Integrand size = 13, antiderivative size = 80 \[ \int x^{5/2} \Gamma (2,a+b x) \, dx=\frac {2}{7} x^{7/2} \Gamma (2,a+b x)-\frac {2 a e^{-a} \sqrt {x} \Gamma \left (\frac {9}{2},b x\right )}{7 b^3 \sqrt {b x}}-\frac {2 e^{-a} \sqrt {x} \Gamma \left (\frac {11}{2},b x\right )}{7 b^3 \sqrt {b x}} \] Output:
2/7*x^(7/2)*exp(-b*x-a)*(b*x+a+1)-2/7*a*x^(1/2)*((b*x)^(7/2)*exp(-b*x)+7/2 *(b*x)^(5/2)*exp(-b*x)+35/4*(b*x)^(3/2)*exp(-b*x)+105/8*(b*x)^(1/2)*exp(-b *x)+105/16*Pi^(1/2)*erfc((b*x)^(1/2)))/b^3/exp(a)/(b*x)^(1/2)-2/7*x^(1/2)* (1048576/61836869254970658257624840625*GAMMA(51/2,b*x)-1048576/61836869254 970658257624840625*(b*x)^(49/2)*exp(-b*x)-524288/1261976923570829760359690 625*(b*x)^(47/2)*exp(-b*x)-262144/26850572841932548092759375*(b*x)^(45/2)* exp(-b*x)-131072/596679396487389957616875*(b*x)^(43/2)*exp(-b*x)-65536/138 76265034590464130625*(b*x)^(41/2)*exp(-b*x)-32768/338445488648547905625*(b *x)^(39/2)*exp(-b*x)-16384/8678089452526869375*(b*x)^(37/2)*exp(-b*x)-8192 /234542958176401875*(b*x)^(35/2)*exp(-b*x)-4096/6701227376468625*(b*x)^(33 /2)*exp(-b*x)-2048/203067496256625*(b*x)^(31/2)*exp(-b*x)-1024/65505643953 75*(b*x)^(29/2)*exp(-b*x)-512/225881530875*(b*x)^(27/2)*exp(-b*x)-256/8365 982625*(b*x)^(25/2)*exp(-b*x)-128/334639305*(b*x)^(23/2)*exp(-b*x)-64/1454 9535*(b*x)^(21/2)*exp(-b*x)-32/692835*(b*x)^(19/2)*exp(-b*x)-16/36465*(b*x )^(17/2)*exp(-b*x)-8/2145*(b*x)^(15/2)*exp(-b*x)-4/143*(b*x)^(13/2)*exp(-b *x)-2/11*(b*x)^(11/2)*exp(-b*x))/b^3/exp(a)/(b*x)^(1/2)
Time = 0.09 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.68 \[ \int x^{5/2} \Gamma (2,a+b x) \, dx=\frac {e^{-a} \left (105 (9+2 a) \sqrt {\pi } \text {erf}\left (\sqrt {b} \sqrt {x}\right )-2 \sqrt {b} e^{-b x} \sqrt {x} \left (945+630 b x+252 b^2 x^2+72 b^3 x^3+16 b^4 x^4+2 a \left (105+70 b x+28 b^2 x^2+8 b^3 x^3\right )-16 b^3 e^{a+b x} x^3 \Gamma (2,a+b x)\right )\right )}{112 b^{7/2}} \] Input:
Integrate[x^(5/2)*Gamma[2, a + b*x],x]
Output:
(105*(9 + 2*a)*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]] - (2*Sqrt[b]*Sqrt[x]*(945 + 6 30*b*x + 252*b^2*x^2 + 72*b^3*x^3 + 16*b^4*x^4 + 2*a*(105 + 70*b*x + 28*b^ 2*x^2 + 8*b^3*x^3) - 16*b^3*E^(a + b*x)*x^3*Gamma[2, a + b*x]))/E^(b*x))/( 112*b^(7/2)*E^a)
Leaf count is larger than twice the leaf count of optimal. \(281\) vs. \(2(80)=160\).
