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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.12, antiderivative size = 71, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2177, 2182, 2178} \[ \frac {1}{8} b^2 e^{-\frac {b x}{2}} \sqrt {e^{a+b x}} \text {Ei}\left (\frac {b x}{2}\right )-\frac {\sqrt {e^{a+b x}}}{2 x^2}-\frac {b \sqrt {e^{a+b x}}}{4 x} \]

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))

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

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^3} \, dx &=-\frac {\sqrt {e^{a+b x}}}{2 x^2}+\frac {1}{4} b \int \frac {\sqrt {e^{a+b x}}}{x^2} \, dx\\ &=-\frac {\sqrt {e^{a+b x}}}{2 x^2}-\frac {b \sqrt {e^{a+b x}}}{4 x}+\frac {1}{8} b^2 \int \frac {\sqrt {e^{a+b x}}}{x} \, dx\\ &=-\frac {\sqrt {e^{a+b x}}}{2 x^2}-\frac {b \sqrt {e^{a+b x}}}{4 x}+\frac {1}{8} \left (b^2 e^{\frac {1}{2} (-a-b x)} \sqrt {e^{a+b x}}\right ) \int \frac {e^{\frac {1}{2} (a+b x)}}{x} \, dx\\ &=-\frac {\sqrt {e^{a+b x}}}{2 x^2}-\frac {b \sqrt {e^{a+b x}}}{4 x}+\frac {1}{8} b^2 e^{-\frac {b x}{2}} \sqrt {e^{a+b x}} \text {Ei}\left (\frac {b x}{2}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 56, normalized size = 0.79 \[ \frac {e^{-\frac {b x}{2}} \sqrt {e^{a+b x}} \left (b^2 x^2 \text {Ei}\left (\frac {b x}{2}\right )-2 e^{\frac {b x}{2}} (b x+2)\right )}{8 x^2} \]

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)

________________________________________________________________________________________

fricas [A]  time = 0.46, size = 38, normalized size = 0.54 \[ \frac {b^{2} x^{2} {\rm Ei}\left (\frac {1}{2} \, b x\right ) e^{\left (\frac {1}{2} \, a\right )} - 2 \, {\left (b x + 2\right )} e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{8 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

giac [A]  time = 0.36, size = 46, normalized size = 0.65 \[ \frac {b^{2} x^{2} {\rm Ei}\left (\frac {1}{2} \, b x\right ) e^{\left (\frac {1}{2} \, a\right )} - 2 \, b x e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )} - 4 \, e^{\left (\frac {1}{2} \, b x + \frac {1}{2} \, a\right )}}{8 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

________________________________________________________________________________________

maple [B]  time = 0.05, size = 155, normalized size = 2.18 \[ \frac {\left (-\frac {\Ei \left (1, -\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}\right )}{2}+\frac {\ln \relax (x )}{2}+\frac {\ln \left (-b \,{\mathrm e}^{\frac {a}{2}}\right )}{2}-\frac {\ln \left (-\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}\right )}{2}-\frac {2 \,{\mathrm e}^{-\frac {a}{2}}}{b x}+\frac {\left (\frac {9 b^{2} x^{2} {\mathrm e}^{a}}{4}+6 b x \,{\mathrm e}^{\frac {a}{2}}+6\right ) {\mathrm e}^{-a}}{3 b^{2} x^{2}}-\frac {2 \,{\mathrm e}^{-a}}{b^{2} x^{2}}-\frac {2 \left (\frac {3 b x \,{\mathrm e}^{\frac {a}{2}}}{2}+3\right ) {\mathrm e}^{\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}-a}}{3 b^{2} x^{2}}-\frac {3}{4}-\frac {\ln \relax (2)}{2}\right ) b^{2} {\mathrm e}^{-\frac {b x \,{\mathrm e}^{\frac {a}{2}}}{2}+a} \sqrt {{\mathrm e}^{b x +a}}}{4} \]

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

________________________________________________________________________________________

maxima [A]  time = 1.24, size = 15, normalized size = 0.21 \[ -\frac {1}{4} \, b^{2} e^{\left (\frac {1}{2} \, a\right )} \Gamma \left (-2, -\frac {1}{2} \, b x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(exp(b*x+a)^(1/2)/x^3,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {{\mathrm {e}}^{a+b\,x}}}{x^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(a + b*x)^(1/2)/x^3,x)

[Out]

int(exp(a + b*x)^(1/2)/x^3, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e^{a} e^{b x}}}{x^{3}}\, dx \]

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)

________________________________________________________________________________________