Integrand size = 15, antiderivative size = 137 \[ \int (c+d x)^2 \Gamma (3,a+b x) \, dx=\frac {(c+d x)^3 \Gamma (3,a+b x)}{3 d}-\frac {(b c-a d)^2 e^{-a+\frac {b c}{d}} \Gamma \left (4,\frac {b (c+d x)}{d}\right )}{3 b^3}+\frac {2 d (b c-a d) e^{-a+\frac {b c}{d}} \Gamma \left (5,\frac {b (c+d x)}{d}\right )}{3 b^3}-\frac {d^2 e^{-a+\frac {b c}{d}} \Gamma \left (6,\frac {b (c+d x)}{d}\right )}{3 b^3} \] Output:
2/3*(d*x+c)^3*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2)/d-2*(-a*d+b*c)^2*exp(-a+ b*c/d)*exp(-b*(d*x+c)/d)*(1+b*(d*x+c)/d+1/2*b^2*(d*x+c)^2/d^2+1/6*b^3*(d*x +c)^3/d^3)/b^3+16*d*(-a*d+b*c)*exp(-a+b*c/d)*exp(-b*(d*x+c)/d)*(1+b*(d*x+c )/d+1/2*b^2*(d*x+c)^2/d^2+1/6*b^3*(d*x+c)^3/d^3+1/24*b^4*(d*x+c)^4/d^4)/b^ 3-40*d^2*exp(-a+b*c/d)*exp(-b*(d*x+c)/d)*(1+b*(d*x+c)/d+1/2*b^2*(d*x+c)^2/ d^2+1/6*b^3*(d*x+c)^3/d^3+1/24*b^4*(d*x+c)^4/d^4+1/120*b^5*(d*x+c)^5/d^5)/ b^3
Time = 0.10 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.84 \[ \int (c+d x)^2 \Gamma (3,a+b x) \, dx=\frac {e^{-a-b x} \left (-6 \left (20+8 a+a^2\right ) d^2-6 b d \left (\left (12+6 a+a^2\right ) c+\left (20+8 a+a^2\right ) d x\right )-b^5 x^3 \left (3 c^2+3 c d x+d^2 x^2\right )-b^4 x^2 \left ((9+6 a) c^2+6 (2+a) c d x+(5+2 a) d^2 x^2\right )-3 b^2 \left (\left (6+4 a+a^2\right ) c^2+2 \left (12+6 a+a^2\right ) c d x+\left (20+8 a+a^2\right ) d^2 x^2\right )-b^3 x \left (3 \left (6+4 a+a^2\right ) c^2+3 \left (12+6 a+a^2\right ) c d x+\left (20+8 a+a^2\right ) d^2 x^2\right )+b^3 e^{a+b x} x \left (3 c^2+3 c d x+d^2 x^2\right ) \Gamma (3,a+b x)\right )}{3 b^3} \] Input:
Integrate[(c + d*x)^2*Gamma[3, a + b*x],x]
Output:
(E^(-a - b*x)*(-6*(20 + 8*a + a^2)*d^2 - 6*b*d*((12 + 6*a + a^2)*c + (20 + 8*a + a^2)*d*x) - b^5*x^3*(3*c^2 + 3*c*d*x + d^2*x^2) - b^4*x^2*((9 + 6*a )*c^2 + 6*(2 + a)*c*d*x + (5 + 2*a)*d^2*x^2) - 3*b^2*((6 + 4*a + a^2)*c^2 + 2*(12 + 6*a + a^2)*c*d*x + (20 + 8*a + a^2)*d^2*x^2) - b^3*x*(3*(6 + 4*a + a^2)*c^2 + 3*(12 + 6*a + a^2)*c*d*x + (20 + 8*a + a^2)*d^2*x^2) + b^3*E ^(a + b*x)*x*(3*c^2 + 3*c*d*x + d^2*x^2)*Gamma[3, a + b*x]))/(3*b^3)
Leaf count is larger than twice the leaf count of optimal. \(436\) vs. \(2(137)=274\).
