Integrand size = 15, antiderivative size = 150 \[ \int (c+d x)^m \Gamma (2,a+b x) \, dx=\frac {(c+d x)^{1+m} \Gamma (2,a+b x)}{d (1+m)}+\frac {(b c-a d) e^{-a+\frac {b c}{d}} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (2+m,\frac {b (c+d x)}{d}\right )}{b d (1+m)}-\frac {e^{-a+\frac {b c}{d}} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (3+m,\frac {b (c+d x)}{d}\right )}{b (1+m)} \] Output:
(d*x+c)^(1+m)*exp(-b*x-a)*(b*x+a+1)/d/(1+m)+(-a*d+b*c)*exp(-a+b*c/d)*(d*x+ c)^m*GAMMA(2+m,b*(d*x+c)/d)/b/d/(1+m)/((b*(d*x+c)/d)^m)-exp(-a+b*c/d)*(d*x +c)^m*GAMMA(3+m,b*(d*x+c)/d)/b/(1+m)/((b*(d*x+c)/d)^m)
Time = 0.16 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.78 \[ \int (c+d x)^m \Gamma (2,a+b x) \, dx=\frac {e^{-a} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \left (b e^a \left (b \left (\frac {c}{d}+x\right )\right )^m (c+d x) \Gamma (2,a+b x)+e^{\frac {b c}{d}} \left ((b c-a d) \Gamma \left (2+m,\frac {b (c+d x)}{d}\right )-d \Gamma \left (3+m,\frac {b (c+d x)}{d}\right )\right )\right )}{b d (1+m)} \] Input:
Integrate[(c + d*x)^m*Gamma[2, a + b*x],x]
Output:
((c + d*x)^m*(b*E^a*(b*(c/d + x))^m*(c + d*x)*Gamma[2, a + b*x] + E^((b*c) /d)*((b*c - a*d)*Gamma[2 + m, (b*(c + d*x))/d] - d*Gamma[3 + m, (b*(c + d* x))/d])))/(b*d*E^a*(1 + m)*((b*(c + d*x))/d)^m)
Time = 0.52 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.99, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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.
\(\displaystyle \int \Gamma (2,a+b x) (c+d x)^m \, dx\) |
\(\Big \downarrow \) 7119 |
\(\displaystyle \frac {b \int e^{-a-b x} (a+b x) (c+d x)^{m+1}dx}{d (m+1)}+\frac {\Gamma (2,a+b x) (c+d x)^{m+1}}{d (m+1)}\) |
\(\Big \downarrow \) 2629 |
\(\displaystyle \frac {b \int \left (\frac {(a d-b c) e^{-a-b x} (c+d x)^{m+1}}{d}+\frac {b e^{-a-b x} (c+d x)^{m+2}}{d}\right )dx}{d (m+1)}+\frac {\Gamma (2,a+b x) (c+d x)^{m+1}}{d (m+1)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {b \left (\frac {(b c-a d) e^{\frac {b c}{d}-a} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (m+2,\frac {b (c+d x)}{d}\right )}{b^2}-\frac {d e^{\frac {b c}{d}-a} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (m+3,\frac {b (c+d x)}{d}\right )}{b^2}\right )}{d (m+1)}+\frac {\Gamma (2,a+b x) (c+d x)^{m+1}}{d (m+1)}\) |
Input:
Int[(c + d*x)^m*Gamma[2, a + b*x],x]
Output:
((c + d*x)^(1 + m)*Gamma[2, a + b*x])/(d*(1 + m)) + (b*(((b*c - a*d)*E^(-a + (b*c)/d)*(c + d*x)^m*Gamma[2 + m, (b*(c + d*x))/d])/(b^2*((b*(c + d*x)) /d)^m) - (d*E^(-a + (b*c)/d)*(c + d*x)^m*Gamma[3 + m, (b*(c + d*x))/d])/(b ^2*((b*(c + d*x))/d)^m)))/(d*(1 + m))
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]
\[\int \left (d x +c \right )^{m} {\mathrm e}^{-b x -a} \left (b x +a +1\right )d x\]
Input:
int((d*x+c)^m*exp(-b*x-a)*(b*x+a+1),x)
Output:
int((d*x+c)^m*exp(-b*x-a)*(b*x+a+1),x)
