Optimal. Leaf size=54 \[ -\frac{a^2}{12 x^{12}}-\frac{2 a c+b^2}{8 x^8}-\frac{a b}{5 x^{10}}-\frac{b c}{3 x^6}-\frac{c^2}{4 x^4} \]
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Rubi [A] time = 0.0353243, antiderivative size = 54, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111, Rules used = {1114, 698} \[ -\frac{a^2}{12 x^{12}}-\frac{2 a c+b^2}{8 x^8}-\frac{a b}{5 x^{10}}-\frac{b c}{3 x^6}-\frac{c^2}{4 x^4} \]
Antiderivative was successfully verified.
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Rule 1114
Rule 698
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2+c x^4\right )^2}{x^{13}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\left (a+b x+c x^2\right )^2}{x^7} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{a^2}{x^7}+\frac{2 a b}{x^6}+\frac{b^2+2 a c}{x^5}+\frac{2 b c}{x^4}+\frac{c^2}{x^3}\right ) \, dx,x,x^2\right )\\ &=-\frac{a^2}{12 x^{12}}-\frac{a b}{5 x^{10}}-\frac{b^2+2 a c}{8 x^8}-\frac{b c}{3 x^6}-\frac{c^2}{4 x^4}\\ \end{align*}
Mathematica [A] time = 0.0153282, size = 50, normalized size = 0.93 \[ -\frac{10 a^2+24 a b x^2+30 a c x^4+15 b^2 x^4+40 b c x^6+30 c^2 x^8}{120 x^{12}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 45, normalized size = 0.8 \begin{align*} -{\frac{ab}{5\,{x}^{10}}}-{\frac{{c}^{2}}{4\,{x}^{4}}}-{\frac{2\,ac+{b}^{2}}{8\,{x}^{8}}}-{\frac{bc}{3\,{x}^{6}}}-{\frac{{a}^{2}}{12\,{x}^{12}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956137, size = 62, normalized size = 1.15 \begin{align*} -\frac{30 \, c^{2} x^{8} + 40 \, b c x^{6} + 15 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 24 \, a b x^{2} + 10 \, a^{2}}{120 \, x^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58857, size = 113, normalized size = 2.09 \begin{align*} -\frac{30 \, c^{2} x^{8} + 40 \, b c x^{6} + 15 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 24 \, a b x^{2} + 10 \, a^{2}}{120 \, x^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.80806, size = 49, normalized size = 0.91 \begin{align*} - \frac{10 a^{2} + 24 a b x^{2} + 40 b c x^{6} + 30 c^{2} x^{8} + x^{4} \left (30 a c + 15 b^{2}\right )}{120 x^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1095, size = 65, normalized size = 1.2 \begin{align*} -\frac{30 \, c^{2} x^{8} + 40 \, b c x^{6} + 15 \, b^{2} x^{4} + 30 \, a c x^{4} + 24 \, a b x^{2} + 10 \, a^{2}}{120 \, x^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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