Time = 0.95 (sec) , antiderivative size = 436, normalized size of antiderivative = 3.18, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {7119, 2626, 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.
| \(\displaystyle \int (c+d x)^2 \Gamma (3,a+b x) \, dx\) |
| \(\Big \downarrow \) 7119 |
| \(\displaystyle \frac {b \int e^{-a-b x} (a+b x)^2 (c+d x)^3dx}{3 d}+\frac {(c+d x)^3 \Gamma (3,a+b x)}{3 d}\) |
| \(\Big \downarrow \) 2626 |
| \(\displaystyle \frac {b \int \left (\frac {b^2 e^{-a-b x} (c+d x)^5}{d^2}-\frac {2 b (b c-a d) e^{-a-b x} (c+d x)^4}{d^2}+\frac {(a d-b c)^2 e^{-a-b x} (c+d x)^3}{d^2}\right )dx}{3 d}+\frac {(c+d x)^3 \Gamma (3,a+b x)}{3 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (\frac {48 d^2 e^{-a-b x} (b c-a d)}{b^4}-\frac {6 d e^{-a-b x} (b c-a d)^2}{b^4}-\frac {120 d^3 e^{-a-b x}}{b^4}-\frac {120 d^2 e^{-a-b x} (c+d x)}{b^3}-\frac {6 e^{-a-b x} (c+d x) (b c-a d)^2}{b^3}+\frac {48 d e^{-a-b x} (c+d x) (b c-a d)}{b^3}-\frac {3 e^{-a-b x} (c+d x)^2 (b c-a d)^2}{b^2 d}-\frac {60 d e^{-a-b x} (c+d x)^2}{b^2}+\frac {24 e^{-a-b x} (c+d x)^2 (b c-a d)}{b^2}-\frac {b e^{-a-b x} (c+d x)^5}{d^2}+\frac {2 e^{-a-b x} (c+d x)^4 (b c-a d)}{d^2}-\frac {e^{-a-b x} (c+d x)^3 (b c-a d)^2}{b d^2}-\frac {5 e^{-a-b x} (c+d x)^4}{d}+\frac {8 e^{-a-b x} (c+d x)^3 (b c-a d)}{b d}-\frac {20 e^{-a-b x} (c+d x)^3}{b}\right )}{3 d}+\frac {(c+d x)^3 \Gamma (3,a+b x)}{3 d}\) |
Input:
Int[(c + d*x)^2*Gamma[3, a + b*x],x]
Output:
(b*((-120*d^3*E^(-a - b*x))/b^4 + (48*d^2*(b*c - a*d)*E^(-a - b*x))/b^4 - (6*d*(b*c - a*d)^2*E^(-a - b*x))/b^4 - (120*d^2*E^(-a - b*x)*(c + d*x))/b^ 3 + (48*d*(b*c - a*d)*E^(-a - b*x)*(c + d*x))/b^3 - (6*(b*c - a*d)^2*E^(-a - b*x)*(c + d*x))/b^3 - (60*d*E^(-a - b*x)*(c + d*x)^2)/b^2 + (24*(b*c - a*d)*E^(-a - b*x)*(c + d*x)^2)/b^2 - (3*(b*c - a*d)^2*E^(-a - b*x)*(c + d* x)^2)/(b^2*d) - (20*E^(-a - b*x)*(c + d*x)^3)/b + (8*(b*c - a*d)*E^(-a - b *x)*(c + d*x)^3)/(b*d) - ((b*c - a*d)^2*E^(-a - b*x)*(c + d*x)^3)/(b*d^2) - (5*E^(-a - b*x)*(c + d*x)^4)/d + (2*(b*c - a*d)*E^(-a - b*x)*(c + d*x)^4 )/d^2 - (b*E^(-a - b*x)*(c + d*x)^5)/d^2))/(3*d) + ((c + d*x)^3*Gamma[3, a + b*x])/(3*d)
Int[(F_)^(v_)*(Px_), x_Symbol] :> Int[ExpandIntegrand[F^v, Px, x], x] /; Fr eeQ[F, 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 = 0.71 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.84
| method | result | size |
| norman | \(\left (-2 a \,d^{2}-2 b c d -6 d^{2}\right ) x^{3} {\mathrm e}^{-b x -a}-\frac {\left (a^{2} c^{2} b^{2}+2 a^{2} b c d +4 a \,c^{2} b^{2}+2 a^{2} d^{2}+12 a b c d +6 b^{2} c^{2}+16 a \,d^{2}+24 b c d +40 d^{2}\right ) {\mathrm e}^{-b x -a}}{b^{3}}-b \,d^{2} x^{4} {\mathrm e}^{-b x -a}-\frac {\left (a^{2} d^{2}+4 a b c d +b^{2} c^{2}+8 a \,d^{2}+10 b c d +20 d^{2}\right ) x^{2} {\mathrm e}^{-b x -a}}{b}-\frac {2 \left (a^{2} b c d +a \,c^{2} b^{2}+a^{2} d^{2}+6 a b c d +2 b^{2} c^{2}+8 a \,d^{2}+12 b c d +20 d^{2}\right ) x \,{\mathrm e}^{-b x -a}}{b^{2}}\) | \(252\) |
