3.97 \(\int \frac{\sqrt{e^{a+b x}}}{x^3} \, dx\)

Optimal. Leaf size=71 \[ \frac{1}{8} b^2 e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{ExpIntegralEi}\left (\frac{b x}{2}\right )-\frac{\sqrt{e^{a+b x}}}{2 x^2}-\frac{b \sqrt{e^{a+b x}}}{4 x} \]

[Out]

-Sqrt[E^(a + b*x)]/(2*x^2) - (b*Sqrt[E^(a + b*x)])/(4*x) + (b^2*Sqrt[E^(a + b*x)
]*ExpIntegralEi[(b*x)/2])/(8*E^((b*x)/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.179693, 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 \[ \frac{1}{8} b^2 e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{ExpIntegralEi}\left (\frac{b x}{2}\right )-\frac{\sqrt{e^{a+b x}}}{2 x^2}-\frac{b \sqrt{e^{a+b x}}}{4 x} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[E^(a + b*x)]/x^3,x]

[Out]

-Sqrt[E^(a + b*x)]/(2*x^2) - (b*Sqrt[E^(a + b*x)])/(4*x) + (b^2*Sqrt[E^(a + b*x)
]*ExpIntegralEi[(b*x)/2])/(8*E^((b*x)/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.4221, size = 68, normalized size = 0.96 \[ \frac{b^{2} e^{\frac{a}{2}} e^{- \frac{a}{2} - \frac{b x}{2}} \sqrt{e^{a + b x}} \operatorname{Ei}{\left (\frac{b x}{2} \right )}}{8} - \frac{b \sqrt{e^{a + b x}}}{4 x} - \frac{\sqrt{e^{a + b x}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(exp(b*x+a)**(1/2)/x**3,x)

[Out]

b**2*exp(a/2)*exp(-a/2 - b*x/2)*sqrt(exp(a + b*x))*Ei(b*x/2)/8 - b*sqrt(exp(a +
b*x))/(4*x) - sqrt(exp(a + b*x))/(2*x**2)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0313286, size = 56, normalized size = 0.79 \[ \frac{e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \left (b^2 x^2 \text{ExpIntegralEi}\left (\frac{b x}{2}\right )-2 e^{\frac{b x}{2}} (b x+2)\right )}{8 x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[E^(a + b*x)]/x^3,x]

[Out]

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

_______________________________________________________________________________________

Maple [B]  time = 0.043, size = 155, normalized size = 2.2 \[{\frac{{b}^{2}}{4}\sqrt{{{\rm e}^{bx+a}}}{{\rm e}^{a-{\frac{bx}{2}{{\rm e}^{{\frac{a}{2}}}}}}} \left ( -2\,{\frac{{{\rm e}^{-a}}}{{x}^{2}{b}^{2}}}-2\,{\frac{{{\rm e}^{-a/2}}}{bx}}-{\frac{3}{4}}+{\frac{\ln \left ( x \right ) }{2}}-{\frac{\ln \left ( 2 \right ) }{2}}+{\frac{1}{2}\ln \left ( -b{{\rm e}^{{\frac{a}{2}}}} \right ) }+{\frac{{{\rm e}^{-a}}}{3\,{x}^{2}{b}^{2}} \left ({\frac{9\,{b}^{2}{x}^{2}{{\rm e}^{a}}}{4}}+6\,bx{{\rm e}^{a/2}}+6 \right ) }-{\frac{2}{3\,{x}^{2}{b}^{2}}{{\rm e}^{-a+{\frac{bx}{2}{{\rm e}^{{\frac{a}{2}}}}}}} \left ({\frac{3\,bx}{2}{{\rm e}^{{\frac{a}{2}}}}}+3 \right ) }-{\frac{1}{2}\ln \left ( -{\frac{bx}{2}{{\rm e}^{{\frac{a}{2}}}}} \right ) }-{\frac{1}{2}{\it Ei} \left ( 1,-{\frac{bx}{2}{{\rm e}^{{\frac{a}{2}}}}} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(exp(b*x+a)^(1/2)/x^3,x)

[Out]

1/4*exp(b*x+a)^(1/2)*exp(a-1/2*b*x*exp(1/2*a))*b^2*(-2/x^2/b^2*exp(-a)-2/x/b*exp
(-1/2*a)-3/4+1/2*ln(x)-1/2*ln(2)+1/2*ln(-b*exp(1/2*a))+1/3/b^2/x^2*exp(-a)*(9/4*
b^2*x^2*exp(a)+6*b*x*exp(1/2*a)+6)-2/3/b^2/x^2*exp(-a+1/2*b*x*exp(1/2*a))*(3/2*b
*x*exp(1/2*a)+3)-1/2*ln(-1/2*b*x*exp(1/2*a))-1/2*Ei(1,-1/2*b*x*exp(1/2*a)))

_______________________________________________________________________________________

Maxima [A]  time = 0.831516, size = 20, normalized size = 0.28 \[ -\frac{1}{4} \, b^{2} e^{\left (\frac{1}{2} \, a\right )} \Gamma \left (-2, -\frac{1}{2} \, b x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(1/2*b*x + 1/2*a)/x^3,x, algorithm="maxima")

[Out]

-1/4*b^2*e^(1/2*a)*gamma(-2, -1/2*b*x)

_______________________________________________________________________________________

Fricas [A]  time = 0.244367, size = 51, normalized size = 0.72 \[ \frac{b^{2} x^{2}{\rm Ei}\left (\frac{1}{2} \, b x\right ) e^{\left (\frac{1}{2} \, a\right )} - 2 \,{\left (b x + 2\right )} e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{8 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(1/2*b*x + 1/2*a)/x^3,x, algorithm="fricas")

[Out]

1/8*(b^2*x^2*Ei(1/2*b*x)*e^(1/2*a) - 2*(b*x + 2)*e^(1/2*b*x + 1/2*a))/x^2

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{e^{a} e^{b x}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(exp(b*x+a)**(1/2)/x**3,x)

[Out]

Integral(sqrt(exp(a)*exp(b*x))/x**3, x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.238296, size = 62, normalized size = 0.87 \[ \frac{b^{2} x^{2}{\rm Ei}\left (\frac{1}{2} \, b x\right ) e^{\left (\frac{1}{2} \, a\right )} - 2 \, b x e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )} - 4 \, e^{\left (\frac{1}{2} \, b x + \frac{1}{2} \, a\right )}}{8 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(e^(1/2*b*x + 1/2*a)/x^3,x, algorithm="giac")

[Out]

1/8*(b^2*x^2*Ei(1/2*b*x)*e^(1/2*a) - 2*b*x*e^(1/2*b*x + 1/2*a) - 4*e^(1/2*b*x +
1/2*a))/x^2