3.96 \(\int \frac{\sqrt{e^{a+b x}}}{x^2} \, dx\)

Optimal. Leaf size=48 \[ \frac{1}{2} b e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{ExpIntegralEi}\left (\frac{b x}{2}\right )-\frac{\sqrt{e^{a+b x}}}{x} \]

[Out]

-(Sqrt[E^(a + b*x)]/x) + (b*Sqrt[E^(a + b*x)]*ExpIntegralEi[(b*x)/2])/(2*E^((b*x
)/2))

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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]

[Out]

-(Sqrt[E^(a + b*x)]/x) + (b*Sqrt[E^(a + b*x)]*ExpIntegralEi[(b*x)/2])/(2*E^((b*x
)/2))

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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]

b*exp(a/2)*exp(-a/2 - b*x/2)*sqrt(exp(a + b*x))*Ei(b*x/2)/2 - sqrt(exp(a + b*x))
/x

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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]

(Sqrt[E^(a + b*x)]*(-2*E^((b*x)/2) + b*x*ExpIntegralEi[(b*x)/2]))/(2*E^((b*x)/2)
*x)

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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]

-1/2*exp(b*x+a)^(1/2)*exp(1/2*a-1/2*b*x*exp(1/2*a))*b*(2/x/b*exp(-1/2*a)+1-ln(x)
+ln(2)-ln(-b*exp(1/2*a))-1/b/x*exp(-1/2*a)*(b*x*exp(1/2*a)+2)+2/b/x*exp(-1/2*a+1
/2*b*x*exp(1/2*a))+ln(-1/2*b*x*exp(1/2*a))+Ei(1,-1/2*b*x*exp(1/2*a)))

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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")

[Out]

1/2*b*e^(1/2*a)*gamma(-1, -1/2*b*x)

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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]

1/2*(b*x*Ei(1/2*b*x)*e^(1/2*a) - 2*e^(1/2*b*x + 1/2*a))/x

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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]

Integral(sqrt(exp(a)*exp(b*x))/x**2, x)

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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]

1/2*(b*x*Ei(1/2*b*x)*e^(1/2*a) - 2*e^(1/2*b*x + 1/2*a))/x