Integrand size = 13, antiderivative size = 93 \[ \int (c+d x) \Gamma (2,a+b x) \, dx=\frac {(c+d x)^2 \Gamma (2,a+b x)}{2 d}+\frac {(b c-a d) e^{-a+\frac {b c}{d}} \Gamma \left (3,\frac {b (c+d x)}{d}\right )}{2 b^2}-\frac {d e^{-a+\frac {b c}{d}} \Gamma \left (4,\frac {b (c+d x)}{d}\right )}{2 b^2} \] Output:
1/2*(d*x+c)^2*exp(-b*x-a)*(b*x+a+1)/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)/b^2-3*d*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^2
Time = 0.06 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.06 \[ \int (c+d x) \Gamma (2,a+b x) \, dx=\frac {e^{-a-b x} \left (-2 (3+a) d-b^3 x^2 (2 c+d x)-2 b ((2+a) c+(3+a) d x)-b^2 x (2 (2+a) c+(3+a) d x)+b^2 e^{a+b x} x (2 c+d x) \Gamma (2,a+b x)\right )}{2 b^2} \] Input:
Integrate[(c + d*x)*Gamma[2, a + b*x],x]
Output:
(E^(-a - b*x)*(-2*(3 + a)*d - b^3*x^2*(2*c + d*x) - 2*b*((2 + a)*c + (3 + a)*d*x) - b^2*x*(2*(2 + a)*c + (3 + a)*d*x) + b^2*E^(a + b*x)*x*(2*c + d*x )*Gamma[2, a + b*x]))/(2*b^2)
Leaf count is larger than twice the leaf count of optimal. \(198\) vs. \(2(93)=186\).
Time = 0.58 (sec) , antiderivative size = 198, normalized size of antiderivative = 2.13, 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.
\(\displaystyle \int (c+d x) \Gamma (2,a+b x) \, dx\) |
\(\Big \downarrow \) 7119 |
\(\displaystyle \frac {b \int e^{-a-b x} (a+b x) (c+d x)^2dx}{2 d}+\frac {(c+d x)^2 \Gamma (2,a+b x)}{2 d}\) |
\(\Big \downarrow \) 2626 |
\(\displaystyle \frac {b \int \left (\frac {b e^{-a-b x} (c+d x)^3}{d}+\frac {(a d-b c) e^{-a-b x} (c+d x)^2}{d}\right )dx}{2 d}+\frac {(c+d x)^2 \Gamma (2,a+b x)}{2 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \left (\frac {2 d e^{-a-b x} (b c-a d)}{b^3}-\frac {6 d^2 e^{-a-b x}}{b^3}-\frac {6 d e^{-a-b x} (c+d x)}{b^2}+\frac {2 e^{-a-b x} (c+d x) (b c-a d)}{b^2}-\frac {e^{-a-b x} (c+d x)^3}{d}+\frac {e^{-a-b x} (c+d x)^2 (b c-a d)}{b d}-\frac {3 e^{-a-b x} (c+d x)^2}{b}\right )}{2 d}+\frac {(c+d x)^2 \Gamma (2,a+b x)}{2 d}\) |
Input:
Int[(c + d*x)*Gamma[2, a + b*x],x]
Output:
(b*((-6*d^2*E^(-a - b*x))/b^3 + (2*d*(b*c - a*d)*E^(-a - b*x))/b^3 - (6*d* E^(-a - b*x)*(c + d*x))/b^2 + (2*(b*c - a*d)*E^(-a - b*x)*(c + d*x))/b^2 - (3*E^(-a - b*x)*(c + d*x)^2)/b + ((b*c - a*d)*E^(-a - b*x)*(c + d*x)^2)/( b*d) - (E^(-a - b*x)*(c + d*x)^3)/d))/(2*d) + ((c + d*x)^2*Gamma[2, 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 = 0.44 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.58
method | result | size |
gosper | \(-\frac {\left (b^{2} d \,x^{2}+a b d x +b^{2} c x +a b c +3 b d x +a d +2 c b +3 d \right ) {\mathrm e}^{-b x -a}}{b^{2}}\) | \(54\) |
risch | \(-\frac {\left (b^{2} d \,x^{2}+a b d x +b^{2} c x +a b c +3 b d x +a d +2 c b +3 d \right ) {\mathrm e}^{-b x -a}}{b^{2}}\) | \(54\) |
