Optimal. Leaf size=145 \[ \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+8 b n \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {\log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-16 b^2 n^2 \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right ) \]
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Rubi [A] time = 0.19, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2375, 2337, 2374, 2383, 6589} \[ -2 \text {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+8 b n \text {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-16 b^2 n^2 \text {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right )+\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-\frac {\log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n} \]
Antiderivative was successfully verified.
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Rule 2337
Rule 2374
Rule 2375
Rule 2383
Rule 6589
Rubi steps
\begin {align*} \int \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx &=\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-\frac {f \int \frac {\left (a+b \log \left (c x^n\right )\right )^3}{\left (e+f \sqrt {x}\right ) \sqrt {x}} \, dx}{6 b n}\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}+\int \frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{x} \, dx\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-2 \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+(4 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )}{x} \, dx\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-2 \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+8 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )-\left (8 b^2 n^2\right ) \int \frac {\text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )}{x} \, dx\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{3 b n}-2 \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+8 b n \left (a+b \log \left (c x^n\right )\right ) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )-16 b^2 n^2 \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )\\ \end {align*}
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Mathematica [A] time = 0.25, size = 263, normalized size = 1.81 \[ \frac {1}{3} \left (\log (x) \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (-3 b n \log (x) \left (a+b \log \left (c x^n\right )\right )+3 \left (a+b \log \left (c x^n\right )\right )^2+b^2 n^2 \log ^2(x)\right )-3 b n \left (-8 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )+4 \log (x) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+\log ^2(x) \log \left (\frac {f \sqrt {x}}{e}+1\right )\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-3 \left (2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+\log (x) \log \left (\frac {f \sqrt {x}}{e}+1\right )\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2-b^2 n^2 \left (48 \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )+6 \log ^2(x) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )-24 \log (x) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )+\log ^3(x) \log \left (\frac {f \sqrt {x}}{e}+1\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}\right )} \log \left (d f \sqrt {x} + d e\right )}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{2} \ln \left (\left (f \sqrt {x}+e \right ) d \right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{2} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^2}{x} \,d x \]
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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