Integrand size = 15, antiderivative size = 48 \[ \int \frac {\sqrt {e^{a+b x}}}{x^2} \, dx=-\frac {\sqrt {e^{a+b x}}}{x}+\frac {1}{2} b e^{-\frac {b x}{2}} \sqrt {e^{a+b x}} \operatorname {ExpIntegralEi}\left (\frac {b x}{2}\right ) \]
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Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2208, 2213, 2209} \[ \int \frac {\sqrt {e^{a+b x}}}{x^2} \, dx=\frac {1}{2} b e^{-\frac {b x}{2}} \sqrt {e^{a+b x}} \operatorname {ExpIntegralEi}\left (\frac {b x}{2}\right )-\frac {\sqrt {e^{a+b x}}}{x} \]
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Rule 2208
Rule 2209
Rule 2213
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.10 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.98 \[ \int \frac {\sqrt {e^{a+b x}}}{x^2} \, dx=\frac {e^{-\frac {b x}{2}} \sqrt {e^{a+b x}} \left (-2 e^{\frac {b x}{2}}+b x \operatorname {ExpIntegralEi}\left (\frac {b x}{2}\right )\right )}{2 x} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(115\) vs. \(2(37)=74\).
Time = 0.02 (sec) , antiderivative size = 116, normalized size of antiderivative = 2.42
method | result | size |
meijerg | \(-\frac {\sqrt {{\mathrm e}^{b x +a}}\, {\mathrm e}^{\frac {a}{2}-\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}} b \left (\frac {2 \,{\mathrm e}^{-\frac {a}{2}}}{x b}+1-\ln \left (x \right )+\ln \left (2\right )-\ln \left (-b \,{\mathrm e}^{\frac {a}{2}}\right )-\frac {{\mathrm e}^{-\frac {a}{2}} \left (2+b x \,{\mathrm e}^{\frac {a}{2}}\right )}{b x}+\frac {2 \,{\mathrm e}^{-\frac {a}{2}+\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}}}{b x}+\ln \left (-\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}\right )+\operatorname {Ei}_{1}\left (-\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}\right )\right )}{2}\) | \(116\) |
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none
Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {e^{a+b x}}}{x^2} \, dx=\frac {b x {\rm Ei}\left (\frac {1}{2} \, b x\right ) e^{\left (\frac {1}{2} \, a\right )} - 2 \, e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{2 \, x} \]
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\[ \int \frac {\sqrt {e^{a+b x}}}{x^2} \, dx=\int \frac {\sqrt {e^{a} e^{b x}}}{x^{2}}\, dx \]
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none
Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.27 \[ \int \frac {\sqrt {e^{a+b x}}}{x^2} \, dx=\frac {1}{2} \, b e^{\left (\frac {1}{2} \, a\right )} \Gamma \left (-1, -\frac {1}{2} \, b x\right ) \]
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none
Time = 0.35 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {e^{a+b x}}}{x^2} \, dx=\frac {b x {\rm Ei}\left (\frac {1}{2} \, b x\right ) e^{\left (\frac {1}{2} \, a\right )} - 2 \, e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{2 \, x} \]
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Timed out. \[ \int \frac {\sqrt {e^{a+b x}}}{x^2} \, dx=\int \frac {\sqrt {{\mathrm {e}}^{a+b\,x}}}{x^2} \,d x \]
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