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

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

[Out]

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

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Rubi [A]  time = 0.0583287, 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, Rules used = {2182, 2178} \[ e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{Ei}\left (\frac{b x}{2}\right ) \]

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)

Rule 2182

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Dist[(b*F^(g*(e +
f*x)))^n/F^(g*n*(e + f*x)), Int[(c + d*x)^m*F^(g*n*(e + f*x)), x], x] /; FreeQ[{F, b, c, d, e, f, g, m, n}, x]

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rubi steps

\begin{align*} \int \frac{\sqrt{e^{a+b x}}}{x} \, dx &=\left (e^{\frac{1}{2} (-a-b x)} \sqrt{e^{a+b x}}\right ) \int \frac{e^{\frac{1}{2} (a+b x)}}{x} \, dx\\ &=e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{Ei}\left (\frac{b x}{2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0156963, size = 27, normalized size = 1. \[ e^{-\frac{b x}{2}} \sqrt{e^{a+b x}} \text{Ei}\left (\frac{b x}{2}\right ) \]

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 = 0.039, size = 57, normalized size = 2.1 \begin{align*} \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 ) \end{align*}

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 = 1.17256, size = 14, normalized size = 0.52 \begin{align*}{\rm Ei}\left (\frac{1}{2} \, b x\right ) e^{\left (\frac{1}{2} \, a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)^(1/2)/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.49306, size = 31, normalized size = 1.15 \begin{align*}{\rm Ei}\left (\frac{1}{2} \, b x\right ) e^{\left (\frac{1}{2} \, a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)^(1/2)/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. \begin{align*} \int \frac{\sqrt{e^{a} e^{b x}}}{x}\, dx \end{align*}

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 [A]  time = 1.36466, size = 14, normalized size = 0.52 \begin{align*}{\rm Ei}\left (\frac{1}{2} \, b x\right ) e^{\left (\frac{1}{2} \, a\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)^(1/2)/x,x, algorithm="giac")

[Out]

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