Integrand size = 15, antiderivative size = 140 \[ \int (c+d x)^3 \Gamma (3,a+b x) \, dx=\frac {(c+d x)^4 \Gamma (3,a+b x)}{4 d}-\frac {d (b c-a d)^2 e^{-a+\frac {b c}{d}} \Gamma \left (5,\frac {b (c+d x)}{d}\right )}{4 b^4}+\frac {d^2 (b c-a d) e^{-a+\frac {b c}{d}} \Gamma \left (6,\frac {b (c+d x)}{d}\right )}{2 b^4}-\frac {d^3 e^{-a+\frac {b c}{d}} \Gamma \left (7,\frac {b (c+d x)}{d}\right )}{4 b^4} \] Output:
1/2*(d*x+c)^4*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2)/d-6*d*(-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+1/24*b^4*(d*x+c)^4/d^4)/b^4+60*d^2*(-a*d+b*c)*exp(-a+b*c/d)*ex p(-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^4-180*d^3*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+1/720*b^6*(d*x+c)^6/d^6 )/b^4
Leaf count is larger than twice the leaf count of optimal. \(361\) vs. \(2(140)=280\).
Time = 0.15 (sec) , antiderivative size = 361, normalized size of antiderivative = 2.58 \[ \int (c+d x)^3 \Gamma (3,a+b x) \, dx=\frac {e^{-a-b x} \left (-24 \left (30+10 a+a^2\right ) d^3-24 b d^2 \left (\left (20+8 a+a^2\right ) c+\left (30+10 a+a^2\right ) d x\right )-4 b^3 (c+d x) \left (\left (6+4 a+a^2\right ) c^2+2 \left (15+7 a+a^2\right ) c d x+\left (30+10 a+a^2\right ) d^2 x^2\right )-12 b^2 d \left (\left (12+6 a+a^2\right ) c^2+2 \left (20+8 a+a^2\right ) c d x+\left (30+10 a+a^2\right ) d^2 x^2\right )-b^6 x^3 \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )-2 b^5 x^2 \left ((6+4 a) c^3+6 (2+a) c^2 d x+2 (5+2 a) c d^2 x^2+(3+a) d^3 x^3\right )-b^4 x \left (4 \left (6+4 a+a^2\right ) c^3+6 \left (12+6 a+a^2\right ) c^2 d x+4 \left (20+8 a+a^2\right ) c d^2 x^2+\left (30+10 a+a^2\right ) d^3 x^3\right )+b^4 e^{a+b x} x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right ) \Gamma (3,a+b x)\right )}{4 b^4} \] Input:
Integrate[(c + d*x)^3*Gamma[3, a + b*x],x]
Output:
(E^(-a - b*x)*(-24*(30 + 10*a + a^2)*d^3 - 24*b*d^2*((20 + 8*a + a^2)*c + (30 + 10*a + a^2)*d*x) - 4*b^3*(c + d*x)*((6 + 4*a + a^2)*c^2 + 2*(15 + 7* a + a^2)*c*d*x + (30 + 10*a + a^2)*d^2*x^2) - 12*b^2*d*((12 + 6*a + a^2)*c ^2 + 2*(20 + 8*a + a^2)*c*d*x + (30 + 10*a + a^2)*d^2*x^2) - b^6*x^3*(4*c^ 3 + 6*c^2*d*x + 4*c*d^2*x^2 + d^3*x^3) - 2*b^5*x^2*((6 + 4*a)*c^3 + 6*(2 + a)*c^2*d*x + 2*(5 + 2*a)*c*d^2*x^2 + (3 + a)*d^3*x^3) - b^4*x*(4*(6 + 4*a + a^2)*c^3 + 6*(12 + 6*a + a^2)*c^2*d*x + 4*(20 + 8*a + a^2)*c*d^2*x^2 + (30 + 10*a + a^2)*d^3*x^3) + b^4*E^(a + b*x)*x*(4*c^3 + 6*c^2*d*x + 4*c*d^ 2*x^2 + d^3*x^3)*Gamma[3, a + b*x]))/(4*b^4)
Leaf count is larger than twice the leaf count of optimal. \(529\) vs. \(2(140)=280\).
Time = 1.17 (sec) , antiderivative size = 529, normalized size of antiderivative = 3.78, 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.
