Optimal. Leaf size=77 \[ \frac {4 \sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{3 c e}-\frac {2 (d+e x)}{3 e \log ^{\frac {3}{2}}(c (d+e x))}-\frac {4 (d+e x)}{3 e \sqrt {\log (c (d+e x))}} \]
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Rubi [A]
time = 0.03, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {2436, 2334,
2336, 2211, 2235} \begin {gather*} \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 {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2334
Rule 2336
Rule 2436
Rubi steps
\begin {align*} \int \frac {1}{\log ^{\frac {5}{2}}(c (d+e x))} \, dx &=\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*}
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Mathematica [A]
time = 0.06, size = 72, normalized size = 0.94 \begin {gather*} -\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))} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \frac {1}{\ln \left (c \left (e x +d \right )\right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.32, size = 47, normalized size = 0.61 \begin {gather*} -\frac {\left (-\log \left (c x e + c d\right )\right )^{\frac {3}{2}} e^{\left (-1\right )} \Gamma \left (-\frac {3}{2}, -\log \left (c x e + c d\right )\right )}{c \log \left (c x e + c d\right )^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 113, normalized size = 1.47 \begin {gather*} \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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