Optimal. Leaf size=134 \[ -\frac{a^2 e \log \left (a+c x^4\right )}{4 c^2 \left (a e^2+c d^2\right )}+\frac{a^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{3/2} \left (a e^2+c d^2\right )}+\frac{d^4 \log \left (d+e x^2\right )}{2 e^3 \left (a e^2+c d^2\right )}-\frac{d x^2}{2 c e^2}+\frac{x^4}{4 c e} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.178927, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {1252, 1629, 635, 205, 260} \[ -\frac{a^2 e \log \left (a+c x^4\right )}{4 c^2 \left (a e^2+c d^2\right )}+\frac{a^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{3/2} \left (a e^2+c d^2\right )}+\frac{d^4 \log \left (d+e x^2\right )}{2 e^3 \left (a e^2+c d^2\right )}-\frac{d x^2}{2 c e^2}+\frac{x^4}{4 c e} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 1252
Rule 1629
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{x^9}{\left (d+e x^2\right ) \left (a+c x^4\right )} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{(d+e x) \left (a+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{d}{c e^2}+\frac{x}{c e}+\frac{d^4}{e^2 \left (c d^2+a e^2\right ) (d+e x)}+\frac{a^2 (d-e x)}{c \left (c d^2+a e^2\right ) \left (a+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{d x^2}{2 c e^2}+\frac{x^4}{4 c e}+\frac{d^4 \log \left (d+e x^2\right )}{2 e^3 \left (c d^2+a e^2\right )}+\frac{a^2 \operatorname{Subst}\left (\int \frac{d-e x}{a+c x^2} \, dx,x,x^2\right )}{2 c \left (c d^2+a e^2\right )}\\ &=-\frac{d x^2}{2 c e^2}+\frac{x^4}{4 c e}+\frac{d^4 \log \left (d+e x^2\right )}{2 e^3 \left (c d^2+a e^2\right )}+\frac{\left (a^2 d\right ) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{2 c \left (c d^2+a e^2\right )}-\frac{\left (a^2 e\right ) \operatorname{Subst}\left (\int \frac{x}{a+c x^2} \, dx,x,x^2\right )}{2 c \left (c d^2+a e^2\right )}\\ &=-\frac{d x^2}{2 c e^2}+\frac{x^4}{4 c e}+\frac{a^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{3/2} \left (c d^2+a e^2\right )}+\frac{d^4 \log \left (d+e x^2\right )}{2 e^3 \left (c d^2+a e^2\right )}-\frac{a^2 e \log \left (a+c x^4\right )}{4 c^2 \left (c d^2+a e^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0611195, size = 134, normalized size = 1. \[ -\frac{a^2 e \log \left (a+c x^4\right )}{4 c^2 \left (a e^2+c d^2\right )}+\frac{a^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{3/2} \left (a e^2+c d^2\right )}+\frac{d^4 \log \left (d+e x^2\right )}{2 e^3 \left (a e^2+c d^2\right )}-\frac{d x^2}{2 c e^2}+\frac{x^4}{4 c e} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.011, size = 122, normalized size = 0.9 \begin{align*}{\frac{{x}^{4}}{4\,ce}}-{\frac{d{x}^{2}}{2\,c{e}^{2}}}-{\frac{{a}^{2}e\ln \left ( c{x}^{4}+a \right ) }{4\,{c}^{2} \left ( a{e}^{2}+c{d}^{2} \right ) }}+{\frac{d{a}^{2}}{ \left ( 2\,a{e}^{2}+2\,c{d}^{2} \right ) c}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{d}^{4}\ln \left ( e{x}^{2}+d \right ) }{2\,{e}^{3} \left ( a{e}^{2}+c{d}^{2} \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 24.9758, size = 563, normalized size = 4.2 \begin{align*} \left [\frac{a c d e^{3} \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{4} + 2 \, c x^{2} \sqrt{-\frac{a}{c}} - a}{c x^{4} + a}\right ) - a^{2} e^{4} \log \left (c x^{4} + a\right ) + 2 \, c^{2} d^{4} \log \left (e x^{2} + d\right ) +{\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{4} - 2 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x^{2}}{4 \,{\left (c^{3} d^{2} e^{3} + a c^{2} e^{5}\right )}}, \frac{2 \, a c d e^{3} \sqrt{\frac{a}{c}} \arctan \left (\frac{c x^{2} \sqrt{\frac{a}{c}}}{a}\right ) - a^{2} e^{4} \log \left (c x^{4} + a\right ) + 2 \, c^{2} d^{4} \log \left (e x^{2} + d\right ) +{\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{4} - 2 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x^{2}}{4 \,{\left (c^{3} d^{2} e^{3} + a c^{2} e^{5}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.09969, size = 163, normalized size = 1.22 \begin{align*} \frac{d^{4} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \,{\left (c d^{2} e^{3} + a e^{5}\right )}} - \frac{a^{2} e \log \left (c x^{4} + a\right )}{4 \,{\left (c^{3} d^{2} + a c^{2} e^{2}\right )}} + \frac{a^{2} d \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{2 \,{\left (c^{2} d^{2} + a c e^{2}\right )} \sqrt{a c}} + \frac{{\left (c x^{4} e - 2 \, c d x^{2}\right )} e^{\left (-2\right )}}{4 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]