| \(\displaystyle \int (c+d x)^3 \Gamma (3,a+b x) \, dx\) |
| \(\Big \downarrow \) 7119 |
| \(\displaystyle \frac {b \int e^{-a-b x} (a+b x)^2 (c+d x)^4dx}{4 d}+\frac {(c+d x)^4 \Gamma (3,a+b x)}{4 d}\) |
| \(\Big \downarrow \) 2626 |
| \(\displaystyle \frac {b \int \left (\frac {b^2 e^{-a-b x} (c+d x)^6}{d^2}-\frac {2 b (b c-a d) e^{-a-b x} (c+d x)^5}{d^2}+\frac {(a d-b c)^2 e^{-a-b x} (c+d x)^4}{d^2}\right )dx}{4 d}+\frac {(c+d x)^4 \Gamma (3,a+b x)}{4 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (\frac {240 d^3 e^{-a-b x} (b c-a d)}{b^5}-\frac {24 d^2 e^{-a-b x} (b c-a d)^2}{b^5}-\frac {720 d^4 e^{-a-b x}}{b^5}-\frac {720 d^3 e^{-a-b x} (c+d x)}{b^4}+\frac {240 d^2 e^{-a-b x} (c+d x) (b c-a d)}{b^4}-\frac {24 d e^{-a-b x} (c+d x) (b c-a d)^2}{b^4}-\frac {360 d^2 e^{-a-b x} (c+d x)^2}{b^3}-\frac {12 e^{-a-b x} (c+d x)^2 (b c-a d)^2}{b^3}+\frac {120 d e^{-a-b x} (c+d x)^2 (b c-a d)}{b^3}-\frac {4 e^{-a-b x} (c+d x)^3 (b c-a d)^2}{b^2 d}-\frac {120 d e^{-a-b x} (c+d x)^3}{b^2}+\frac {40 e^{-a-b x} (c+d x)^3 (b c-a d)}{b^2}-\frac {b e^{-a-b x} (c+d x)^6}{d^2}+\frac {2 e^{-a-b x} (c+d x)^5 (b c-a d)}{d^2}-\frac {e^{-a-b x} (c+d x)^4 (b c-a d)^2}{b d^2}-\frac {6 e^{-a-b x} (c+d x)^5}{d}+\frac {10 e^{-a-b x} (c+d x)^4 (b c-a d)}{b d}-\frac {30 e^{-a-b x} (c+d x)^4}{b}\right )}{4 d}+\frac {(c+d x)^4 \Gamma (3,a+b x)}{4 d}\) |
Input:
Int[(c + d*x)^3*Gamma[3, a + b*x],x]
Output:
(b*((-720*d^4*E^(-a - b*x))/b^5 + (240*d^3*(b*c - a*d)*E^(-a - b*x))/b^5 - (24*d^2*(b*c - a*d)^2*E^(-a - b*x))/b^5 - (720*d^3*E^(-a - b*x)*(c + d*x) )/b^4 + (240*d^2*(b*c - a*d)*E^(-a - b*x)*(c + d*x))/b^4 - (24*d*(b*c - a* d)^2*E^(-a - b*x)*(c + d*x))/b^4 - (360*d^2*E^(-a - b*x)*(c + d*x)^2)/b^3 + (120*d*(b*c - a*d)*E^(-a - b*x)*(c + d*x)^2)/b^3 - (12*(b*c - a*d)^2*E^( -a - b*x)*(c + d*x)^2)/b^3 - (120*d*E^(-a - b*x)*(c + d*x)^3)/b^2 + (40*(b *c - a*d)*E^(-a - b*x)*(c + d*x)^3)/b^2 - (4*(b*c - a*d)^2*E^(-a - b*x)*(c + d*x)^3)/(b^2*d) - (30*E^(-a - b*x)*(c + d*x)^4)/b + (10*(b*c - a*d)*E^( -a - b*x)*(c + d*x)^4)/(b*d) - ((b*c - a*d)^2*E^(-a - b*x)*(c + d*x)^4)/(b *d^2) - (6*E^(-a - b*x)*(c + d*x)^5)/d + (2*(b*c - a*d)*E^(-a - b*x)*(c + d*x)^5)/d^2 - (b*E^(-a - b*x)*(c + d*x)^6)/d^2))/(4*d) + ((c + d*x)^4*Gamm a[3, a + b*x])/(4*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 = 0.77 (sec) , antiderivative size = 421, normalized size of antiderivative = 3.01
| method | result | size |
| norman | \(\left (-2 a \,d^{3}-3 d^{2} c b -7 d^{3}\right ) x^{4} {\mathrm e}^{-b x -a}-\frac {\left (a^{2} c^{3} b^{3}+3 a^{2} b^{2} c^{2} d +4 a \,b^{3} c^{3}+6 a^{2} b c \,d^{2}+18 c^{2} d a \,b^{2}+6 b^{3} c^{3}+6 a^{2} d^{3}+48 a b c \,d^{2}+36 b^{2} c^{2} d +60 a \,d^{3}+120 d^{2} c b +180 d^{3}\right ) {\mathrm e}^{-b x -a}}{b^{4}}-d^{3} b \,x^{5} {\mathrm e}^{-b x -a}-\frac {\left (3 a^{2} b c \,d^{2}+6 c^{2} d a \,b^{2}+b^{3} c^{3}+3 a^{2} d^{3}+24 a b c \,d^{2}+15 b^{2} c^{2} d +30 a \,d^{3}+60 d^{2} c b +90 d^{3}\right ) x^{2} {\mathrm e}^{-b x -a}}{b^{2}}-\frac {\left (3 a^{2} b^{2} c^{2} d +2 a \,b^{3} c^{3}+6 a^{2} b c \,d^{2}+18 c^{2} d a \,b^{2}+4 b^{3} c^{3}+6 a^{2} d^{3}+48 a b c \,d^{2}+36 b^{2} c^{2} d +60 a \,d^{3}+120 d^{2} c b +180 d^{3}\right ) x \,{\mathrm e}^{-b x -a}}{b^{3}}-\frac {d \left (a^{2} d^{2}+6 a b c d +3 b^{2} c^{2}+10 a \,d^{2}+18 b c d +30 d^{2}\right ) x^{3} {\mathrm e}^{-b x -a}}{b}\) | \(421\) |
