Optimal. Leaf size=50 \[ -\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{2 c e}+\frac {(d+e x) \sqrt {\log (c (d+e x))}}{e} \]
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Rubi [A]
time = 0.02, antiderivative size = 50, normalized size of antiderivative = 1.00, 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 = {2436, 2333,
2336, 2211, 2235} \begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 2211
Rule 2235
Rule 2333
Rule 2336
Rule 2436
Rubi steps
\begin {align*} \int \sqrt {\log (c (d+e x))} \, dx &=\frac {\text {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 {\text {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 {\text {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 {\text {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*}
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Mathematica [A]
time = 0.01, size = 50, normalized size = 1.00 \begin {gather*} -\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 {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \sqrt {\ln \left (c \left (e x +d \right )\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains complex when optimal does not.
time = 0.28, size = 53, normalized size = 1.06 \begin {gather*} -\frac {-i \, \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {\log \left (c x e + c d\right )}\right ) e^{\left (-1\right )} - 2 \, {\left (c x e + c d\right )} e^{\left (-1\right )} \sqrt {\log \left (c x e + c d\right )}}{2 \, c} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs.
\(2 (41) = 82\).
time = 1.14, size = 90, normalized size = 1.80 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [C] Result contains complex when optimal does not.
time = 3.34, size = 53, normalized size = 1.06 \begin {gather*} -\frac {i \, \sqrt {\pi } \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.18, size = 46, normalized size = 0.92 \begin {gather*} \frac {\sqrt {\ln \left (c\,\left (d+e\,x\right )\right )}\,\left (d+e\,x\right )}{e}+\frac {\sqrt {\pi }\,\mathrm {erf}\left (\sqrt {\ln \left (c\,\left (d+e\,x\right )\right )}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,c\,e} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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