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 = 0.06, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 2178
Rule 2182
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*}
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Mathematica [A] time = 0.02, size = 27, normalized size = 1.00 \[ e^{-\frac {b x}{2}} \sqrt {e^{a+b x}} \text {Ei}\left (\frac {b x}{2}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 10, normalized size = 0.37 \[ {\rm Ei}\left (\frac {1}{2} \, b x\right ) e^{\left (\frac {1}{2} \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.36, size = 10, normalized size = 0.37 \[ {\rm Ei}\left (\frac {1}{2} \, b x\right ) e^{\left (\frac {1}{2} \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 57, normalized size = 2.11 \[ \left (-\Ei \left (1, -\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}\right )+\ln \relax (x )+\ln \left (-b \,{\mathrm e}^{\frac {a}{2}}\right )-\ln \left (-\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}\right )-\ln \relax (2)\right ) {\mathrm e}^{-\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}} \sqrt {{\mathrm e}^{b x +a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.23, size = 10, normalized size = 0.37 \[ {\rm Ei}\left (\frac {1}{2} \, b x\right ) e^{\left (\frac {1}{2} \, a\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {{\mathrm {e}}^{a+b\,x}}}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e^{a} e^{b x}}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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