Integrand size = 23, antiderivative size = 152 \[ \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}} \, dx=\frac {2 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{\sqrt {d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}-\frac {4 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{\sqrt {d}}-\frac {2 b n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{\sqrt {d}} \]
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Time = 0.14 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {65, 214, 2390, 12, 6131, 6055, 2449, 2352} \[ \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}+\frac {2 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{\sqrt {d}}-\frac {4 b n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{\sqrt {d}}-\frac {2 b n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{\sqrt {d}} \]
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Rule 12
Rule 65
Rule 214
Rule 2352
Rule 2390
Rule 2449
Rule 6055
Rule 6131
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}-(b n) \int -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d} x} \, dx \\ & = -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}+\frac {(2 b n) \int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x} \, dx}{\sqrt {d}} \\ & = -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}+\frac {(4 b n) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{-d+x^2} \, dx,x,\sqrt {d+e x}\right )}{\sqrt {d}} \\ & = \frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{\sqrt {d}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}-\frac {(4 b n) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{1-\frac {x}{\sqrt {d}}} \, dx,x,\sqrt {d+e x}\right )}{d} \\ & = \frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{\sqrt {d}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}-\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{\sqrt {d}}+\frac {(4 b n) \text {Subst}\left (\int \frac {\log \left (\frac {2}{1-\frac {x}{\sqrt {d}}}\right )}{1-\frac {x^2}{d}} \, dx,x,\sqrt {d+e x}\right )}{d} \\ & = \frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{\sqrt {d}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}-\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{\sqrt {d}}-\frac {(4 b n) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {d+e x}}{\sqrt {d}}}\right )}{\sqrt {d}} \\ & = \frac {2 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{\sqrt {d}}-\frac {2 \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}-\frac {4 b n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{\sqrt {d}}-\frac {2 b n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {d+e x}}{\sqrt {d}}}\right )}{\sqrt {d}} \\ \end{align*}
Time = 0.07 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.64 \[ \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}} \, dx=\frac {2 \left (a+b \log \left (c x^n\right )\right ) \log \left (\sqrt {d}-\sqrt {d+e x}\right )-2 \left (a+b \log \left (c x^n\right )\right ) \log \left (\sqrt {d}+\sqrt {d+e x}\right )-b n \left (\log \left (\sqrt {d}-\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}-\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )+b n \left (\log \left (\sqrt {d}+\sqrt {d+e x}\right ) \left (\log \left (\sqrt {d}+\sqrt {d+e x}\right )+2 \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )\right )+2 \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )\right )}{2 \sqrt {d}} \]
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\[\int \frac {a +b \ln \left (c \,x^{n}\right )}{x \sqrt {e x +d}}d x\]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{\sqrt {e x + d} x} \,d x } \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}} \, dx=\int \frac {a + b \log {\left (c x^{n} \right )}}{x \sqrt {d + e x}}\, dx \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{\sqrt {e x + d} x} \,d x } \]
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\[ \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}} \, dx=\int { \frac {b \log \left (c x^{n}\right ) + a}{\sqrt {e x + d} x} \,d x } \]
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Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{x \sqrt {d+e x}} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{x\,\sqrt {d+e\,x}} \,d x \]
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