Optimal. Leaf size=115 \[ -\frac {2 b e p \log \left (-\frac {e}{d (f+g x)}\right ) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )}{d g}+\frac {(e+d (f+g x)) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2}{d g}-\frac {2 b^2 e p^2 \text {Li}_2\left (1+\frac {e}{d (f+g x)}\right )}{d g} \]
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Rubi [A]
time = 0.06, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {2533, 2499,
2504, 2441, 2352} \begin {gather*} -\frac {2 b^2 e p^2 \text {PolyLog}\left (2,\frac {e}{d (f+g x)}+1\right )}{d g}-\frac {2 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 )}{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 )^2}{d g} \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2441
Rule 2499
Rule 2504
Rule 2533
Rubi steps
\begin {align*} \int \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2 \, dx &=\frac {\text {Subst}\left (\int \left (a+b \log \left (c \left (d+\frac {e}{x}\right )^p\right )\right )^2 \, 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 )^2}{d g}+\frac {(2 b e p) \text {Subst}\left (\int \frac {a+b \log \left (c \left (d+\frac {e}{x}\right )^p\right )}{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 )^2}{d g}-\frac {(2 b e p) \text {Subst}\left (\int \frac {a+b \log \left (c (d+e x)^p\right )}{x} \, dx,x,\frac {1}{f+g x}\right )}{d g}\\ &=-\frac {2 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 )}{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 )^2}{d g}+\frac {\left (2 b^2 e^2 p^2\right ) \text {Subst}\left (\int \frac {\log \left (-\frac {e x}{d}\right )}{d+e x} \, dx,x,\frac {1}{f+g x}\right )}{d g}\\ &=-\frac {2 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 )}{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 )^2}{d g}-\frac {2 b^2 e p^2 \text {Li}_2\left (1+\frac {e}{d (f+g x)}\right )}{d g}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 219, normalized size = 1.90 \begin {gather*} x \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )^2-\frac {b p \left (2 d f \log (f+g x) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right )-2 (e+d f) \left (a+b \log \left (c \left (d+\frac {e}{f+g x}\right )^p\right )\right ) \log (e+d (f+g x))+b d f p \left (\log (f+g x) \left (\log (f+g x)-2 \log \left (\frac {e+d f+d g x}{e}\right )\right )-2 \text {Li}_2\left (-\frac {d (f+g x)}{e}\right )\right )-b (e+d f) p \left (\left (2 \log \left (-\frac {d (f+g x)}{e}\right )-\log (e+d f+d g x)\right ) \log (e+d (f+g x))+2 \text {Li}_2\left (\frac {e+d f+d g x}{e}\right )\right )\right )}{d g} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (a +b \ln \left (c \left (d +\frac {e}{g x +f}\right )^{p}\right )\right )^{2}\, dx\]
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 )^{2}\, 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 )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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