orering | \(-\frac {\left (b^{2} d \,x^{2}+a b d x +b^{2} c x +a b c +3 b d x +a d +2 c b +3 d \right ) {\mathrm e}^{-b x -a}}{b^{2}}\) | \(54\) |
norman | \(-d \,x^{2} {\mathrm e}^{-b x -a}-\frac {\left (a b c +a d +2 c b +3 d \right ) {\mathrm e}^{-b x -a}}{b^{2}}-\frac {\left (a d +c b +3 d \right ) x \,{\mathrm e}^{-b x -a}}{b}\) | \(71\) |
parallelrisch | \(-\frac {d \,x^{2} {\mathrm e}^{-b x -a} b^{2}+x \,{\mathrm e}^{-b x -a} a b d +x \,{\mathrm e}^{-b x -a} b^{2} c +3 \,{\mathrm e}^{-b x -a} b d x +{\mathrm e}^{-b x -a} a b c +{\mathrm e}^{-b x -a} a d +2 \,{\mathrm e}^{-b x -a} b c +3 \,{\mathrm e}^{-b x -a} d}{b^{2}}\) | \(117\) |
meijerg | \(\frac {d \,{\mathrm e}^{-a} \left (2-\frac {\left (3 b^{2} x^{2}+6 b x +6\right ) {\mathrm e}^{-b x}}{3}\right )}{b^{2}}+\frac {d \,{\mathrm e}^{-a} a \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}+\frac {{\mathrm e}^{-a} d \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b^{2}}+\frac {c \,{\mathrm e}^{-a} \left (1-\frac {\left (2 b x +2\right ) {\mathrm e}^{-b x}}{2}\right )}{b}+\frac {c \,{\mathrm e}^{-a} a \left (1-{\mathrm e}^{-b x}\right )}{b}+\frac {c \,{\mathrm e}^{-a} \left (1-{\mathrm e}^{-b x}\right )}{b}\) | \(144\) |
parts | \(-d \,x^{2} {\mathrm e}^{-b x -a}-\frac {{\mathrm e}^{-b x -a} d x a}{b}-{\mathrm e}^{-b x -a} c x -\frac {{\mathrm e}^{-b x -a} a c}{b}-\frac {d x \,{\mathrm e}^{-b x -a}}{b}-\frac {{\mathrm e}^{-b x -a} c}{b}-\frac {{\mathrm e}^{-b x -a} d +{\mathrm e}^{-b x -a} b c -{\mathrm e}^{-b x -a} a d -2 d \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b^{2}}\) | \(169\) |
derivativedivides | \(-\frac {{\mathrm e}^{-b x -a} c +\frac {d \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b}+\frac {d a \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}-c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )-\frac {{\mathrm e}^{-b x -a} d a}{b}-\frac {d \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}}{b}\) | \(195\) |
default | \(-\frac {{\mathrm e}^{-b x -a} c +\frac {d \left (\left (-b x -a \right )^{2} {\mathrm e}^{-b x -a}-2 \left (-b x -a \right ) {\mathrm e}^{-b x -a}+2 \,{\mathrm e}^{-b x -a}\right )}{b}+\frac {d a \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}-c \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )-\frac {{\mathrm e}^{-b x -a} d a}{b}-\frac {d \left (\left (-b x -a \right ) {\mathrm e}^{-b x -a}-{\mathrm e}^{-b x -a}\right )}{b}}{b}\) | \(195\) |
Input:
int((d*x+c)*exp(-b*x-a)*(b*x+a+1),x,method=_RETURNVERBOSE)
Output:
-(b^2*d*x^2+a*b*d*x+b^2*c*x+a*b*c+3*b*d*x+a*d+2*b*c+3*d)*exp(-b*x-a)/b^2
