Optimal. Leaf size=151 \[ \frac{\sqrt{c} d \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a e+c d x^2}{4 a \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{e^3 \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2}-\frac{e^3 \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2} \]
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Rubi [A] time = 0.180638, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1248, 741, 801, 635, 205, 260} \[ \frac{\sqrt{c} d \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \left (a e^2+c d^2\right )^2}+\frac{a e+c d x^2}{4 a \left (a+c x^4\right ) \left (a e^2+c d^2\right )}+\frac{e^3 \log \left (d+e x^2\right )}{2 \left (a e^2+c d^2\right )^2}-\frac{e^3 \log \left (a+c x^4\right )}{4 \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Rule 1248
Rule 741
Rule 801
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{x}{\left (d+e x^2\right ) \left (a+c x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{(d+e x) \left (a+c x^2\right )^2} \, dx,x,x^2\right )\\ &=\frac{a e+c d x^2}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \frac{-c d^2-2 a e^2-c d e x}{(d+e x) \left (a+c x^2\right )} \, dx,x,x^2\right )}{4 a \left (c d^2+a e^2\right )}\\ &=\frac{a e+c d x^2}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}-\frac{\operatorname{Subst}\left (\int \left (-\frac{2 a e^4}{\left (c d^2+a e^2\right ) (d+e x)}-\frac{c \left (c d^3+3 a d e^2-2 a e^3 x\right )}{\left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx,x,x^2\right )}{4 a \left (c d^2+a e^2\right )}\\ &=\frac{a e+c d x^2}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac{e^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}+\frac{c \operatorname{Subst}\left (\int \frac{c d^3+3 a d e^2-2 a e^3 x}{a+c x^2} \, dx,x,x^2\right )}{4 a \left (c d^2+a e^2\right )^2}\\ &=\frac{a e+c d x^2}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac{e^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}-\frac{\left (c e^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 (c d \left (c d^2+3 a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{4 a \left (c d^2+a e^2\right )^2}\\ &=\frac{a e+c d x^2}{4 a \left (c d^2+a e^2\right ) \left (a+c x^4\right )}+\frac{\sqrt{c} d \left (c d^2+3 a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 a^{3/2} \left (c d^2+a e^2\right )^2}+\frac{e^3 \log \left (d+e x^2\right )}{2 \left (c d^2+a e^2\right )^2}-\frac{e^3 \log \left (a+c x^4\right )}{4 \left (c d^2+a e^2\right )^2}\\ \end{align*}
Mathematica [A] time = 0.138577, size = 117, normalized size = 0.77 \[ \frac{\frac{\sqrt{c} d \left (3 a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{a^{3/2}}+\frac{\left (a e^2+c d^2\right ) \left (a e+c d x^2\right )}{a \left (a+c x^4\right )}-e^3 \log \left (a+c x^4\right )+2 e^3 \log \left (d+e x^2\right )}{4 \left (a e^2+c d^2\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.016, size = 255, normalized size = 1.7 \begin{align*}{\frac{{x}^{2}cd{e}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{{c}^{2}{d}^{3}{x}^{2}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) a}}+{\frac{{e}^{3}a}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{e{d}^{2}c}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{{e}^{3}\ln \left ( c{x}^{4}+a \right ) }{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}}}+{\frac{3\,cd{e}^{2}}{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{{c}^{2}{d}^{3}}{4\, \left ( a{e}^{2}+c{d}^{2} \right ) ^{2}a}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{e}^{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 = 39.4054, size = 925, normalized size = 6.13 \begin{align*} \left [\frac{2 \, a c d^{2} e + 2 \, a^{2} e^{3} + 2 \,{\left (c^{2} d^{3} + a c d e^{2}\right )} x^{2} +{\left (a c d^{3} + 3 \, a^{2} d e^{2} +{\left (c^{2} d^{3} + 3 \, a c d e^{2}\right )} x^{4}\right )} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{4} + 2 \, a x^{2} \sqrt{-\frac{c}{a}} - a}{c x^{4} + a}\right ) - 2 \,{\left (a c e^{3} x^{4} + a^{2} e^{3}\right )} \log \left (c x^{4} + a\right ) + 4 \,{\left (a c e^{3} x^{4} + a^{2} e^{3}\right )} \log \left (e x^{2} + d\right )}{8 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} +{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{4}\right )}}, \frac{a c d^{2} e + a^{2} e^{3} +{\left (c^{2} d^{3} + a c d e^{2}\right )} x^{2} -{\left (a c d^{3} + 3 \, a^{2} d e^{2} +{\left (c^{2} d^{3} + 3 \, a c d e^{2}\right )} x^{4}\right )} \sqrt{\frac{c}{a}} \arctan \left (\frac{a \sqrt{\frac{c}{a}}}{c x^{2}}\right ) -{\left (a c e^{3} x^{4} + a^{2} e^{3}\right )} \log \left (c x^{4} + a\right ) + 2 \,{\left (a c e^{3} x^{4} + a^{2} e^{3}\right )} \log \left (e x^{2} + d\right )}{4 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} +{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c 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 = 1.09404, size = 269, normalized size = 1.78 \begin{align*} -\frac{e^{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{e^{4} \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{{\left (c^{2} d^{3} + 3 \, a c d e^{2}\right )} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \,{\left (a c^{2} d^{4} + 2 \, a^{2} c d^{2} e^{2} + a^{3} e^{4}\right )} \sqrt{a c}} + \frac{a c d^{2} e +{\left (c^{2} d^{3} + a c d e^{2}\right )} x^{2} + a^{2} e^{3}}{4 \,{\left (c x^{4} + a\right )}{\left (c d^{2} + a e^{2}\right )}^{2} a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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