Optimal. Leaf size=27 \[ e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{Ei}\left (\frac{b x}{2}\right ) \]
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Rubi [A] time = 0.0583287, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {2182, 2178} \[ e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{Ei}\left (\frac{b x}{2}\right ) \]
Antiderivative was successfully verified.
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Rule 2182
Rule 2178
Rubi steps
\begin{align*} \int \frac{\sqrt{e^{a+b x}}}{x} \, dx &=\left (e^{\frac{1}{2} (-a-b x)} \sqrt{e^{a+b x}}\right ) \int \frac{e^{\frac{1}{2} (a+b x)}}{x} \, dx\\ &=e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{Ei}\left (\frac{b x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0156963, size = 27, normalized size = 1. \[ e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{Ei}\left (\frac{b x}{2}\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.039, size = 57, normalized size = 2.1 \begin{align*} \sqrt{{{\rm e}^{bx+a}}}{{\rm e}^{-{\frac{bx}{2}{{\rm e}^{{\frac{a}{2}}}}}}} \left ( \ln \left ( x \right ) -\ln \left ( 2 \right ) +\ln \left ( -b{{\rm e}^{{\frac{a}{2}}}} \right ) -\ln \left ( -{\frac{bx}{2}{{\rm e}^{{\frac{a}{2}}}}} \right ) -{\it Ei} \left ( 1,-{\frac{bx}{2}{{\rm e}^{{\frac{a}{2}}}}} \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.17256, size = 14, normalized size = 0.52 \begin{align*}{\rm Ei}\left (\frac{1}{2} \, b x\right ) e^{\left (\frac{1}{2} \, a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49306, size = 31, normalized size = 1.15 \begin{align*}{\rm Ei}\left (\frac{1}{2} \, b x\right ) e^{\left (\frac{1}{2} \, a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e^{a} e^{b x}}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36466, size = 14, normalized size = 0.52 \begin{align*}{\rm Ei}\left (\frac{1}{2} \, b x\right ) e^{\left (\frac{1}{2} \, a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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