Optimal. Leaf size=20 \[ \frac {1}{3} \left (a x+b \log ^2\left (c x^n\right )\right )^3 \]
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Rubi [A] time = 0.15, antiderivative size = 20, normalized size of antiderivative = 1.00, 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} \[ \frac {1}{3} \left (a x+b \log ^2\left (c x^n\right )\right )^3 \]
Antiderivative was successfully verified.
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Rule 2544
Rule 2561
Rubi steps
\begin {align*} \int \left (\frac {a}{x^2}+\frac {2 b n \log \left (c x^n\right )}{x^3}\right ) \left (a x^2+b x \log ^2\left (c x^n\right )\right )^2 \, dx &=\int \frac {\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 )^2}{x^3} \, dx\\ &=\int \frac {\left (a x+2 b n \log \left (c x^n\right )\right ) \left (a x+b \log ^2\left (c x^n\right )\right )^2}{x} \, dx\\ &=\frac {1}{3} \left (a x+b \log ^2\left (c x^n\right )\right )^3\\ \end {align*}
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Mathematica [A] time = 0.02, size = 20, normalized size = 1.00 \[ \frac {1}{3} \left (a x+b \log ^2\left (c x^n\right )\right )^3 \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 195, normalized size = 9.75 \[ \frac {1}{3} \, b^{3} n^{6} \log \relax (x)^{6} + 2 \, b^{3} n^{5} \log \relax (c) \log \relax (x)^{5} + a b^{2} x \log \relax (c)^{4} + a^{2} b x^{2} \log \relax (c)^{2} + \frac {1}{3} \, a^{3} x^{3} + {\left (5 \, b^{3} n^{4} \log \relax (c)^{2} + a b^{2} n^{4} x\right )} \log \relax (x)^{4} + \frac {4}{3} \, {\left (5 \, b^{3} n^{3} \log \relax (c)^{3} + 3 \, a b^{2} n^{3} x \log \relax (c)\right )} \log \relax (x)^{3} + {\left (5 \, b^{3} n^{2} \log \relax (c)^{4} + 6 \, a b^{2} n^{2} x \log \relax (c)^{2} + a^{2} b n^{2} x^{2}\right )} \log \relax (x)^{2} + 2 \, {\left (b^{3} n \log \relax (c)^{5} + 2 \, a b^{2} n x \log \relax (c)^{3} + a^{2} b n x^{2} \log \relax (c)\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 198, normalized size = 9.90 \[ \frac {1}{3} \, b^{3} n^{6} \log \relax (x)^{6} + 2 \, b^{3} n^{5} \log \relax (c) \log \relax (x)^{5} + 2 \, b^{3} n \log \relax (c)^{5} \log \relax (x) + a b^{2} x \log \relax (c)^{4} + a^{2} b x^{2} \log \relax (c)^{2} + \frac {1}{3} \, a^{3} x^{3} + {\left (5 \, b^{3} n^{4} \log \relax (c)^{2} + a b^{2} n^{4} x\right )} \log \relax (x)^{4} + \frac {4}{3} \, {\left (5 \, b^{3} n^{3} \log \relax (c)^{3} + 3 \, a b^{2} n^{3} x \log \relax (c)\right )} \log \relax (x)^{3} + {\left (5 \, b^{3} n^{2} \log \relax (c)^{4} + 6 \, a b^{2} n^{2} x \log \relax (c)^{2} + a^{2} b n^{2} x^{2}\right )} \log \relax (x)^{2} + 2 \, {\left (2 \, a b^{2} n x \log \relax (c)^{3} + a^{2} b n x^{2} \log \relax (c)\right )} \log \relax (x) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 2.89, size = 16321, normalized size = 816.05 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.49, size = 211, normalized size = 10.55 \[ \frac {1}{3} \, b^{3} \log \left (c x^{n}\right )^{6} + 4 \, a b^{2} n x \log \left (c x^{n}\right )^{3} + a b^{2} x \log \left (c x^{n}\right )^{4} - \frac {1}{2} \, a^{2} b n^{2} x^{2} + a^{2} b n x^{2} \log \left (c x^{n}\right ) + a^{2} b x^{2} \log \left (c x^{n}\right )^{2} + \frac {1}{3} \, a^{3} x^{3} - 12 \, {\left (n x \log \left (c x^{n}\right )^{2} + 2 \, {\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} n\right )} a b^{2} n + \frac {1}{2} \, {\left (n^{2} x^{2} - 2 \, n x^{2} \log \left (c x^{n}\right )\right )} a^{2} b - 4 \, {\left (n x \log \left (c x^{n}\right )^{3} - 3 \, {\left (n x \log \left (c x^{n}\right )^{2} + 2 \, {\left (n^{2} x - n x \log \left (c x^{n}\right )\right )} n\right )} n\right )} a b^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.33, size = 52, normalized size = 2.60 \[ \frac {a^3\,x^3}{3}+a^2\,b\,x^2\,{\ln \left (c\,x^n\right )}^2+a\,b^2\,x\,{\ln \left (c\,x^n\right )}^4+\frac {b^3\,{\ln \left (c\,x^n\right )}^6}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.32, size = 221, normalized size = 11.05 \[ \frac {a^{3} x^{3}}{3} + a^{2} b n^{2} x^{2} \log {\relax (x )}^{2} - a^{2} b n^{2} x^{2} \log {\relax (x )} + 2 a^{2} b n x^{2} \log {\relax (c )} \log {\relax (x )} - a^{2} b n x^{2} \log {\relax (c )} + a^{2} b n x^{2} \log {\left (c x^{n} \right )} + a^{2} b x^{2} \log {\relax (c )}^{2} + a b^{2} n^{4} x \log {\relax (x )}^{4} + 4 a b^{2} n^{3} x \log {\relax (c )} \log {\relax (x )}^{3} + 6 a b^{2} n^{2} x \log {\relax (c )}^{2} \log {\relax (x )}^{2} + 4 a b^{2} n x \log {\relax (c )}^{3} \log {\relax (x )} + a b^{2} x \log {\relax (c )}^{4} - 2 b^{3} n \left (\begin {cases} - \log {\relax (c )}^{5} \log {\relax (x )} & \text {for}\: n = 0 \\- \frac {\log {\left (c x^{n} \right )}^{6}}{6 n} & \text {otherwise} \end {cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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