Integrand size = 22, antiderivative size = 139 \[ \int \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=-\frac {\sqrt {b} e^{-\frac {a}{b m n}} \sqrt {m} \sqrt {n} \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )}{2 f}+\frac {(e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{f} \]
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Time = 0.15 (sec) , antiderivative size = 139, 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, 2333, 2337, 2211, 2235, 2495} \[ \int \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\frac {(e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{f}-\frac {\sqrt {\pi } \sqrt {b} \sqrt {m} \sqrt {n} (e+f x) e^{-\frac {a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )}{2 f} \]
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Rule 2211
Rule 2235
Rule 2333
Rule 2337
Rule 2436
Rule 2495
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )} \, 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 \sqrt {a+b \log \left (c d^n x^{m n}\right )} \, 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 {(e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{f}-\text {Subst}\left (\frac {(b m n) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c d^n x^{m n}\right )}} \, dx,x,e+f x\right )}{2 f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = \frac {(e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{f}-\text {Subst}\left (\frac {\left (b (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 )}{2 f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = \frac {(e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{f}-\text {Subst}\left (\frac {\left ((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 )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = -\frac {\sqrt {b} e^{-\frac {a}{b m n}} \sqrt {m} \sqrt {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 )}{2 f}+\frac {(e+f x) \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{f} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.96 \[ \int \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\frac {(e+f x) \left (-\sqrt {b} e^{-\frac {a}{b m n}} \sqrt {m} \sqrt {n} \sqrt {\pi } \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )+2 \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}\right )}{2 f} \]
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\[\int \sqrt {a +b \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )}d x\]
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Exception generated. \[ \int \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\int \sqrt {a + b \log {\left (c \left (d \left (e + f x\right )^{m}\right )^{n} \right )}}\, dx \]
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\[ \int \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\int { \sqrt {b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a} \,d x } \]
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\[ \int \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\int { \sqrt {b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a} \,d x } \]
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Timed out. \[ \int \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )} \, dx=\int \sqrt {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )} \,d x \]
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