Integrand size = 12, antiderivative size = 77 \[ \int \frac {1}{\log ^{\frac {5}{2}}(c (d+e x))} \, dx=\frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{3 c e}-\frac {2 (d+e x)}{3 e \log ^{\frac {3}{2}}(c (d+e x))}-\frac {4 (d+e x)}{3 e \sqrt {\log (c (d+e x))}} \]
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Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2436, 2334, 2336, 2211, 2235} \[ \int \frac {1}{\log ^{\frac {5}{2}}(c (d+e x))} \, dx=\frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{3 c e}-\frac {2 (d+e x)}{3 e \log ^{\frac {3}{2}}(c (d+e x))}-\frac {4 (d+e x)}{3 e \sqrt {\log (c (d+e x))}} \]
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Rule 2211
Rule 2235
Rule 2334
Rule 2336
Rule 2436
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{\log ^{\frac {5}{2}}(c x)} \, dx,x,d+e x\right )}{e} \\ & = -\frac {2 (d+e x)}{3 e \log ^{\frac {3}{2}}(c (d+e x))}+\frac {2 \text {Subst}\left (\int \frac {1}{\log ^{\frac {3}{2}}(c x)} \, dx,x,d+e x\right )}{3 e} \\ & = -\frac {2 (d+e x)}{3 e \log ^{\frac {3}{2}}(c (d+e x))}-\frac {4 (d+e x)}{3 e \sqrt {\log (c (d+e x))}}+\frac {4 \text {Subst}\left (\int \frac {1}{\sqrt {\log (c x)}} \, dx,x,d+e x\right )}{3 e} \\ & = -\frac {2 (d+e x)}{3 e \log ^{\frac {3}{2}}(c (d+e x))}-\frac {4 (d+e x)}{3 e \sqrt {\log (c (d+e x))}}+\frac {4 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\log (c (d+e x))\right )}{3 c e} \\ & = -\frac {2 (d+e x)}{3 e \log ^{\frac {3}{2}}(c (d+e x))}-\frac {4 (d+e x)}{3 e \sqrt {\log (c (d+e x))}}+\frac {8 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\log (c (d+e x))}\right )}{3 c e} \\ & = \frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{3 c e}-\frac {2 (d+e x)}{3 e \log ^{\frac {3}{2}}(c (d+e x))}-\frac {4 (d+e x)}{3 e \sqrt {\log (c (d+e x))}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\log ^{\frac {5}{2}}(c (d+e x))} \, dx=-\frac {2 \left (2 \Gamma \left (\frac {1}{2},-\log (c (d+e x))\right ) (-\log (c (d+e x)))^{3/2}+c (d+e x) (1+2 \log (c (d+e x)))\right )}{3 c e \log ^{\frac {3}{2}}(c (d+e x))} \]
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\[\int \frac {1}{\ln \left (c \left (e x +d \right )\right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {1}{\log ^{\frac {5}{2}}(c (d+e x))} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{\log ^{\frac {5}{2}}(c (d+e x))} \, dx=\text {Timed out} \]
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none
Time = 0.24 (sec) , antiderivative size = 45, normalized size of antiderivative = 0.58 \[ \int \frac {1}{\log ^{\frac {5}{2}}(c (d+e x))} \, dx=-\frac {\left (-\log \left (c e x + c d\right )\right )^{\frac {3}{2}} \Gamma \left (-\frac {3}{2}, -\log \left (c e x + c d\right )\right )}{c e \log \left (c e x + c d\right )^{\frac {3}{2}}} \]
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\[ \int \frac {1}{\log ^{\frac {5}{2}}(c (d+e x))} \, dx=\int { \frac {1}{\log \left ({\left (e x + d\right )} c\right )^{\frac {5}{2}}} \,d x } \]
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Time = 1.31 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.47 \[ \int \frac {1}{\log ^{\frac {5}{2}}(c (d+e x))} \, dx=\frac {4\,\sqrt {\pi }\,{\left (-\ln \left (c\,\left (d+e\,x\right )\right )\right )}^{5/2}\,\mathrm {erfc}\left (\sqrt {-\ln \left (c\,\left (d+e\,x\right )\right )}\right )}{3\,c\,e\,{\ln \left (c\,\left (d+e\,x\right )\right )}^{5/2}}-\frac {4\,d\,{\ln \left (c\,\left (d+e\,x\right )\right )}^2+2\,d\,\ln \left (c\,\left (d+e\,x\right )\right )+2\,e\,x\,\ln \left (c\,\left (d+e\,x\right )\right )+4\,e\,x\,{\ln \left (c\,\left (d+e\,x\right )\right )}^2}{3\,e\,{\ln \left (c\,\left (d+e\,x\right )\right )}^{5/2}} \]
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