| gosper | \(-\frac {{\mathrm e}^{-b x -a} \left (d^{3} b^{5} x^{5}+2 a \,b^{4} d^{3} x^{4}+3 b^{5} c \,d^{2} x^{4}+a^{2} b^{3} d^{3} x^{3}+6 a \,b^{4} c \,d^{2} x^{3}+3 b^{5} c^{2} d \,x^{3}+7 b^{4} d^{3} x^{4}+3 a^{2} b^{3} c \,d^{2} x^{2}+6 a \,b^{4} c^{2} d \,x^{2}+10 a \,b^{3} d^{3} x^{3}+b^{5} c^{3} x^{2}+18 b^{4} c \,d^{2} x^{3}+3 a^{2} b^{3} c^{2} d x +3 a^{2} b^{2} d^{3} x^{2}+2 a \,b^{4} c^{3} x +24 a \,b^{3} c \,d^{2} x^{2}+15 b^{4} c^{2} d \,x^{2}+30 b^{3} d^{3} x^{3}+a^{2} c^{3} b^{3}+6 a^{2} b^{2} c \,d^{2} x +18 a \,b^{3} c^{2} d x +30 a \,b^{2} d^{3} x^{2}+4 b^{4} c^{3} x +60 b^{3} c \,d^{2} x^{2}+3 a^{2} b^{2} c^{2} d +6 a^{2} b \,d^{3} x +4 a \,b^{3} c^{3}+48 a \,b^{2} c \,d^{2} x +36 b^{3} c^{2} d x +90 b^{2} d^{3} x^{2}+6 a^{2} b c \,d^{2}+18 c^{2} d a \,b^{2}+60 a b \,d^{3} x +6 b^{3} c^{3}+120 b^{2} c \,d^{2} x +6 a^{2} d^{3}+48 a b c \,d^{2}+36 b^{2} c^{2} d +180 b \,d^{3} x +60 a \,d^{3}+120 d^{2} c b +180 d^{3}\right )}{b^{4}}\) | \(460\) |
| risch | \(-\frac {{\mathrm e}^{-b x -a} \left (d^{3} b^{5} x^{5}+2 a \,b^{4} d^{3} x^{4}+3 b^{5} c \,d^{2} x^{4}+a^{2} b^{3} d^{3} x^{3}+6 a \,b^{4} c \,d^{2} x^{3}+3 b^{5} c^{2} d \,x^{3}+7 b^{4} d^{3} x^{4}+3 a^{2} b^{3} c \,d^{2} x^{2}+6 a \,b^{4} c^{2} d \,x^{2}+10 a \,b^{3} d^{3} x^{3}+b^{5} c^{3} x^{2}+18 b^{4} c \,d^{2} x^{3}+3 a^{2} b^{3} c^{2} d x +3 a^{2} b^{2} d^{3} x^{2}+2 a \,b^{4} c^{3} x +24 a \,b^{3} c \,d^{2} x^{2}+15 b^{4} c^{2} d \,x^{2}+30 b^{3} d^{3} x^{3}+a^{2} c^{3} b^{3}+6 a^{2} b^{2} c \,d^{2} x +18 a \,b^{3} c^{2} d x +30 a \,b^{2} d^{3} x^{2}+4 b^{4} c^{3} x +60 b^{3} c \,d^{2} x^{2}+3 a^{2} b^{2} c^{2} d +6 a^{2} b \,d^{3} x +4 a \,b^{3} c^{3}+48 a \,b^{2} c \,d^{2} x +36 b^{3} c^{2} d x +90 b^{2} d^{3} x^{2}+6 a^{2} b c \,d^{2}+18 c^{2} d a \,b^{2}+60 a b \,d^{3} x +6 b^{3} c^{3}+120 b^{2} c \,d^{2} x +6 a^{2} d^{3}+48 a b c \,d^{2}+36 b^{2} c^{2} d +180 b \,d^{3} x +60 a \,d^{3}+120 d^{2} c b +180 d^{3}\right )}{b^{4}}\) | \(460\) |
| orering | \(-\frac {2 \left (d^{3} b^{5} x^{5}+2 a \,b^{4} d^{3} x^{4}+3 b^{5} c \,d^{2} x^{4}+a^{2} b^{3} d^{3} x^{3}+6 a \,b^{4} c \,d^{2} x^{3}+3 b^{5} c^{2} d \,x^{3}+7 b^{4} d^{3} x^{4}+3 a^{2} b^{3} c \,d^{2} x^{2}+6 a \,b^{4} c^{2} d \,x^{2}+10 a \,b^{3} d^{3} x^{3}+b^{5} c^{3} x^{2}+18 b^{4} c \,d^{2} x^{3}+3 a^{2} b^{3} c^{2} d x +3 a^{2} b^{2} d^{3} x^{2}+2 a \,b^{4} c^{3} x +24 a \,b^{3} c \,d^{2} x^{2}+15 b^{4} c^{2} d \,x^{2}+30 b^{3} d^{3} x^{3}+a^{2} c^{3} b^{3}+6 a^{2} b^{2} c \,d^{2} x +18 a \,b^{3} c^{2} d x +30 a \,b^{2} d^{3} x^{2}+4 b^{4} c^{3} x +60 b^{3} c \,d^{2} x^{2}+3 a^{2} b^{2} c^{2} d +6 a^{2} b \,d^{3} x +4 a \,b^{3} c^{3}+48 a \,b^{2} c \,d^{2} x +36 b^{3} c^{2} d x +90 b^{2} d^{3} x^{2}+6 a^{2} b c \,d^{2}+18 c^{2} d a \,b^{2}+60 a b \,d^{3} x +6 b^{3} c^{3}+120 b^{2} c \,d^{2} x +6 a^{2} d^{3}+48 a b c \,d^{2}+36 b^{2} c^{2} d +180 b \,d^{3} x +60 a \,d^{3}+120 d^{2} c b +180 d^{3}\right ) {\mathrm e}^{-b x -a} \left (1+b x +a +\frac {\left (b x +a \right )^{2}}{2}\right )}{b^{4} \left (b^{2} x^{2}+2 b x a +a^{2}+2 b x +2 a +2\right )}\) | \(501\) |
| meijerg | \(\frac {2 \,{\mathrm e}^{-a} d^{3} \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 {2 d^{3} {\mathrm e}^{-a} 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^{4}}+\frac {d^{3} {\mathrm e}^{-a} a^{2} \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 {2 d^{3} {\mathrm e}^{-a} \left (a b +b \right ) \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^{5}}+\frac {d^{3} {\mathrm e}^{-a} \left (120-\frac {\left (6 b^{5} x^{5}+30 b^{4} x^{4}+120 b^{3} x^{3}+360 b^{2} x^{2}+720 b x +720\right ) {\mathrm e}^{-b x}}{6}\right )}{b^{4}}+\frac {6 \,{\mathrm e}^{-a} d^{2} c \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b^{3}}+\frac {6 d^{2} c \,{\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 {3 d^{2} c \,{\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 {6 d^{2} c \,{\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 {3 d^{2} c \,{\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 {6 \,{\mathrm e}^{-a} d \,c^{2} \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}+\frac {6 d \,c^{2} {\mathrm e}^{-a} a \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}+\frac {3 d \,c^{2} {\mathrm e}^{-a} a^{2} \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}+\frac {6 c^{2} 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 {3 d \,c^{2} {\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 \,{\mathrm e}^{-a} c^{3} \left (1-{\mathrm e}^{-b x}\right )}{b}+\frac {2 c^{3} {\mathrm e}^{-a} a \left (1-{\mathrm e}^{-b x}\right )}{b}+\frac {c^{3} {\mathrm e}^{-a} a^{2} \left (1-{\mathrm e}^{-b x}\right )}{b}+\frac {2 c^{3} {\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^{3} {\mathrm e}^{-a} \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b}\) | \(766\) |
| parallelrisch | \(-\frac {24 x^{2} {\mathrm e}^{-b x -a} a \,b^{3} c \,d^{2}+18 x \,{\mathrm e}^{-b x -a} a \,b^{3} c^{2} d +48 x \,{\mathrm e}^{-b x -a} a \,b^{2} c \,d^{2}+6 x \,{\mathrm e}^{-b x -a} a^{2} b^{2} c \,d^{2}+6 x^{3} {\mathrm e}^{-b x -a} a \,b^{4} c \,d^{2}+3 x^{2} {\mathrm e}^{-b x -a} a^{2} b^{3} c \,d^{2}+6 x^{2} {\mathrm e}^{-b x -a} a \,b^{4} c^{2} d +3 x \,{\mathrm e}^{-b x -a} a^{2} b^{3} c^{2} d +30 x^{3} {\mathrm e}^{-b x -a} d^{3} b^{3}+90 x^{2} {\mathrm e}^{-b x -a} b^{2} d^{3}+180 x \,{\mathrm e}^{-b x -a} b \,d^{3}+6 \,{\mathrm e}^{-b x -a} a^{2} d^{3}+60 \,{\mathrm e}^{-b x -a} a \,d^{3}+36 \,{\mathrm e}^{-b x -a} b^{2} c^{2} d +18 x^{3} {\mathrm e}^{-b x -a} b^{4} c \,d^{2}+15 x^{2} {\mathrm e}^{-b x -a} b^{4} c^{2} d +120 \,{\mathrm e}^{-b x -a} b c \,d^{2}+180 \,{\mathrm e}^{-b x -a} d^{3}+7 d^{3} {\mathrm e}^{-b x -a} x^{4} b^{4}+4 x \,{\mathrm e}^{-b x -a} b^{4} c^{3}+2 x^{4} {\mathrm e}^{-b x -a} a \,b^{4} d^{3}+3 x^{4} {\mathrm e}^{-b x -a} b^{5} c \,d^{2}+x^{3} {\mathrm e}^{-b x -a} a^{2} b^{3} d^{3}+3 x^{3} {\mathrm e}^{-b x -a} b^{5} c^{2} d +2 x \,{\mathrm e}^{-b x -a} a \,b^{4} c^{3}+6 \,{\mathrm e}^{-b x -a} b^{3} c^{3}+60 x^{2} {\mathrm e}^{-b x -a} b^{3} c \,d^{2}+36 x \,{\mathrm e}^{-b x -a} b^{3} c^{2} d +120 x \,{\mathrm e}^{-b x -a} b^{2} c \,d^{2}+d^{3} b^{5} {\mathrm e}^{-b x -a} x^{5}+x^{2} {\mathrm e}^{-b x -a} b^{5} c^{3}+{\mathrm e}^{-b x -a} a^{2} b^{3} c^{3}+4 \,{\mathrm e}^{-b x -a} a \,b^{3} c^{3}+10 x^{3} {\mathrm e}^{-b x -a} a \,b^{3} d^{3}+3 x^{2} {\mathrm e}^{-b x -a} a^{2} b^{2} d^{3}+30 x^{2} {\mathrm e}^{-b x -a} a \,b^{2} d^{3}+6 x \,{\mathrm e}^{-b x -a} a^{2} b \,d^{3}+3 \,{\mathrm e}^{-b x -a} a^{2} b^{2} c^{2} d +60 x \,{\mathrm e}^{-b x -a} a b \,d^{3}+6 \,{\mathrm e}^{-b x -a} a^{2} b c \,d^{2}+18 \,{\mathrm e}^{-b x -a} a \,b^{2} c^{2} d +48 \,{\mathrm e}^{-b x -a} a b c \,d^{2}}{b^{4}}\) | \(829\) |
| parts | \(\text {Expression too large to display}\) | \(1648\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1818\) |
| default | \(\text {Expression too large to display}\) | \(1818\) |
Input:
int(2*(d*x+c)^3*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2),x,method=_RETURNVERBOS E)
Output:
(-2*a*d^3-3*b*c*d^2-7*d^3)*x^4*exp(-b*x-a)-(a^2*b^3*c^3+3*a^2*b^2*c^2*d+4* a*b^3*c^3+6*a^2*b*c*d^2+18*a*b^2*c^2*d+6*b^3*c^3+6*a^2*d^3+48*a*b*c*d^2+36 *b^2*c^2*d+60*a*d^3+120*b*c*d^2+180*d^3)/b^4*exp(-b*x-a)-d^3*b*x^5*exp(-b* x-a)-(3*a^2*b*c*d^2+6*a*b^2*c^2*d+b^3*c^3+3*a^2*d^3+24*a*b*c*d^2+15*b^2*c^ 2*d+30*a*d^3+60*b*c*d^2+90*d^3)/b^2*x^2*exp(-b*x-a)-(3*a^2*b^2*c^2*d+2*a*b ^3*c^3+6*a^2*b*c*d^2+18*a*b^2*c^2*d+4*b^3*c^3+6*a^2*d^3+48*a*b*c*d^2+36*b^ 2*c^2*d+60*a*d^3+120*b*c*d^2+180*d^3)/b^3*x*exp(-b*x-a)-d*(a^2*d^2+6*a*b*c *d+3*b^2*c^2+10*a*d^2+18*b*c*d+30*d^2)/b*x^3*exp(-b*x-a)
Leaf count of result is larger than twice the leaf count of optimal. 401 vs. \(2 (129) = 258\).
Time = 0.10 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.86 \[ \int (c+d x)^3 \Gamma (3,a+b x) \, dx=-\frac {{\left (b^{6} d^{3} x^{6} + 4 \, {\left (a^{2} + 4 \, a + 6\right )} b^{3} c^{3} + 12 \, {\left (a^{2} + 6 \, a + 12\right )} b^{2} c^{2} d + 2 \, {\left (2 \, b^{6} c d^{2} + {\left (a + 3\right )} b^{5} d^{3}\right )} x^{5} + 24 \, {\left (a^{2} + 8 \, a + 20\right )} b c d^{2} + {\left (6 \, b^{6} c^{2} d + 4 \, {\left (2 \, a + 5\right )} b^{5} c d^{2} + {\left (a^{2} + 10 \, a + 30\right )} b^{4} d^{3}\right )} x^{4} + 24 \, {\left (a^{2} + 10 \, a + 30\right )} d^{3} + 4 \, {\left (b^{6} c^{3} + 3 \, {\left (a + 2\right )} b^{5} c^{2} d + {\left (a^{2} + 8 \, a + 20\right )} b^{4} c d^{2} + {\left (a^{2} + 10 \, a + 30\right )} b^{3} d^{3}\right )} x^{3} + 2 \, {\left (2 \, {\left (2 \, a + 3\right )} b^{5} c^{3} + 3 \, {\left (a^{2} + 6 \, a + 12\right )} b^{4} c^{2} d + 6 \, {\left (a^{2} + 8 \, a + 20\right )} b^{3} c d^{2} + 6 \, {\left (a^{2} + 10 \, a + 30\right )} b^{2} d^{3}\right )} x^{2} + 4 \, {\left ({\left (a^{2} + 4 \, a + 6\right )} b^{4} c^{3} + 3 \, {\left (a^{2} + 6 \, a + 12\right )} b^{3} c^{2} d + 6 \, {\left (a^{2} + 8 \, a + 20\right )} b^{2} c d^{2} + 6 \, {\left (a^{2} + 10 \, a + 30\right )} b d^{3}\right )} x\right )} e^{\left (-b x - a\right )} - {\left (b^{4} d^{3} x^{4} + 4 \, b^{4} c d^{2} x^{3} + 6 \, b^{4} c^{2} d x^{2} + 4 \, b^{4} c^{3} x\right )} \Gamma \left (3, b x + a\right )}{4 \, b^{4}} \] Input:
integrate((d*x+c)^3*gamma(3,b*x+a),x, algorithm="fricas")
Output:
-1/4*((b^6*d^3*x^6 + 4*(a^2 + 4*a + 6)*b^3*c^3 + 12*(a^2 + 6*a + 12)*b^2*c ^2*d + 2*(2*b^6*c*d^2 + (a + 3)*b^5*d^3)*x^5 + 24*(a^2 + 8*a + 20)*b*c*d^2 + (6*b^6*c^2*d + 4*(2*a + 5)*b^5*c*d^2 + (a^2 + 10*a + 30)*b^4*d^3)*x^4 + 24*(a^2 + 10*a + 30)*d^3 + 4*(b^6*c^3 + 3*(a + 2)*b^5*c^2*d + (a^2 + 8*a + 20)*b^4*c*d^2 + (a^2 + 10*a + 30)*b^3*d^3)*x^3 + 2*(2*(2*a + 3)*b^5*c^3 + 3*(a^2 + 6*a + 12)*b^4*c^2*d + 6*(a^2 + 8*a + 20)*b^3*c*d^2 + 6*(a^2 + 1 0*a + 30)*b^2*d^3)*x^2 + 4*((a^2 + 4*a + 6)*b^4*c^3 + 3*(a^2 + 6*a + 12)*b ^3*c^2*d + 6*(a^2 + 8*a + 20)*b^2*c*d^2 + 6*(a^2 + 10*a + 30)*b*d^3)*x)*e^ (-b*x - a) - (b^4*d^3*x^4 + 4*b^4*c*d^2*x^3 + 6*b^4*c^2*d*x^2 + 4*b^4*c^3* x)*gamma(3, b*x + a))/b^4
Leaf count of result is larger than twice the leaf count of optimal. 741 vs. \(2 (342) = 684\).
Time = 0.20 (sec) , antiderivative size = 741, normalized size of antiderivative = 5.29 \[ \int (c+d x)^3 \Gamma (3,a+b x) \, dx=\begin {cases} \frac {\left (- a^{2} b^{3} c^{3} - 3 a^{2} b^{3} c^{2} d x - 3 a^{2} b^{3} c d^{2} x^{2} - a^{2} b^{3} d^{3} x^{3} - 3 a^{2} b^{2} c^{2} d - 6 a^{2} b^{2} c d^{2} x - 3 a^{2} b^{2} d^{3} x^{2} - 6 a^{2} b c d^{2} - 6 a^{2} b d^{3} x - 6 a^{2} d^{3} - 2 a b^{4} c^{3} x - 6 a b^{4} c^{2} d x^{2} - 6 a b^{4} c d^{2} x^{3} - 2 a b^{4} d^{3} x^{4} - 4 a b^{3} c^{3} - 18 a b^{3} c^{2} d x - 24 a b^{3} c d^{2} x^{2} - 10 a b^{3} d^{3} x^{3} - 18 a b^{2} c^{2} d - 48 a b^{2} c d^{2} x - 30 a b^{2} d^{3} x^{2} - 48 a b c d^{2} - 60 a b d^{3} x - 60 a d^{3} - b^{5} c^{3} x^{2} - 3 b^{5} c^{2} d x^{3} - 3 b^{5} c d^{2} x^{4} - b^{5} d^{3} x^{5} - 4 b^{4} c^{3} x - 15 b^{4} c^{2} d x^{2} - 18 b^{4} c d^{2} x^{3} - 7 b^{4} d^{3} x^{4} - 6 b^{3} c^{3} - 36 b^{3} c^{2} d x - 60 b^{3} c d^{2} x^{2} - 30 b^{3} d^{3} x^{3} - 36 b^{2} c^{2} d - 120 b^{2} c d^{2} x - 90 b^{2} d^{3} x^{2} - 120 b c d^{2} - 180 b d^{3} x - 180 d^{3}\right ) e^{- a - b x}}{b^{4}} & \text {for}\: b^{4} \neq 0 \\\frac {b^{2} d^{3} x^{6}}{6} + x^{5} \cdot \left (\frac {2 a b d^{3}}{5} + \frac {3 b^{2} c d^{2}}{5} + \frac {2 b d^{3}}{5}\right ) + x^{4} \left (\frac {a^{2} d^{3}}{4} + \frac {3 a b c d^{2}}{2} + \frac {a d^{3}}{2} + \frac {3 b^{2} c^{2} d}{4} + \frac {3 b c d^{2}}{2} + \frac {d^{3}}{2}\right ) + x^{3} \left (a^{2} c d^{2} + 2 a b c^{2} d + 2 a c d^{2} + \frac {b^{2} c^{3}}{3} + 2 b c^{2} d + 2 c d^{2}\right ) + x^{2} \cdot \left (\frac {3 a^{2} c^{2} d}{2} + a b c^{3} + 3 a c^{2} d + b c^{3} + 3 c^{2} d\right ) + x \left (a^{2} c^{3} + 2 a c^{3} + 2 c^{3}\right ) & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)**3*uppergamma(3,b*x+a),x)
