\(\int (c+d x)^4 \Gamma (n,a+b x) \, dx\) [188]

Optimal result

Integrand size = 15, antiderivative size = 169 \[ \int (c+d x)^4 \Gamma (n,a+b x) \, dx=-\frac {(b c-a d)^5 \Gamma (n,a+b x)}{5 b^5 d}+\frac {(c+d x)^5 \Gamma (n,a+b x)}{5 d}-\frac {(b c-a d)^4 \Gamma (1+n,a+b x)}{b^5}-\frac {2 d (b c-a d)^3 \Gamma (2+n,a+b x)}{b^5}-\frac {2 d^2 (b c-a d)^2 \Gamma (3+n,a+b x)}{b^5}-\frac {d^3 (b c-a d) \Gamma (4+n,a+b x)}{b^5}-\frac {d^4 \Gamma (5+n,a+b x)}{5 b^5} \] Output:

-1/5*(-a*d+b*c)^5*GAMMA(n,b*x+a)/b^5/d+1/5*(d*x+c)^5*GAMMA(n,b*x+a)/d-(-a* 
d+b*c)^4*GAMMA(1+n,b*x+a)/b^5-2*d*(-a*d+b*c)^3*GAMMA(2+n,b*x+a)/b^5-2*d^2* 
(-a*d+b*c)^2*GAMMA(3+n,b*x+a)/b^5-d^3*(-a*d+b*c)*GAMMA(4+n,b*x+a)/b^5-1/5* 
d^4*GAMMA(5+n,b*x+a)/b^5
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.28 \[ \int (c+d x)^4 \Gamma (n,a+b x) \, dx=\frac {\left (5 a b^4 c^4-10 a^2 b^3 c^3 d+10 a^3 b^2 c^2 d^2-5 a^4 b c d^3+a^5 d^4+b^5 x \left (5 c^4+10 c^3 d x+10 c^2 d^2 x^2+5 c d^3 x^3+d^4 x^4\right )\right ) \Gamma (n,a+b x)-5 (b c-a d)^4 \Gamma (1+n,a+b x)+d \left (-10 (b c-a d)^3 \Gamma (2+n,a+b x)-d \left (10 (b c-a d)^2 \Gamma (3+n,a+b x)+d (5 (b c-a d) \Gamma (4+n,a+b x)+d \Gamma (5+n,a+b x))\right )\right )}{5 b^5} \] Input:

Integrate[(c + d*x)^4*Gamma[n, a + b*x],x]
 

Output:

((5*a*b^4*c^4 - 10*a^2*b^3*c^3*d + 10*a^3*b^2*c^2*d^2 - 5*a^4*b*c*d^3 + a^ 
5*d^4 + b^5*x*(5*c^4 + 10*c^3*d*x + 10*c^2*d^2*x^2 + 5*c*d^3*x^3 + d^4*x^4 
))*Gamma[n, a + b*x] - 5*(b*c - a*d)^4*Gamma[1 + n, a + b*x] + d*(-10*(b*c 
 - a*d)^3*Gamma[2 + n, a + b*x] - d*(10*(b*c - a*d)^2*Gamma[3 + n, a + b*x 
] + d*(5*(b*c - a*d)*Gamma[4 + n, a + b*x] + d*Gamma[5 + n, a + b*x]))))/( 
5*b^5)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.03, 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.

\(\displaystyle \int (c+d x)^4 \Gamma (n,a+b x) \, dx\)

\(\Big \downarrow \) 7119

\(\displaystyle \frac {b \int e^{-a-b x} (a+b x)^{n-1} (c+d x)^5dx}{5 d}+\frac {(c+d x)^5 \Gamma (n,a+b x)}{5 d}\)

