Optimal. Leaf size=20 \[ -\frac{1}{2 \left (a x+b \log ^2\left (c x^n\right )\right )^2} \]
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Rubi [A] time = 0.159982, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {2561, 2544} \[ -\frac{1}{2 \left (a x+b \log ^2\left (c x^n\right )\right )^2} \]
Antiderivative was successfully verified.
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Rule 2561
Rule 2544
Rubi steps
\begin{align*} \int \frac{a x^3+2 b n x^2 \log \left (c x^n\right )}{\left (a x^2+b x \log ^2\left (c x^n\right )\right )^3} \, dx &=\int \frac{x^2 \left (a x+2 b n \log \left (c x^n\right )\right )}{\left (a x^2+b x \log ^2\left (c x^n\right )\right )^3} \, dx\\ &=\int \frac{a x+2 b n \log \left (c x^n\right )}{x \left (a x+b \log ^2\left (c x^n\right )\right )^3} \, dx\\ &=-\frac{1}{2 \left (a x+b \log ^2\left (c x^n\right )\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0141217, size = 20, normalized size = 1. \[ -\frac{1}{2 \left (a x+b \log ^2\left (c x^n\right )\right )^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.085, size = 451, normalized size = 22.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.42503, size = 128, normalized size = 6.4 \begin{align*} -\frac{1}{2 \,{\left (b^{2} \log \left (c\right )^{4} + 4 \, b^{2} \log \left (c\right ) \log \left (x^{n}\right )^{3} + b^{2} \log \left (x^{n}\right )^{4} + 2 \, a b x \log \left (c\right )^{2} + a^{2} x^{2} + 2 \,{\left (3 \, b^{2} \log \left (c\right )^{2} + a b x\right )} \log \left (x^{n}\right )^{2} + 4 \,{\left (b^{2} \log \left (c\right )^{3} + a b x \log \left (c\right )\right )} \log \left (x^{n}\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02759, size = 254, normalized size = 12.7 \begin{align*} -\frac{1}{2 \,{\left (b^{2} n^{4} \log \left (x\right )^{4} + 4 \, b^{2} n^{3} \log \left (c\right ) \log \left (x\right )^{3} + b^{2} \log \left (c\right )^{4} + 2 \, a b x \log \left (c\right )^{2} + a^{2} x^{2} + 2 \,{\left (3 \, b^{2} n^{2} \log \left (c\right )^{2} + a b n^{2} x\right )} \log \left (x\right )^{2} + 4 \,{\left (b^{2} n \log \left (c\right )^{3} + a b n x \log \left (c\right )\right )} \log \left (x\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [B] time = 1.27848, size = 413, normalized size = 20.65 \begin{align*} -\frac{4 \, a b n^{2} x + a^{2} x^{2}}{2 \,{\left (4 \, a b^{3} n^{6} x \log \left (x\right )^{4} + 16 \, a b^{3} n^{5} x \log \left (c\right ) \log \left (x\right )^{3} + a^{2} b^{2} n^{4} x^{2} \log \left (x\right )^{4} + 24 \, a b^{3} n^{4} x \log \left (c\right )^{2} \log \left (x\right )^{2} + 4 \, a^{2} b^{2} n^{3} x^{2} \log \left (c\right ) \log \left (x\right )^{3} + 16 \, a b^{3} n^{3} x \log \left (c\right )^{3} \log \left (x\right ) + 8 \, a^{2} b^{2} n^{4} x^{2} \log \left (x\right )^{2} + 6 \, a^{2} b^{2} n^{2} x^{2} \log \left (c\right )^{2} \log \left (x\right )^{2} + 4 \, a b^{3} n^{2} x \log \left (c\right )^{4} + 16 \, a^{2} b^{2} n^{3} x^{2} \log \left (c\right ) \log \left (x\right ) + 4 \, a^{2} b^{2} n x^{2} \log \left (c\right )^{3} \log \left (x\right ) + 2 \, a^{3} b n^{2} x^{3} \log \left (x\right )^{2} + 8 \, a^{2} b^{2} n^{2} x^{2} \log \left (c\right )^{2} + a^{2} b^{2} x^{2} \log \left (c\right )^{4} + 4 \, a^{3} b n x^{3} \log \left (c\right ) \log \left (x\right ) + 4 \, a^{3} b n^{2} x^{3} + 2 \, a^{3} b x^{3} \log \left (c\right )^{2} + a^{4} x^{4}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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