Optimal. Leaf size=135 \[ 2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \log \left (-\frac {e \sqrt {x}}{d}\right )+6 b n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )-12 b^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \text {Li}_3\left (1+\frac {e \sqrt {x}}{d}\right )+12 b^3 n^3 \text {Li}_4\left (1+\frac {e \sqrt {x}}{d}\right ) \]
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Rubi [A]
time = 0.14, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2504, 2443,
2481, 2421, 2430, 6724} \begin {gather*} -12 b^2 n^2 \text {PolyLog}\left (3,\frac {e \sqrt {x}}{d}+1\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )+6 b n \text {PolyLog}\left (2,\frac {e \sqrt {x}}{d}+1\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2+12 b^3 n^3 \text {PolyLog}\left (4,\frac {e \sqrt {x}}{d}+1\right )+2 \log \left (-\frac {e \sqrt {x}}{d}\right ) \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \end {gather*}
Antiderivative was successfully verified.
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Rule 2421
Rule 2430
Rule 2443
Rule 2481
Rule 2504
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3}{x} \, dx &=2 \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^n\right )\right )^3}{x} \, dx,x,\sqrt {x}\right )\\ &=2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \log \left (-\frac {e \sqrt {x}}{d}\right )-(6 b e n) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )^2}{d+e x} \, dx,x,\sqrt {x}\right )\\ &=2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \log \left (-\frac {e \sqrt {x}}{d}\right )-(6 b n) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right )^2 \log \left (-\frac {e \left (-\frac {d}{e}+\frac {x}{e}\right )}{d}\right )}{x} \, dx,x,d+e \sqrt {x}\right )\\ &=2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \log \left (-\frac {e \sqrt {x}}{d}\right )+6 b n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )-\left (12 b^2 n^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^n\right )\right ) \text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+e \sqrt {x}\right )\\ &=2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \log \left (-\frac {e \sqrt {x}}{d}\right )+6 b n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )-12 b^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \text {Li}_3\left (1+\frac {e \sqrt {x}}{d}\right )+\left (12 b^3 n^3\right ) \text {Subst}\left (\int \frac {\text {Li}_3\left (\frac {x}{d}\right )}{x} \, dx,x,d+e \sqrt {x}\right )\\ &=2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \log \left (-\frac {e \sqrt {x}}{d}\right )+6 b n \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )-12 b^2 n^2 \left (a+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \text {Li}_3\left (1+\frac {e \sqrt {x}}{d}\right )+12 b^3 n^3 \text {Li}_4\left (1+\frac {e \sqrt {x}}{d}\right )\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(333\) vs. \(2(135)=270\).
time = 0.09, size = 333, normalized size = 2.47 \begin {gather*} \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^3 \log (x)+3 b n \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right )^2 \left (\left (\log \left (d+e \sqrt {x}\right )-\log \left (1+\frac {e \sqrt {x}}{d}\right )\right ) \log (x)-2 \text {Li}_2\left (-\frac {e \sqrt {x}}{d}\right )\right )+6 b^2 n^2 \left (a-b n \log \left (d+e \sqrt {x}\right )+b \log \left (c \left (d+e \sqrt {x}\right )^n\right )\right ) \left (\log ^2\left (d+e \sqrt {x}\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+2 \log \left (d+e \sqrt {x}\right ) \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )-2 \text {Li}_3\left (1+\frac {e \sqrt {x}}{d}\right )\right )+2 b^3 n^3 \left (\log ^3\left (d+e \sqrt {x}\right ) \log \left (-\frac {e \sqrt {x}}{d}\right )+3 \log ^2\left (d+e \sqrt {x}\right ) \text {Li}_2\left (1+\frac {e \sqrt {x}}{d}\right )-6 \log \left (d+e \sqrt {x}\right ) \text {Li}_3\left (1+\frac {e \sqrt {x}}{d}\right )+6 \text {Li}_4\left (1+\frac {e \sqrt {x}}{d}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \ln \left (c \left (d +e \sqrt {x}\right )^{n}\right )\right )^{3}}{x}\, dx\]
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \log {\left (c \left (d + e \sqrt {x}\right )^{n} \right )}\right )^{3}}{x}\, dx \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 {{\left (a+b\,\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^n\right )\right )}^3}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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