Integrand size = 13, antiderivative size = 84 \[ \int (c+d x) \Gamma (3,a+b x) \, dx=-\frac {(b c-a d)^2 \Gamma (3,a+b x)}{2 b^2 d}+\frac {(c+d x)^2 \Gamma (3,a+b x)}{2 d}-\frac {(b c-a d) \Gamma (4,a+b x)}{b^2}-\frac {d \Gamma (5,a+b x)}{2 b^2} \] Output:
-(-a*d+b*c)^2*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2)/b^2/d+(d*x+c)^2*exp(-b*x -a)*(1+b*x+a+1/2*(b*x+a)^2)/d-6*(-a*d+b*c)*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a )^2+1/6*(b*x+a)^3)/b^2-12*d*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2+1/6*(b*x+a) ^3+1/24*(b*x+a)^4)/b^2
Time = 0.08 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.74 \[ \int (c+d x) \Gamma (3,a+b x) \, dx=\frac {e^{-a-b x} \left (-2 \left (12+6 a+a^2\right ) d-b^4 x^3 (2 c+d x)-2 b^3 x^2 ((3+2 a) c+(2+a) d x)-2 b \left (\left (6+4 a+a^2\right ) c+\left (12+6 a+a^2\right ) d x\right )-b^2 x \left (2 \left (6+4 a+a^2\right ) c+\left (12+6 a+a^2\right ) d x\right )+b^2 e^{a+b x} x (2 c+d x) \Gamma (3,a+b x)\right )}{2 b^2} \] Input:
Integrate[(c + d*x)*Gamma[3, a + b*x],x]
Output:
(E^(-a - b*x)*(-2*(12 + 6*a + a^2)*d - b^4*x^3*(2*c + d*x) - 2*b^3*x^2*((3 + 2*a)*c + (2 + a)*d*x) - 2*b*((6 + 4*a + a^2)*c + (12 + 6*a + a^2)*d*x) - b^2*x*(2*(6 + 4*a + a^2)*c + (12 + 6*a + a^2)*d*x) + b^2*E^(a + b*x)*x*( 2*c + d*x)*Gamma[3, a + b*x]))/(2*b^2)
Leaf count is larger than twice the leaf count of optimal. \(349\) vs. \(2(84)=168\).
Time = 0.81 (sec) , antiderivative size = 349, normalized size of antiderivative = 4.15, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, 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.
| \(\displaystyle \int (c+d x) \Gamma (3,a+b x) \, dx\) |
| \(\Big \downarrow \) 7119 |
| \(\displaystyle \frac {b \int e^{-a-b x} (a+b x)^2 (c+d x)^2dx}{2 d}+\frac {(c+d x)^2 \Gamma (3,a+b x)}{2 d}\) |
| \(\Big \downarrow \) 2626 |
| \(\displaystyle \frac {b \int \left (\frac {d^2 e^{-a-b x} (a+b x)^4}{b^2}+\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^3}{b^2}+\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^2}{b^2}\right )dx}{2 d}+\frac {(c+d x)^2 \Gamma (3,a+b x)}{2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {b \left (-\frac {2 d e^{-a-b x} (a+b x)^3 (b c-a d)}{b^3}-\frac {e^{-a-b x} (a+b x)^2 (b c-a d)^2}{b^3}-\frac {6 d e^{-a-b x} (a+b x)^2 (b c-a d)}{b^3}-\frac {2 e^{-a-b x} (a+b x) (b c-a d)^2}{b^3}-\frac {12 d e^{-a-b x} (a+b x) (b c-a d)}{b^3}-\frac {2 e^{-a-b x} (b c-a d)^2}{b^3}-\frac {12 d e^{-a-b x} (b c-a d)}{b^3}-\frac {d^2 e^{-a-b x} (a+b x)^4}{b^3}-\frac {4 d^2 e^{-a-b x} (a+b x)^3}{b^3}-\frac {12 d^2 e^{-a-b x} (a+b x)^2}{b^3}-\frac {24 d^2 e^{-a-b x} (a+b x)}{b^3}-\frac {24 d^2 e^{-a-b x}}{b^3}\right )}{2 d}+\frac {(c+d x)^2 \Gamma (3,a+b x)}{2 d}\) |
Input:
Int[(c + d*x)*Gamma[3, a + b*x],x]
Output:
(b*((-24*d^2*E^(-a - b*x))/b^3 - (12*d*(b*c - a*d)*E^(-a - b*x))/b^3 - (2* (b*c - a*d)^2*E^(-a - b*x))/b^3 - (24*d^2*E^(-a - b*x)*(a + b*x))/b^3 - (1 2*d*(b*c - a*d)*E^(-a - b*x)*(a + b*x))/b^3 - (2*(b*c - a*d)^2*E^(-a - b*x )*(a + b*x))/b^3 - (12*d^2*E^(-a - b*x)*(a + b*x)^2)/b^3 - (6*d*(b*c - a*d )*E^(-a - b*x)*(a + b*x)^2)/b^3 - ((b*c - a*d)^2*E^(-a - b*x)*(a + b*x)^2) /b^3 - (4*d^2*E^(-a - b*x)*(a + b*x)^3)/b^3 - (2*d*(b*c - a*d)*E^(-a - b*x )*(a + b*x)^3)/b^3 - (d^2*E^(-a - b*x)*(a + b*x)^4)/b^3))/(2*d) + ((c + d* x)^2*Gamma[3, a + b*x])/(2*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 = 0.55 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.32
