Optimal. Leaf size=48 \[ \frac {1}{2} b e^{-\frac {b x}{2}} \sqrt {e^{a+b x}} \text {Ei}\left (\frac {b x}{2}\right )-\frac {\sqrt {e^{a+b x}}}{x} \]
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Rubi [A] time = 0.09, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2177, 2182, 2178} \[ \frac {1}{2} b e^{-\frac {b x}{2}} \sqrt {e^{a+b x}} \text {Ei}\left (\frac {b x}{2}\right )-\frac {\sqrt {e^{a+b x}}}{x} \]
Antiderivative was successfully verified.
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Rule 2177
Rule 2178
Rule 2182
Rubi steps
\begin {align*} \int \frac {\sqrt {e^{a+b x}}}{x^2} \, dx &=-\frac {\sqrt {e^{a+b x}}}{x}+\frac {1}{2} b \int \frac {\sqrt {e^{a+b x}}}{x} \, dx\\ &=-\frac {\sqrt {e^{a+b x}}}{x}+\frac {1}{2} \left (b e^{\frac {1}{2} (-a-b x)} \sqrt {e^{a+b x}}\right ) \int \frac {e^{\frac {1}{2} (a+b x)}}{x} \, dx\\ &=-\frac {\sqrt {e^{a+b x}}}{x}+\frac {1}{2} b 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 = 0.03, size = 47, normalized size = 0.98 \[ \frac {e^{-\frac {b x}{2}} \sqrt {e^{a+b x}} \left (b x \text {Ei}\left (\frac {b x}{2}\right )-2 e^{\frac {b x}{2}}\right )}{2 x} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 29, normalized size = 0.60 \[ \frac {b x {\rm Ei}\left (\frac {1}{2} \, b x\right ) e^{\left (\frac {1}{2} \, a\right )} - 2 \, e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.41, size = 29, normalized size = 0.60 \[ \frac {b x {\rm Ei}\left (\frac {1}{2} \, b x\right ) e^{\left (\frac {1}{2} \, a\right )} - 2 \, e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 116, normalized size = 2.42 \[ -\frac {\left (\Ei \left (1, -\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}\right )-\ln \relax (x )-\ln \left (-b \,{\mathrm e}^{\frac {a}{2}}\right )+\ln \left (-\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}\right )-\frac {\left (b x \,{\mathrm e}^{\frac {a}{2}}+2\right ) {\mathrm e}^{-\frac {a}{2}}}{b x}+\frac {2 \,{\mathrm e}^{-\frac {a}{2}}}{b x}+\frac {2 \,{\mathrm e}^{\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}-\frac {a}{2}}}{b x}+1+\ln \relax (2)\right ) b \,{\mathrm e}^{-\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}+\frac {a}{2}} \sqrt {{\mathrm e}^{b x +a}}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.24, size = 13, normalized size = 0.27 \[ \frac {1}{2} \, b e^{\left (\frac {1}{2} \, a\right )} \Gamma \left (-1, -\frac {1}{2} \, b x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {{\mathrm {e}}^{a+b\,x}}}{x^2} \,d x \]
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 \[ \int \frac {\sqrt {e^{a} e^{b x}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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