Integrand size = 13, antiderivative size = 80 \[ \int \sqrt {x} \Gamma (2,a+b x) \, dx=\frac {2}{3} x^{3/2} \Gamma (2,a+b x)-\frac {2 a e^{-a} \sqrt {x} \Gamma \left (\frac {5}{2},b x\right )}{3 b \sqrt {b x}}-\frac {2 e^{-a} \sqrt {x} \Gamma \left (\frac {7}{2},b x\right )}{3 b \sqrt {b x}} \] Output:
2/3*x^(3/2)*exp(-b*x-a)*(b*x+a+1)-2/3*a*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)))/b/exp(a)/(b*x)^(1/2 )-2/3*x^(1/2)*((b*x)^(5/2)*exp(-b*x)+5/2*(b*x)^(3/2)*exp(-b*x)+15/4*(b*x)^ (1/2)*exp(-b*x)+15/8*Pi^(1/2)*erfc((b*x)^(1/2)))/b/exp(a)/(b*x)^(1/2)
Time = 0.06 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.21 \[ \int \sqrt {x} \Gamma (2,a+b x) \, dx=\frac {e^{-a} \left (3 (5+2 a) \sqrt {\pi } \text {erf}\left (\sqrt {b} \sqrt {x}\right )-2 \sqrt {b} e^{-b x} \sqrt {x} \left (15+6 a+10 b x+4 a b x+4 b^2 x^2-4 b e^{a+b x} x \Gamma (2,a+b x)\right )\right )}{12 b^{3/2}} \] Input:
Integrate[Sqrt[x]*Gamma[2, a + b*x],x]
Output:
(3*(5 + 2*a)*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]] - (2*Sqrt[b]*Sqrt[x]*(15 + 6*a + 10*b*x + 4*a*b*x + 4*b^2*x^2 - 4*b*E^(a + b*x)*x*Gamma[2, a + b*x]))/E^( b*x))/(12*b^(3/2)*E^a)
Leaf count is larger than twice the leaf count of optimal. \(191\) vs. \(2(80)=160\).
Time = 0.51 (sec) , antiderivative size = 191, normalized size of antiderivative = 2.39, 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.
| \(\displaystyle \int \sqrt {x} \Gamma (2,a+b x) \, dx\) |
| \(\Big \downarrow \) 7119 |
| \(\displaystyle \frac {2}{3} b \int e^{-a-b x} x^{3/2} (a+b x)dx+\frac {2}{3} x^{3/2} \Gamma (2,a+b x)\) |
| \(\Big \downarrow \) 2629 |
| \(\displaystyle \frac {2}{3} b \int \left (b e^{-a-b x} x^{5/2}+a e^{-a-b x} x^{3/2}\right )dx+\frac {2}{3} x^{3/2} \Gamma (2,a+b x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{3} b \left (\frac {3 \sqrt {\pi } a e^{-a} \text {erf}\left (\sqrt {b} \sqrt {x}\right )}{4 b^{5/2}}+\frac {15 \sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} \sqrt {x}\right )}{8 b^{5/2}}-\frac {3 a \sqrt {x} e^{-a-b x}}{2 b^2}-\frac {15 \sqrt {x} e^{-a-b x}}{4 b^2}+x^{5/2} \left (-e^{-a-b x}\right )-\frac {a x^{3/2} e^{-a-b x}}{b}-\frac {5 x^{3/2} e^{-a-b x}}{2 b}\right )+\frac {2}{3} x^{3/2} \Gamma (2,a+b x)\) |
Input:
Int[Sqrt[x]*Gamma[2, a + b*x],x]
Output:
(2*b*((-15*E^(-a - b*x)*Sqrt[x])/(4*b^2) - (3*a*E^(-a - b*x)*Sqrt[x])/(2*b ^2) - (5*E^(-a - b*x)*x^(3/2))/(2*b) - (a*E^(-a - b*x)*x^(3/2))/b - E^(-a - b*x)*x^(5/2) + (15*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/(8*b^(5/2)*E^a) + (3*a *Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/(4*b^(5/2)*E^a)))/3 + (2*x^(3/2)*Gamma[2, a + b*x])/3
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 = 0.29 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.42
