )}-\frac{d^3 \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2}+\frac{d^3 \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1252
Rule 1647
Rule 801
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{x^7}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^3}{(d+e x) \left (a+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{-\frac{a^2 d e}{c d^2+a e^2}-\frac{a \left (2 c d^2+a e^2\right ) x}{c d^2+a e^2}}{(d+e x) \left (a+c x^2\right )} \, dx,x,x^2\right )}{4 a c}\\ &=\frac{a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \left (\frac{2 a c d^3 e}{\left (c d^2+a e^2\right )^2 (d+e x)}-\frac{a \left (3 a c d^2 e+a^2 e^3+2 c^2 d^3 x\right )}{\left (c d^2+a e^2\right )^2 \left (a+c x^2\right )}\right ) \, dx,x,x^2\right )}{4 a c}\\ &=\frac{a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac{d^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac{\operatorname{Subst}\left (\int \frac{3 a c d^2 e+a^2 e^3+2 c^2 d^3 x}{a+c x^2} \, dx,x,x^2\right )}{4 c \left (c d^2+a e^2\right )^2}\\ &=\frac{a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac{d^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac{\left (c d^3\right ) \operatorname{Subst}\left (\int \frac{x}{a+c x^2} \, dx,x,x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac{\left (a e \left (3 c d^2+a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{4 c \left (c d^2+a e^2\right )^2}\\ &=\frac{a \left (d-e x^2\right )}{4 c \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac{\sqrt{a} e \left (3 c d^2+a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{3/2} \left (c d^2+a e^2\right )^2}-\frac{d^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac{d^3 \log \left (a+c x^4\right )}{4 \left (c d^2+a e^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.115127, size = 142, normalized size = 0.95 \[ \frac{\sqrt{c} \left (a \left (d-e x^2\right ) \left (a e^2+c d^2\right )-2 c d^3 \left (a+c x^4\right ) \log \left (d+e x^2\right )+c d^3 \left (a+c x^4\right ) \log \left (a+c x^4\right )\right )+\sqrt{a} e \left (a+c x^4\right ) \left (a e^2+3 c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{3/2} \left (a+c x^4\right ) \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.017, size = 260, normalized size = 1.7 \begin{align*} -{\frac{{a}^{2}{e}^{3}{x}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) c}}-{\frac{ae{x}^{2}{d}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{{a}^{2}d{e}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) c}}+{\frac{a{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{{d}^{3}\ln \left ( c{x}^{4}+a \right ) }{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{{a}^{2}{e}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}c}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,e{d}^{2}a}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{{d}^{3}\ln \left ( e{x}^{2}+d \right ) }{2\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{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 = 39.6776, size = 922, normalized size = 6.15 \begin{align*} \left [\frac{2 \, a c d^{3} + 2 \, a^{2} d e^{2} - 2 \,{\left (a c d^{2} e + a^{2} e^{3}\right )} x^{2} +{\left (3 \, a c d^{2} e + a^{2} e^{3} +{\left (3 \, c^{2} d^{2} e + a c e^{3}\right )} x^{4}\right )} \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{4} + 2 \, c x^{2} \sqrt{-\frac{a}{c}} - a}{c x^{4} + a}\right ) + 2 \,{\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (c x^{4} + a\right ) - 4 \,{\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (e x^{2} + d\right )}{8 \,{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4} +{\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{4}\right )}}, \frac{a c d^{3} + a^{2} d e^{2} -{\left (a c d^{2} e + a^{2} e^{3}\right )} x^{2} +{\left (3 \, a c d^{2} e + a^{2} e^{3} +{\left (3 \, c^{2} d^{2} e + a c e^{3}\right )} x^{4}\right )} \sqrt{\frac{a}{c}} \arctan \left (\frac{c x^{2} \sqrt{\frac{a}{c}}}{a}\right ) +{\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (c x^{4} + a\right ) - 2 \,{\left (c^{2} d^{3} x^{4} + a c d^{3}\right )} \log \left (e x^{2} + d\right )}{4 \,{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4} +{\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{4}\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 = 1.10852, size = 301, normalized size = 2.01 \begin{align*} -\frac{d^{3} e \log \left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c^{2} d^{4} e + 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )}} + \frac{d^{3} \log \left (c x^{4} + a\right )}{4 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )}} + \frac{{\left (3 \, a c d^{2} e + a^{2} e^{3}\right )} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \,{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \sqrt{a c}} - \frac{c^{2} d^{3} x^{4} + a c d^{2} x^{2} e + a^{2} x^{2} e^{3} - a^{2} d e^{2}}{4 \,{\left (c^{3} d^{4} + 2 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )}{\left (c x^{4} + a\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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