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

Optimal. Leaf size=27 \[ e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{ExpIntegralEi}\left (\frac{b x}{2}\right ) \]

[Out]

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

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Rubi [A]  time = 0.088036, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{ExpIntegralEi}\left (\frac{b x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[E^(a + b*x)]/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 5.4813, size = 32, normalized size = 1.19 \[ e^{\frac{a}{2}} e^{- \frac{a}{2} - \frac{b x}{2}} \sqrt{e^{a + b x}} \operatorname{Ei}{\left (\frac{b x}{2} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(b*x+a)**(1/2)/x,x)

[Out]

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

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Mathematica [A]  time = 0.00654141, size = 27, normalized size = 1. \[ e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{ExpIntegralEi}\left (\frac{b x}{2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[E^(a + b*x)]/x,x]

[Out]

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

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Maple [B]  time = 0.088, size = 57, normalized size = 2.1 \[ \sqrt{{{\rm e}^{bx+a}}}{{\rm e}^{-{\frac{bx}{2}{{\rm e}^{{\frac{a}{2}}}}}}} \left ( \ln \left ( x \right ) -\ln \left ( 2 \right ) +\ln \left ( -b{{\rm e}^{{\frac{a}{2}}}} \right ) -\ln \left ( -{\frac{bx}{2}{{\rm e}^{{\frac{a}{2}}}}} \right ) -{\it Ei} \left ( 1,-{\frac{bx}{2}{{\rm e}^{{\frac{a}{2}}}}} \right ) \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(b*x+a)^(1/2)/x,x)

[Out]

exp(b*x+a)^(1/2)*exp(-1/2*b*x*exp(1/2*a))*(ln(x)-ln(2)+ln(-b*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 = 0.848026, size = 14, normalized size = 0.52 \[{\rm Ei}\left (\frac{1}{2} \, b x\right ) e^{\left (\frac{1}{2} \, a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(1/2*b*x + 1/2*a)/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.262248, size = 14, normalized size = 0.52 \[{\rm Ei}\left (\frac{1}{2} \, b x\right ) e^{\left (\frac{1}{2} \, a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(1/2*b*x + 1/2*a)/x,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e^{a} e^{b x}}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(b*x+a)**(1/2)/x,x)

[Out]

Integral(sqrt(exp(a)*exp(b*x))/x, x)

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GIAC/XCAS [A]  time = 0.231435, size = 14, normalized size = 0.52 \[{\rm Ei}\left (\frac{1}{2} \, b x\right ) e^{\left (\frac{1}{2} \, a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(1/2*b*x + 1/2*a)/x,x, algorithm="giac")

[Out]

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