| method | result | size |
| gosper | \(-\frac {{\mathrm e}^{-b x -a} \left (b^{3} d \,x^{3}+2 a \,b^{2} d \,x^{2}+b^{3} c \,x^{2}+b \,a^{2} d x +2 a \,b^{2} c x +5 b^{2} d \,x^{2}+b \,a^{2} c +6 a b d x +4 b^{2} c x +d \,a^{2}+4 a b c +12 b d x +6 a d +6 c b +12 d \right )}{b^{2}}\) | \(111\) |
| risch | \(-\frac {{\mathrm e}^{-b x -a} \left (b^{3} d \,x^{3}+2 a \,b^{2} d \,x^{2}+b^{3} c \,x^{2}+b \,a^{2} d x +2 a \,b^{2} c x +5 b^{2} d \,x^{2}+b \,a^{2} c +6 a b d x +4 b^{2} c x +d \,a^{2}+4 a b c +12 b d x +6 a d +6 c b +12 d \right )}{b^{2}}\) | \(111\) |
| norman | \(\left (-2 a d -c b -5 d \right ) x^{2} {\mathrm e}^{-b x -a}-\frac {\left (b \,a^{2} c +d \,a^{2}+4 a b c +6 a d +6 c b +12 d \right ) {\mathrm e}^{-b x -a}}{b^{2}}-b d \,x^{3} {\mathrm e}^{-b x -a}-\frac {\left (d \,a^{2}+2 a b c +6 a d +4 c b +12 d \right ) x \,{\mathrm e}^{-b x -a}}{b}\) | \(122\) |
| orering | \(-\frac {2 \left (b^{3} d \,x^{3}+2 a \,b^{2} d \,x^{2}+b^{3} c \,x^{2}+b \,a^{2} d x +2 a \,b^{2} c x +5 b^{2} d \,x^{2}+b \,a^{2} c +6 a b d x +4 b^{2} c x +d \,a^{2}+4 a b c +12 b d x +6 a d +6 c b +12 d \right ) {\mathrm e}^{-b x -a} \left (1+b x +a +\frac {\left (b x +a \right )^{2}}{2}\right )}{b^{2} \left (b^{2} x^{2}+2 b x a +a^{2}+2 b x +2 a +2\right )}\) | \(152\) |
| parallelrisch | \(-\frac {b^{3} d \,{\mathrm e}^{-b x -a} x^{3}+2 x^{2} {\mathrm e}^{-b x -a} a \,b^{2} d +x^{2} {\mathrm e}^{-b x -a} b^{3} c +5 d \,x^{2} {\mathrm e}^{-b x -a} b^{2}+x \,{\mathrm e}^{-b x -a} a^{2} b d +2 x \,{\mathrm e}^{-b x -a} a \,b^{2} c +6 x \,{\mathrm e}^{-b x -a} a b d +4 x \,{\mathrm e}^{-b x -a} b^{2} c +{\mathrm e}^{-b x -a} a^{2} b c +12 \,{\mathrm e}^{-b x -a} b d x +{\mathrm e}^{-b x -a} a^{2} d +4 \,{\mathrm e}^{-b x -a} a b c +6 \,{\mathrm e}^{-b x -a} a d +6 \,{\mathrm e}^{-b x -a} b c +12 \,{\mathrm e}^{-b x -a} d}{b^{2}}\) | \(237\) |
| meijerg | \(\frac {2 d \,{\mathrm e}^{-a} \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}+\frac {2 d \,{\mathrm e}^{-a} a \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}+\frac {d \,{\mathrm e}^{-a} a^{2} \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}+\frac {d \,{\mathrm e}^{-a} \left (6-\frac {\left (4 b^{3} x^{3}+12 b^{2} x^{2}+24 b x +24\right ) {\mathrm e}^{-b x}}{4}\right )}{b^{2}}+\frac {2 d \,{\mathrm e}^{-a} \left (a b +b \right ) \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b^{3}}+\frac {2 c \,{\mathrm e}^{-a} \left (1-{\mathrm e}^{-b x}\right )}{b}+\frac {2 c \,{\mathrm e}^{-a} a \left (1-{\mathrm e}^{-b x}\right )}{b}+\frac {c \,{\mathrm e}^{-a} a^{2} \left (1-{\mathrm e}^{-b x}\right )}{b}+\frac {c \,{\mathrm e}^{-a} \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b}+\frac {2 c \,{\mathrm e}^{-a} \left (a b +b \right ) \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}\) | \(280\) |
