Integrand size = 24, antiderivative size = 181 \[ \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\frac {e^{-\frac {a}{b n}} (e f-d g) \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^2 \sqrt {n}}+\frac {e^{-\frac {2 a}{b n}} g \sqrt {\frac {\pi }{2}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^2 \sqrt {n}} \]
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Time = 0.19 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2448, 2436, 2337, 2211, 2235, 2437, 2347} \[ \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\frac {\sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^2 \sqrt {n}}+\frac {\sqrt {\frac {\pi }{2}} g e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^2 \sqrt {n}} \]
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Rule 2211
Rule 2235
Rule 2337
Rule 2347
Rule 2436
Rule 2437
Rule 2448
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {e f-d g}{e \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {g (d+e x)}{e \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right ) \, dx \\ & = \frac {g \int \frac {d+e x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{e}+\frac {(e f-d g) \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx}{e} \\ & = \frac {g \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^2}+\frac {(e f-d g) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{e^2} \\ & = \frac {\left (g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int \frac {e^{\frac {2 x}{n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c (d+e x)^n\right )\right )}{e^2 n}+\frac {\left ((e f-d g) (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 )}{e^2 n} \\ & = \frac {\left (2 g (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n}\right ) \text {Subst}\left (\int e^{-\frac {2 a}{b n}+\frac {2 x^2}{b n}} \, dx,x,\sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{b e^2 n}+\frac {\left (2 (e f-d g) (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 e^2 n} \\ & = \frac {e^{-\frac {a}{b n}} (e f-d g) \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 )}{\sqrt {b} e^2 \sqrt {n}}+\frac {e^{-\frac {2 a}{b n}} g \sqrt {\frac {\pi }{2}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^2 \sqrt {n}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.91 \[ \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\frac {e^{-\frac {2 a}{b n}} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (2 e^{\frac {a}{b n}} (e f-d g) \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+\sqrt {2} g (d+e x) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )\right )}{2 \sqrt {b} e^2 \sqrt {n}} \]
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\[\int \frac {g x +f}{\sqrt {a +b \ln \left (c \left (e x +d \right )^{n}\right )}}d x\]
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Exception generated. \[ \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int \frac {f + g x}{\sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}}\, dx \]
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\[ \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int { \frac {g x + f}{\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}} \,d x } \]
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\[ \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int { \frac {g x + f}{\sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a}} \,d x } \]
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Timed out. \[ \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}} \, dx=\int \frac {f+g\,x}{\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}} \,d x \]
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