| gosper | \(-\frac {{\mathrm e}^{-b x -a} \left (b^{4} d^{2} x^{4}+2 a \,b^{3} d^{2} x^{3}+2 b^{4} c d \,x^{3}+a^{2} b^{2} d^{2} x^{2}+4 a \,b^{3} c d \,x^{2}+b^{4} c^{2} x^{2}+6 b^{3} d^{2} x^{3}+2 a^{2} b^{2} c d x +2 a \,b^{3} c^{2} x +8 a \,b^{2} d^{2} x^{2}+10 b^{3} c d \,x^{2}+a^{2} c^{2} b^{2}+2 a^{2} b \,d^{2} x +12 a \,b^{2} c d x +4 c^{2} x \,b^{3}+20 b^{2} d^{2} x^{2}+2 a^{2} b c d +4 a \,c^{2} b^{2}+16 a b \,d^{2} x +24 b^{2} c d x +2 a^{2} d^{2}+12 a b c d +6 b^{2} c^{2}+40 b \,d^{2} x +16 a \,d^{2}+24 b c d +40 d^{2}\right )}{b^{3}}\) | \(263\) |
| risch | \(-\frac {{\mathrm e}^{-b x -a} \left (b^{4} d^{2} x^{4}+2 a \,b^{3} d^{2} x^{3}+2 b^{4} c d \,x^{3}+a^{2} b^{2} d^{2} x^{2}+4 a \,b^{3} c d \,x^{2}+b^{4} c^{2} x^{2}+6 b^{3} d^{2} x^{3}+2 a^{2} b^{2} c d x +2 a \,b^{3} c^{2} x +8 a \,b^{2} d^{2} x^{2}+10 b^{3} c d \,x^{2}+a^{2} c^{2} b^{2}+2 a^{2} b \,d^{2} x +12 a \,b^{2} c d x +4 c^{2} x \,b^{3}+20 b^{2} d^{2} x^{2}+2 a^{2} b c d +4 a \,c^{2} b^{2}+16 a b \,d^{2} x +24 b^{2} c d x +2 a^{2} d^{2}+12 a b c d +6 b^{2} c^{2}+40 b \,d^{2} x +16 a \,d^{2}+24 b c d +40 d^{2}\right )}{b^{3}}\) | \(263\) |
| orering | \(-\frac {2 \left (b^{4} d^{2} x^{4}+2 a \,b^{3} d^{2} x^{3}+2 b^{4} c d \,x^{3}+a^{2} b^{2} d^{2} x^{2}+4 a \,b^{3} c d \,x^{2}+b^{4} c^{2} x^{2}+6 b^{3} d^{2} x^{3}+2 a^{2} b^{2} c d x +2 a \,b^{3} c^{2} x +8 a \,b^{2} d^{2} x^{2}+10 b^{3} c d \,x^{2}+a^{2} c^{2} b^{2}+2 a^{2} b \,d^{2} x +12 a \,b^{2} c d x +4 c^{2} x \,b^{3}+20 b^{2} d^{2} x^{2}+2 a^{2} b c d +4 a \,c^{2} b^{2}+16 a b \,d^{2} x +24 b^{2} c d x +2 a^{2} d^{2}+12 a b c d +6 b^{2} c^{2}+40 b \,d^{2} x +16 a \,d^{2}+24 b c d +40 d^{2}\right ) {\mathrm e}^{-b x -a} \left (1+b x +a +\frac {\left (b x +a \right )^{2}}{2}\right )}{b^{3} \left (b^{2} x^{2}+2 b x a +a^{2}+2 b x +2 a +2\right )}\) | \(304\) |
| parallelrisch | \(-\frac {12 x \,{\mathrm e}^{-b x -a} a \,b^{2} c d +4 x^{2} {\mathrm e}^{-b x -a} a \,b^{3} c d +2 x \,{\mathrm e}^{-b x -a} a^{2} b^{2} c d +10 x^{2} {\mathrm e}^{-b x -a} b^{3} c d +20 x^{2} {\mathrm e}^{-b x -a} d^{2} b^{2}+40 x \,{\mathrm e}^{-b x -a} b \,d^{2}+24 \,{\mathrm e}^{-b x -a} b c d +b^{4} d^{2} {\mathrm e}^{-b x -a} x^{4}+x^{2} {\mathrm e}^{-b x -a} b^{4} c^{2}+{\mathrm e}^{-b x -a} a^{2} b^{2} c^{2}+2 x^{3} {\mathrm e}^{-b x -a} a \,b^{3} d^{2}+2 x^{3} {\mathrm e}^{-b x -a} b^{4} c d +x^{2} {\mathrm e}^{-b x -a} a^{2} b^{2} d^{2}+2 x \,{\mathrm e}^{-b x -a} a \,b^{3} c^{2}+4 x \,{\mathrm e}^{-b x -a} b^{3} c^{2}+40 \,{\mathrm e}^{-b x -a} d^{2}+24 x \,{\mathrm e}^{-b x -a} b^{2} c d +8 x^{2} {\mathrm e}^{-b x -a} a \,b^{2} d^{2}+2 x \,{\mathrm e}^{-b x -a} a^{2} b \,d^{2}+16 x \,{\mathrm e}^{-b x -a} a b \,d^{2}+2 \,{\mathrm e}^{-b x -a} a^{2} b c d +12 \,{\mathrm e}^{-b x -a} a b c d +6 d^{2} {\mathrm e}^{-b x -a} x^{3} b^{3}+4 \,{\mathrm e}^{-b x -a} a \,b^{2} c^{2}+6 \,{\mathrm e}^{-b x -a} b^{2} c^{2}+2 \,{\mathrm e}^{-b x -a} a^{2} d^{2}+16 \,{\mathrm e}^{-b x -a} a \,d^{2}}{b^{3}}\) | \(497\) |