\(\Big \downarrow \) 2629

\(\displaystyle \frac {b \int \left (\frac {(b c-a d)^5 e^{-a-b x} (a+b x)^{n-1}}{b^5}+\frac {5 d (b c-a d)^4 e^{-a-b x} (a+b x)^n}{b^5}+\frac {10 d^2 (b c-a d)^3 e^{-a-b x} (a+b x)^{n+1}}{b^5}+\frac {10 d^3 (b c-a d)^2 e^{-a-b x} (a+b x)^{n+2}}{b^5}+\frac {5 d^4 (b c-a d) e^{-a-b x} (a+b x)^{n+3}}{b^5}+\frac {d^5 e^{-a-b x} (a+b x)^{n+4}}{b^5}\right )dx}{5 d}+\frac {(c+d x)^5 \Gamma (n,a+b x)}{5 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {b \left (-\frac {5 d^4 (b c-a d) \Gamma (n+4,a+b x)}{b^6}-\frac {10 d^3 (b c-a d)^2 \Gamma (n+3,a+b x)}{b^6}-\frac {10 d^2 (b c-a d)^3 \Gamma (n+2,a+b x)}{b^6}-\frac {5 d (b c-a d)^4 \Gamma (n+1,a+b x)}{b^6}-\frac {(b c-a d)^5 \Gamma (n,a+b x)}{b^6}-\frac {d^5 \Gamma (n+5,a+b x)}{b^6}\right )}{5 d}+\frac {(c+d x)^5 \Gamma (n,a+b x)}{5 d}\)

Input:

Int[(c + d*x)^4*Gamma[n, a + b*x],x]
 

Output:

((c + d*x)^5*Gamma[n, a + b*x])/(5*d) + (b*(-(((b*c - a*d)^5*Gamma[n, a + 
b*x])/b^6) - (5*d*(b*c - a*d)^4*Gamma[1 + n, a + b*x])/b^6 - (10*d^2*(b*c 
- a*d)^3*Gamma[2 + n, a + b*x])/b^6 - (10*d^3*(b*c - a*d)^2*Gamma[3 + n, a 
 + b*x])/b^6 - (5*d^4*(b*c - a*d)*Gamma[4 + n, a + b*x])/b^6 - (d^5*Gamma[ 
5 + n, a + b*x])/b^6))/(5*d)
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3716\) vs. \(2(163)=326\).

Time = 17.76 (sec) , antiderivative size = 3717, normalized size of antiderivative = 21.99

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(3717\)

Input:

int((d*x+c)^4*GAMMA(n,b*x+a),x,method=_RETURNVERBOSE)
 

Output:

-1/5*(5*x^4*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^5*c*d^3+x^4*exp(-b*x-a)*(b*x+a) 
^(-1+n)*a*b^4*d^4*n-x^3*exp(-b*x-a)*(b*x+a)^(-1+n)*a^2*b^3*d^4*n+10*x^3*ex 
p(-b*x-a)*(b*x+a)^(-1+n)*a*b^5*c^2*d^2+x^3*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^ 
3*d^4*n^2+15*x^3*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^4*c*d^3+7*x^3*exp(-b*x-a)* 
(b*x+a)^(-1+n)*a*b^3*d^4*n+x^2*exp(-b*x-a)*(b*x+a)^(-1+n)*a^3*b^2*d^4*n-2* 
x^2*exp(-b*x-a)*(b*x+a)^(-1+n)*a^2*b^2*d^4*n^2+10*x^2*exp(-b*x-a)*(b*x+a)^ 
(-1+n)*a*b^5*c^3*d+x^2*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^2*d^4*n^3+5*x^2*exp( 
-b*x-a)*(b*x+a)^(-1+n)*a^2*b^3*c*d^3+50*x*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b*d 
^4*n+55*exp(-b*x-a)*(b*x+a)^(-1+n)*a^2*b*c*d^3*n-35*exp(-b*x-a)*(b*x+a)^(- 
1+n)*a^3*b*c*d^3*n+30*exp(-b*x-a)*(b*x+a)^(-1+n)*a^2*b^2*c^2*d^2*n+30*exp( 
-b*x-a)*(b*x+a)^(-1+n)*a^2*b*c*d^3*n^2+30*x*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b 
^2*c*d^3-4*x^2*exp(-b*x-a)*(b*x+a)^(-1+n)*a^2*b^2*d^4*n+20*x^2*exp(-b*x-a) 
*(b*x+a)^(-1+n)*a*b^4*c^2*d^2+9*x^2*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^2*d^4*n 
^2-x*exp(-b*x-a)*(b*x+a)^(-1+n)*a^4*b*d^4*n+3*x*exp(-b*x-a)*(b*x+a)^(-1+n) 
*a^3*b*d^4*n^2-3*x*exp(-b*x-a)*(b*x+a)^(-1+n)*a^2*b*d^4*n^3+x*exp(-b*x-a)* 
(b*x+a)^(-1+n)*a*b*d^4*n^4+30*x^2*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^3*c*d^3+2 
6*x^2*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^2*d^4*n-5*x*exp(-b*x-a)*(b*x+a)^(-1+n 
)*a^3*b^2*c*d^3+x*exp(-b*x-a)*(b*x+a)^(-1+n)*a^3*b*d^4*n+10*x*exp(-b*x-a)* 
(b*x+a)^(-1+n)*a^2*b^3*c^2*d^2-12*x*exp(-b*x-a)*(b*x+a)^(-1+n)*a^2*b*d^4*n 
^2+10*x*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^4*c^3*d+10*x*exp(-b*x-a)*(b*x+a)...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1080 vs. \(2 (163) = 326\).

