Optimal. Leaf size=178 \[ \frac {\log \left (d \left (e+f \sqrt {x}\right )\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-\frac {\log \left (1+\frac {f \sqrt {x}}{e}\right ) \left (a+b \log \left (c x^n\right )\right )^4}{4 b n}-2 \left (a+b \log \left (c x^n\right )\right )^3 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )+12 b n \left (a+b \log \left (c x^n\right )\right )^2 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )-48 b^2 n^2 \left (a+b \log \left (c x^n\right )\right ) \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )+96 b^3 n^3 \text {Li}_5\left (-\frac {f \sqrt {x}}{e}\right ) \]
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Rubi [A]
time = 0.16, 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 = {2422, 2375,
2421, 2430, 6724} \begin {gather*} -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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2375
Rule 2421
Rule 2422
Rule 2430
Rule 6724
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] Leaf count is larger than twice the leaf count of optimal. \(403\) vs. \(2(178)=356\).
time = 0.24, size = 403, normalized size = 2.26 \begin {gather*} \frac {1}{8} \left (-2 \log \left (d \left (e+f \sqrt {x}\right )\right ) \log (x) \left (b^3 n^3 \log ^3(x)-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\right )-8 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^3 \left (\log \left (1+\frac {f \sqrt {x}}{e}\right ) \log (x)+2 \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )\right )-12 b n \left (a-b n \log (x)+b \log \left (c x^n\right )\right )^2 \left (\log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^2(x)+4 \log (x) \text {Li}_2\left (-\frac {f \sqrt {x}}{e}\right )-8 \text {Li}_3\left (-\frac {f \sqrt {x}}{e}\right )\right )-8 b^2 n^2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right ) \left (\log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^3(x)+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 )+48 \text {Li}_4\left (-\frac {f \sqrt {x}}{e}\right )\right )-2 b^3 n^3 \left (\log \left (1+\frac {f \sqrt {x}}{e}\right ) \log ^4(x)+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 )-384 \text {Li}_5\left (-\frac {f \sqrt {x}}{e}\right )\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right )^{3} \ln \left (d \left (e +f \sqrt {x}\right )\right )}{x}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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