Optimal. Leaf size=50 \[ \frac{(d+e x) \sqrt{\log (c (d+e x))}}{e}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{2 c e} \]
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Rubi [A] time = 0.0313145, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{(d+e x) \sqrt{\log (c (d+e x))}}{e}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{2 c e} \]
Antiderivative was successfully verified.
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Rule 2389
Rule 2296
Rule 2299
Rule 2180
Rule 2204
Rubi steps
\begin{align*} \int \sqrt{\log (c (d+e x))} \, dx &=\frac{\operatorname{Subst}\left (\int \sqrt{\log (c x)} \, dx,x,d+e x\right )}{e}\\ &=\frac{(d+e x) \sqrt{\log (c (d+e x))}}{e}-\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{\log (c x)}} \, dx,x,d+e x\right )}{2 e}\\ &=\frac{(d+e x) \sqrt{\log (c (d+e x))}}{e}-\frac{\operatorname{Subst}\left (\int \frac{e^x}{\sqrt{x}} \, dx,x,\log (c (d+e x))\right )}{2 c e}\\ &=\frac{(d+e x) \sqrt{\log (c (d+e x))}}{e}-\frac{\operatorname{Subst}\left (\int e^{x^2} \, dx,x,\sqrt{\log (c (d+e x))}\right )}{c e}\\ &=-\frac{\sqrt{\pi } \text{erfi}\left (\sqrt{\log (c (d+e x))}\right )}{2 c e}+\frac{(d+e x) \sqrt{\log (c (d+e x))}}{e}\\ \end{align*}
Mathematica [A] time = 0.0085198, size = 50, normalized size = 1. \[ \frac{(d+e x) \sqrt{\log (c (d+e x))}}{e}-\frac{\sqrt{\pi } \text{Erfi}\left (\sqrt{\log (c (d+e x))}\right )}{2 c e} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.342, size = 0, normalized size = 0. \begin{align*} \int \sqrt{\ln \left ( c \left ( ex+d \right ) \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] time = 1.2155, size = 66, normalized size = 1.32 \begin{align*} -\frac{-i \, \sqrt{\pi } \operatorname{erf}\left (i \, \sqrt{\log \left (c e x + c d\right )}\right ) - 2 \,{\left (c e x + c d\right )} \sqrt{\log \left (c e x + c d\right )}}{2 \, 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 [A] time = 4.11946, size = 90, normalized size = 1.8 \begin{align*} \begin{cases} \tilde{\infty } x & \text{for}\: c = 0 \\x \sqrt{\log{\left (c d \right )}} & \text{for}\: e = 0 \\\frac{\left (\sqrt{- \log{\left (c d + c e x \right )}} \left (c d + c e x\right ) + \frac{\sqrt{\pi } \operatorname{erfc}{\left (\sqrt{- \log{\left (c d + c e x \right )}} \right )}}{2}\right ) \sqrt{\log{\left (c d + c e x \right )}}}{c e \sqrt{- \log{\left (c d + c e x \right )}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31059, size = 74, normalized size = 1.48 \begin{align*} -\frac{\sqrt{\pi } i \operatorname{erf}\left (-i \sqrt{\log \left (c x e + c d\right )}\right ) e^{\left (-1\right )}}{2 \, c} + \frac{{\left (c x e + c d\right )} e^{\left (-1\right )} \sqrt{\log \left (c x e + c d\right )}}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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