Output:
Piecewise(((-a**2*b**3*c**3 - 3*a**2*b**3*c**2*d*x - 3*a**2*b**3*c*d**2*x* *2 - a**2*b**3*d**3*x**3 - 3*a**2*b**2*c**2*d - 6*a**2*b**2*c*d**2*x - 3*a **2*b**2*d**3*x**2 - 6*a**2*b*c*d**2 - 6*a**2*b*d**3*x - 6*a**2*d**3 - 2*a *b**4*c**3*x - 6*a*b**4*c**2*d*x**2 - 6*a*b**4*c*d**2*x**3 - 2*a*b**4*d**3 *x**4 - 4*a*b**3*c**3 - 18*a*b**3*c**2*d*x - 24*a*b**3*c*d**2*x**2 - 10*a* b**3*d**3*x**3 - 18*a*b**2*c**2*d - 48*a*b**2*c*d**2*x - 30*a*b**2*d**3*x* *2 - 48*a*b*c*d**2 - 60*a*b*d**3*x - 60*a*d**3 - b**5*c**3*x**2 - 3*b**5*c **2*d*x**3 - 3*b**5*c*d**2*x**4 - b**5*d**3*x**5 - 4*b**4*c**3*x - 15*b**4 *c**2*d*x**2 - 18*b**4*c*d**2*x**3 - 7*b**4*d**3*x**4 - 6*b**3*c**3 - 36*b **3*c**2*d*x - 60*b**3*c*d**2*x**2 - 30*b**3*d**3*x**3 - 36*b**2*c**2*d - 120*b**2*c*d**2*x - 90*b**2*d**3*x**2 - 120*b*c*d**2 - 180*b*d**3*x - 180* d**3)*exp(-a - b*x)/b**4, Ne(b**4, 0)), (b**2*d**3*x**6/6 + x**5*(2*a*b*d* *3/5 + 3*b**2*c*d**2/5 + 2*b*d**3/5) + x**4*(a**2*d**3/4 + 3*a*b*c*d**2/2 + a*d**3/2 + 3*b**2*c**2*d/4 + 3*b*c*d**2/2 + d**3/2) + x**3*(a**2*c*d**2 + 2*a*b*c**2*d + 2*a*c*d**2 + b**2*c**3/3 + 2*b*c**2*d + 2*c*d**2) + x**2* (3*a**2*c**2*d/2 + a*b*c**3 + 3*a*c**2*d + b*c**3 + 3*c**2*d) + x*(a**2*c* *3 + 2*a*c**3 + 2*c**3), True))
\[ \int (c+d x)^3 \Gamma (3,a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \Gamma \left (3, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)^3*gamma(3,b*x+a),x, algorithm="maxima")
Output:
((b*x + a)*gamma(3, b*x + a) - gamma(4, b*x + a))*c^3/b + integrate(d^3*x^ 3*gamma(3, b*x + a) + 3*c*d^2*x^2*gamma(3, b*x + a) + 3*c^2*d*x*gamma(3, b *x + a), x)
\[ \int (c+d x)^3 \Gamma (3,a+b x) \, dx=\int { {\left (d x + c\right )}^{3} \Gamma \left (3, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)^3*gamma(3,b*x+a),x, algorithm="giac")
Output:
integrate((d*x + c)^3*gamma(3, b*x + a), x)
Time = 0.13 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.94 \[ \int (c+d x)^3 \Gamma (3,a+b x) \, dx=-x^3\,{\mathrm {e}}^{-a-b\,x}\,\left (\frac {a^2\,d^3+10\,a\,d^3+30\,d^3}{b}+18\,c\,d^2+6\,a\,c\,d^2+3\,b\,c^2\,d\right )-\frac {{\mathrm {e}}^{-a-b\,x}\,\left (b^3\,\left (a^2\,c^3+4\,a\,c^3+6\,c^3\right )+b^2\,\left (3\,d\,a^2\,c^2+18\,d\,a\,c^2+36\,d\,c^2\right )+60\,a\,d^3+180\,d^3+b\,\left (6\,c\,a^2\,d^2+48\,c\,a\,d^2+120\,c\,d^2\right )+6\,a^2\,d^3\right )}{b^4}-b\,d^3\,x^5\,{\mathrm {e}}^{-a-b\,x}-d^2\,x^4\,{\mathrm {e}}^{-a-b\,x}\,\left (7\,d+2\,a\,d+3\,b\,c\right )-\frac {x\,{\mathrm {e}}^{-a-b\,x}\,\left (3\,a^2\,b^2\,c^2\,d+6\,a^2\,b\,c\,d^2+6\,a^2\,d^3+2\,a\,b^3\,c^3+18\,a\,b^2\,c^2\,d+48\,a\,b\,c\,d^2+60\,a\,d^3+4\,b^3\,c^3+36\,b^2\,c^2\,d+120\,b\,c\,d^2+180\,d^3\right )}{b^3}-\frac {x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (3\,a^2\,b\,c\,d^2+3\,a^2\,d^3+6\,a\,b^2\,c^2\,d+24\,a\,b\,c\,d^2+30\,a\,d^3+b^3\,c^3+15\,b^2\,c^2\,d+60\,b\,c\,d^2+90\,d^3\right )}{b^2} \] Input:
int(2*exp(- a - b*x)*(c + d*x)^3*(a + b*x + (a + b*x)^2/2 + 1),x)
Output:
- x^3*exp(- a - b*x)*((10*a*d^3 + 30*d^3 + a^2*d^3)/b + 18*c*d^2 + 6*a*c*d ^2 + 3*b*c^2*d) - (exp(- a - b*x)*(b^3*(4*a*c^3 + 6*c^3 + a^2*c^3) + b^2*( 36*c^2*d + 3*a^2*c^2*d + 18*a*c^2*d) + 60*a*d^3 + 180*d^3 + b*(120*c*d^2 + 6*a^2*c*d^2 + 48*a*c*d^2) + 6*a^2*d^3))/b^4 - b*d^3*x^5*exp(- a - b*x) - d^2*x^4*exp(- a - b*x)*(7*d + 2*a*d + 3*b*c) - (x*exp(- a - b*x)*(60*a*d^3 + 180*d^3 + 6*a^2*d^3 + 4*b^3*c^3 + 2*a*b^3*c^3 + 36*b^2*c^2*d + 120*b*c* d^2 + 3*a^2*b^2*c^2*d + 48*a*b*c*d^2 + 18*a*b^2*c^2*d + 6*a^2*b*c*d^2))/b^ 3 - (x^2*exp(- a - b*x)*(30*a*d^3 + 90*d^3 + 3*a^2*d^3 + b^3*c^3 + 15*b^2* c^2*d + 60*b*c*d^2 + 24*a*b*c*d^2 + 6*a*b^2*c^2*d + 3*a^2*b*c*d^2))/b^2
Time = 0.21 (sec) , antiderivative size = 462, normalized size of antiderivative = 3.30 \[ \int (c+d x)^3 \Gamma (3,a+b x) \, dx=\frac {-b^{5} d^{3} x^{5}-2 a \,b^{4} d^{3} x^{4}-3 b^{5} c \,d^{2} x^{4}-a^{2} b^{3} d^{3} x^{3}-6 a \,b^{4} c \,d^{2} x^{3}-3 b^{5} c^{2} d \,x^{3}-7 b^{4} d^{3} x^{4}-3 a^{2} b^{3} c \,d^{2} x^{2}-6 a \,b^{4} c^{2} d \,x^{2}-10 a \,b^{3} d^{3} x^{3}-b^{5} c^{3} x^{2}-18 b^{4} c \,d^{2} x^{3}-3 a^{2} b^{3} c^{2} d x -3 a^{2} b^{2} d^{3} x^{2}-2 a \,b^{4} c^{3} x -24 a \,b^{3} c \,d^{2} x^{2}-15 b^{4} c^{2} d \,x^{2}-30 b^{3} d^{3} x^{3}-a^{2} b^{3} c^{3}-6 a^{2} b^{2} c \,d^{2} x -18 a \,b^{3} c^{2} d x -30 a \,b^{2} d^{3} x^{2}-4 b^{4} c^{3} x -60 b^{3} c \,d^{2} x^{2}-3 a^{2} b^{2} c^{2} d -6 a^{2} b \,d^{3} x -4 a \,b^{3} c^{3}-48 a \,b^{2} c \,d^{2} x -36 b^{3} c^{2} d x -90 b^{2} d^{3} x^{2}-6 a^{2} b c \,d^{2}-18 a \,b^{2} c^{2} d -60 a b \,d^{3} x -6 b^{3} c^{3}-120 b^{2} c \,d^{2} x -6 a^{2} d^{3}-48 a b c \,d^{2}-36 b^{2} c^{2} d -180 b \,d^{3} x -60 a \,d^{3}-120 b c \,d^{2}-180 d^{3}}{e^{b x +a} b^{4}} \] Input:
int(2*(d*x+c)^3*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2),x)
Output:
( - a**2*b**3*c**3 - 3*a**2*b**3*c**2*d*x - 3*a**2*b**3*c*d**2*x**2 - a**2 *b**3*d**3*x**3 - 3*a**2*b**2*c**2*d - 6*a**2*b**2*c*d**2*x - 3*a**2*b**2* d**3*x**2 - 6*a**2*b*c*d**2 - 6*a**2*b*d**3*x - 6*a**2*d**3 - 2*a*b**4*c** 3*x - 6*a*b**4*c**2*d*x**2 - 6*a*b**4*c*d**2*x**3 - 2*a*b**4*d**3*x**4 - 4 *a*b**3*c**3 - 18*a*b**3*c**2*d*x - 24*a*b**3*c*d**2*x**2 - 10*a*b**3*d**3 *x**3 - 18*a*b**2*c**2*d - 48*a*b**2*c*d**2*x - 30*a*b**2*d**3*x**2 - 48*a *b*c*d**2 - 60*a*b*d**3*x - 60*a*d**3 - b**5*c**3*x**2 - 3*b**5*c**2*d*x** 3 - 3*b**5*c*d**2*x**4 - b**5*d**3*x**5 - 4*b**4*c**3*x - 15*b**4*c**2*d*x **2 - 18*b**4*c*d**2*x**3 - 7*b**4*d**3*x**4 - 6*b**3*c**3 - 36*b**3*c**2* d*x - 60*b**3*c*d**2*x**2 - 30*b**3*d**3*x**3 - 36*b**2*c**2*d - 120*b**2* c*d**2*x - 90*b**2*d**3*x**2 - 120*b*c*d**2 - 180*b*d**3*x - 180*d**3)/(e* *(a + b*x)*b**4)