| meijerg | \(\frac {2 d^{2} {\mathrm e}^{-a} \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b^{3}}+\frac {2 d^{2} {\mathrm e}^{-a} a \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b^{3}}+\frac {d^{2} {\mathrm e}^{-a} a^{2} \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b^{3}}+\frac {2 d^{2} {\mathrm e}^{-a} \left (a b +b \right ) \left (6-\frac {\left (4 b^{3} x^{3}+12 b^{2} x^{2}+24 b x +24\right ) {\mathrm e}^{-b x}}{4}\right )}{b^{4}}+\frac {d^{2} {\mathrm e}^{-a} \left (24-\frac {\left (5 b^{4} x^{4}+20 b^{3} x^{3}+60 b^{2} x^{2}+120 b x +120\right ) {\mathrm e}^{-b x}}{5}\right )}{b^{3}}+\frac {4 d c \,{\mathrm e}^{-a} \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}+\frac {4 d c \,{\mathrm e}^{-a} a \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}+\frac {2 d c \,{\mathrm e}^{-a} a^{2} \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}+\frac {4 c d \,{\mathrm e}^{-a} \left (a b +b \right ) \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b^{3}}+\frac {2 d c \,{\mathrm e}^{-a} \left (6-\frac {\left (4 b^{3} x^{3}+12 b^{2} x^{2}+24 b x +24\right ) {\mathrm e}^{-b x}}{4}\right )}{b^{2}}+\frac {2 c^{2} {\mathrm e}^{-a} \left (1-{\mathrm e}^{-b x}\right )}{b}+\frac {2 c^{2} {\mathrm e}^{-a} a \left (1-{\mathrm e}^{-b x}\right )}{b}+\frac {c^{2} {\mathrm e}^{-a} a^{2} \left (1-{\mathrm e}^{-b x}\right )}{b}+\frac {2 c^{2} {\mathrm e}^{-a} \left (a b +b \right ) \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}+\frac {c^{2} {\mathrm e}^{-a} \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b}\) | \(503\) |
| parts | \(-b \,d^{2} x^{4} {\mathrm e}^{-b x -a}-2 \,{\mathrm e}^{-b x -a} a \,d^{2} x^{3}-2 \,{\mathrm e}^{-b x -a} b c d \,x^{3}-\frac {{\mathrm e}^{-b x -a} d^{2} x^{2} a^{2}}{b}-4 \,{\mathrm e}^{-b x -a} a c d \,x^{2}-{\mathrm e}^{-b x -a} b \,c^{2} x^{2}-2 d^{2} x^{3} {\mathrm e}^{-b x -a}-\frac {2 \,{\mathrm e}^{-b x -a} a^{2} c d x}{b}-2 \,{\mathrm e}^{-b x -a} a \,c^{2} x -\frac {2 \,{\mathrm e}^{-b x -a} a \,d^{2} x^{2}}{b}-4 \,{\mathrm e}^{-b x -a} c d \,x^{2}-\frac {{\mathrm e}^{-b x -a} a^{2} c^{2}}{b}-\frac {4 \,{\mathrm e}^{-b x -a} x a d c}{b}-2 \,{\mathrm e}^{-b x -a} c^{2} x -\frac {2 d^{2} x^{2} {\mathrm e}^{-b x -a}}{b}-\frac {2 \,{\mathrm e}^{-b x -a} a \,c^{2}}{b}-\frac {4 \,{\mathrm e}^{-b x -a} c d x}{b}-\frac {2 \,{\mathrm e}^{-b x -a} c^{2}}{b}-\frac {4 \,{\mathrm e}^{-b x -a} c d +2 \,{\mathrm e}^{-b x -a} b \,c^{2}-2 c^{2} b \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )-4 \,{\mathrm e}^{-b x -a} d a c -8 c d \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )+6 c d \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )-\frac {4 \,{\mathrm e}^{-b x -a} d^{2} a}{b}+\frac {2 \,{\mathrm e}^{-b x -a} d^{2} a^{2}}{b}-\frac {4 d^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}+\frac {6 d^{2} \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b}-\frac {4 d^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b}+4 d a c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )-\frac {2 d^{2} a^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}+\frac {8 d^{2} a \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}-\frac {6 d^{2} a \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b}}{b^{2}}\) | \(891\) |