| derivativedivides | \(-\frac {c \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )+2 \,{\mathrm e}^{-b x -a} c -2 c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )-\frac {2 \,{\mathrm e}^{-b x -a} d a}{b}-\frac {2 d \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}+\frac {2 d \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b}-\frac {d \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b}+\frac {2 d a \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}-\frac {d a \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b}}{b}\) | \(387\) |
| default | \(-\frac {c \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )+2 \,{\mathrm e}^{-b x -a} c -2 c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )-\frac {2 \,{\mathrm e}^{-b x -a} d a}{b}-\frac {2 d \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}+\frac {2 d \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b}-\frac {d \left ({\mathrm e}^{-b x -a} \left (-b x -a \right )^{3}-3 \left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}+6 \left (-b x -a \right ) {\mathrm e}^{-b x -a}-6 \,{\mathrm e}^{-b x -a}\right )}{b}+\frac {2 d a \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}-\frac {d a \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b}}{b}\) | \(387\) |
| parts | \(-b d \,x^{3} {\mathrm e}^{-b x -a}-2 \,{\mathrm e}^{-b x -a} a d \,x^{2}-{\mathrm e}^{-b x -a} b c \,x^{2}-\frac {{\mathrm e}^{-b x -a} a^{2} d x}{b}-2 \,{\mathrm e}^{-b x -a} a c x -2 \,{\mathrm e}^{-b x -a} d \,x^{2}-\frac {{\mathrm e}^{-b x -a} a^{2} c}{b}-\frac {2 \,{\mathrm e}^{-b x -a} d x a}{b}-2 \,{\mathrm e}^{-b x -a} c x -\frac {2 \,{\mathrm e}^{-b x -a} a c}{b}-\frac {2 \,{\mathrm e}^{-b x -a} d x}{b}-\frac {2 \,{\mathrm e}^{-b x -a} c}{b}-\frac {2 \,{\mathrm e}^{-b x -a} d +2 \,{\mathrm e}^{-b x -a} b c -2 \,{\mathrm e}^{-b x -a} a d -4 d \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )+3 d \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )-2 c b \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )+2 d a \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b^{2}}\) | \(392\) |
Input:
int(2*(d*x+c)*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2),x,method=_RETURNVERBOSE)
Output:
-exp(-b*x-a)*(b^3*d*x^3+2*a*b^2*d*x^2+b^3*c*x^2+a^2*b*d*x+2*a*b^2*c*x+5*b^ 2*d*x^2+a^2*b*c+6*a*b*d*x+4*b^2*c*x+a^2*d+4*a*b*c+12*b*d*x+6*a*d+6*b*c+12* d)/b^2
Time = 0.09 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.77 \[ \int (c+d x) \Gamma (3,a+b x) \, dx=-\frac {{\left (b^{4} d x^{4} + 2 \, {\left (b^{4} c + {\left (a + 2\right )} b^{3} d\right )} x^{3} + 2 \, {\left (a^{2} + 4 \, a + 6\right )} b c + {\left (2 \, {\left (2 \, a + 3\right )} b^{3} c + {\left (a^{2} + 6 \, a + 12\right )} b^{2} d\right )} x^{2} + 2 \, {\left (a^{2} + 6 \, a + 12\right )} d + 2 \, {\left ({\left (a^{2} + 4 \, a + 6\right )} b^{2} c + {\left (a^{2} + 6 \, a + 12\right )} b d\right )} x\right )} e^{\left (-b x - a\right )} - {\left (b^{2} d x^{2} + 2 \, b^{2} c x\right )} \Gamma \left (3, b x + a\right )}{2 \, b^{2}} \] Input:
integrate((d*x+c)*gamma(3,b*x+a),x, algorithm="fricas")
Output:
