Integrand size = 13, antiderivative size = 69 \[ \int \frac {\Gamma (2,a+b x)}{x^{3/2}} \, dx=\frac {2 a e^{-a} \sqrt {b x} \Gamma \left (\frac {1}{2},b x\right )}{\sqrt {x}}+\frac {2 b e^{-a} \sqrt {x} \Gamma \left (\frac {3}{2},b x\right )}{\sqrt {b x}}-\frac {2 \Gamma (2,a+b x)}{\sqrt {x}} \] Output:
2*a*(b*x)^(1/2)*Pi^(1/2)*erfc((b*x)^(1/2))/exp(a)/x^(1/2)+2*b*x^(1/2)*((b* x)^(1/2)*exp(-b*x)+1/2*Pi^(1/2)*erfc((b*x)^(1/2)))/exp(a)/(b*x)^(1/2)-2*ex p(-b*x-a)*(b*x+a+1)/x^(1/2)
Time = 0.05 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.94 \[ \int \frac {\Gamma (2,a+b x)}{x^{3/2}} \, dx=-\left ((1+2 a) \sqrt {b} e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} \sqrt {x}\right )\right )+\frac {2 \left (b e^{-a-b x} x-\Gamma (2,a+b x)\right )}{\sqrt {x}} \] Input:
Integrate[Gamma[2, a + b*x]/x^(3/2),x]
Output:
-(((1 + 2*a)*Sqrt[b]*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/E^a) + (2*(b*E^(-a - b *x)*x - Gamma[2, a + b*x]))/Sqrt[x]
Time = 0.38 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.39, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {7119, 2629, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\Gamma (2,a+b x)}{x^{3/2}} \, dx\) |
\(\Big \downarrow \) 7119 |
\(\displaystyle -2 b \int \frac {e^{-a-b x} (a+b x)}{\sqrt {x}}dx-\frac {2 \Gamma (2,a+b x)}{\sqrt {x}}\) |
\(\Big \downarrow \) 2629 |
\(\displaystyle -2 b \int \left (\frac {e^{-a-b x} a}{\sqrt {x}}+b e^{-a-b x} \sqrt {x}\right )dx-\frac {2 \Gamma (2,a+b x)}{\sqrt {x}}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -2 b \left (\frac {\sqrt {\pi } a e^{-a} \text {erf}\left (\sqrt {b} \sqrt {x}\right )}{\sqrt {b}}+\frac {\sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} \sqrt {x}\right )}{2 \sqrt {b}}-\sqrt {x} e^{-a-b x}\right )-\frac {2 \Gamma (2,a+b x)}{\sqrt {x}}\) |
Input:
Int[Gamma[2, a + b*x]/x^(3/2),x]
Output:
-2*b*(-(E^(-a - b*x)*Sqrt[x]) + (Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/(2*Sqrt[b] *E^a) + (a*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/(Sqrt[b]*E^a)) - (2*Gamma[2, a + b*x])/Sqrt[x]
Int[(F_)^(v_)*(Px_)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandInte grand[F^v, Px*(d + e*x)^m, x], x] /; FreeQ[{F, d, e, m}, x] && PolynomialQ[ Px, x] && LinearQ[v, x] && !TrueQ[$UseGamma]
Int[Gamma[n_, (a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Block[{$UseGamma = True}, Simp[(c + d*x)^(m + 1)*(Gamma[n, a + b*x]/(d*(m + 1))), x] + Simp[b/(d*(m + 1)) Int[(c + d*x)^(m + 1)*((a + b*x)^(n - 1)/E ^(a + b*x)), x], x]] /; FreeQ[{a, b, c, d, m, n}, x] && (IGtQ[m, 0] || IGtQ [n, 0] || IntegersQ[m, n]) && NeQ[m, -1]
Time = 0.35 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.28
method | result | size |
derivativedivides | \(\sqrt {b}\, {\mathrm e}^{-a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )+2 \,{\mathrm e}^{-a} \left (-\frac {{\mathrm e}^{-b x}}{\sqrt {x}}-\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )+2 \,{\mathrm e}^{-a} a \left (-\frac {{\mathrm e}^{-b x}}{\sqrt {x}}-\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )\) | \(88\) |
default | \(\sqrt {b}\, {\mathrm e}^{-a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )+2 \,{\mathrm e}^{-a} \left (-\frac {{\mathrm e}^{-b x}}{\sqrt {x}}-\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )+2 \,{\mathrm e}^{-a} a \left (-\frac {{\mathrm e}^{-b x}}{\sqrt {x}}-\sqrt {b}\, \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )\) | \(88\) |
meijerg | \(\sqrt {b}\, {\mathrm e}^{-a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )+{\mathrm e}^{-a} a \sqrt {b}\, \left (-\frac {2 \,{\mathrm e}^{-b x}}{\sqrt {x}\, \sqrt {b}}-2 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )+\sqrt {b}\, {\mathrm e}^{-a} \left (-\frac {2 \,{\mathrm e}^{-b x}}{\sqrt {x}\, \sqrt {b}}-2 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )\right )\) | \(92\) |