Time = 0.11 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.03 \[ \int (c+d x)^m \Gamma (2,a+b x) \, dx=-\frac {{\left (d m^{2} - b c + {\left (a + 2\right )} d - {\left (b c - {\left (a + 3\right )} d\right )} m\right )} e^{\left (-\frac {d m \log \left (\frac {b}{d}\right ) - b c + a d}{d}\right )} \Gamma \left (m + 1, \frac {b d x + b c}{d}\right ) + {\left ({\left (b^{2} d x^{2} + {\left (a + 2\right )} b c + b c m + {\left (b^{2} c + {\left (a + 2\right )} b d + b d m\right )} x\right )} e^{\left (-b x - a\right )} - {\left (b d x + b c\right )} \Gamma \left (2, b x + a\right )\right )} {\left (d x + c\right )}^{m}}{b d m + b d} \] Input:
integrate((d*x+c)^m*gamma(2,b*x+a),x, algorithm="fricas")
Output:
-((d*m^2 - b*c + (a + 2)*d - (b*c - (a + 3)*d)*m)*e^(-(d*m*log(b/d) - b*c + a*d)/d)*gamma(m + 1, (b*d*x + b*c)/d) + ((b^2*d*x^2 + (a + 2)*b*c + b*c* m + (b^2*c + (a + 2)*b*d + b*d*m)*x)*e^(-b*x - a) - (b*d*x + b*c)*gamma(2, b*x + a))*(d*x + c)^m)/(b*d*m + b*d)
Exception generated. \[ \int (c+d x)^m \Gamma (2,a+b x) \, dx=\text {Exception raised: TypeError} \] Input:
integrate((d*x+c)**m*uppergamma(2,b*x+a),x)
Output:
Exception raised: TypeError >> cannot determine truth value of Relational
\[ \int (c+d x)^m \Gamma (2,a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \Gamma \left (2, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)^m*gamma(2,b*x+a),x, algorithm="maxima")
Output:
integrate((d*x + c)^m*gamma(2, b*x + a), x)
\[ \int (c+d x)^m \Gamma (2,a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \Gamma \left (2, b x + a\right ) \,d x } \] Input:
integrate((d*x+c)^m*gamma(2,b*x+a),x, algorithm="giac")
Output:
integrate((d*x + c)^m*gamma(2, b*x + a), x)
Timed out. \[ \int (c+d x)^m \Gamma (2,a+b x) \, dx=\int {\mathrm {e}}^{-a-b\,x}\,{\left (c+d\,x\right )}^m\,\left (a+b\,x+1\right ) \,d x \] Input:
int(exp(- a - b*x)*(c + d*x)^m*(a + b*x + 1),x)
Output:
int(exp(- a - b*x)*(c + d*x)^m*(a + b*x + 1), x)
\[ \int (c+d x)^m \Gamma (2,a+b x) \, dx=\frac {-\left (d x +c \right )^{m} a -\left (d x +c \right )^{m} b x -\left (d x +c \right )^{m} m -2 \left (d x +c \right )^{m}+e^{b x} \left (\int \frac {\left (d x +c \right )^{m}}{e^{b x} c +e^{b x} d x}d x \right ) a d m -e^{b x} \left (\int \frac {\left (d x +c \right )^{m}}{e^{b x} c +e^{b x} d x}d x \right ) b c m +e^{b x} \left (\int \frac {\left (d x +c \right )^{m}}{e^{b x} c +e^{b x} d x}d x \right ) d \,m^{2}+2 e^{b x} \left (\int \frac {\left (d x +c \right )^{m}}{e^{b x} c +e^{b x} d x}d x \right ) d m}{e^{b x +a} b} \] Input:
int((d*x+c)^m*exp(-b*x-a)*(b*x+a+1),x)
Output:
( - (c + d*x)**m*a - (c + d*x)**m*b*x - (c + d*x)**m*m - 2*(c + d*x)**m + e**(b*x)*int((c + d*x)**m/(e**(b*x)*c + e**(b*x)*d*x),x)*a*d*m - e**(b*x)* int((c + d*x)**m/(e**(b*x)*c + e**(b*x)*d*x),x)*b*c*m + e**(b*x)*int((c + d*x)**m/(e**(b*x)*c + e**(b*x)*d*x),x)*d*m**2 + 2*e**(b*x)*int((c + d*x)** m/(e**(b*x)*c + e**(b*x)*d*x),x)*d*m)/(e**(a + b*x)*b)