Optimal. Leaf size=48 \[ \frac{1}{2} b e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{ExpIntegralEi}\left (\frac{b x}{2}\right )-\frac{\sqrt{e^{a+b x}}}{x} \]
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Rubi [A] time = 0.133816, antiderivative size = 48, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{1}{2} b e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{ExpIntegralEi}\left (\frac{b x}{2}\right )-\frac{\sqrt{e^{a+b x}}}{x} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[E^(a + b*x)]/x^2,x]
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Rubi in Sympy [A] time = 7.80962, size = 48, normalized size = 1. \[ \frac{b e^{\frac{a}{2}} e^{- \frac{a}{2} - \frac{b x}{2}} \sqrt{e^{a + b x}} \operatorname{Ei}{\left (\frac{b x}{2} \right )}}{2} - \frac{\sqrt{e^{a + b x}}}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(exp(b*x+a)**(1/2)/x**2,x)
[Out]
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Mathematica [A] time = 0.0191843, size = 47, normalized size = 0.98 \[ \frac{e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \left (b x \text{ExpIntegralEi}\left (\frac{b x}{2}\right )-2 e^{\frac{b x}{2}}\right )}{2 x} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[E^(a + b*x)]/x^2,x]
[Out]
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Maple [B] time = 0.033, size = 116, normalized size = 2.4 \[ -{\frac{b}{2}\sqrt{{{\rm e}^{bx+a}}}{{\rm e}^{{\frac{a}{2}}-{\frac{bx}{2}{{\rm e}^{{\frac{a}{2}}}}}}} \left ( 2\,{\frac{{{\rm e}^{-a/2}}}{bx}}+1-\ln \left ( x \right ) +\ln \left ( 2 \right ) -\ln \left ( -b{{\rm e}^{{\frac{a}{2}}}} \right ) -{\frac{1}{bx}{{\rm e}^{-{\frac{a}{2}}}} \left ( bx{{\rm e}^{{\frac{a}{2}}}}+2 \right ) }+2\,{\frac{{{\rm e}^{-a/2+1/2\,bx{{\rm e}^{a/2}}}}}{bx}}+\ln \left ( -{\frac{bx}{2}{{\rm e}^{{\frac{a}{2}}}}} \right ) +{\it Ei} \left ( 1,-{\frac{bx}{2}{{\rm e}^{{\frac{a}{2}}}}} \right ) \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(exp(b*x+a)^(1/2)/x^2,x)
[Out]
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Maxima [A] time = 0.847892, size = 18, normalized size = 0.38 \[ \frac{1}{2} \, b e^{\left (\frac{1}{2} \, a\right )} \Gamma \left (-1, -\frac{1}{2} \, b x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(1/2*b*x + 1/2*a)/x^2,x, algorithm="maxima")
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Fricas [A] time = 0.249588, size = 39, normalized size = 0.81 \[ \frac{b x{\rm Ei}\left (\frac{1}{2} \, b x\right ) e^{\left (\frac{1}{2} \, a\right )} - 2 \, e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(1/2*b*x + 1/2*a)/x^2,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e^{a} e^{b x}}}{x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(exp(b*x+a)**(1/2)/x**2,x)
[Out]
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GIAC/XCAS [A] time = 0.24324, size = 39, normalized size = 0.81 \[ \frac{b x{\rm Ei}\left (\frac{1}{2} \, b x\right ) e^{\left (\frac{1}{2} \, a\right )} - 2 \, e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(e^(1/2*b*x + 1/2*a)/x^2,x, algorithm="giac")
[Out]