Optimal. Leaf size=52 \[ -\frac{a^2}{8 c^3 \left (a+c x^4\right )^2}+\frac{a}{2 c^3 \left (a+c x^4\right )}+\frac{\log \left (a+c x^4\right )}{4 c^3} \]
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Rubi [A] time = 0.0362498, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {266, 43} \[ -\frac{a^2}{8 c^3 \left (a+c x^4\right )^2}+\frac{a}{2 c^3 \left (a+c x^4\right )}+\frac{\log \left (a+c x^4\right )}{4 c^3} \]
Antiderivative was successfully verified.
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Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^{11}}{\left (a+c x^4\right )^3} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x^2}{(a+c x)^3} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{a^2}{c^2 (a+c x)^3}-\frac{2 a}{c^2 (a+c x)^2}+\frac{1}{c^2 (a+c x)}\right ) \, dx,x,x^4\right )\\ &=-\frac{a^2}{8 c^3 \left (a+c x^4\right )^2}+\frac{a}{2 c^3 \left (a+c x^4\right )}+\frac{\log \left (a+c x^4\right )}{4 c^3}\\ \end{align*}
Mathematica [A] time = 0.017793, size = 39, normalized size = 0.75 \[ \frac{\frac{a \left (3 a+4 c x^4\right )}{\left (a+c x^4\right )^2}+2 \log \left (a+c x^4\right )}{8 c^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 47, normalized size = 0.9 \begin{align*} -{\frac{{a}^{2}}{8\,{c}^{3} \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{a}{2\,{c}^{3} \left ( c{x}^{4}+a \right ) }}+{\frac{\ln \left ( c{x}^{4}+a \right ) }{4\,{c}^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.0194, size = 74, normalized size = 1.42 \begin{align*} \frac{4 \, a c x^{4} + 3 \, a^{2}}{8 \,{\left (c^{5} x^{8} + 2 \, a c^{4} x^{4} + a^{2} c^{3}\right )}} + \frac{\log \left (c x^{4} + a\right )}{4 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67201, size = 143, normalized size = 2.75 \begin{align*} \frac{4 \, a c x^{4} + 3 \, a^{2} + 2 \,{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \log \left (c x^{4} + a\right )}{8 \,{\left (c^{5} x^{8} + 2 \, a c^{4} x^{4} + a^{2} c^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.65684, size = 53, normalized size = 1.02 \begin{align*} \frac{3 a^{2} + 4 a c x^{4}}{8 a^{2} c^{3} + 16 a c^{4} x^{4} + 8 c^{5} x^{8}} + \frac{\log{\left (a + c x^{4} \right )}}{4 c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11675, size = 57, normalized size = 1.1 \begin{align*} \frac{\log \left ({\left | c x^{4} + a \right |}\right )}{4 \, c^{3}} - \frac{3 \, c x^{8} + 2 \, a x^{4}}{8 \,{\left (c x^{4} + a\right )}^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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