Integrand size = 22, antiderivative size = 147 \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}} \, dx=\frac {2 e^{-\frac {a}{b m n}} \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )}{b^{3/2} f m^{3/2} n^{3/2}}-\frac {2 (e+f x)}{b f m n \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}} \]
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Time = 0.18 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2436, 2334, 2337, 2211, 2235, 2495} \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}} \, dx=\frac {2 \sqrt {\pi } (e+f x) e^{-\frac {a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )}{b^{3/2} f m^{3/2} n^{3/2}}-\frac {2 (e+f x)}{b f m n \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}} \]
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Rule 2211
Rule 2235
Rule 2334
Rule 2337
Rule 2436
Rule 2495
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{3/2}} \, dx,c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = \text {Subst}\left (\frac {\text {Subst}\left (\int \frac {1}{\left (a+b \log \left (c d^n x^{m n}\right )\right )^{3/2}} \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = -\frac {2 (e+f x)}{b f m n \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}+\text {Subst}\left (\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c d^n x^{m n}\right )}} \, dx,x,e+f x\right )}{b f m n},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = -\frac {2 (e+f x)}{b f m n \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}+\text {Subst}\left (\frac {\left (2 (e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{m n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c d^n (e+f x)^{m n}\right )\right )}{b f m^2 n^2},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = -\frac {2 (e+f x)}{b f m n \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}+\text {Subst}\left (\frac {\left (4 (e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}}\right ) \text {Subst}\left (\int e^{-\frac {a}{b m n}+\frac {x^2}{b m n}} \, dx,x,\sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}\right )}{b^2 f m^2 n^2},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = \frac {2 e^{-\frac {a}{b m n}} \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )}{b^{3/2} f m^{3/2} n^{3/2}}-\frac {2 (e+f x)}{b f m n \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.23 \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}} \, dx=-\frac {2 e^{-\frac {a}{b m n}} (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \left (e^{\frac {a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{\frac {1}{m n}}-\Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right ) \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}}\right )}{b f m n \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}} \]
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\[\int \frac {1}{{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )\right )}^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}} \, dx=\int \frac {1}{\left (a + b \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}} \, dx=\int \frac {1}{{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )\right )}^{3/2}} \,d x \]
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