Optimal. Leaf size=59 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{5/2}}-\frac{x^6}{4 c \left (a+c x^4\right )}+\frac{3 x^2}{4 c^2} \]
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Rubi [A] time = 0.031221, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {275, 288, 321, 205} \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{5/2}}-\frac{x^6}{4 c \left (a+c x^4\right )}+\frac{3 x^2}{4 c^2} \]
Antiderivative was successfully verified.
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Rule 275
Rule 288
Rule 321
Rule 205
Rubi steps
\begin{align*} \int \frac{x^9}{\left (a+c x^4\right )^2} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{\left (a+c x^2\right )^2} \, dx,x,x^2\right )\\ &=-\frac{x^6}{4 c \left (a+c x^4\right )}+\frac{3 \operatorname{Subst}\left (\int \frac{x^2}{a+c x^2} \, dx,x,x^2\right )}{4 c}\\ &=\frac{3 x^2}{4 c^2}-\frac{x^6}{4 c \left (a+c x^4\right )}-\frac{(3 a) \operatorname{Subst}\left (\int \frac{1}{a+c x^2} \, dx,x,x^2\right )}{4 c^2}\\ &=\frac{3 x^2}{4 c^2}-\frac{x^6}{4 c \left (a+c x^4\right )}-\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.042412, size = 60, normalized size = 1.02 \[ \frac{a x^2}{4 c^2 \left (a+c x^4\right )}-\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{5/2}}+\frac{x^2}{2 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 50, normalized size = 0.9 \begin{align*}{\frac{{x}^{2}}{2\,{c}^{2}}}+{\frac{a{x}^{2}}{4\,{c}^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{3\,a}{4\,{c}^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \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 = 1.7415, size = 296, normalized size = 5.02 \begin{align*} \left [\frac{4 \, c x^{6} + 6 \, a x^{2} + 3 \,{\left (c x^{4} + a\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 )}{8 \,{\left (c^{3} x^{4} + a c^{2}\right )}}, \frac{2 \, c x^{6} + 3 \, a x^{2} - 3 \,{\left (c x^{4} + a\right )} \sqrt{\frac{a}{c}} \arctan \left (\frac{c x^{2} \sqrt{\frac{a}{c}}}{a}\right )}{4 \,{\left (c^{3} x^{4} + a c^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.73952, size = 92, normalized size = 1.56 \begin{align*} \frac{a x^{2}}{4 a c^{2} + 4 c^{3} x^{4}} + \frac{3 \sqrt{- \frac{a}{c^{5}}} \log{\left (- c^{2} \sqrt{- \frac{a}{c^{5}}} + x^{2} \right )}}{8} - \frac{3 \sqrt{- \frac{a}{c^{5}}} \log{\left (c^{2} \sqrt{- \frac{a}{c^{5}}} + x^{2} \right )}}{8} + \frac{x^{2}}{2 c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12747, size = 66, normalized size = 1.12 \begin{align*} \frac{a x^{2}}{4 \,{\left (c x^{4} + a\right )} c^{2}} + \frac{x^{2}}{2 \, c^{2}} - \frac{3 \, a \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \, \sqrt{a c} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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