Optimal. Leaf size=40 \[ -\frac{3 b^2 c}{2 x^2}-\frac{b^3}{4 x^4}+3 b c^2 \log (x)+\frac{c^3 x^2}{2} \]
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Rubi [A] time = 0.0256056, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {1584, 266, 43} \[ -\frac{3 b^2 c}{2 x^2}-\frac{b^3}{4 x^4}+3 b c^2 \log (x)+\frac{c^3 x^2}{2} \]
Antiderivative was successfully verified.
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Rule 1584
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (b x^2+c x^4\right )^3}{x^{11}} \, dx &=\int \frac{\left (b+c x^2\right )^3}{x^5} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{(b+c x)^3}{x^3} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (c^3+\frac{b^3}{x^3}+\frac{3 b^2 c}{x^2}+\frac{3 b c^2}{x}\right ) \, dx,x,x^2\right )\\ &=-\frac{b^3}{4 x^4}-\frac{3 b^2 c}{2 x^2}+\frac{c^3 x^2}{2}+3 b c^2 \log (x)\\ \end{align*}
Mathematica [A] time = 0.0042707, size = 40, normalized size = 1. \[ -\frac{3 b^2 c}{2 x^2}-\frac{b^3}{4 x^4}+3 b c^2 \log (x)+\frac{c^3 x^2}{2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 35, normalized size = 0.9 \begin{align*} -{\frac{{b}^{3}}{4\,{x}^{4}}}-{\frac{3\,{b}^{2}c}{2\,{x}^{2}}}+{\frac{{c}^{3}{x}^{2}}{2}}+3\,b{c}^{2}\ln \left ( x \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.934611, size = 50, normalized size = 1.25 \begin{align*} \frac{1}{2} \, c^{3} x^{2} + \frac{3}{2} \, b c^{2} \log \left (x^{2}\right ) - \frac{6 \, b^{2} c x^{2} + b^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.42323, size = 85, normalized size = 2.12 \begin{align*} \frac{2 \, c^{3} x^{6} + 12 \, b c^{2} x^{4} \log \left (x\right ) - 6 \, b^{2} c x^{2} - b^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.369361, size = 36, normalized size = 0.9 \begin{align*} 3 b c^{2} \log{\left (x \right )} + \frac{c^{3} x^{2}}{2} - \frac{b^{3} + 6 b^{2} c x^{2}}{4 x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.23102, size = 62, normalized size = 1.55 \begin{align*} \frac{1}{2} \, c^{3} x^{2} + \frac{3}{2} \, b c^{2} \log \left (x^{2}\right ) - \frac{9 \, b c^{2} x^{4} + 6 \, b^{2} c x^{2} + b^{3}}{4 \, x^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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