Integrand size = 13, antiderivative size = 88 \[ \int (c+d x) \Gamma (n,a+b x) \, dx=-\frac {(b c-a d)^2 \Gamma (n,a+b x)}{2 b^2 d}+\frac {(c+d x)^2 \Gamma (n,a+b x)}{2 d}-\frac {(b c-a d) \Gamma (1+n,a+b x)}{b^2}-\frac {d \Gamma (2+n,a+b x)}{2 b^2} \] Output:
-1/2*(-a*d+b*c)^2*GAMMA(n,b*x+a)/b^2/d+1/2*(d*x+c)^2*GAMMA(n,b*x+a)/d-(-a* d+b*c)*GAMMA(1+n,b*x+a)/b^2-1/2*d*GAMMA(2+n,b*x+a)/b^2
Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.74 \[ \int (c+d x) \Gamma (n,a+b x) \, dx=-\frac {(a+b x) (a d-b (2 c+d x)) \Gamma (n,a+b x)+2 (b c-a d) \Gamma (1+n,a+b x)+d \Gamma (2+n,a+b x)}{2 b^2} \] Input:
Integrate[(c + d*x)*Gamma[n, a + b*x],x]
Output:
-1/2*((a + b*x)*(a*d - b*(2*c + d*x))*Gamma[n, a + b*x] + 2*(b*c - a*d)*Ga mma[1 + n, a + b*x] + d*Gamma[2 + n, a + b*x])/b^2
Time = 0.45 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.06, 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.
\(\displaystyle \int (c+d x) \Gamma (n,a+b x) \, dx\) |
\(\Big \downarrow \) 7119 |
\(\displaystyle \frac {b \int e^{-a-b x} (a+b x)^{n-1} (c+d x)^2dx}{2 d}+\frac {(c+d x)^2 \Gamma (n,a+b x)}{2 d}\) |
\(\Big \downarrow \) 2629 |
\(\displaystyle \frac {b \int \left (\frac {(b c-a d)^2 e^{-a-b x} (a+b x)^{n-1}}{b^2}+\frac {2 d (b c-a d) e^{-a-b x} (a+b x)^n}{b^2}+\frac {d^2 e^{-a-b x} (a+b x)^{n+1}}{b^2}\right )dx}{2 d}+\frac {(c+d x)^2 \Gamma (n,a+b x)}{2 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \left (-\frac {2 d (b c-a d) \Gamma (n+1,a+b x)}{b^3}-\frac {(b c-a d)^2 \Gamma (n,a+b x)}{b^3}-\frac {d^2 \Gamma (n+2,a+b x)}{b^3}\right )}{2 d}+\frac {(c+d x)^2 \Gamma (n,a+b x)}{2 d}\) |
Input:
Int[(c + d*x)*Gamma[n, a + b*x],x]
Output:
((c + d*x)^2*Gamma[n, a + b*x])/(2*d) + (b*(-(((b*c - a*d)^2*Gamma[n, a + b*x])/b^3) - (2*d*(b*c - a*d)*Gamma[1 + n, a + b*x])/b^3 - (d^2*Gamma[2 + n, a + b*x])/b^3))/(2*d)
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]
Leaf count of result is larger than twice the leaf count of optimal. \(314\) vs. \(2(82)=164\).
