\(\int (c+d x)^m \Gamma (3,a+b x) \, dx\) [179]

Optimal result

Integrand size = 15, antiderivative size = 221 \[ \int (c+d x)^m \Gamma (3,a+b x) \, dx=\frac {(c+d x)^{1+m} \Gamma (3,a+b x)}{d (1+m)}-\frac {(b c-a d)^2 e^{-a+\frac {b c}{d}} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (2+m,\frac {b (c+d x)}{d}\right )}{b d^2 (1+m)}+\frac {2 (b c-a d) e^{-a+\frac {b c}{d}} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (3+m,\frac {b (c+d x)}{d}\right )}{b d (1+m)}-\frac {e^{-a+\frac {b c}{d}} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (4+m,\frac {b (c+d x)}{d}\right )}{b (1+m)} \] Output:

2*(d*x+c)^(1+m)*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2)/d/(1+m)-(-a*d+b*c)^2*e 
xp(-a+b*c/d)*(d*x+c)^m*GAMMA(2+m,b*(d*x+c)/d)/b/d^2/(1+m)/((b*(d*x+c)/d)^m 
)+2*(-a*d+b*c)*exp(-a+b*c/d)*(d*x+c)^m*GAMMA(3+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(4+m,b*(d*x+c)/d)/b/(1+m)/((b 
*(d*x+c)/d)^m)
 

Mathematica [A] (verified)

Time = 0.29 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.67 \[ \int (c+d x)^m \Gamma (3,a+b x) \, dx=\frac {e^{-a} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \left (b d e^a \left (b \left (\frac {c}{d}+x\right )\right )^m (c+d x) \Gamma (3,a+b x)-e^{\frac {b c}{d}} \left ((b c-a d)^2 \Gamma \left (2+m,\frac {b (c+d x)}{d}\right )+d \left ((-2 b c+2 a d) \Gamma \left (3+m,\frac {b (c+d x)}{d}\right )+d \Gamma \left (4+m,\frac {b (c+d x)}{d}\right )\right )\right )\right )}{b d^2 (1+m)} \] Input:

Integrate[(c + d*x)^m*Gamma[3, a + b*x],x]
 

Output:

((c + d*x)^m*(b*d*E^a*(b*(c/d + x))^m*(c + d*x)*Gamma[3, a + b*x] - E^((b* 
c)/d)*((b*c - a*d)^2*Gamma[2 + m, (b*(c + d*x))/d] + d*((-2*b*c + 2*a*d)*G 
amma[3 + m, (b*(c + d*x))/d] + d*Gamma[4 + m, (b*(c + d*x))/d]))))/(b*d^2* 
E^a*(1 + m)*((b*(c + d*x))/d)^m)
 

Rubi [A] (verified)

Time = 0.90 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.97, 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.

\(\displaystyle \int \Gamma (3,a+b x) (c+d x)^m \, dx\)

\(\Big \downarrow \) 7119

\(\displaystyle \frac {b \int e^{-a-b x} (a+b x)^2 (c+d x)^{m+1}dx}{d (m+1)}+\frac {\Gamma (3,a+b x) (c+d x)^{m+1}}{d (m+1)}\)

\(\Big \downarrow \) 2629

\(\displaystyle \frac {b \int \left (\frac {(a d-b c)^2 e^{-a-b x} (c+d x)^{m+1}}{d^2}-\frac {2 b (b c-a d) e^{-a-b x} (c+d x)^{m+2}}{d^2}+\frac {b^2 e^{-a-b x} (c+d x)^{m+3}}{d^2}\right )dx}{d (m+1)}+\frac {\Gamma (3,a+b x) (c+d x)^{m+1}}{d (m+1)}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \left (-\frac {(b c-a d)^2 e^{\frac {b c}{d}-a} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (m+2,\frac {b (c+d x)}{d}\right )}{b^2 d}+\frac {2 (b c-a d) e^{\frac {b c}{d}-a} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (m+3,\frac {b (c+d x)}{d}\right )}{b^2}-\frac {d e^{\frac {b c}{d}-a} (c+d x)^m \left (\frac {b (c+d x)}{d}\right )^{-m} \Gamma \left (m+4,\frac {b (c+d x)}{d}\right )}{b^2}\right )}{d (m+1)}+\frac {\Gamma (3,a+b x) (c+d x)^{m+1}}{d (m+1)}\)

Input:

Int[(c + d*x)^m*Gamma[3, a + b*x],x]
 

Output:

((c + d*x)^(1 + m)*Gamma[3, a + b*x])/(d*(1 + m)) + (b*(-(((b*c - a*d)^2*E 
^(-a + (b*c)/d)*(c + d*x)^m*Gamma[2 + m, (b*(c + d*x))/d])/(b^2*d*((b*(c + 
 d*x))/d)^m)) + (2*(b*c - a*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*E^(-a + (b*c)/d)*(c + d*x) 
^m*Gamma[4 + m, (b*(c + d*x))/d])/(b^2*((b*(c + d*x))/d)^m)))/(d*(1 + m))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2629