| derivativedivides | \(-\frac {c^{2} \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )+\frac {d^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{4}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}+\frac {d^{2} a^{2} \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}+2 \,{\mathrm e}^{-b x -a} c^{2}-2 c^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )+\frac {2 \,{\mathrm e}^{-b x -a} d^{2} a^{2}}{b^{2}}+\frac {2 d^{2} \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {2 d^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {4 \,{\mathrm e}^{-b x -a} d a c}{b}-\frac {4 d c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}+\frac {4 d c \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b}-\frac {2 d c \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b}-\frac {2 d^{2} a^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b^{2}}+\frac {4 d^{2} a \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {4 d^{2} a \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}+\frac {2 d^{2} a \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}+\frac {4 d a c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}-\frac {2 d a c \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b}}{b}\) | \(940\) |
| default | \(-\frac {c^{2} \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )+\frac {d^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{4}-4 \,{\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}+12 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-24 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+24 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}+\frac {d^{2} a^{2} \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}+2 \,{\mathrm e}^{-b x -a} c^{2}-2 c^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )+\frac {2 \,{\mathrm e}^{-b x -a} d^{2} a^{2}}{b^{2}}+\frac {2 d^{2} \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {2 d^{2} \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {4 \,{\mathrm e}^{-b x -a} d a c}{b}-\frac {4 d c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}+\frac {4 d c \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b}-\frac {2 d c \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b}-\frac {2 d^{2} a^{2} \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b^{2}}+\frac {4 d^{2} a \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b^{2}}-\frac {4 d^{2} a \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}+\frac {2 d^{2} a \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b^{2}}+\frac {4 d a c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}-\frac {2 d a c \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b}}{b}\) | \(940\) |
Input:
int(2*(d*x+c)^2*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2),x,method=_RETURNVERBOS E)
Output:
(-2*a*d^2-2*b*c*d-6*d^2)*x^3*exp(-b*x-a)-(a^2*b^2*c^2+2*a^2*b*c*d+4*a*b^2* c^2+2*a^2*d^2+12*a*b*c*d+6*b^2*c^2+16*a*d^2+24*b*c*d+40*d^2)/b^3*exp(-b*x- a)-b*d^2*x^4*exp(-b*x-a)-(a^2*d^2+4*a*b*c*d+b^2*c^2+8*a*d^2+10*b*c*d+20*d^ 2)/b*x^2*exp(-b*x-a)-2*(a^2*b*c*d+a*b^2*c^2+a^2*d^2+6*a*b*c*d+2*b^2*c^2+8* a*d^2+12*b*c*d+20*d^2)/b^2*x*exp(-b*x-a)
Leaf count of result is larger than twice the leaf count of optimal. 265 vs. \(2 (126) = 252\).
