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 = 0.16, 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} \[ -\frac {1}{2 \left (a x+b \log ^2\left (c x^n\right )\right )^2} \]
Antiderivative was successfully verified.
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Rule 2544
Rule 2561
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*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.00 \[ -\frac {1}{2 \left (a x+b \log ^2\left (c x^n\right )\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 101, normalized size = 5.05 \[ -\frac {1}{2 \, {\left (b^{2} n^{4} \log \relax (x)^{4} + 4 \, b^{2} n^{3} \log \relax (c) \log \relax (x)^{3} + b^{2} \log \relax (c)^{4} + 2 \, a b x \log \relax (c)^{2} + a^{2} x^{2} + 2 \, {\left (3 \, b^{2} n^{2} \log \relax (c)^{2} + a b n^{2} x\right )} \log \relax (x)^{2} + 4 \, {\left (b^{2} n \log \relax (c)^{3} + a b n x \log \relax (c)\right )} \log \relax (x)\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 306, normalized size = 15.30 \[ -\frac {4 \, a b n^{2} x + a^{2} x^{2}}{2 \, {\left (4 \, a b^{3} n^{6} x \log \relax (x)^{4} + 16 \, a b^{3} n^{5} x \log \relax (c) \log \relax (x)^{3} + a^{2} b^{2} n^{4} x^{2} \log \relax (x)^{4} + 24 \, a b^{3} n^{4} x \log \relax (c)^{2} \log \relax (x)^{2} + 4 \, a^{2} b^{2} n^{3} x^{2} \log \relax (c) \log \relax (x)^{3} + 16 \, a b^{3} n^{3} x \log \relax (c)^{3} \log \relax (x) + 8 \, a^{2} b^{2} n^{4} x^{2} \log \relax (x)^{2} + 6 \, a^{2} b^{2} n^{2} x^{2} \log \relax (c)^{2} \log \relax (x)^{2} + 4 \, a b^{3} n^{2} x \log \relax (c)^{4} + 16 \, a^{2} b^{2} n^{3} x^{2} \log \relax (c) \log \relax (x) + 4 \, a^{2} b^{2} n x^{2} \log \relax (c)^{3} \log \relax (x) + 2 \, a^{3} b n^{2} x^{3} \log \relax (x)^{2} + 8 \, a^{2} b^{2} n^{2} x^{2} \log \relax (c)^{2} + a^{2} b^{2} x^{2} \log \relax (c)^{4} + 4 \, a^{3} b n x^{3} \log \relax (c) \log \relax (x) + 4 \, a^{3} b n^{2} x^{3} + 2 \, a^{3} b x^{3} \log \relax (c)^{2} + a^{4} x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 447, normalized size = 22.35 \[ -\frac {8}{\left (\pi ^{2} b \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{2}-2 \pi ^{2} b \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+\pi ^{2} b \mathrm {csgn}\left (i c \right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}-2 \pi ^{2} b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{3}+4 \pi ^{2} b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{4}-2 \pi ^{2} b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}+\pi ^{2} b \mathrm {csgn}\left (i x^{n}\right )^{2} \mathrm {csgn}\left (i c \,x^{n}\right )^{4}-2 \pi ^{2} b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{5}+\pi ^{2} b \mathrm {csgn}\left (i c \,x^{n}\right )^{6}+4 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \relax (c )+4 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right ) \ln \left (x^{n}\right )-4 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )-4 i \pi b \,\mathrm {csgn}\left (i c \right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (x^{n}\right )-4 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \relax (c )-4 i \pi b \,\mathrm {csgn}\left (i x^{n}\right ) \mathrm {csgn}\left (i c \,x^{n}\right )^{2} \ln \left (x^{n}\right )+4 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \relax (c )+4 i \pi b \mathrm {csgn}\left (i c \,x^{n}\right )^{3} \ln \left (x^{n}\right )-4 b \ln \relax (c )^{2}-8 b \ln \relax (c ) \ln \left (x^{n}\right )-4 b \ln \left (x^{n}\right )^{2}-4 a x \right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.85, size = 95, normalized size = 4.75 \[ -\frac {1}{2 \, {\left (b^{2} \log \relax (c)^{4} + 4 \, b^{2} \log \relax (c) \log \left (x^{n}\right )^{3} + b^{2} \log \left (x^{n}\right )^{4} + 2 \, a b x \log \relax (c)^{2} + a^{2} x^{2} + 2 \, {\left (3 \, b^{2} \log \relax (c)^{2} + a b x\right )} \log \left (x^{n}\right )^{2} + 4 \, {\left (b^{2} \log \relax (c)^{3} + a b x \log \relax (c)\right )} \log \left (x^{n}\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.27, size = 39, normalized size = 1.95 \[ -\frac {1}{2\,a^2\,x^2+4\,a\,b\,x\,{\ln \left (c\,x^n\right )}^2+2\,b^2\,{\ln \left (c\,x^n\right )}^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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