Time = 0.68 (sec) , antiderivative size = 281, normalized size of antiderivative = 3.51, 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.
| \(\displaystyle \int x^{5/2} \Gamma (2,a+b x) \, dx\) |
| \(\Big \downarrow \) 7119 |
| \(\displaystyle \frac {2}{7} b \int e^{-a-b x} x^{7/2} (a+b x)dx+\frac {2}{7} x^{7/2} \Gamma (2,a+b x)\) |
| \(\Big \downarrow \) 2629 |
| \(\displaystyle \frac {2}{7} b \int \left (b e^{-a-b x} x^{9/2}+a e^{-a-b x} x^{7/2}\right )dx+\frac {2}{7} x^{7/2} \Gamma (2,a+b x)\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {2}{7} b \left (\frac {105 \sqrt {\pi } a e^{-a} \text {erf}\left (\sqrt {b} \sqrt {x}\right )}{16 b^{9/2}}+\frac {945 \sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} \sqrt {x}\right )}{32 b^{9/2}}-\frac {105 a \sqrt {x} e^{-a-b x}}{8 b^4}-\frac {945 \sqrt {x} e^{-a-b x}}{16 b^4}-\frac {35 a x^{3/2} e^{-a-b x}}{4 b^3}-\frac {315 x^{3/2} e^{-a-b x}}{8 b^3}-\frac {7 a x^{5/2} e^{-a-b x}}{2 b^2}-\frac {63 x^{5/2} e^{-a-b x}}{4 b^2}+x^{9/2} \left (-e^{-a-b x}\right )-\frac {a x^{7/2} e^{-a-b x}}{b}-\frac {9 x^{7/2} e^{-a-b x}}{2 b}\right )+\frac {2}{7} x^{7/2} \Gamma (2,a+b x)\) |
Input:
Int[x^(5/2)*Gamma[2, a + b*x],x]
Output:
(2*b*((-945*E^(-a - b*x)*Sqrt[x])/(16*b^4) - (105*a*E^(-a - b*x)*Sqrt[x])/ (8*b^4) - (315*E^(-a - b*x)*x^(3/2))/(8*b^3) - (35*a*E^(-a - b*x)*x^(3/2)) /(4*b^3) - (63*E^(-a - b*x)*x^(5/2))/(4*b^2) - (7*a*E^(-a - b*x)*x^(5/2))/ (2*b^2) - (9*E^(-a - b*x)*x^(7/2))/(2*b) - (a*E^(-a - b*x)*x^(7/2))/b - E^ (-a - b*x)*x^(9/2) + (945*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/(32*b^(9/2)*E^a) + (105*a*Sqrt[Pi]*Erf[Sqrt[b]*Sqrt[x]])/(16*b^(9/2)*E^a)))/7 + (2*x^(7/2)* Gamma[2, a + b*x])/7
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 = 0.32 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.98
| method | result | size |
| meijerg | \(\frac {{\mathrm e}^{-a} \left (-\frac {\sqrt {x}\, \sqrt {b}\, \left (72 b^{3} x^{3}+252 b^{2} x^{2}+630 b x +945\right ) {\mathrm e}^{-b x}}{72}+\frac {105 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{16}\right )}{b^{\frac {7}{2}}}+\frac {{\mathrm e}^{-a} a \left (-\frac {\sqrt {x}\, \sqrt {b}\, \left (28 b^{2} x^{2}+70 b x +105\right ) {\mathrm e}^{-b x}}{28}+\frac {15 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8}\right )}{b^{\frac {7}{2}}}+\frac {{\mathrm e}^{-a} \left (-\frac {\sqrt {x}\, \sqrt {b}\, \left (28 b^{2} x^{2}+70 b x +105\right ) {\mathrm e}^{-b x}}{28}+\frac {15 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8}\right )}{b^{\frac {7}{2}}}\) | \(158\) |