| method | result | size |
| meijerg | \(\frac {{\mathrm e}^{-a} \left (-\frac {\sqrt {x}\, \sqrt {b}\, \left (10 b x +15\right ) {\mathrm e}^{-b x}}{10}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{4}\right )}{b^{\frac {3}{2}}}+\frac {{\mathrm e}^{-a} a \left (-\sqrt {x}\, \sqrt {b}\, {\mathrm e}^{-b x}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{2}\right )}{b^{\frac {3}{2}}}+\frac {{\mathrm e}^{-a} \left (-\sqrt {x}\, \sqrt {b}\, {\mathrm e}^{-b x}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{2}\right )}{b^{\frac {3}{2}}}\) | \(114\) |
| derivativedivides | \(2 \,{\mathrm e}^{-a} \left (-\frac {\sqrt {x}\, {\mathrm e}^{-b x}}{2 b}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{4 b^{\frac {3}{2}}}\right )+2 \,{\mathrm e}^{-a} a \left (-\frac {\sqrt {x}\, {\mathrm e}^{-b x}}{2 b}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{4 b^{\frac {3}{2}}}\right )+2 \,{\mathrm e}^{-a} b \left (-\frac {x^{\frac {3}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}}{b}\right )\) | \(131\) |
| default | \(2 \,{\mathrm e}^{-a} \left (-\frac {\sqrt {x}\, {\mathrm e}^{-b x}}{2 b}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{4 b^{\frac {3}{2}}}\right )+2 \,{\mathrm e}^{-a} a \left (-\frac {\sqrt {x}\, {\mathrm e}^{-b x}}{2 b}+\frac {\sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{4 b^{\frac {3}{2}}}\right )+2 \,{\mathrm e}^{-a} b \left (-\frac {x^{\frac {3}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}}{b}\right )\) | \(131\) |
Input:
int(x^(1/2)*exp(-b*x-a)*(b*x+a+1),x,method=_RETURNVERBOSE)
Output:
1/b^(3/2)*exp(-a)*(-1/10*x^(1/2)*b^(1/2)*(10*b*x+15)*exp(-b*x)+3/4*Pi^(1/2 )*erf(b^(1/2)*x^(1/2)))+1/b^(3/2)*exp(-a)*a*(-x^(1/2)*b^(1/2)*exp(-b*x)+1/ 2*Pi^(1/2)*erf(b^(1/2)*x^(1/2)))+1/b^(3/2)*exp(-a)*(-x^(1/2)*b^(1/2)*exp(- b*x)+1/2*Pi^(1/2)*erf(b^(1/2)*x^(1/2)))
Time = 0.11 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.11 \[ \int \sqrt {x} \Gamma (2,a+b x) \, dx=\frac {3 \, \sqrt {\pi } {\left (2 \, a + 5\right )} \sqrt {b} \operatorname {erf}\left (\sqrt {b} \sqrt {x}\right ) e^{\left (-a\right )} + 2 \, {\left (4 \, b^{2} x \Gamma \left (2, b x + a\right ) - {\left (4 \, b^{3} x^{2} + 2 \, {\left (2 \, a + 5\right )} b^{2} x + 3 \, {\left (2 \, a + 5\right )} b\right )} e^{\left (-b x - a\right )}\right )} \sqrt {x}}{12 \, b^{2}} \] Input:
integrate(x^(1/2)*gamma(2,b*x+a),x, algorithm="fricas")
Output:
1/12*(3*sqrt(pi)*(2*a + 5)*sqrt(b)*erf(sqrt(b)*sqrt(x))*e^(-a) + 2*(4*b^2* x*gamma(2, b*x + a) - (4*b^3*x^2 + 2*(2*a + 5)*b^2*x + 3*(2*a + 5)*b)*e^(- b*x - a))*sqrt(x))/b^2