-1/2*((b^4*d*x^4 + 2*(b^4*c + (a + 2)*b^3*d)*x^3 + 2*(a^2 + 4*a + 6)*b*c + (2*(2*a + 3)*b^3*c + (a^2 + 6*a + 12)*b^2*d)*x^2 + 2*(a^2 + 6*a + 12)*d + 2*((a^2 + 4*a + 6)*b^2*c + (a^2 + 6*a + 12)*b*d)*x)*e^(-b*x - a) - (b^2*d *x^2 + 2*b^2*c*x)*gamma(3, b*x + a))/b^2
Time = 0.10 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.40 \[ \int (c+d x) \Gamma (3,a+b x) \, dx=\begin {cases} \frac {\left (- a^{2} b c - a^{2} b d x - a^{2} d - 2 a b^{2} c x - 2 a b^{2} d x^{2} - 4 a b c - 6 a b d x - 6 a d - b^{3} c x^{2} - b^{3} d x^{3} - 4 b^{2} c x - 5 b^{2} d x^{2} - 6 b c - 12 b d x - 12 d\right ) e^{- a - b x}}{b^{2}} & \text {for}\: b^{2} \neq 0 \\\frac {b^{2} d x^{4}}{4} + x^{3} \cdot \left (\frac {2 a b d}{3} + \frac {b^{2} c}{3} + \frac {2 b d}{3}\right ) + x^{2} \left (\frac {a^{2} d}{2} + a b c + a d + b c + d\right ) + x \left (a^{2} c + 2 a c + 2 c\right ) & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)*uppergamma(3,b*x+a),x)
Output:
Piecewise(((-a**2*b*c - a**2*b*d*x - a**2*d - 2*a*b**2*c*x - 2*a*b**2*d*x* *2 - 4*a*b*c - 6*a*b*d*x - 6*a*d - b**3*c*x**2 - b**3*d*x**3 - 4*b**2*c*x - 5*b**2*d*x**2 - 6*b*c - 12*b*d*x - 12*d)*exp(-a - b*x)/b**2, Ne(b**2, 0) ), (b**2*d*x**4/4 + x**3*(2*a*b*d/3 + b**2*c/3 + 2*b*d/3) + x**2*(a**2*d/2 + a*b*c + a*d + b*c + d) + x*(a**2*c + 2*a*c + 2*c), True))
\[ \int (c+d x) \Gamma (3,a+b x) \, dx=\int { {\left (d x + c\right )} \Gamma \left (3, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)*gamma(3,b*x+a),x, algorithm="maxima")
Output:
d*integrate(x*gamma(3, b*x + a), x) + ((b*x + a)*gamma(3, b*x + a) - gamma (4, b*x + a))*c/b
\[ \int (c+d x) \Gamma (3,a+b x) \, dx=\int { {\left (d x + c\right )} \Gamma \left (3, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)*gamma(3,b*x+a),x, algorithm="giac")
Output:
integrate((d*x + c)*gamma(3, b*x + a), x)
Time = 0.09 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.44 \[ \int (c+d x) \Gamma (3,a+b x) \, dx=-\frac {{\mathrm {e}}^{-a-b\,x}\,\left (12\,d+6\,a\,d+a^2\,d+b\,\left (c\,a^2+4\,c\,a+6\,c\right )\right )}{b^2}-x^2\,{\mathrm {e}}^{-a-b\,x}\,\left (5\,d+2\,a\,d+b\,c\right )-x\,{\mathrm {e}}^{-a-b\,x}\,\left (4\,c+2\,a\,c+\frac {d\,a^2+6\,d\,a+12\,d}{b}\right )-b\,d\,x^3\,{\mathrm {e}}^{-a-b\,x} \] Input:
int(2*exp(- a - b*x)*(c + d*x)*(a + b*x + (a + b*x)^2/2 + 1),x)
Output:
- (exp(- a - b*x)*(12*d + 6*a*d + a^2*d + b*(6*c + 4*a*c + a^2*c)))/b^2 - x^2*exp(- a - b*x)*(5*d + 2*a*d + b*c) - x*exp(- a - b*x)*(4*c + 2*a*c + ( 12*d + 6*a*d + a^2*d)/b) - b*d*x^3*exp(- a - b*x)
Time = 0.23 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.36 \[ \int (c+d x) \Gamma (3,a+b x) \, dx=\frac {-b^{3} d \,x^{3}-2 a \,b^{2} d \,x^{2}-b^{3} c \,x^{2}-a^{2} b d x -2 a \,b^{2} c x -5 b^{2} d \,x^{2}-a^{2} b c -6 a b d x -4 b^{2} c x -a^{2} d -4 a b c -12 b d x -6 a d -6 b c -12 d}{e^{b x +a} b^{2}} \] Input:
int(2*(d*x+c)*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2),x)
Output:
( - a**2*b*c - a**2*b*d*x - a**2*d - 2*a*b**2*c*x - 2*a*b**2*d*x**2 - 4*a* b*c - 6*a*b*d*x - 6*a*d - b**3*c*x**2 - b**3*d*x**3 - 4*b**2*c*x - 5*b**2* d*x**2 - 6*b*c - 12*b*d*x - 12*d)/(e**(a + b*x)*b**2)