Integrand size = 24, antiderivative size = 255 \[ \int (f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=-\frac {\sqrt {b} e^{-\frac {a}{b n}} (e f-d g) \sqrt {n} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 e^2}-\frac {\sqrt {b} e^{-\frac {2 a}{b n}} g \sqrt {n} \sqrt {\frac {\pi }{2}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{4 e^2}+\frac {(e f-d g) (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^2}+\frac {g (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e^2} \]
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Time = 0.24 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2448, 2436, 2333, 2337, 2211, 2235, 2437, 2342, 2347} \[ \int (f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=-\frac {\sqrt {\pi } \sqrt {b} \sqrt {n} e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{2 e^2}-\frac {\sqrt {\frac {\pi }{2}} \sqrt {b} g \sqrt {n} e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{4 e^2}+\frac {(d+e x) (e f-d g) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^2}+\frac {g (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e^2} \]
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Rule 2211
Rule 2235
Rule 2333
Rule 2337
Rule 2342
Rule 2347
Rule 2436
Rule 2437
Rule 2448
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(e f-d g) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e}+\frac {g (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e}\right ) \, dx \\ & = \frac {g \int (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx}{e}+\frac {(e f-d g) \int \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx}{e} \\ & = \frac {g \text {Subst}\left (\int 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 \sqrt {a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{e^2} \\ & = \frac {(e f-d g) (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^2}+\frac {g (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e^2}-\frac {(b g n) \text {Subst}\left (\int \frac {x}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{4 e^2}-\frac {(b (e f-d g) n) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{2 e^2} \\ & = \frac {(e f-d g) (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^2}+\frac {g (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e^2}-\frac {\left (b 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 )}{4 e^2}-\frac {\left (b (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 )}{2 e^2} \\ & = \frac {(e f-d g) (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^2}+\frac {g (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e^2}-\frac {\left (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 )}{2 e^2}-\frac {\left ((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 )}{e^2} \\ & = -\frac {\sqrt {b} e^{-\frac {a}{b n}} (e f-d g) \sqrt {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 )}{2 e^2}-\frac {\sqrt {b} e^{-\frac {2 a}{b n}} g \sqrt {n} \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 )}{4 e^2}+\frac {(e f-d g) (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{e^2}+\frac {g (d+e x)^2 \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e^2} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.92 \[ \int (f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=-\frac {e^{-\frac {2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (4 \sqrt {b} e^{\frac {a}{b n}} (e f-d g) \sqrt {n} \sqrt {\pi } \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+\sqrt {b} g \sqrt {n} \sqrt {2 \pi } (d+e x) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )-4 e^{\frac {2 a}{b n}} \left (c (d+e x)^n\right )^{2/n} (2 e f-d g+e g x) \sqrt {a+b \log \left (c (d+e x)^n\right )}\right )}{8 e^2} \]
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\[\int \left (g x +f \right ) \sqrt {a +b \ln \left (c \left (e x +d \right )^{n}\right )}d x\]
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Exception generated. \[ \int (f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\text {Exception raised: TypeError} \]
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\[ \int (f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int \sqrt {a + b \log {\left (c \left (d + e x\right )^{n} \right )}} \left (f + g x\right )\, dx \]
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\[ \int (f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int { {\left (g x + f\right )} \sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]
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\[ \int (f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int { {\left (g x + f\right )} \sqrt {b \log \left ({\left (e x + d\right )}^{n} c\right ) + a} \,d x } \]
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Timed out. \[ \int (f+g x) \sqrt {a+b \log \left (c (d+e x)^n\right )} \, dx=\int \left (f+g\,x\right )\,\sqrt {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )} \,d x \]
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