Optimal. Leaf size=256 \[ \frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{\sqrt {e} \sqrt {e f-d g}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e} \sqrt {e f-d g}}-\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{\sqrt {e} \sqrt {e f-d g}}-\frac {2 b n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{\sqrt {e} \sqrt {e f-d g}} \]
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Rubi [A]
time = 0.48, antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 11, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {2458, 65, 214,
2390, 12, 1601, 6873, 6131, 6055, 2449, 2352} \begin {gather*} -\frac {2 b n \text {PolyLog}\left (2,1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{\sqrt {e} \sqrt {e f-d g}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e} \sqrt {e f-d g}}+\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{\sqrt {e} \sqrt {e f-d g}}-\frac {4 b n \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right ) \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{\sqrt {e} \sqrt {e f-d g}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 214
Rule 1601
Rule 2352
Rule 2390
Rule 2449
Rule 2458
Rule 6055
Rule 6131
Rule 6873
Rubi steps
\begin {align*} \int \frac {a+b \log \left (c (d+e x)^n\right )}{(d+e x) \sqrt {f+g x}} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \log \left (c x^n\right )}{x \sqrt {\frac {e f-d g}{e}+\frac {g x}{e}}} \, dx,x,d+e x\right )}{e}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e} \sqrt {e f-d g}}-\frac {(b n) \text {Subst}\left (\int -\frac {2 \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g x}{e}}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g} x} \, dx,x,d+e x\right )}{e}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e} \sqrt {e f-d g}}+\frac {(2 b n) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f-\frac {d g}{e}+\frac {g x}{e}}}{\sqrt {e f-d g}}\right )}{x} \, dx,x,d+e x\right )}{\sqrt {e} \sqrt {e f-d g}}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e} \sqrt {e f-d g}}+\frac {\left (4 b \sqrt {e} n\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{d g+e \left (-f+x^2\right )} \, dx,x,\sqrt {f+g x}\right )}{\sqrt {e f-d g}}\\ &=-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e} \sqrt {e f-d g}}+\frac {\left (4 b \sqrt {e} n\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{-e f+d g+e x^2} \, dx,x,\sqrt {f+g x}\right )}{\sqrt {e f-d g}}\\ &=\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{\sqrt {e} \sqrt {e f-d g}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e} \sqrt {e f-d g}}-\frac {(4 b n) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {e} x}{\sqrt {e f-d g}}\right )}{1-\frac {\sqrt {e} x}{\sqrt {e f-d g}}} \, dx,x,\sqrt {f+g x}\right )}{e f-d g}\\ &=\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{\sqrt {e} \sqrt {e f-d g}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e} \sqrt {e f-d g}}-\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{\sqrt {e} \sqrt {e f-d g}}+\frac {(4 b n) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {\sqrt {e} x}{\sqrt {e f-d g}}}\right )}{1-\frac {e x^2}{e f-d g}} \, dx,x,\sqrt {f+g x}\right )}{e f-d g}\\ &=\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{\sqrt {e} \sqrt {e f-d g}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e} \sqrt {e f-d g}}-\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{\sqrt {e} \sqrt {e f-d g}}-\frac {(4 b n) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{\sqrt {e} \sqrt {e f-d g}}\\ &=\frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )^2}{\sqrt {e} \sqrt {e f-d g}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e} \sqrt {e f-d g}}-\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \log \left (\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{\sqrt {e} \sqrt {e f-d g}}-\frac {2 b n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}}\right )}{\sqrt {e} \sqrt {e f-d g}}\\ \end {align*}
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Mathematica [A]
time = 2.99, size = 314, normalized size = 1.23 \begin {gather*} \frac {-\frac {2 \sqrt {e} \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right ) \left (a-b n \log (d+e x)+b \log \left (c (d+e x)^n\right )\right )}{\sqrt {e f-d g}}+b n \left (-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {f+g x}}{\sqrt {f-\frac {d g}{e}}}\right ) \left (\log (d+e x)-\log \left (\frac {g (d+e x)}{-e f+d g}\right )\right )}{\sqrt {f-\frac {d g}{e}}}+\frac {\sqrt {\frac {e (f+g x)}{e f-d g}} \left (\log ^2\left (\frac {g (d+e x)}{-e f+d g}\right )-4 \log \left (\frac {g (d+e x)}{-e f+d g}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {\frac {e (f+g x)}{e f-d g}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {\frac {e (f+g x)}{e f-d g}}\right )\right )-4 \text {Li}_2\left (\frac {1}{2}-\frac {1}{2} \sqrt {\frac {e (f+g x)}{e f-d g}}\right )\right )}{2 \sqrt {f+g x}}\right )}{e} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.27, size = 0, normalized size = 0.00 \[\int \frac {a +b \ln \left (c \left (e x +d \right )^{n}\right )}{\left (e x +d \right ) \sqrt {g x +f}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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 = -0.00 \begin {gather*} \int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{\sqrt {f+g\,x}\,\left (d+e\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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