Time = 0.13 (sec) , antiderivative size = 1080, normalized size of antiderivative = 6.39 \[ \int (c+d x)^4 \Gamma (n,a+b x) \, dx=\text {Too large to display} \] Input:

integrate((d*x+c)^4*gamma(n,b*x+a),x, algorithm="fricas")
 

Output:

-1/5*((b^5*d^4*x^5 + 5*a*b^4*c^4 + a*d^4*n^4 - 10*(a^2 - a)*b^3*c^3*d + 10 
*(a^3 - a^2 + 2*a)*b^2*c^2*d^2 - 5*(a^4 - a^3 + 2*a^2 - 6*a)*b*c*d^3 + (a^ 
5 - a^4 + 2*a^3 - 6*a^2 + 24*a)*d^4 + (5*b^5*c*d^3 + b^4*d^4*n + 4*b^4*d^4 
)*x^4 + (5*a*b*c*d^3 - 2*(2*a^2 - 5*a)*d^4)*n^3 + (10*b^5*c^2*d^2 + b^3*d^ 
4*n^2 + 15*b^4*c*d^3 + (a + 12)*b^3*d^4 + (5*b^4*c*d^3 - (a - 7)*b^3*d^4)* 
n)*x^3 + (10*a*b^2*c^2*d^2 - 15*(a^2 - 2*a)*b*c*d^3 + (6*a^3 - 21*a^2 + 35 
*a)*d^4)*n^2 + (10*b^5*c^3*d + b^2*d^4*n^3 + 20*b^4*c^2*d^2 + 5*(a + 6)*b^ 
3*c*d^3 - (a^2 - 6*a - 24)*b^2*d^4 + (5*b^3*c*d^3 - (2*a - 9)*b^2*d^4)*n^2 
 + (10*b^4*c^2*d^2 - 5*(a - 5)*b^3*c*d^3 + (a^2 - 4*a + 26)*b^2*d^4)*n)*x^ 
2 + (10*a*b^3*c^3*d - 10*(2*a^2 - 3*a)*b^2*c^2*d^2 + 5*(3*a^3 - 7*a^2 + 11 
*a)*b*c*d^3 - (4*a^4 - 12*a^3 + 29*a^2 - 50*a)*d^4)*n + (5*b^5*c^4 + b*d^4 
*n^4 + 10*b^4*c^3*d + 10*(a + 2)*b^3*c^2*d^2 - 5*(a^2 - 4*a - 6)*b^2*c*d^3 
 + (a^3 - 4*a^2 + 18*a + 24)*b*d^4 + (5*b^2*c*d^3 - (3*a - 10)*b*d^4)*n^3 
+ (10*b^3*c^2*d^2 - 10*(a - 3)*b^2*c*d^3 + (3*a^2 - 12*a + 35)*b*d^4)*n^2 
+ (10*b^4*c^3*d - 10*(a - 3)*b^3*c^2*d^2 + 5*(a^2 - 2*a + 11)*b^2*c*d^3 - 
(a^3 - a^2 + 3*a - 50)*b*d^4)*n)*x)*(b*x + a)^(n - 1)*e^(-b*x - a) - (b^5* 
d^4*x^5 + 5*b^5*c*d^3*x^4 + 10*b^5*c^2*d^2*x^3 + 10*b^5*c^3*d*x^2 + 5*b^5* 
c^4*x + 5*a*b^4*c^4 - 10*a^2*b^3*c^3*d + 10*a^3*b^2*c^2*d^2 - 5*a^4*b*c*d^ 
3 + a^5*d^4 - d^4*n^5 - 5*(b*c*d^3 - (a - 2)*d^4)*n^4 - 5*(2*b^2*c^2*d^2 - 
 2*(2*a - 3)*b*c*d^3 + (2*a^2 - 6*a + 7)*d^4)*n^3 - 5*(2*b^3*c^3*d - 6*...
 