Time = 0.74 (sec) , antiderivative size = 315, normalized size of antiderivative = 3.58
method | result | size |
parallelrisch | \(-\frac {x^{2} {\mathrm e}^{-b x -a} \left (b x +a \right )^{-1+n} a \,b^{2} d -x^{2} \Gamma \left (n , b x +a \right ) a \,b^{2} d +2 x \,{\mathrm e}^{-b x -a} \left (b x +a \right )^{-1+n} a \,b^{2} c +x \,{\mathrm e}^{-b x -a} \left (b x +a \right )^{-1+n} a b d n +x \,{\mathrm e}^{-b x -a} \left (b x +a \right )^{-1+n} a b d -2 x \Gamma \left (n , b x +a \right ) a \,b^{2} c -{\mathrm e}^{-b x -a} \left (b x +a \right )^{-1+n} a^{3} d +2 \,{\mathrm e}^{-b x -a} \left (b x +a \right )^{-1+n} a^{2} b c +{\mathrm e}^{-b x -a} \left (b x +a \right )^{-1+n} a^{2} d n +{\mathrm e}^{-b x -a} \left (b x +a \right )^{-1+n} a^{2} d +\Gamma \left (n , b x +a \right ) a^{3} d -2 \Gamma \left (n , b x +a \right ) a^{2} b c -2 \Gamma \left (n , b x +a \right ) a^{2} d n +2 \Gamma \left (n , b x +a \right ) a b c n +\Gamma \left (n , b x +a \right ) a d \,n^{2}+\Gamma \left (n , b x +a \right ) a d n}{2 a \,b^{2}}\) | \(315\) |
Input:
int((d*x+c)*GAMMA(n,b*x+a),x,method=_RETURNVERBOSE)
Output:
-1/2*(x^2*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^2*d-x^2*GAMMA(n,b*x+a)*a*b^2*d+2* x*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^2*c+x*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b*d*n+ x*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b*d-2*x*GAMMA(n,b*x+a)*a*b^2*c-exp(-b*x-a)* (b*x+a)^(-1+n)*a^3*d+2*exp(-b*x-a)*(b*x+a)^(-1+n)*a^2*b*c+exp(-b*x-a)*(b*x +a)^(-1+n)*a^2*d*n+exp(-b*x-a)*(b*x+a)^(-1+n)*a^2*d+GAMMA(n,b*x+a)*a^3*d-2 *GAMMA(n,b*x+a)*a^2*b*c-2*GAMMA(n,b*x+a)*a^2*d*n+2*GAMMA(n,b*x+a)*a*b*c*n+ GAMMA(n,b*x+a)*a*d*n^2+GAMMA(n,b*x+a)*a*d*n)/a/b^2
Time = 0.10 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.44 \[ \int (c+d x) \Gamma (n,a+b x) \, dx=-\frac {{\left (b^{2} d x^{2} + 2 \, a b c + a d n - {\left (a^{2} - a\right )} d + {\left (2 \, b^{2} c + b d n + b d\right )} x\right )} {\left (b x + a\right )}^{n - 1} e^{\left (-b x - a\right )} - {\left (b^{2} d x^{2} + 2 \, b^{2} c x + 2 \, a b c - a^{2} d - d n^{2} - {\left (2 \, b c - {\left (2 \, a - 1\right )} d\right )} n\right )} \Gamma \left (n, b x + a\right )}{2 \, b^{2}} \] Input:
integrate((d*x+c)*gamma(n,b*x+a),x, algorithm="fricas")
Output:
-1/2*((b^2*d*x^2 + 2*a*b*c + a*d*n - (a^2 - a)*d + (2*b^2*c + b*d*n + b*d) *x)*(b*x + a)^(n - 1)*e^(-b*x - a) - (b^2*d*x^2 + 2*b^2*c*x + 2*a*b*c - a^ 2*d - d*n^2 - (2*b*c - (2*a - 1)*d)*n)*gamma(n, b*x + a))/b^2
\[ \int (c+d x) \Gamma (n,a+b x) \, dx=\int \left (c + d x\right ) \Gamma \left (n, a + b x\right )\, dx \] Input:
integrate((d*x+c)*uppergamma(n,b*x+a),x)
Output:
Integral((c + d*x)*uppergamma(n, a + b*x), x)
\[ \int (c+d x) \Gamma (n,a+b x) \, dx=\int { {\left (d x + c\right )} \Gamma \left (n, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)*gamma(n,b*x+a),x, algorithm="maxima")
Output:
d*integrate(x*gamma(n, b*x + a), x) + ((b*x + a)*gamma(n, b*x + a) - gamma (n + 1, b*x + a))*c/b
\[ \int (c+d x) \Gamma (n,a+b x) \, dx=\int { {\left (d x + c\right )} \Gamma \left (n, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)*gamma(n,b*x+a),x, algorithm="giac")
Output:
integrate((d*x + c)*gamma(n, b*x + a), x)
Timed out. \[ \int (c+d x) \Gamma (n,a+b x) \, dx=\int \Gamma \left (n,a+b\,x\right )\,\left (c+d\,x\right ) \,d x \] Input:
int(igamma(n, a + b*x)*(c + d*x),x)
Output:
int(igamma(n, a + b*x)*(c + d*x), x)
\[ \int (c+d x) \Gamma (n,a+b x) \, dx=\left (\int \gamma \left (n , b x +a \right )d x \right ) c +\left (\int \gamma \left (n , b x +a \right ) x d x \right ) d \] Input:
int((d*x+c)*GAMMA(n,b*x+a),x)
Output:
int(gamma(n,a + b*x),x)*c + int(gamma(n,a + b*x)*x,x)*d