Input:
int(exp(-b*x-a)*(b*x+a+1)/x^(3/2),x,method=_RETURNVERBOSE)
Output:
1/exp(a)*b^(1/2)*Pi^(1/2)*erf(b^(1/2)*x^(1/2))+2/exp(a)*(-1/x^(1/2)*exp(-b *x)-b^(1/2)*Pi^(1/2)*erf(b^(1/2)*x^(1/2)))+2/exp(a)*a*(-1/x^(1/2)*exp(-b*x )-b^(1/2)*Pi^(1/2)*erf(b^(1/2)*x^(1/2)))
Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.84 \[ \int \frac {\Gamma (2,a+b x)}{x^{3/2}} \, dx=-\frac {\sqrt {\pi } {\left (2 \, a + 1\right )} \sqrt {b} x \operatorname {erf}\left (\sqrt {b} \sqrt {x}\right ) e^{\left (-a\right )} - 2 \, {\left (b x e^{\left (-b x - a\right )} - \Gamma \left (2, b x + a\right )\right )} \sqrt {x}}{x} \] Input:
integrate(gamma(2,b*x+a)/x^(3/2),x, algorithm="fricas")
Output:
-(sqrt(pi)*(2*a + 1)*sqrt(b)*x*erf(sqrt(b)*sqrt(x))*e^(-a) - 2*(b*x*e^(-b* x - a) - gamma(2, b*x + a))*sqrt(x))/x
Time = 3.16 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.68 \[ \int \frac {\Gamma (2,a+b x)}{x^{3/2}} \, dx=\left (- \frac {a \left (b x\right )^{\frac {3}{2}} \left (- 2 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {b x} \right )} + \frac {2 e^{- b x}}{\sqrt {b x}}\right )}{b x^{\frac {3}{2}}} - \frac {\sqrt {\pi } \sqrt {b x} \operatorname {erfc}{\left (\sqrt {b x} \right )}}{\sqrt {x}} - \frac {\left (b x\right )^{\frac {3}{2}} \left (- 2 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {b x} \right )} + \frac {2 e^{- b x}}{\sqrt {b x}}\right )}{b x^{\frac {3}{2}}}\right ) e^{- a} \] Input:
integrate(uppergamma(2,b*x+a)/x**(3/2),x)
Output:
(-a*(b*x)**(3/2)*(-2*sqrt(pi)*erfc(sqrt(b*x)) + 2*exp(-b*x)/sqrt(b*x))/(b* x**(3/2)) - sqrt(pi)*sqrt(b*x)*erfc(sqrt(b*x))/sqrt(x) - (b*x)**(3/2)*(-2* sqrt(pi)*erfc(sqrt(b*x)) + 2*exp(-b*x)/sqrt(b*x))/(b*x**(3/2)))*exp(-a)
\[ \int \frac {\Gamma (2,a+b x)}{x^{3/2}} \, dx=\int { \frac {\Gamma \left (2, b x + a\right )}{x^{\frac {3}{2}}} \,d x } \] Input:
integrate(gamma(2,b*x+a)/x^(3/2),x, algorithm="maxima")
Output:
integrate(gamma(2, b*x + a)/x^(3/2), x)
\[ \int \frac {\Gamma (2,a+b x)}{x^{3/2}} \, dx=\int { \frac {\Gamma \left (2, b x + a\right )}{x^{\frac {3}{2}}} \,d x } \] Input:
integrate(gamma(2,b*x+a)/x^(3/2),x, algorithm="giac")
Output:
integrate(gamma(2, b*x + a)/x^(3/2), x)
Time = 0.27 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.42 \[ \int \frac {\Gamma (2,a+b x)}{x^{3/2}} \, dx=\frac {2\,\sqrt {\pi }\,{\mathrm {e}}^{-a}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )\,\sqrt {b\,x}}{\sqrt {x}}-\frac {2\,a\,{\mathrm {e}}^{-a-b\,x}}{\sqrt {x}}-\frac {{\mathrm {e}}^{-a}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )\,\sqrt {\pi \,b\,x}}{\sqrt {x}}-\frac {2\,{\mathrm {e}}^{-a-b\,x}}{\sqrt {x}}+\frac {2\,a\,\sqrt {\pi }\,{\mathrm {e}}^{-a}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )\,\sqrt {b\,x}}{\sqrt {x}} \] Input:
int((exp(- a - b*x)*(a + b*x + 1))/x^(3/2),x)
Output:
(2*pi^(1/2)*exp(-a)*erfc((b*x)^(1/2))*(b*x)^(1/2))/x^(1/2) - (2*a*exp(- a - b*x))/x^(1/2) - (exp(-a)*erfc((b*x)^(1/2))*(b*x*pi)^(1/2))/x^(1/2) - (2* exp(- a - b*x))/x^(1/2) + (2*a*pi^(1/2)*exp(-a)*erfc((b*x)^(1/2))*(b*x)^(1 /2))/x^(1/2)
\[ \int \frac {\Gamma (2,a+b x)}{x^{3/2}} \, dx=\frac {2 \sqrt {x}\, e^{b x} \left (\int \frac {\sqrt {x}}{e^{b x} x^{2}}d x \right ) a +\sqrt {x}\, e^{b x} \left (\int \frac {\sqrt {x}}{e^{b x} x^{2}}d x \right )-2}{2 \sqrt {x}\, e^{b x +a}} \] Input:
int(exp(-b*x-a)*(b*x+a+1)/x^(3/2),x)
Output:
(2*sqrt(x)*e**(b*x)*int(sqrt(x)/(e**(b*x)*x**2),x)*a + sqrt(x)*e**(b*x)*in t(sqrt(x)/(e**(b*x)*x**2),x) - 2)/(2*sqrt(x)*e**(a + b*x))