Sympy [F(-1)]

Timed out. \[ \int (c+d x)^4 \Gamma (n,a+b x) \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**4*uppergamma(n,b*x+a),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int (c+d x)^4 \Gamma (n,a+b x) \, dx=\int { {\left (d x + c\right )}^{4} \Gamma \left (n, b x + a\right ) \,d x } \] Input:

integrate((d*x+c)^4*gamma(n,b*x+a),x, algorithm="maxima")
 

Output:

((b*x + a)*gamma(n, b*x + a) - gamma(n + 1, b*x + a))*c^4/b + integrate(d^ 
4*x^4*gamma(n, b*x + a) + 4*c*d^3*x^3*gamma(n, b*x + a) + 6*c^2*d^2*x^2*ga 
mma(n, b*x + a) + 4*c^3*d*x*gamma(n, b*x + a), x)
 

Giac [F]

\[ \int (c+d x)^4 \Gamma (n,a+b x) \, dx=\int { {\left (d x + c\right )}^{4} \Gamma \left (n, b x + a\right ) \,d x } \] Input:

integrate((d*x+c)^4*gamma(n,b*x+a),x, algorithm="giac")
 

Output:

integrate((d*x + c)^4*gamma(n, b*x + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^4 \Gamma (n,a+b x) \, dx=\int \Gamma \left (n,a+b\,x\right )\,{\left (c+d\,x\right )}^4 \,d x \] Input:

int(igamma(n, a + b*x)*(c + d*x)^4,x)
 

Output:

int(igamma(n, a + b*x)*(c + d*x)^4, x)
 

Reduce [F]

\[ \int (c+d x)^4 \Gamma (n,a+b x) \, dx=\left (\int \gamma \left (n , b x +a \right )d x \right ) c^{4}+\left (\int \gamma \left (n , b x +a \right ) x^{4}d x \right ) d^{4}+4 \left (\int \gamma \left (n , b x +a \right ) x^{3}d x \right ) c \,d^{3}+6 \left (\int \gamma \left (n , b x +a \right ) x^{2}d x \right ) c^{2} d^{2}+4 \left (\int \gamma \left (n , b x +a \right ) x d x \right ) c^{3} d \] Input:

int((d*x+c)^4*GAMMA(n,b*x+a),x)
                                                                                    
                                                                                    
 

Output:

int(gamma(n,a + b*x),x)*c**4 + int(gamma(n,a + b*x)*x**4,x)*d**4 + 4*int(g 
amma(n,a + b*x)*x**3,x)*c*d**3 + 6*int(gamma(n,a + b*x)*x**2,x)*c**2*d**2 
+ 4*int(gamma(n,a + b*x)*x,x)*c**3*d