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]
 

rule 7119

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]
 

Maple [F]

\[\int 2 \left (d x +c \right )^{m} {\mathrm e}^{-b x -a} \left (1+b x +a +\frac {\left (b x +a \right )^{2}}{2}\right )d x\]

Input:

int(2*(d*x+c)^m*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2),x)
 

Output:

int(2*(d*x+c)^m*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2),x)
 

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.42 \[ \int (c+d x)^m \Gamma (3,a+b x) \, dx=-\frac {{\left (d^{2} m^{3} + b^{2} c^{2} - 2 \, {\left (a + 2\right )} b c d + {\left (a^{2} + 4 \, a + 6\right )} d^{2} - 2 \, {\left (b c d - {\left (a + 3\right )} d^{2}\right )} m^{2} + {\left (b^{2} c^{2} - 2 \, {\left (a + 3\right )} b c d + {\left (a^{2} + 6 \, a + 11\right )} d^{2}\right )} m\right )} e^{\left (-\frac {d m \log \left (\frac {b}{d}\right ) - b c + a d}{d}\right )} \Gamma \left (m + 1, \frac {b d x + b c}{d}\right ) + {\left ({\left (b^{3} d^{2} x^{3} + b c d m^{2} - b^{2} c^{2} + {\left (a^{2} + 4 \, a + 6\right )} b c d + {\left (b^{3} c d + {\left (2 \, a + 3\right )} b^{2} d^{2} + b^{2} d^{2} m\right )} x^{2} - {\left (b^{2} c^{2} - {\left (2 \, a + 5\right )} b c d\right )} m + {\left (2 \, {\left (a + 1\right )} b^{2} c d + {\left (2 \, a + 5\right )} b d^{2} m + b d^{2} m^{2} + {\left (a^{2} + 4 \, a + 6\right )} b d^{2}\right )} x\right )} e^{\left (-b x - a\right )} - {\left (b d^{2} x + b c d\right )} \Gamma \left (3, b x + a\right )\right )} {\left (d x + c\right )}^{m}}{b d^{2} m + b d^{2}} \] Input:

integrate((d*x+c)^m*gamma(3,b*x+a),x, algorithm="fricas")
 

Output:

-((d^2*m^3 + b^2*c^2 - 2*(a + 2)*b*c*d + (a^2 + 4*a + 6)*d^2 - 2*(b*c*d - 
(a + 3)*d^2)*m^2 + (b^2*c^2 - 2*(a + 3)*b*c*d + (a^2 + 6*a + 11)*d^2)*m)*e 
^(-(d*m*log(b/d) - b*c + a*d)/d)*gamma(m + 1, (b*d*x + b*c)/d) + ((b^3*d^2 
*x^3 + b*c*d*m^2 - b^2*c^2 + (a^2 + 4*a + 6)*b*c*d + (b^3*c*d + (2*a + 3)* 
b^2*d^2 + b^2*d^2*m)*x^2 - (b^2*c^2 - (2*a + 5)*b*c*d)*m + (2*(a + 1)*b^2* 
c*d + (2*a + 5)*b*d^2*m + b*d^2*m^2 + (a^2 + 4*a + 6)*b*d^2)*x)*e^(-b*x - 
a) - (b*d^2*x + b*c*d)*gamma(3, b*x + a))*(d*x + c)^m)/(b*d^2*m + b*d^2)
 

Sympy [F(-1)]

Timed out. \[ \int (c+d x)^m \Gamma (3,a+b x) \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**m*uppergamma(3,b*x+a),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (c+d x)^m \Gamma (3,a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \Gamma \left (3, b x + a\right ) \,d x } \] Input:

integrate((d*x+c)^m*gamma(3,b*x+a),x, algorithm="maxima")
 

Output:

integrate((d*x + c)^m*gamma(3, b*x + a), x)
 

Giac [F]

\[ \int (c+d x)^m \Gamma (3,a+b x) \, dx=\int { {\left (d x + c\right )}^{m} \Gamma \left (3, b x + a\right ) \,d x } \] Input:

integrate((d*x+c)^m*gamma(3,b*x+a),x, algorithm="giac")
 

Output:

integrate((d*x + c)^m*gamma(3, b*x + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^m \Gamma (3,a+b x) \, dx=\int 2\,{\mathrm {e}}^{-a-b\,x}\,{\left (c+d\,x\right )}^m\,\left (a+b\,x+\frac {{\left (a+b\,x\right )}^2}{2}+1\right ) \,d x \] Input:

int(2*exp(- a - b*x)*(c + d*x)^m*(a + b*x + (a + b*x)^2/2 + 1),x)
 

