Optimal. Leaf size=178 \[ -48 b^2 n^2 \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3+12 b n \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2-\frac {\log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}+96 b^3 n^3 \text {Li}_5\left (-\frac {f \sqrt {x}}{e}\right ) \]
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Rubi [A] time = 0.23, antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -48 b^2 n^2 \text {PolyLog}\left (4,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )-2 \text {PolyLog}\left (2,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3+12 b n \text {PolyLog}\left (3,-\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^2+96 b^3 n^3 \text {PolyLog}\left (5,-\frac {f \sqrt {x}}{e}\right )+\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-\frac {\log \left (\frac {f \sqrt {x}}{e}+1\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n} \]
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 )^3}{x} \, dx &=\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-\frac {f \int \frac {\left (a+b \log \left (c x^n\right )\right )^4}{\left (e+f \sqrt {x}\right ) \sqrt {x}} \, dx}{8 b n}\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}+\int \frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^3}{x} \, dx\\ &=\frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-2 \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+(6 b n) \int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \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 )^4}{4 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-2 \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+12 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )-\left (24 b^2 n^2\right ) \int \frac {\left (a+b \log \left (c x^n\right )\right ) \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 )^4}{4 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-2 \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+12 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )-48 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )+\left (48 b^3 n^3\right ) \int \frac {\text {Li}_4\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 )^4}{4 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-2 \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+12 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )-48 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )+96 b^3 n^3 \text {Li}_5\left (-\frac {f \sqrt {x}}{e}\right )\\ \end {align*}
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Mathematica [B] time = 0.42, size = 403, normalized size = 2.26 \[ \frac {1}{8} \left (-8 b^2 n^2 \left (48 \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )+6 \log ^2(x) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )-24 \log (x) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )+\log ^3(x) \log \left (\frac {f \sqrt {x}}{e}+1\right )\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )-2 \log (x) \log \left (d \left (e+f \sqrt {x}\right )\right ) \left (-4 b^2 n^2 \log ^2(x) \left (a+b \log \left (c x^n\right )\right )+6 b n \log (x) \left (a+b \log \left (c x^n\right )\right )^2-4 \left (a+b \log \left (c x^n\right )\right )^3+b^3 n^3 \log ^3(x)\right )-12 b n \left (-8 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )+4 \log (x) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+\log ^2(x) \log \left (\frac {f \sqrt {x}}{e}+1\right )\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^2-8 \left (2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+\log (x) \log \left (\frac {f \sqrt {x}}{e}+1\right )\right ) \left (a+b \log \left (c x^n\right )-b n \log (x)\right )^3-2 b^3 n^3 \left (-384 \text {Li}_5\left (-\frac {f \sqrt {x}}{e}\right )+8 \log ^3(x) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )-48 \log ^2(x) \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )+192 \log (x) \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )+\log ^4(x) \log \left (\frac {f \sqrt {x}}{e}+1\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b^{3} \log \left (c x^{n}\right )^{3} + 3 \, a b^{2} \log \left (c x^{n}\right )^{2} + 3 \, a^{2} b \log \left (c x^{n}\right ) + a^{3}\right )} \log \left (d f \sqrt {x} + d e\right )}{x}, x\right ) \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.13, size = 0, normalized size = 0.00 \[ \int \frac {\left (b \ln \left (c \,x^{n}\right )+a \right )^{3} \ln \left (\left (f \sqrt {x}+e \right ) d \right )}{x}\, dx \]
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 \[ \int \frac {{\left (b \log \left (c x^{n}\right ) + a\right )}^{3} \log \left ({\left (f \sqrt {x} + e\right )} d\right )}{x}\,{d x} \]
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 \[ \int \frac {\ln \left (d\,\left (e+f\,\sqrt {x}\right )\right )\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^3}{x} \,d x \]
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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