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]

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

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Rubi [A]  time = 0.06, antiderivative size = 27, normalized size of antiderivative = 1.00, 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 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

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]

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*}

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Mathematica [A]  time = 0.02, size = 27, normalized size = 1.00 \[ e^{-\frac {b x}{2}} \sqrt {e^{a+b x}} \text {Ei}\left (\frac {b x}{2}\right ) \]

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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fricas [A]  time = 0.40, size = 10, normalized size = 0.37 \[ {\rm Ei}\left (\frac {1}{2} \, b x\right ) e^{\left (\frac {1}{2} \, a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.36, size = 10, normalized size = 0.37 \[ {\rm Ei}\left (\frac {1}{2} \, b x\right ) e^{\left (\frac {1}{2} \, a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.06, size = 57, normalized size = 2.11 \[ \left (-\Ei \left (1, -\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}\right )+\ln \relax (x )+\ln \left (-b \,{\mathrm e}^{\frac {a}{2}}\right )-\ln \left (-\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}\right )-\ln \relax (2)\right ) {\mathrm e}^{-\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}} \sqrt {{\mathrm e}^{b x +a}} \]

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

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maxima [A]  time = 1.23, size = 10, normalized size = 0.37 \[ {\rm Ei}\left (\frac {1}{2} \, b x\right ) e^{\left (\frac {1}{2} \, a\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {{\mathrm {e}}^{a+b\,x}}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a + b*x)^(1/2)/x,x)

[Out]

int(exp(a + b*x)^(1/2)/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e^{a} e^{b x}}}{x}\, dx \]

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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