| derivativedivides | \(2 \,{\mathrm e}^{-a} \left (-\frac {x^{\frac {5}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {-\frac {5 x^{\frac {3}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {5 \left (-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}}{b}\right )+2 \,{\mathrm e}^{-a} a \left (-\frac {x^{\frac {5}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {-\frac {5 x^{\frac {3}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {5 \left (-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}}{b}\right )+2 \,{\mathrm e}^{-a} b \left (-\frac {x^{\frac {7}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {-\frac {7 x^{\frac {5}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {7 \left (-\frac {5 x^{\frac {3}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {5 \left (-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )}{2 b}}{b}\right )\) | \(245\) |
| default | \(2 \,{\mathrm e}^{-a} \left (-\frac {x^{\frac {5}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {-\frac {5 x^{\frac {3}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {5 \left (-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}}{b}\right )+2 \,{\mathrm e}^{-a} a \left (-\frac {x^{\frac {5}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {-\frac {5 x^{\frac {3}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {5 \left (-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}}{b}\right )+2 \,{\mathrm e}^{-a} b \left (-\frac {x^{\frac {7}{2}} {\mathrm e}^{-b x}}{2 b}+\frac {-\frac {7 x^{\frac {5}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {7 \left (-\frac {5 x^{\frac {3}{2}} {\mathrm e}^{-b x}}{4 b}+\frac {5 \left (-\frac {3 \sqrt {x}\, {\mathrm e}^{-b x}}{4 b}+\frac {3 \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {b}\, \sqrt {x}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b}\right )}{2 b}}{b}\right )\) | \(245\) |
Input:
int(x^(5/2)*exp(-b*x-a)*(b*x+a+1),x,method=_RETURNVERBOSE)
Output:
1/b^(7/2)*exp(-a)*(-1/72*x^(1/2)*b^(1/2)*(72*b^3*x^3+252*b^2*x^2+630*b*x+9 45)*exp(-b*x)+105/16*Pi^(1/2)*erf(b^(1/2)*x^(1/2)))+1/b^(7/2)*exp(-a)*a*(- 1/28*x^(1/2)*b^(1/2)*(28*b^2*x^2+70*b*x+105)*exp(-b*x)+15/8*Pi^(1/2)*erf(b ^(1/2)*x^(1/2)))+1/b^(7/2)*exp(-a)*(-1/28*x^(1/2)*b^(1/2)*(28*b^2*x^2+70*b *x+105)*exp(-b*x)+15/8*Pi^(1/2)*erf(b^(1/2)*x^(1/2)))
Time = 0.11 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.46 \[ \int x^{5/2} \Gamma (2,a+b x) \, dx=\frac {105 \, \sqrt {\pi } {\left (2 \, a + 9\right )} \sqrt {b} \operatorname {erf}\left (\sqrt {b} \sqrt {x}\right ) e^{\left (-a\right )} + 2 \, {\left (16 \, b^{4} x^{3} \Gamma \left (2, b x + a\right ) - {\left (16 \, b^{5} x^{4} + 8 \, {\left (2 \, a + 9\right )} b^{4} x^{3} + 28 \, {\left (2 \, a + 9\right )} b^{3} x^{2} + 70 \, {\left (2 \, a + 9\right )} b^{2} x + 105 \, {\left (2 \, a + 9\right )} b\right )} e^{\left (-b x - a\right )}\right )} \sqrt {x}}{112 \, b^{4}} \] Input:
integrate(x^(5/2)*gamma(2,b*x+a),x, algorithm="fricas")
Output:
1/112*(105*sqrt(pi)*(2*a + 9)*sqrt(b)*erf(sqrt(b)*sqrt(x))*e^(-a) + 2*(16* b^4*x^3*gamma(2, b*x + a) - (16*b^5*x^4 + 8*(2*a + 9)*b^4*x^3 + 28*(2*a + 9)*b^3*x^2 + 70*(2*a + 9)*b^2*x + 105*(2*a + 9)*b)*e^(-b*x - a))*sqrt(x))/ b^4
Timed out. \[ \int x^{5/2} \Gamma (2,a+b x) \, dx=\text {Timed out} \] Input:
integrate(x**(5/2)*uppergamma(2,b*x+a),x)
Output:
Timed out
\[ \int x^{5/2} \Gamma (2,a+b x) \, dx=\int { x^{\frac {5}{2}} \Gamma \left (2, b x + a\right ) \,d x } \] Input:
integrate(x^(5/2)*gamma(2,b*x+a),x, algorithm="maxima")
Output:
integrate(x^(5/2)*gamma(2, b*x + a), x)
\[ \int x^{5/2} \Gamma (2,a+b x) \, dx=\int { x^{\frac {5}{2}} \Gamma \left (2, b x + a\right ) \,d x } \] Input:
integrate(x^(5/2)*gamma(2,b*x+a),x, algorithm="giac")
Output:
integrate(x^(5/2)*gamma(2, b*x + a), x)
Time = 0.28 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.09 \[ \int x^{5/2} \Gamma (2,a+b x) \, dx=-x^{7/2}\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}-\frac {135\,\sqrt {x}\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}}{8\,b^3}-\frac {45\,x^{3/2}\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}}{4\,b^2}-\frac {9\,x^{5/2}\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}}{2\,b}-\frac {15\,a\,\sqrt {x}\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}}{4\,b^3}-\frac {5\,a\,x^{3/2}\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}}{2\,b^2}-\frac {a\,x^{5/2}\,{\mathrm {e}}^{-a}\,{\mathrm {e}}^{-b\,x}}{b}-\frac {135\,x^{7/2}\,\sqrt {\pi }\,{\mathrm {e}}^{-a}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )}{16\,{\left (b\,x\right )}^{7/2}}-\frac {15\,a\,x^{7/2}\,\sqrt {\pi }\,{\mathrm {e}}^{-a}\,\mathrm {erfc}\left (\sqrt {b\,x}\right )}{8\,{\left (b\,x\right )}^{7/2}} \] Input:
int(x^(5/2)*exp(- a - b*x)*(a + b*x + 1),x)
Output:
- x^(7/2)*exp(-a)*exp(-b*x) - (135*x^(1/2)*exp(-a)*exp(-b*x))/(8*b^3) - (4 5*x^(3/2)*exp(-a)*exp(-b*x))/(4*b^2) - (9*x^(5/2)*exp(-a)*exp(-b*x))/(2*b) - (15*a*x^(1/2)*exp(-a)*exp(-b*x))/(4*b^3) - (5*a*x^(3/2)*exp(-a)*exp(-b* x))/(2*b^2) - (a*x^(5/2)*exp(-a)*exp(-b*x))/b - (135*x^(7/2)*pi^(1/2)*exp( -a)*erfc((b*x)^(1/2)))/(16*(b*x)^(7/2)) - (15*a*x^(7/2)*pi^(1/2)*exp(-a)*e rfc((b*x)^(1/2)))/(8*(b*x)^(7/2))
\[ \int x^{5/2} \Gamma (2,a+b x) \, dx=\frac {30 e^{b x} \left (\int \frac {\sqrt {x}}{e^{b x} x}d x \right ) a +135 e^{b x} \left (\int \frac {\sqrt {x}}{e^{b x} x}d x \right )-16 \sqrt {x}\, a \,b^{2} x^{2}-40 \sqrt {x}\, a b x -60 \sqrt {x}\, a -16 \sqrt {x}\, b^{3} x^{3}-72 \sqrt {x}\, b^{2} x^{2}-180 \sqrt {x}\, b x -270 \sqrt {x}}{16 e^{b x +a} b^{3}} \] Input:
int(x^(5/2)*exp(-b*x-a)*(b*x+a+1),x)
Output:
(30*e**(b*x)*int(sqrt(x)/(e**(b*x)*x),x)*a + 135*e**(b*x)*int(sqrt(x)/(e** (b*x)*x),x) - 16*sqrt(x)*a*b**2*x**2 - 40*sqrt(x)*a*b*x - 60*sqrt(x)*a - 1 6*sqrt(x)*b**3*x**3 - 72*sqrt(x)*b**2*x**2 - 180*sqrt(x)*b*x - 270*sqrt(x) )/(16*e**(a + b*x)*b**3)