Optimal. Leaf size=74 \[ \frac{3 \sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{4 c e}+\frac{(d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{e}-\frac{3 (d+e x) \sqrt{\log (c (d+e x))}}{2 e} \]
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Rubi [A] time = 0.037502, antiderivative size = 74, 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, 2296, 2299, 2180, 2204} \[ \frac{3 \sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{4 c e}+\frac{(d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{e}-\frac{3 (d+e x) \sqrt{\log (c (d+e x))}}{2 e} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2296
Rule 2299
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \log ^{\frac{3}{2}}(c (d+e x)) \, dx &=\frac{\operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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 \operatorname{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*}
Mathematica [A] time = 0.0098714, size = 63, normalized size = 0.85 \[ \frac{3 \sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )+2 c (d+e x) \sqrt{\log (c (d+e x))} (2 \log (c (d+e x))-3)}{4 c e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.276, size = 0, normalized size = 0. \begin{align*} \int \left ( \ln \left ( c \left ( ex+d \right ) \right ) \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.17343, size = 88, normalized size = 1.19 \begin{align*} \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} \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 \log \left ({\left (e x + d\right )} c\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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