Optimal. Leaf size=71 \[ \frac{1}{8} b^2 e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{Ei}\left (\frac{b x}{2}\right )-\frac{\sqrt{e^{a+b x}}}{2 x^2}-\frac{b \sqrt{e^{a+b x}}}{4 x} \]
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Rubi [A] time = 0.119316, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2177, 2182, 2178} \[ \frac{1}{8} b^2 e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{Ei}\left (\frac{b x}{2}\right )-\frac{\sqrt{e^{a+b x}}}{2 x^2}-\frac{b \sqrt{e^{a+b x}}}{4 x} \]
Antiderivative was successfully verified.
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Rule 2177
Rule 2182
Rule 2178
Rubi steps
\begin{align*} \int \frac{\sqrt{e^{a+b x}}}{x^3} \, dx &=-\frac{\sqrt{e^{a+b x}}}{2 x^2}+\frac{1}{4} b \int \frac{\sqrt{e^{a+b x}}}{x^2} \, dx\\ &=-\frac{\sqrt{e^{a+b x}}}{2 x^2}-\frac{b \sqrt{e^{a+b x}}}{4 x}+\frac{1}{8} b^2 \int \frac{\sqrt{e^{a+b x}}}{x} \, dx\\ &=-\frac{\sqrt{e^{a+b x}}}{2 x^2}-\frac{b \sqrt{e^{a+b x}}}{4 x}+\frac{1}{8} \left (b^2 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}}}{2 x^2}-\frac{b \sqrt{e^{a+b x}}}{4 x}+\frac{1}{8} b^2 e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{Ei}\left (\frac{b x}{2}\right )\\ \end{align*}
Mathematica [A] time = 0.0423023, size = 56, normalized size = 0.79 \[ \frac{e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \left (b^2 x^2 \text{Ei}\left (\frac{b x}{2}\right )-2 e^{\frac{b x}{2}} (b x+2)\right )}{8 x^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.028, size = 155, normalized size = 2.2 \begin{align*}{\frac{{b}^{2}}{4}\sqrt{{{\rm e}^{bx+a}}}{{\rm e}^{a-{\frac{bx}{2}{{\rm e}^{{\frac{a}{2}}}}}}} \left ( -2\,{\frac{{{\rm e}^{-a}}}{{x}^{2}{b}^{2}}}-2\,{\frac{{{\rm e}^{-a/2}}}{bx}}-{\frac{3}{4}}+{\frac{\ln \left ( x \right ) }{2}}-{\frac{\ln \left ( 2 \right ) }{2}}+{\frac{1}{2}\ln \left ( -b{{\rm e}^{{\frac{a}{2}}}} \right ) }+{\frac{{{\rm e}^{-a}}}{3\,{x}^{2}{b}^{2}} \left ({\frac{9\,{b}^{2}{x}^{2}{{\rm e}^{a}}}{4}}+6\,bx{{\rm e}^{a/2}}+6 \right ) }-{\frac{2}{3\,{x}^{2}{b}^{2}}{{\rm e}^{-a+{\frac{bx}{2}{{\rm e}^{{\frac{a}{2}}}}}}} \left ({\frac{3\,bx}{2}{{\rm e}^{{\frac{a}{2}}}}}+3 \right ) }-{\frac{1}{2}\ln \left ( -{\frac{bx}{2}{{\rm e}^{{\frac{a}{2}}}}} \right ) }-{\frac{1}{2}{\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 = 1.14584, size = 20, normalized size = 0.28 \begin{align*} -\frac{1}{4} \, b^{2} e^{\left (\frac{1}{2} \, a\right )} \Gamma \left (-2, -\frac{1}{2} \, b x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50765, size = 101, normalized size = 1.42 \begin{align*} \frac{b^{2} x^{2}{\rm Ei}\left (\frac{1}{2} \, b x\right ) e^{\left (\frac{1}{2} \, a\right )} - 2 \,{\left (b x + 2\right )} e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{8 \, x^{2}} \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^{3}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.28556, size = 62, normalized size = 0.87 \begin{align*} \frac{b^{2} x^{2}{\rm Ei}\left (\frac{1}{2} \, b x\right ) e^{\left (\frac{1}{2} \, a\right )} - 2 \, b x e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )} - 4 \, e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{8 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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