Time = 0.09 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.93 \[ \int (c+d x)^2 \Gamma (3,a+b x) \, dx=-\frac {{\left (b^{5} d^{2} x^{5} + 3 \, {\left (a^{2} + 4 \, a + 6\right )} b^{2} c^{2} + {\left (3 \, b^{5} c d + {\left (2 \, a + 5\right )} b^{4} d^{2}\right )} x^{4} + 6 \, {\left (a^{2} + 6 \, a + 12\right )} b c d + {\left (3 \, b^{5} c^{2} + 6 \, {\left (a + 2\right )} b^{4} c d + {\left (a^{2} + 8 \, a + 20\right )} b^{3} d^{2}\right )} x^{3} + 6 \, {\left (a^{2} + 8 \, a + 20\right )} d^{2} + 3 \, {\left ({\left (2 \, a + 3\right )} b^{4} c^{2} + {\left (a^{2} + 6 \, a + 12\right )} b^{3} c d + {\left (a^{2} + 8 \, a + 20\right )} b^{2} d^{2}\right )} x^{2} + 3 \, {\left ({\left (a^{2} + 4 \, a + 6\right )} b^{3} c^{2} + 2 \, {\left (a^{2} + 6 \, a + 12\right )} b^{2} c d + 2 \, {\left (a^{2} + 8 \, a + 20\right )} b d^{2}\right )} x\right )} e^{\left (-b x - a\right )} - {\left (b^{3} d^{2} x^{3} + 3 \, b^{3} c d x^{2} + 3 \, b^{3} c^{2} x\right )} \Gamma \left (3, b x + a\right )}{3 \, b^{3}} \] Input:
integrate((d*x+c)^2*gamma(3,b*x+a),x, algorithm="fricas")
Output:
-1/3*((b^5*d^2*x^5 + 3*(a^2 + 4*a + 6)*b^2*c^2 + (3*b^5*c*d + (2*a + 5)*b^ 4*d^2)*x^4 + 6*(a^2 + 6*a + 12)*b*c*d + (3*b^5*c^2 + 6*(a + 2)*b^4*c*d + ( a^2 + 8*a + 20)*b^3*d^2)*x^3 + 6*(a^2 + 8*a + 20)*d^2 + 3*((2*a + 3)*b^4*c ^2 + (a^2 + 6*a + 12)*b^3*c*d + (a^2 + 8*a + 20)*b^2*d^2)*x^2 + 3*((a^2 + 4*a + 6)*b^3*c^2 + 2*(a^2 + 6*a + 12)*b^2*c*d + 2*(a^2 + 8*a + 20)*b*d^2)* x)*e^(-b*x - a) - (b^3*d^2*x^3 + 3*b^3*c*d*x^2 + 3*b^3*c^2*x)*gamma(3, b*x + a))/b^3
Time = 0.15 (sec) , antiderivative size = 447, normalized size of antiderivative = 3.26 \[ \int (c+d x)^2 \Gamma (3,a+b x) \, dx=\begin {cases} \frac {\left (- a^{2} b^{2} c^{2} - 2 a^{2} b^{2} c d x - a^{2} b^{2} d^{2} x^{2} - 2 a^{2} b c d - 2 a^{2} b d^{2} x - 2 a^{2} d^{2} - 2 a b^{3} c^{2} x - 4 a b^{3} c d x^{2} - 2 a b^{3} d^{2} x^{3} - 4 a b^{2} c^{2} - 12 a b^{2} c d x - 8 a b^{2} d^{2} x^{2} - 12 a b c d - 16 a b d^{2} x - 16 a d^{2} - b^{4} c^{2} x^{2} - 2 b^{4} c d x^{3} - b^{4} d^{2} x^{4} - 4 b^{3} c^{2} x - 10 b^{3} c d x^{2} - 6 b^{3} d^{2} x^{3} - 6 b^{2} c^{2} - 