Integrand size = 13, antiderivative size = 68 \[ \int \frac {\Gamma (2,a+b x)}{\sqrt {x}} \, dx=-\frac {2 a e^{-a} \sqrt {x} \Gamma \left (\frac {3}{2},b x\right )}{\sqrt {b x}}+2 \sqrt {x} \Gamma (2,a+b x)-\frac {2 e^{-a} \sqrt {x} \Gamma \left (\frac {5}{2},b x\right )}{\sqrt {b x}} \] Output:
-2*a*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*x^(1/2)*exp(-b*x-a)*(b*x+a+1)-2*x^(1/2)*((b*x)^(3/2)*exp(-b *x)+3/2*(b*x)^(1/2)*exp(-b*x)+3/4*Pi^(1/2)*erfc((b*x)^(1/2)))/exp(a)/(b*x) ^(1/2)
Time = 0.05 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.12 \[ \int \frac {\Gamma (2,a+b x)}{\sqrt {x}} \, dx=\frac {1}{2} e^{-a} \left (\frac {(3+2 a) \sqrt {\pi } \text {erf}\left (\sqrt {b} \sqrt {x}\right )}{\sqrt {b}}-2 e^{-b x} \sqrt {x} \left (3+2 a+2 b x-2 e^{a+b x} \Gamma (2,a+b x)\right )\right ) \] Input:
Integrate[Gamma[2, a + b*x]/Sqrt[x],x]
Output:
(((3 + 2*a)*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/Sqrt[b] - (2*Sqrt[x]*(3 + 2*a + 2*b*x - 2*E^(a + b*x)*Gamma[2, a + b*x]))/E^(b*x))/(2*E^a)
Leaf count is larger than twice the leaf count of optimal. \(142\) vs. \(2(68)=136\).
Time = 0.45 (sec) , antiderivative size = 142, normalized size of antiderivative = 2.09, 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.
\(\displaystyle \int \frac {\Gamma (2,a+b x)}{\sqrt {x}} \, dx\) |
\(\Big \downarrow \) 7119 |
\(\displaystyle 2 b \int e^{-a-b x} \sqrt {x} (a+b x)dx+2 \sqrt {x} \Gamma (2,a+b x)\) |
\(\Big \downarrow \) 2629 |
\(\displaystyle 2 b \int \left (b e^{-a-b x} x^{3/2}+a e^{-a-b x} \sqrt {x}\right )dx+2 \sqrt {x} \Gamma (2,a+b x)\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 b \left (\frac {\sqrt {\pi } a e^{-a} \text {erf}\left (\sqrt {b} \sqrt {x}\right )}{2 b^{3/2}}+\frac {3 \sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} \sqrt {x}\right )}{4 b^{3/2}}+x^{3/2} \left (-e^{-a-b x}\right )-\frac {a \sqrt {x} e^{-a-b x}}{b}-\frac {3 \sqrt {x} e^{-a-b x}}{2 b}\right )+2 \sqrt {x} \Gamma (2,a+b x)\) |
Input:
Int[Gamma[2, a + b*x]/Sqrt[x],x]
Output:
2*b*((-3*E^(-a - b*x)*Sqrt[x])/(2*b) - (a*E^(-a - b*x)*Sqrt[x])/b - E^(-a - b*x)*x^(3/2) + (3*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/(4*b^(3/2)*E^a) + (a*Sq rt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/(2*b^(3/2)*E^a)) + 2*Sqrt[x]*Gamma[2, a + b*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 = 0.31 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.12
method | result | size |
meijerg | \(\frac {{\mathrm e}^{-a} \left (-\sqrt {x}\, \sqrt {b}\, {\mathrm e}^{-b x}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{2}\right )}{\sqrt {b}}+\frac {{\mathrm e}^{-a} a \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{\sqrt {b}}+\frac {{\mathrm e}^{-a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{\sqrt {b}}\) | \(76\) |
derivativedivides | \(\frac {{\mathrm e}^{-a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{\sqrt {b}}+\frac {{\mathrm e}^{-a} a \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{\sqrt {b}}+2 \,{\mathrm e}^{-a} b \left (-\frac {\sqrt {x}\, {\mathrm e}^{-b x}}{2 b}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{4 b^{\frac {3}{2}}}\right )\) | \(78\) |
default | \(\frac {{\mathrm e}^{-a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{\sqrt {b}}+\frac {{\mathrm e}^{-a} a \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{\sqrt {b}}+2 \,{\mathrm e}^{-a} b \left (-\frac {\sqrt {x}\, {\mathrm e}^{-b x}}{2 b}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{4 b^{\frac {3}{2}}}\right )\) | \(78\) |
