Optimal. Leaf size=168 \[ -\frac {3 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}-\frac {6 b^2 e p^2 \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right ) \text {Li}_2\left (1+\frac {e}{d (f+g x)}\right )}{d g}+\frac {6 b^3 e p^3 \text {Li}_3\left (1+\frac {e}{d (f+g x)}\right )}{d g} \]
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Rubi [A]
time = 0.12, antiderivative size = 168, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {2533, 2499,
2504, 2443, 2481, 2421, 6724} \begin {gather*} -\frac {6 b^2 e p^2 \text {PolyLog}\left (2,\frac {e}{d (f+g x)}+1\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )}{d g}+\frac {6 b^3 e p^3 \text {PolyLog}\left (3,\frac {e}{d (f+g x)}+1\right )}{d g}-\frac {3 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac {(d (f+g x)+e) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g} \end {gather*}
Antiderivative was successfully verified.
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Rule 2421
Rule 2443
Rule 2481
Rule 2499
Rule 2504
Rule 2533
Rule 6724
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3 \, dx &=\frac {\text {Subst}\left (\int \left (a+b \log \left (c \left (d+\frac {e}{x}\right )^p\right )\right )^3 \, dx,x,f+g x\right )}{g}\\ &=\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac {(3 b e p) \text {Subst}\left (\int \frac {\left (a+b \log \left (c \left (d+\frac {e}{x}\right )^p\right )\right )^2}{x} \, dx,x,f+g x\right )}{d g}\\ &=\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}-\frac {(3 b e p) \text {Subst}\left (\int \frac {\left (a+b \log \left (c (d+e x)^p\right )\right )^2}{x} \, dx,x,\frac {1}{f+g x}\right )}{d g}\\ &=-\frac {3 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac {\left (6 b^2 e^2 p^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right ) \left (a+b \log \left (c (d+e x)^p\right )\right )}{d+e x} \, dx,x,\frac {1}{f+g x}\right )}{d g}\\ &=-\frac {3 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}+\frac {\left (6 b^2 e p^2\right ) \text {Subst}\left (\int \frac {\left (a+b \log \left (c x^p\right )\right ) \log \left (-\frac {e \left (-\frac {d}{e}+\frac {x}{e}\right )}{d}\right )}{x} \, dx,x,d+\frac {e}{f+g x}\right )}{d g}\\ &=-\frac {3 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}-\frac {6 b^2 e p^2 \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right ) \text {Li}_2\left (\frac {d+\frac {e}{f+g x}}{d}\right )}{d g}+\frac {\left (6 b^3 e p^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {x}{d}\right )}{x} \, dx,x,d+\frac {e}{f+g x}\right )}{d g}\\ &=-\frac {3 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}+\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3}{d g}-\frac {6 b^2 e p^2 \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right ) \text {Li}_2\left (\frac {d+\frac {e}{f+g x}}{d}\right )}{d g}+\frac {6 b^3 e p^3 \text {Li}_3\left (\frac {d+\frac {e}{f+g x}}{d}\right )}{d g}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(415\) vs. \(2(168)=336\).
time = 0.32, size = 415, normalized size = 2.47 \begin {gather*} \frac {3 b d p (f+g x) \log \left (d+\frac {e}{f+g x}\right ) \left (a-b p \log \left (d+\frac {e}{f+g x}\right )+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2+d (f+g x) \left (a-b p \log \left (d+\frac {e}{f+g x}\right )+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^3+3 b e p \left (a-b p \log \left (d+\frac {e}{f+g x}\right )+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2 \log (e+d (f+g x))+3 b^2 p^2 \left (a-b p \log \left (d+\frac {e}{f+g x}\right )+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right ) \left (d (f+g x) \log ^2\left (d+\frac {e}{f+g x}\right )+e \left (2 \log \left (-\frac {d (f+g x)}{e}\right )-\log (e+d f+d g x)+2 \log \left (d+\frac {e}{f+g x}\right )\right ) \log (e+d (f+g x))+2 e \text {Li}_2\left (1+\frac {d (f+g x)}{e}\right )\right )+b^3 p^3 \left (\log ^2\left (d+\frac {e}{f+g x}\right ) \left (-3 e \log \left (-\frac {e}{d f+d g x}\right )+(e+d f+d g x) \log \left (d+\frac {e}{f+g x}\right )\right )-6 e \log \left (d+\frac {e}{f+g x}\right ) \text {Li}_2\left (1+\frac {e}{d f+d g x}\right )+6 e \text {Li}_3\left (1+\frac {e}{d f+d g x}\right )\right )}{d g} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \left (d +\frac {e}{g x +f}\right )^{p}\right )\right )^{3}\, dx\]
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 \left (a + b \log {\left (c \left (d + \frac {e}{f + g x}\right )^{p} \right )}\right )^{3}\, 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 {\left (a+b\,\ln \left (c\,{\left (d+\frac {e}{f+g\,x}\right )}^p\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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