24 b^{2} c d x - 20 b^{2} d^{2} x^{2} - 24 b c d - 40 b d^{2} x - 40 d^{2}\right ) e^{- a - b x}}{b^{3}} & \text {for}\: b^{3} \neq 0 \\\frac {b^{2} d^{2} x^{5}}{5} + x^{4} \left (\frac {a b d^{2}}{2} + \frac {b^{2} c d}{2} + \frac {b d^{2}}{2}\right ) + x^{3} \left (\frac {a^{2} d^{2}}{3} + \frac {4 a b c d}{3} + \frac {2 a d^{2}}{3} + \frac {b^{2} c^{2}}{3} + \frac {4 b c d}{3} + \frac {2 d^{2}}{3}\right ) + x^{2} \left (a^{2} c d + a b c^{2} + 2 a c d + b c^{2} + 2 c d\right ) + x \left (a^{2} c^{2} + 2 a c^{2} + 2 c^{2}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**2*uppergamma(3,b*x+a),x)
Output:
Piecewise(((-a**2*b**2*c**2 - 2*a**2*b**2*c*d*x - a**2*b**2*d**2*x**2 - 2* a**2*b*c*d - 2*a**2*b*d**2*x - 2*a**2*d**2 - 2*a*b**3*c**2*x - 4*a*b**3*c* d*x**2 - 2*a*b**3*d**2*x**3 - 4*a*b**2*c**2 - 12*a*b**2*c*d*x - 8*a*b**2*d **2*x**2 - 12*a*b*c*d - 16*a*b*d**2*x - 16*a*d**2 - b**4*c**2*x**2 - 2*b** 4*c*d*x**3 - b**4*d**2*x**4 - 4*b**3*c**2*x - 10*b**3*c*d*x**2 - 6*b**3*d* *2*x**3 - 6*b**2*c**2 - 24*b**2*c*d*x - 20*b**2*d**2*x**2 - 24*b*c*d - 40* b*d**2*x - 40*d**2)*exp(-a - b*x)/b**3, Ne(b**3, 0)), (b**2*d**2*x**5/5 + x**4*(a*b*d**2/2 + b**2*c*d/2 + b*d**2/2) + x**3*(a**2*d**2/3 + 4*a*b*c*d/ 3 + 2*a*d**2/3 + b**2*c**2/3 + 4*b*c*d/3 + 2*d**2/3) + x**2*(a**2*c*d + a* b*c**2 + 2*a*c*d + b*c**2 + 2*c*d) + x*(a**2*c**2 + 2*a*c**2 + 2*c**2), Tr ue))
\[ \int (c+d x)^2 \Gamma (3,a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \Gamma \left (3, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)^2*gamma(3,b*x+a),x, algorithm="maxima")
Output:
((b*x + a)*gamma(3, b*x + a) - gamma(4, b*x + a))*c^2/b + integrate(d^2*x^ 2*gamma(3, b*x + a) + 2*c*d*x*gamma(3, b*x + a), x)
\[ \int (c+d x)^2 \Gamma (3,a+b x) \, dx=\int { {\left (d x + c\right )}^{2} \Gamma \left (3, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)^2*gamma(3,b*x+a),x, algorithm="giac")
Output:
integrate((d*x + c)^2*gamma(3, b*x + a), x)
