Integrand size = 23, antiderivative size = 259 \[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-4 b e n \sqrt {d+e x}-\frac {b d n \sqrt {d+e x}}{x}+3 b \sqrt {d} e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+3 b \sqrt {d} e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+3 e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-6 b \sqrt {d} e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-3 b \sqrt {d} e n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right ) \]
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Time = 0.24 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {43, 52, 65, 214, 2392, 14, 6131, 6055, 2449, 2352} \[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=-3 \sqrt {d} e \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-\frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}+3 e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )+3 b \sqrt {d} e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+3 b \sqrt {d} e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-6 b \sqrt {d} e n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \log \left (\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-3 b \sqrt {d} e n \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {d}}{\sqrt {d}-\sqrt {d+e x}}\right )-4 b e n \sqrt {d+e x}-\frac {b d n \sqrt {d+e x}}{x} \]
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Rule 14
Rule 43
Rule 52
Rule 65
Rule 214
Rule 2352
Rule 2392
Rule 2449
Rule 6055
Rule 6131
Rubi steps \begin{align*} \text {integral}& = 3 e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \frac {-\left ((d-2 e x) \sqrt {d+e x}\right )-3 \sqrt {d} e x \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x^2} \, dx \\ & = 3 e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-(b n) \int \left (-\frac {d \sqrt {d+e x}}{x^2}+\frac {2 e \sqrt {d+e x}}{x}-\frac {3 \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x}\right ) \, dx \\ & = 3 e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+(b d n) \int \frac {\sqrt {d+e x}}{x^2} \, dx-(2 b e n) \int \frac {\sqrt {d+e x}}{x} \, dx+\left (3 b \sqrt {d} e n\right ) \int \frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{x} \, dx \\ & = -4 b e n \sqrt {d+e x}-\frac {b d n \sqrt {d+e x}}{x}+3 e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+\left (6 b \sqrt {d} e 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}\right )+\frac {1}{2} (b d e n) \int \frac {1}{x \sqrt {d+e x}} \, dx-(2 b d e n) \int \frac {1}{x \sqrt {d+e x}} \, dx \\ & = -4 b e n \sqrt {d+e x}-\frac {b d n \sqrt {d+e x}}{x}+3 b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+3 e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )+(b d n) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )-(4 b d n) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )-(6 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 ) \\ & = -4 b e n \sqrt {d+e x}-\frac {b d n \sqrt {d+e x}}{x}+3 b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+3 b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+3 e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-6 b \sqrt {d} 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 )+(6 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 ) \\ & = -4 b e n \sqrt {d+e x}-\frac {b d n \sqrt {d+e x}}{x}+3 b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+3 b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+3 e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-6 b \sqrt {d} 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 )-\left (6 b \sqrt {d} e n\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-\frac {\sqrt {d+e x}}{\sqrt {d}}}\right ) \\ & = -4 b e n \sqrt {d+e x}-\frac {b d n \sqrt {d+e x}}{x}+3 b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+3 b \sqrt {d} e n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )^2+3 e \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )-\frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x}-3 \sqrt {d} e \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) \left (a+b \log \left (c x^n\right )\right )-6 b \sqrt {d} 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 )-3 b \sqrt {d} e n \text {Li}_2\left (1-\frac {2}{1-\frac {\sqrt {d+e x}}{\sqrt {d}}}\right ) \\ \end{align*}
Time = 0.22 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.85 \[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\frac {-4 a d \sqrt {d+e x}-4 b d n \sqrt {d+e x}+8 a e x \sqrt {d+e x}-16 b e n x \sqrt {d+e x}+12 b \sqrt {d} e n x \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )-4 b d \sqrt {d+e x} \log \left (c x^n\right )+8 b e x \sqrt {d+e x} \log \left (c x^n\right )+6 a \sqrt {d} e x \log \left (\sqrt {d}-\sqrt {d+e x}\right )+6 b \sqrt {d} e x \log \left (c x^n\right ) \log \left (\sqrt {d}-\sqrt {d+e x}\right )-3 b \sqrt {d} e n x \log ^2\left (\sqrt {d}-\sqrt {d+e x}\right )-6 a \sqrt {d} e x \log \left (\sqrt {d}+\sqrt {d+e x}\right )-6 b \sqrt {d} e x \log \left (c x^n\right ) \log \left (\sqrt {d}+\sqrt {d+e x}\right )+3 b \sqrt {d} e n x \log ^2\left (\sqrt {d}+\sqrt {d+e x}\right )+6 b \sqrt {d} e n x \log \left (\sqrt {d}+\sqrt {d+e x}\right ) \log \left (\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )-6 b \sqrt {d} e n x \log \left (\sqrt {d}-\sqrt {d+e x}\right ) \log \left (\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )-6 b \sqrt {d} e n x \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {\sqrt {d+e x}}{2 \sqrt {d}}\right )+6 b \sqrt {d} e n x \operatorname {PolyLog}\left (2,\frac {1}{2} \left (1+\frac {\sqrt {d+e x}}{\sqrt {d}}\right )\right )}{4 x} \]
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\[\int \frac {\left (e x +d \right )^{\frac {3}{2}} \left (a +b \ln \left (c \,x^{n}\right )\right )}{x^{2}}d x\]
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\[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{2}} \,d x } \]
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\[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {\left (a + b \log {\left (c x^{n} \right )}\right ) \left (d + e x\right )^{\frac {3}{2}}}{x^{2}}\, dx \]
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\[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{2}} \,d x } \]
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\[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {3}{2}} {\left (b \log \left (c x^{n}\right ) + a\right )}}{x^{2}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^{3/2} \left (a+b \log \left (c x^n\right )\right )}{x^2} \, dx=\int \frac {\left (a+b\,\ln \left (c\,x^n\right )\right )\,{\left (d+e\,x\right )}^{3/2}}{x^2} \,d x \]
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