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 = 0.03, 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 = {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 2180
Rule 2204
Rule 2296
Rule 2299
Rule 2389
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*}
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Mathematica [A] time = 0.01, size = 50, normalized size = 1.00 \[ \frac {(d+e x) \sqrt {\log (c (d+e x))}}{e}-\frac {\sqrt {\pi } \text {erfi}\left (\sqrt {\log (c (d+e x))}\right )}{2 c e} \]
Antiderivative was successfully verified.
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fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 55, normalized size = 1.10 \[ -\frac {\sqrt {\pi } i \operatorname {erf}\left (-i \sqrt {\log \left (c x e + c d\right )}\right ) e^{\left (-1\right )}}{2 \, c} + \frac {{\left (c x e + c d\right )} e^{\left (-1\right )} \sqrt {\log \left (c x e + c d\right )}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.19, size = 0, normalized size = 0.00 \[ \int \sqrt {\ln \left (\left (e x +d \right ) c \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.62, size = 49, normalized size = 0.98 \[ -\frac {-i \, \sqrt {\pi } \operatorname {erf}\left (i \, \sqrt {\log \left (c e x + c d\right )}\right ) - 2 \, {\left (c e x + c d\right )} \sqrt {\log \left (c e x + c d\right )}}{2 \, c e} \]
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 \[ \frac {\sqrt {\ln \left (c\,\left (d+e\,x\right )\right )}\,\left (d+e\,x\right )}{e}+\frac {\sqrt {\pi }\,\mathrm {erf}\left (\sqrt {\ln \left (c\,\left (d+e\,x\right )\right )}\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{2\,c\,e} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.24, size = 90, normalized size = 1.80 \[ \begin {cases} \tilde {\infty } x & \text {for}\: c = 0 \\x \sqrt {\log {\left (c d \right )}} & \text {for}\: e = 0 \\\frac {\left (\sqrt {- \log {\left (c d + c e x \right )}} \left (c d + c e x\right ) + \frac {\sqrt {\pi } \operatorname {erfc}{\left (\sqrt {- \log {\left (c d + c e x \right )}} \right )}}{2}\right ) \sqrt {\log {\left (c d + c e x \right )}}}{c e \sqrt {- \log {\left (c d + c e x \right )}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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