Integrand size = 12, antiderivative size = 74 \[ \int \log ^{\frac {3}{2}}(c (d+e x)) \, dx=\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{4 c e}-\frac {3 (d+e x) \sqrt {\log (c (d+e x))}}{2 e}+\frac {(d+e x) \log ^{\frac {3}{2}}(c (d+e x))}{e} \]
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Time = 0.03 (sec) , antiderivative size = 74, 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, 2333, 2336, 2211, 2235} \[ \int \log ^{\frac {3}{2}}(c (d+e x)) \, dx=\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{4 c e}+\frac {(d+e x) \log ^{\frac {3}{2}}(c (d+e x))}{e}-\frac {3 (d+e x) \sqrt {\log (c (d+e x))}}{2 e} \]
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Rule 2211
Rule 2235
Rule 2333
Rule 2336
Rule 2436
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \log ^{\frac {3}{2}}(c x) \, dx,x,d+e x\right )}{e} \\ & = \frac {(d+e x) \log ^{\frac {3}{2}}(c (d+e x))}{e}-\frac {3 \text {Subst}\left (\int \sqrt {\log (c x)} \, dx,x,d+e x\right )}{2 e} \\ & = -\frac {3 (d+e x) \sqrt {\log (c (d+e x))}}{2 e}+\frac {(d+e x) \log ^{\frac {3}{2}}(c (d+e x))}{e}+\frac {3 \text {Subst}\left (\int \frac {1}{\sqrt {\log (c x)}} \, dx,x,d+e x\right )}{4 e} \\ & = -\frac {3 (d+e x) \sqrt {\log (c (d+e x))}}{2 e}+\frac {(d+e x) \log ^{\frac {3}{2}}(c (d+e x))}{e}+\frac {3 \text {Subst}\left (\int \frac {e^x}{\sqrt {x}} \, dx,x,\log (c (d+e x))\right )}{4 c e} \\ & = -\frac {3 (d+e x) \sqrt {\log (c (d+e x))}}{2 e}+\frac {(d+e x) \log ^{\frac {3}{2}}(c (d+e x))}{e}+\frac {3 \text {Subst}\left (\int e^{x^2} \, dx,x,\sqrt {\log (c (d+e x))}\right )}{2 c e} \\ & = \frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{4 c e}-\frac {3 (d+e x) \sqrt {\log (c (d+e x))}}{2 e}+\frac {(d+e x) \log ^{\frac {3}{2}}(c (d+e x))}{e} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.85 \[ \int \log ^{\frac {3}{2}}(c (d+e x)) \, dx=\frac {3 \sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )+2 c (d+e x) \sqrt {\log (c (d+e x))} (-3+2 \log (c (d+e x)))}{4 c e} \]
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\[\int \ln \left (c \left (e x +d \right )\right )^{\frac {3}{2}}d x\]
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Exception generated. \[ \int \log ^{\frac {3}{2}}(c (d+e x)) \, dx=\text {Exception raised: TypeError} \]
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Time = 55.01 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.42 \[ \int \log ^{\frac {3}{2}}(c (d+e x)) \, dx=\begin {cases} \tilde {\infty } x & \text {for}\: c = 0 \\x \log {\left (c d \right )}^{\frac {3}{2}} & \text {for}\: e = 0 \\\frac {\left (- \sqrt {- \log {\left (c d + c e x \right )}} \left (c d + c e x\right ) \left (\log {\left (c d + c e x \right )} - \frac {3}{2}\right ) + \frac {3 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- \log {\left (c d + c e x \right )}} \right )}}{4}\right ) \log {\left (c d + c e x \right )}^{\frac {3}{2}}}{c e \left (- \log {\left (c d + c e x \right )}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
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Result contains complex when optimal does not.
Time = 0.21 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.88 \[ \int \log ^{\frac {3}{2}}(c (d+e x)) \, dx=\frac {2 \, {\left (c e x + c d\right )} {\left (2 \, \log \left (c e x + c d\right )^{\frac {3}{2}} - 3 \, \sqrt {\log \left (c e x + c d\right )}\right )} - 3 i \, \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {\log \left (c e x + c d\right )}\right )}{4 \, c e} \]
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\[ \int \log ^{\frac {3}{2}}(c (d+e x)) \, dx=\int { \log \left ({\left (e x + d\right )} c\right )^{\frac {3}{2}} \,d x } \]
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Time = 1.27 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.11 \[ \int \log ^{\frac {3}{2}}(c (d+e x)) \, dx=\frac {{\ln \left (c\,\left (d+e\,x\right )\right )}^{3/2}\,\left (\frac {3\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {-\ln \left (c\,\left (d+e\,x\right )\right )}\right )}{4}+c\,\left (\frac {3\,\sqrt {-\ln \left (c\,\left (d+e\,x\right )\right )}}{2}+{\left (-\ln \left (c\,\left (d+e\,x\right )\right )\right )}^{3/2}\right )\,\left (d+e\,x\right )\right )}{c\,e\,{\left (-\ln \left (c\,\left (d+e\,x\right )\right )\right )}^{3/2}} \]
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