Time = 0.09 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.74 \[ \int (c+d x)^2 \Gamma (3,a+b x) \, dx=-x\,{\mathrm {e}}^{-a-b\,x}\,\left (2\,a\,c^2+\frac {16\,a\,d^2+b\,\left (2\,c\,d\,a^2+12\,c\,d\,a+24\,c\,d\right )+40\,d^2+2\,a^2\,d^2}{b^2}+4\,c^2\right )-\frac {{\mathrm {e}}^{-a-b\,x}\,\left (b^2\,\left (a^2\,c^2+4\,a\,c^2+6\,c^2\right )+16\,a\,d^2+b\,\left (2\,c\,d\,a^2+12\,c\,d\,a+24\,c\,d\right )+40\,d^2+2\,a^2\,d^2\right )}{b^3}-x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (\frac {a^2\,d^2+8\,a\,d^2+20\,d^2}{b}+10\,c\,d+b\,c^2+4\,a\,c\,d\right )-b\,d^2\,x^4\,{\mathrm {e}}^{-a-b\,x}-2\,d\,x^3\,{\mathrm {e}}^{-a-b\,x}\,\left (3\,d+a\,d+b\,c\right ) \] Input:
int(2*exp(- a - b*x)*(c + d*x)^2*(a + b*x + (a + b*x)^2/2 + 1),x)
Output:
- x*exp(- a - b*x)*(2*a*c^2 + (16*a*d^2 + b*(24*c*d + 12*a*c*d + 2*a^2*c*d ) + 40*d^2 + 2*a^2*d^2)/b^2 + 4*c^2) - (exp(- a - b*x)*(b^2*(4*a*c^2 + 6*c ^2 + a^2*c^2) + 16*a*d^2 + b*(24*c*d + 12*a*c*d + 2*a^2*c*d) + 40*d^2 + 2* a^2*d^2))/b^3 - x^2*exp(- a - b*x)*((8*a*d^2 + 20*d^2 + a^2*d^2)/b + 10*c* d + b*c^2 + 4*a*c*d) - b*d^2*x^4*exp(- a - b*x) - 2*d*x^3*exp(- a - b*x)*( 3*d + a*d + b*c)
Time = 0.23 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.93 \[ \int (c+d x)^2 \Gamma (3,a+b x) \, dx=\frac {-b^{4} d^{2} x^{4}-2 a \,b^{3} d^{2} x^{3}-2 b^{4} c d \,x^{3}-a^{2} b^{2} d^{2} x^{2}-4 a \,b^{3} c d \,x^{2}-b^{4} c^{2} x^{2}-6 b^{3} d^{2} x^{3}-2 a^{2} b^{2} c d x -2 a \,b^{3} c^{2} x -8 a \,b^{2} d^{2} x^{2}-10 b^{3} c d \,x^{2}-a^{2} b^{2} c^{2}-2 a^{2} b \,d^{2} x -12 a \,b^{2} c d x -4 b^{3} c^{2} x -20 b^{2} d^{2} x^{2}-2 a^{2} b c d -4 a \,b^{2} c^{2}-16 a b \,d^{2} x -24 b^{2} c d x -2 a^{2} d^{2}-12 a b c d -6 b^{2} c^{2}-40 b \,d^{2} x -16 a \,d^{2}-24 b c d -40 d^{2}}{e^{b x +a} b^{3}} \] Input:
int(2*(d*x+c)^2*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2),x)
Output:
( - a**2*b**2*c**2 - 2*a**2*b**2*c*d*x - a**2*b**2*d**2*x**2 - 2*a**2*b*c* d - 2*a**2*b*d**2*x - 2*a**2*d**2 - 2*a*b**3*c**2*x - 4*a*b**3*c*d*x**2 - 2*a*b**3*d**2*x**3 - 4*a*b**2*c**2 - 12*a*b**2*c*d*x - 8*a*b**2*d**2*x**2 - 12*a*b*c*d - 16*a*b*d**2*x - 16*a*d**2 - b**4*c**2*x**2 - 2*b**4*c*d*x** 3 - b**4*d**2*x**4 - 4*b**3*c**2*x - 10*b**3*c*d*x**2 - 6*b**3*d**2*x**3 - 6*b**2*c**2 - 24*b**2*c*d*x - 20*b**2*d**2*x**2 - 24*b*c*d - 40*b*d**2*x - 40*d**2)/(e**(a + b*x)*b**3)