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 = 0.038312, antiderivative size = 77, normalized size of antiderivative = 1., 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 = {2389, 2297, 2299, 2180, 2204} \[ \frac{4 \sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{3 c e}-\frac{2 (d+e x)}{3 e \log ^{\frac{3}{2}}(c (d+e x))}-\frac{4 (d+e x)}{3 e \sqrt{\log (c (d+e x))}} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2297
Rule 2299
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \frac{1}{\log ^{\frac{5}{2}}(c (d+e x))} \, dx &=\frac{\operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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*}
Mathematica [A] time = 0.0291489, size = 72, normalized size = 0.94 \[ -\frac{2 \left (2 (-\log (c (d+e x)))^{3/2} \text{Gamma}\left (\frac{1}{2},-\log (c (d+e x))\right )+c (d+e x) (2 \log (c (d+e x))+1)\right )}{3 c e \log ^{\frac{3}{2}}(c (d+e x))} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.272, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( c \left ( ex+d \right ) \right ) \right ) ^{-{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.2611, size = 61, normalized size = 0.79 \begin{align*} -\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}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\log \left ({\left (e x + d\right )} c\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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