Integrand size = 25, antiderivative size = 284 \[ \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=-\frac {16 b d n \sqrt {d+e x^r}}{3 r^2}-\frac {4 b n \left (d+e x^r\right )^{3/2}}{9 r^2}+\frac {16 b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{3 r^2}+\frac {2 b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{r^2}+\frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{r^2}-\frac {2 b d^{3/2} n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{r^2} \]
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Time = 0.29 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {272, 52, 65, 214, 2390, 6131, 6055, 2449, 2352} \[ \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\frac {2}{3} \left (-\frac {3 d^{3/2} \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}+\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}\right ) \left (a+b \log \left (c x^n\right )\right )+\frac {2 b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{r^2}+\frac {16 b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{3 r^2}-\frac {4 b d^{3/2} n \text {arctanh}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{r^2}-\frac {2 b d^{3/2} n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {e x^r+d}}\right )}{r^2}-\frac {4 b n \left (d+e x^r\right )^{3/2}}{9 r^2}-\frac {16 b d n \sqrt {d+e x^r}}{3 r^2} \]
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Rule 52
Rule 65
Rule 214
Rule 272
Rule 2352
Rule 2390
Rule 2449
Rule 6055
Rule 6131
Rubi steps \begin{align*} \text {integral}& = \frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (\frac {2 d \sqrt {d+e x^r}}{r x}+\frac {2 \left (d+e x^r\right )^{3/2}}{3 r x}-\frac {2 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r x}\right ) \, dx \\ & = \frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(2 b n) \int \frac {\left (d+e x^r\right )^{3/2}}{x} \, dx}{3 r}-\frac {(2 b d n) \int \frac {\sqrt {d+e x^r}}{x} \, dx}{r}+\frac {\left (2 b d^{3/2} n\right ) \int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{x} \, dx}{r} \\ & = \frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(2 b n) \text {Subst}\left (\int \frac {(d+e x)^{3/2}}{x} \, dx,x,x^r\right )}{3 r^2}-\frac {(2 b d n) \text {Subst}\left (\int \frac {\sqrt {d+e x}}{x} \, dx,x,x^r\right )}{r^2}+\frac {\left (2 b d^{3/2} n\right ) \text {Subst}\left (\int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x} \, dx,x,x^r\right )}{r^2} \\ & = -\frac {4 b d n \sqrt {d+e x^r}}{r^2}-\frac {4 b n \left (d+e x^r\right )^{3/2}}{9 r^2}+\frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(2 b d n) \text {Subst}\left (\int \frac {\sqrt {d+e x}}{x} \, dx,x,x^r\right )}{3 r^2}+\frac {\left (4 b d^{3/2} n\right ) \text {Subst}\left (\int \frac {x \tanh ^{-1}\left (\frac {x}{\sqrt {d}}\right )}{-d+x^2} \, dx,x,\sqrt {d+e x^r}\right )}{r^2}-\frac {\left (2 b d^2 n\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^r\right )}{r^2} \\ & = -\frac {16 b d n \sqrt {d+e x^r}}{3 r^2}-\frac {4 b n \left (d+e x^r\right )^{3/2}}{9 r^2}+\frac {2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{r^2}+\frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(4 b d 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^r}\right )}{r^2}-\frac {\left (2 b d^2 n\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {d+e x}} \, dx,x,x^r\right )}{3 r^2}-\frac {\left (4 b d^2 n\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^r}\right )}{e r^2} \\ & = -\frac {16 b d n \sqrt {d+e x^r}}{3 r^2}-\frac {4 b n \left (d+e x^r\right )^{3/2}}{9 r^2}+\frac {4 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r^2}+\frac {2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{r^2}+\frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{r^2}+\frac {(4 b d 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^r}\right )}{r^2}-\frac {\left (4 b d^2 n\right ) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x^r}\right )}{3 e r^2} \\ & = -\frac {16 b d n \sqrt {d+e x^r}}{3 r^2}-\frac {4 b n \left (d+e x^r\right )^{3/2}}{9 r^2}+\frac {16 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{3 r^2}+\frac {2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{r^2}+\frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{r^2}-\frac {\left (4 b d^{3/2} n\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {d+e x^r}}{\sqrt {d}}}\right )}{r^2} \\ & = -\frac {16 b d n \sqrt {d+e x^r}}{3 r^2}-\frac {4 b n \left (d+e x^r\right )^{3/2}}{9 r^2}+\frac {16 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{3 r^2}+\frac {2 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )^2}{r^2}+\frac {2}{3} \left (\frac {3 d \sqrt {d+e x^r}}{r}+\frac {\left (d+e x^r\right )^{3/2}}{r}-\frac {3 d^{3/2} \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right )}{r}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {4 b d^{3/2} n \tanh ^{-1}\left (\frac {\sqrt {d+e x^r}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x^r}}\right )}{r^2}-\frac {2 b d^{3/2} n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {d+e x^r}}{\sqrt {d}}}\right )}{r^2} \\ \end{align*}
\[ \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx \]
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\[\int \frac {\left (d +e \,x^{r}\right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{x}d x\]
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Exception generated. \[ \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \left (d + e x^{r}\right )^{\frac {3}{2}}}{x}\, dx \]
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\[ \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \]
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\[ \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int { \frac {{\left (e x^{r} + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x} \,d x } \]
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Timed out. \[ \int \frac {\left (d+e x^r\right )^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x} \, dx=\int \frac {{\left (d+e\,x^r\right )}^{3/2}\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{x} \,d x \]
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