Time = 8.24 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.70 \[ \int \sqrt {x} \Gamma (2,a+b x) \, dx=\left (- \frac {a \sqrt {x} \left (\sqrt {b x} e^{- b x} + \frac {\sqrt {\pi } \operatorname {erfc}{\left (\sqrt {b x} \right )}}{2}\right )}{b \sqrt {b x}} - \frac {x^{\frac {3}{2}} \left (- \sqrt {b x} \left (- b x - \frac {3}{2}\right ) e^{- b x} + \frac {3 \sqrt {\pi } \operatorname {erfc}{\left (\sqrt {b x} \right )}}{4}\right )}{\left (b x\right )^{\frac {3}{2}}} - \frac {\sqrt {x} \left (\sqrt {b x} e^{- b x} + \frac {\sqrt {\pi } \operatorname {erfc}{\left (\sqrt {b x} \right )}}{2}\right )}{b \sqrt {b x}}\right ) e^{- a} \] Input:
integrate(x**(1/2)*uppergamma(2,b*x+a),x)
Output:
(-a*sqrt(x)*(sqrt(b*x)*exp(-b*x) + sqrt(pi)*erfc(sqrt(b*x))/2)/(b*sqrt(b*x )) - x**(3/2)*(-sqrt(b*x)*(-b*x - 3/2)*exp(-b*x) + 3*sqrt(pi)*erfc(sqrt(b* x))/4)/(b*x)**(3/2) - sqrt(x)*(sqrt(b*x)*exp(-b*x) + sqrt(pi)*erfc(sqrt(b* x))/2)/(b*sqrt(b*x)))*exp(-a)
\[ \int \sqrt {x} \Gamma (2,a+b x) \, dx=\int { \sqrt {x} \Gamma \left (2, b x + a\right ) \,d x } \] Input:
integrate(x^(1/2)*gamma(2,b*x+a),x, algorithm="maxima")
Output:
integrate(sqrt(x)*gamma(2, b*x + a), x)
\[ \int \sqrt {x} \Gamma (2,a+b x) \, dx=\int { \sqrt {x} \Gamma \left (2, b x + a\right ) \,d x } \] Input:
integrate(x^(1/2)*gamma(2,b*x+a),x, algorithm="giac")
Output:
integrate(sqrt(x)*gamma(2, b*x + a), x)
Time = 0.23 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.68 \[ \int \sqrt {x} \Gamma (2,a+b x) \, dx=-\frac {\sqrt {x}\,{\mathrm {e}}^{-a-b\,x}}{b}-\frac {a\,\sqrt {x}\,{\mathrm {e}}^{-a-b\,x}}{b}-\frac {x^{3/2}\,{\mathrm {e}}^{-a}\,\left (\frac {3\,\sqrt {\pi }\,\mathrm {erfc}\left (\sqrt {b\,x}\right )}{4}+{\mathrm {e}}^{-b\,x}\,\left (\frac {3\,\sqrt {b\,x}}{2}+{\left (b\,x\right )}^{3/2}\right )\right )}{{\left (b\,x\right )}^{3/2}}-\frac {\sqrt {x}\,\sqrt {\pi }\,{\mathrm {e}}^{-a}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )}{2\,b\,\sqrt {b\,x}}-\frac {a\,\sqrt {x}\,\sqrt {\pi }\,{\mathrm {e}}^{-a}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )}{2\,b\,\sqrt {b\,x}} \] Input:
int(x^(1/2)*exp(- a - b*x)*(a + b*x + 1),x)
Output:
- (x^(1/2)*exp(- a - b*x))/b - (a*x^(1/2)*exp(- a - b*x))/b - (x^(3/2)*exp (-a)*((3*pi^(1/2)*erfc((b*x)^(1/2)))/4 + exp(-b*x)*((3*(b*x)^(1/2))/2 + (b *x)^(3/2))))/(b*x)^(3/2) - (x^(1/2)*pi^(1/2)*exp(-a)*erfc((b*x)^(1/2)))/(2 *b*(b*x)^(1/2)) - (a*x^(1/2)*pi^(1/2)*exp(-a)*erfc((b*x)^(1/2)))/(2*b*(b*x )^(1/2))
\[ \int \sqrt {x} \Gamma (2,a+b x) \, dx=\frac {2 e^{b x} \left (\int \frac {\sqrt {x}}{e^{b x} x}d x \right ) a +5 e^{b x} \left (\int \frac {\sqrt {x}}{e^{b x} x}d x \right )-4 \sqrt {x}\, a -4 \sqrt {x}\, b x -10 \sqrt {x}}{4 e^{b x +a} b} \] Input:
int(x^(1/2)*exp(-b*x-a)*(b*x+a+1),x)
Output:
(2*e**(b*x)*int(sqrt(x)/(e**(b*x)*x),x)*a + 5*e**(b*x)*int(sqrt(x)/(e**(b* x)*x),x) - 4*sqrt(x)*a - 4*sqrt(x)*b*x - 10*sqrt(x))/(4*e**(a + b*x)*b)