Output:

int(2*exp(- a - b*x)*(c + d*x)^m*(a + b*x + (a + b*x)^2/2 + 1), x)
 

Reduce [F]

\[ \int (c+d x)^m \Gamma (3,a+b x) \, dx=\frac {-\left (d x +c \right )^{m} a^{2} d -2 \left (d x +c \right )^{m} a b d x -2 \left (d x +c \right )^{m} a d m -4 \left (d x +c \right )^{m} a d -\left (d x +c \right )^{m} b^{2} d \,x^{2}+\left (d x +c \right )^{m} b c m -\left (d x +c \right )^{m} b d m x -4 \left (d x +c \right )^{m} b d x -\left (d x +c \right )^{m} d \,m^{2}-5 \left (d x +c \right )^{m} d m -6 \left (d x +c \right )^{m} d +e^{b x} \left (\int \frac {\left (d x +c \right )^{m}}{e^{b x} c +e^{b x} d x}d x \right ) a^{2} d^{2} m -2 e^{b x} \left (\int \frac {\left (d x +c \right )^{m}}{e^{b x} c +e^{b x} d x}d x \right ) a b c d m +2 e^{b x} \left (\int \frac {\left (d x +c \right )^{m}}{e^{b x} c +e^{b x} d x}d x \right ) a \,d^{2} m^{2}+4 e^{b x} \left (\int \frac {\left (d x +c \right )^{m}}{e^{b x} c +e^{b x} d x}d x \right ) a \,d^{2} m +e^{b x} \left (\int \frac {\left (d x +c \right )^{m}}{e^{b x} c +e^{b x} d x}d x \right ) b^{2} c^{2} m -2 e^{b x} \left (\int \frac {\left (d x +c \right )^{m}}{e^{b x} c +e^{b x} d x}d x \right ) b c d \,m^{2}-4 e^{b x} \left (\int \frac {\left (d x +c \right )^{m}}{e^{b x} c +e^{b x} d x}d x \right ) b c d m +e^{b x} \left (\int \frac {\left (d x +c \right )^{m}}{e^{b x} c +e^{b x} d x}d x \right ) d^{2} m^{3}+5 e^{b x} \left (\int \frac {\left (d x +c \right )^{m}}{e^{b x} c +e^{b x} d x}d x \right ) d^{2} m^{2}+6 e^{b x} \left (\int \frac {\left (d x +c \right )^{m}}{e^{b x} c +e^{b x} d x}d x \right ) d^{2} m}{e^{b x +a} b d} \] Input:

int(2*(d*x+c)^m*exp(-b*x-a)*(1+b*x+a+1/2*(b*x+a)^2),x)
 

Output:

( - (c + d*x)**m*a**2*d - 2*(c + d*x)**m*a*b*d*x - 2*(c + d*x)**m*a*d*m - 
4*(c + d*x)**m*a*d - (c + d*x)**m*b**2*d*x**2 + (c + d*x)**m*b*c*m - (c + 
d*x)**m*b*d*m*x - 4*(c + d*x)**m*b*d*x - (c + d*x)**m*d*m**2 - 5*(c + d*x) 
**m*d*m - 6*(c + d*x)**m*d + e**(b*x)*int((c + d*x)**m/(e**(b*x)*c + e**(b 
*x)*d*x),x)*a**2*d**2*m - 2*e**(b*x)*int((c + d*x)**m/(e**(b*x)*c + e**(b* 
x)*d*x),x)*a*b*c*d*m + 2*e**(b*x)*int((c + d*x)**m/(e**(b*x)*c + e**(b*x)* 
d*x),x)*a*d**2*m**2 + 4*e**(b*x)*int((c + d*x)**m/(e**(b*x)*c + e**(b*x)*d 
*x),x)*a*d**2*m + e**(b*x)*int((c + d*x)**m/(e**(b*x)*c + e**(b*x)*d*x),x) 
*b**2*c**2*m - 2*e**(b*x)*int((c + d*x)**m/(e**(b*x)*c + e**(b*x)*d*x),x)* 
b*c*d*m**2 - 4*e**(b*x)*int((c + d*x)**m/(e**(b*x)*c + e**(b*x)*d*x),x)*b* 
c*d*m + e**(b*x)*int((c + d*x)**m/(e**(b*x)*c + e**(b*x)*d*x),x)*d**2*m**3 
 + 5*e**(b*x)*int((c + d*x)**m/(e**(b*x)*c + e**(b*x)*d*x),x)*d**2*m**2 + 
6*e**(b*x)*int((c + d*x)**m/(e**(b*x)*c + e**(b*x)*d*x),x)*d**2*m)/(e**(a 
+ b*x)*b*d)