Input:
int(exp(-b*x-a)*(b*x+a+1)/x^(1/2),x,method=_RETURNVERBOSE)
Output:
1/b^(1/2)*exp(-a)*(-x^(1/2)*b^(1/2)*exp(-b*x)+1/2*Pi^(1/2)*erf(b^(1/2)*x^( 1/2)))+exp(-a)*a/b^(1/2)*Pi^(1/2)*erf(b^(1/2)*x^(1/2))+1/b^(1/2)*exp(-a)*P i^(1/2)*erf(b^(1/2)*x^(1/2))
Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.03 \[ \int \frac {\Gamma (2,a+b x)}{\sqrt {x}} \, dx=\frac {\sqrt {\pi } {\left (2 \, a + 3\right )} \sqrt {b} \operatorname {erf}\left (\sqrt {b} \sqrt {x}\right ) e^{\left (-a\right )} - 2 \, {\left ({\left (2 \, b^{2} x + {\left (2 \, a + 3\right )} b\right )} e^{\left (-b x - a\right )} - 2 \, b \Gamma \left (2, b x + a\right )\right )} \sqrt {x}}{2 \, b} \] Input:
integrate(gamma(2,b*x+a)/x^(1/2),x, algorithm="fricas")
Output:
1/2*(sqrt(pi)*(2*a + 3)*sqrt(b)*erf(sqrt(b)*sqrt(x))*e^(-a) - 2*((2*b^2*x + (2*a + 3)*b)*e^(-b*x - a) - 2*b*gamma(2, b*x + a))*sqrt(x))/b
Time = 3.81 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.46 \[ \int \frac {\Gamma (2,a+b x)}{\sqrt {x}} \, dx=\left (- \frac {\sqrt {\pi } a \sqrt {b x} \operatorname {erfc}{\left (\sqrt {b x} \right )}}{b \sqrt {x}} - \frac {\sqrt {x} \left (\sqrt {b x} e^{- b x} + \frac {\sqrt {\pi } \operatorname {erfc}{\left (\sqrt {b x} \right )}}{2}\right )}{\sqrt {b x}} - \frac {\sqrt {\pi } \sqrt {b x} \operatorname {erfc}{\left (\sqrt {b x} \right )}}{b \sqrt {x}}\right ) e^{- a} \] Input:
integrate(uppergamma(2,b*x+a)/x**(1/2),x)
Output:
(-sqrt(pi)*a*sqrt(b*x)*erfc(sqrt(b*x))/(b*sqrt(x)) - sqrt(x)*(sqrt(b*x)*ex p(-b*x) + sqrt(pi)*erfc(sqrt(b*x))/2)/sqrt(b*x) - sqrt(pi)*sqrt(b*x)*erfc( sqrt(b*x))/(b*sqrt(x)))*exp(-a)
\[ \int \frac {\Gamma (2,a+b x)}{\sqrt {x}} \, dx=\int { \frac {\Gamma \left (2, b x + a\right )}{\sqrt {x}} \,d x } \] Input:
integrate(gamma(2,b*x+a)/x^(1/2),x, algorithm="maxima")
Output:
integrate(gamma(2, b*x + a)/sqrt(x), x)
\[ \int \frac {\Gamma (2,a+b x)}{\sqrt {x}} \, dx=\int { \frac {\Gamma \left (2, b x + a\right )}{\sqrt {x}} \,d x } \] Input:
integrate(gamma(2,b*x+a)/x^(1/2),x, algorithm="giac")
Output:
integrate(gamma(2, b*x + a)/sqrt(x), x)
Time = 0.25 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.28 \[ \int \frac {\Gamma (2,a+b x)}{\sqrt {x}} \, dx=-\sqrt {x}\,{\mathrm {e}}^{-a-b\,x}-\frac {{\mathrm {e}}^{-a}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )\,\sqrt {\pi \,b\,x}}{b\,\sqrt {x}}-\frac {\sqrt {x}\,\sqrt {\pi }\,{\mathrm {e}}^{-a}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )}{2\,\sqrt {b\,x}}-\frac {a\,{\mathrm {e}}^{-a}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )\,\sqrt {\pi \,b\,x}}{b\,\sqrt {x}} \] Input:
int((exp(- a - b*x)*(a + b*x + 1))/x^(1/2),x)
Output:
- x^(1/2)*exp(- a - b*x) - (exp(-a)*erfc((b*x)^(1/2))*(b*x*pi)^(1/2))/(b*x ^(1/2)) - (x^(1/2)*pi^(1/2)*exp(-a)*erfc((b*x)^(1/2)))/(2*(b*x)^(1/2)) - ( a*exp(-a)*erfc((b*x)^(1/2))*(b*x*pi)^(1/2))/(b*x^(1/2))
\[ \int \frac {\Gamma (2,a+b x)}{\sqrt {x}} \, dx=\frac {2 e^{b x} \left (\int \frac {\sqrt {x}}{e^{b x} x}d x \right ) a +3 e^{b x} \left (\int \frac {\sqrt {x}}{e^{b x} x}d x \right )-2 \sqrt {x}}{2 e^{b x +a}} \] Input:
int(exp(-b*x-a)*(b*x+a+1)/x^(1/2),x)
Output:
(2*e**(b*x)*int(sqrt(x)/(e**(b*x)*x),x)*a + 3*e**(b*x)*int(sqrt(x)/(e**(b* x)*x),x) - 2*sqrt(x))/(2*e**(a + b*x))