Integrand size = 18, antiderivative size = 116 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\frac {2 e^{-\frac {a}{b n}} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e n^{3/2}}-\frac {2 (d+e x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]
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Time = 0.07 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2436, 2334, 2337, 2211, 2235} \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\frac {2 \sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e n^{3/2}}-\frac {2 (d+e x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]
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Rule 2211
Rule 2235
Rule 2334
Rule 2337
Rule 2436
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\left (a+b \log \left (c x^n\right )\right )^{3/2}} \, dx,x,d+e x\right )}{e} \\ & = -\frac {2 (d+e x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{b e n} \\ & = -\frac {2 (d+e x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (2 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{b e n^2} \\ & = -\frac {2 (d+e x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {\left (4 (d+e x) \left (c (d+e x)^n\right )^{-1/n}\right ) \text {Subst}\left (\int e^{-\frac {a}{b n}+\frac {x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{b^2 e n^2} \\ & = \frac {2 e^{-\frac {a}{b n}} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e n^{3/2}}-\frac {2 (d+e x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}} \\ \end{align*}
Time = 0.09 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=-\frac {2 e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}}-\Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]
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\[\int \frac {1}{{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{3/2}} \,d x \]
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