Time = 0.09 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.08 \[ \int (c+d x) \Gamma (2,a+b x) \, dx=-\frac {{\left (b^{3} d x^{3} + 2 \, {\left (a + 2\right )} b c + {\left (2 \, b^{3} c + {\left (a + 3\right )} b^{2} d\right )} x^{2} + 2 \, {\left (a + 3\right )} d + 2 \, {\left ({\left (a + 2\right )} b^{2} c + {\left (a + 3\right )} b d\right )} x\right )} e^{\left (-b x - a\right )} - {\left (b^{2} d x^{2} + 2 \, b^{2} c x\right )} \Gamma \left (2, b x + a\right )}{2 \, b^{2}} \] Input:
integrate((d*x+c)*gamma(2,b*x+a),x, algorithm="fricas")
Output:
-1/2*((b^3*d*x^3 + 2*(a + 2)*b*c + (2*b^3*c + (a + 3)*b^2*d)*x^2 + 2*(a + 3)*d + 2*((a + 2)*b^2*c + (a + 3)*b*d)*x)*e^(-b*x - a) - (b^2*d*x^2 + 2*b^ 2*c*x)*gamma(2, b*x + a))/b^2
Time = 0.06 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.99 \[ \int (c+d x) \Gamma (2,a+b x) \, dx=\begin {cases} \frac {\left (- a b c - a b d x - a d - b^{2} c x - b^{2} d x^{2} - 2 b c - 3 b d x - 3 d\right ) e^{- a - b x}}{b^{2}} & \text {for}\: b^{2} \neq 0 \\\frac {b d x^{3}}{3} + x^{2} \left (\frac {a d}{2} + \frac {b c}{2} + \frac {d}{2}\right ) + x \left (a c + c\right ) & \text {otherwise} \end {cases} \] Input:
integrate((d*x+c)*uppergamma(2,b*x+a),x)
Output:
Piecewise(((-a*b*c - a*b*d*x - a*d - b**2*c*x - b**2*d*x**2 - 2*b*c - 3*b* d*x - 3*d)*exp(-a - b*x)/b**2, Ne(b**2, 0)), (b*d*x**3/3 + x**2*(a*d/2 + b *c/2 + d/2) + x*(a*c + c), True))
\[ \int (c+d x) \Gamma (2,a+b x) \, dx=\int { {\left (d x + c\right )} \Gamma \left (2, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)*gamma(2,b*x+a),x, algorithm="maxima")
Output:
d*integrate(x*gamma(2, b*x + a), x) + ((b*x + a)*gamma(2, b*x + a) - gamma (3, b*x + a))*c/b
\[ \int (c+d x) \Gamma (2,a+b x) \, dx=\int { {\left (d x + c\right )} \Gamma \left (2, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)*gamma(2,b*x+a),x, algorithm="giac")
Output:
integrate((d*x + c)*gamma(2, b*x + a), x)
Time = 0.10 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.75 \[ \int (c+d x) \Gamma (2,a+b x) \, dx=-{\mathrm {e}}^{-a-b\,x}\,\left (d\,x^2+c\,x\right )-\frac {{\mathrm {e}}^{-a-b\,x}\,\left (3\,d+a\,d\right )+b\,{\mathrm {e}}^{-a-b\,x}\,\left (2\,c+a\,c+3\,d\,x+a\,d\,x\right )}{b^2} \] Input:
int(exp(- a - b*x)*(c + d*x)*(a + b*x + 1),x)
Output:
- exp(- a - b*x)*(c*x + d*x^2) - (exp(- a - b*x)*(3*d + a*d) + b*exp(- a - b*x)*(2*c + a*c + 3*d*x + a*d*x))/b^2
Time = 0.20 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.61 \[ \int (c+d x) \Gamma (2,a+b x) \, dx=\frac {-b^{2} d \,x^{2}-a b d x -b^{2} c x -a b c -3 b d x -a d -2 b c -3 d}{e^{b x +a} b^{2}} \] Input:
int((d*x+c)*exp(-b*x-a)*(b*x+a+1),x)
Output:
( - a*b*c - a*b*d*x - a*d - b**2*c*x - b**2*d*x**2 - 2*b*c - 3*b*d*x - 3*d )/(e**(a + b*x)*b**2)