Integrand size = 18, antiderivative size = 143 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \, dx=\frac {3 b^{3/2} e^{-\frac {a}{b n}} n^{3/2} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{4 e}-\frac {3 b n (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e} \]
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Time = 0.08 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2436, 2333, 2337, 2211, 2235} \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \, dx=\frac {3 \sqrt {\pi } b^{3/2} n^{3/2} e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{4 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}-\frac {3 b n (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e} \]
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Rule 2211
Rule 2235
Rule 2333
Rule 2337
Rule 2436
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (a+b \log \left (c x^n\right )\right )^{3/2} \, dx,x,d+e x\right )}{e} \\ & = \frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}-\frac {(3 b n) \text {Subst}\left (\int \sqrt {a+b \log \left (c x^n\right )} \, dx,x,d+e x\right )}{2 e} \\ & = -\frac {3 b n (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}+\frac {\left (3 b^2 n^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c x^n\right )}} \, dx,x,d+e x\right )}{4 e} \\ & = -\frac {3 b n (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}+\frac {\left (3 b^2 n (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 )}{4 e} \\ & = -\frac {3 b n (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e}+\frac {\left (3 b n (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 )}{2 e} \\ & = \frac {3 b^{3/2} e^{-\frac {a}{b n}} n^{3/2} \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 )}{4 e}-\frac {3 b n (d+e x) \sqrt {a+b \log \left (c (d+e x)^n\right )}}{2 e}+\frac {(d+e x) \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}{e} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.89 \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \, dx=\frac {(d+e x) \left (3 b^{3/2} e^{-\frac {a}{b n}} n^{3/2} \sqrt {\pi } \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )+2 \sqrt {a+b \log \left (c (d+e x)^n\right )} \left (2 a-3 b n+2 b \log \left (c (d+e x)^n\right )\right )\right )}{4 e} \]
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\[\int {\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{\frac {3}{2}}d x\]
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Exception generated. \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \, dx=\int \left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {3}{2}}\, dx \]
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\[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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\[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \, dx=\int { {\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2} \, dx=\int {\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{3/2} \,d x \]
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