Optimal. Leaf size=98 \[ -\frac{15 \sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{8 c e}+\frac{(d+e x) \log ^{\frac{5}{2}}(c (d+e x))}{e}-\frac{5 (d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{2 e}+\frac{15 (d+e x) \sqrt{\log (c (d+e x))}}{4 e} \]
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Rubi [A] time = 0.0495554, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{15 \sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{8 c e}+\frac{(d+e x) \log ^{\frac{5}{2}}(c (d+e x))}{e}-\frac{5 (d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{2 e}+\frac{15 (d+e x) \sqrt{\log (c (d+e x))}}{4 e} \]
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{5}{2}}(c (d+e x)) \, dx &=\frac{\operatorname{Subst}\left (\int \log ^{\frac{5}{2}}(c x) \, dx,x,d+e x\right )}{e}\\ &=\frac{(d+e x) \log ^{\frac{5}{2}}(c (d+e x))}{e}-\frac{5 \operatorname{Subst}\left (\int \log ^{\frac{3}{2}}(c x) \, dx,x,d+e x\right )}{2 e}\\ &=-\frac{5 (d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{2 e}+\frac{(d+e x) \log ^{\frac{5}{2}}(c (d+e x))}{e}+\frac{15 \operatorname{Subst}\left (\int \sqrt{\log (c x)} \, dx,x,d+e x\right )}{4 e}\\ &=\frac{15 (d+e x) \sqrt{\log (c (d+e x))}}{4 e}-\frac{5 (d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{2 e}+\frac{(d+e x) \log ^{\frac{5}{2}}(c (d+e x))}{e}-\frac{15 \operatorname{Subst}\left (\int \frac{1}{\sqrt{\log (c x)}} \, dx,x,d+e x\right )}{8 e}\\ &=\frac{15 (d+e x) \sqrt{\log (c (d+e x))}}{4 e}-\frac{5 (d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{2 e}+\frac{(d+e x) \log ^{\frac{5}{2}}(c (d+e x))}{e}-\frac{15 \operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\log (c (d+e x))\right )}{8 c e}\\ &=\frac{15 (d+e x) \sqrt{\log (c (d+e x))}}{4 e}-\frac{5 (d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{2 e}+\frac{(d+e x) \log ^{\frac{5}{2}}(c (d+e x))}{e}-\frac{15 \operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\log (c (d+e x))}\right )}{4 c e}\\ &=-\frac{15 \sqrt{\pi } \text{erfi}\left (\sqrt{\log (c (d+e x))}\right )}{8 c e}+\frac{15 (d+e x) \sqrt{\log (c (d+e x))}}{4 e}-\frac{5 (d+e x) \log ^{\frac{3}{2}}(c (d+e x))}{2 e}+\frac{(d+e x) \log ^{\frac{5}{2}}(c (d+e x))}{e}\\ \end{align*}
Mathematica [A] time = 0.0128011, size = 75, normalized size = 0.77 \[ \frac{2 c (d+e x) \sqrt{\log (c (d+e x))} \left (4 \log ^2(c (d+e x))-10 \log (c (d+e x))+15\right )-15 \sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{8 c e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.327, 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 [C] time = 1.1516, size = 105, normalized size = 1.07 \begin{align*} \frac{2 \,{\left (c e x + c d\right )}{\left (4 \, \log \left (c e x + c d\right )^{\frac{5}{2}} - 10 \, \log \left (c e x + c d\right )^{\frac{3}{2}} + 15 \, \sqrt{\log \left (c e x + c d\right )}\right )} + 15 i \, \sqrt{\pi } \operatorname{erf}\left (i \, \sqrt{\log \left (c e x + c d\right )}\right )}{8 \, 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{5}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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