Integrand size = 23, antiderivative size = 221 \[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {b n \sqrt {d+e x}}{x}-\frac {b e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}+\frac {b e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{\sqrt {d}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}-\frac {2 b e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{\sqrt {d}}-\frac {b e n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{\sqrt {d}} \]
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Time = 0.20 (sec) , antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.391, Rules used = {43, 65, 214, 2392, 14, 6131, 6055, 2449, 2352} \[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}+\frac {b e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{\sqrt {d}}-\frac {b e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}-\frac {2 b e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{\sqrt {d}}-\frac {b e n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )}{\sqrt {d}}-\frac {b n \sqrt {d+e x}}{x} \]
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Rule 14
Rule 43
Rule 65
Rule 214
Rule 2352
Rule 2392
Rule 2449
Rule 6055
Rule 6131
Rubi steps \begin{align*} \text {integral}& = -\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {e \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 {-\sqrt {d+e x}-\frac {e x \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}}{x^2} \, dx \\ & = -\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {e \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 \left (-\frac {\sqrt {d+e x}}{x^2}-\frac {e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d} x}\right ) \, dx \\ & = -\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {e \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 {\sqrt {d+e x}}{x^2} \, dx+\frac {(b e n) \int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x} \, dx}{\sqrt {d}} \\ & = -\frac {b n \sqrt {d+e x}}{x}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{\sqrt {d}}+\frac {1}{2} (b e n) \int \frac {1}{x \sqrt {d+e x}} \, dx+\frac {(2 b e 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 {b n \sqrt {d+e x}}{x}+\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{\sqrt {d}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {e \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) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )-\frac {(2 b e 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 {b n \sqrt {d+e x}}{x}-\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}+\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{\sqrt {d}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {e \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 e 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 e 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 {b n \sqrt {d+e x}}{x}-\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}+\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{\sqrt {d}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {e \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 e 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 e 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 {b n \sqrt {d+e x}}{x}-\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{\sqrt {d}}+\frac {b e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2}{\sqrt {d}}-\frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x}-\frac {e \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 e 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 {b e n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {d+e x}}{\sqrt {d}}}\right )}{\sqrt {d}} \\ \end{align*}
Time = 0.20 (sec) , antiderivative size = 392, normalized size of antiderivative = 1.77 \[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-\frac {4 a \sqrt {d} \sqrt {d+e x}+4 b \sqrt {d} n \sqrt {d+e x}+4 b e n x \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+4 b \sqrt {d} \sqrt {d+e x} \log \left (c x^n\right )-2 a e x \log \left (\sqrt {d}-\sqrt {d+e x}\right )-2 b e x \log \left (c x^n\right ) \log \left (\sqrt {d}-\sqrt {d+e x}\right )+b e n x \log ^2\left (\sqrt {d}-\sqrt {d+e x}\right )+2 a e x \log \left (\sqrt {d}+\sqrt {d+e x}\right )+2 b e x \log \left (c x^n\right ) \log \left (\sqrt {d}+\sqrt {d+e x}\right )-b e n x \log ^2\left (\sqrt {d}+\sqrt {d+e x}\right )-2 b e n x \log \left (\sqrt {d}+\sqrt {d+e x}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )+2 b e n x \log \left (\sqrt {d}-\sqrt {d+e x}\right ) \log \left (\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )+2 b e n x \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )-2 b e n x \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{4 \sqrt {d} x} \]
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\[\int \frac {\left (a +b \ln \left (c \,x^{n}\right )\right ) \sqrt {e x +d}}{x^{2}}d x\]
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\[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{2}} \,d x } \]
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\[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \sqrt {d + e x}}{x^{2}}\, dx \]
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\[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{2}} \,d x } \]
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\[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {\left (a+b\,\ln \left (c\,x^n\right )\right )\,\sqrt {d+e\,x}}{x^2} \,d x \]
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