\(\int (c+d x)^3 \Gamma (n,a+b x) \, dx\) [189]

Optimal result

Integrand size = 15, antiderivative size = 144 \[ \int (c+d x)^3 \Gamma (n,a+b x) \, dx=-\frac {(b c-a d)^4 \Gamma (n,a+b x)}{4 b^4 d}+\frac {(c+d x)^4 \Gamma (n,a+b x)}{4 d}-\frac {(b c-a d)^3 \Gamma (1+n,a+b x)}{b^4}-\frac {3 d (b c-a d)^2 \Gamma (2+n,a+b x)}{2 b^4}-\frac {d^2 (b c-a d) \Gamma (3+n,a+b x)}{b^4}-\frac {d^3 \Gamma (4+n,a+b x)}{4 b^4} \] Output:

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

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.17 \[ \int (c+d x)^3 \Gamma (n,a+b x) \, dx=\frac {\left (4 a b^3 c^3-6 a^2 b^2 c^2 d+4 a^3 b c d^2-a^4 d^3+b^4 x \left (4 c^3+6 c^2 d x+4 c d^2 x^2+d^3 x^3\right )\right ) \Gamma (n,a+b x)-4 (b c-a d)^3 \Gamma (1+n,a+b x)-d \left (6 (b c-a d)^2 \Gamma (2+n,a+b x)+d (4 (b c-a d) \Gamma (3+n,a+b x)+d \Gamma (4+n,a+b x))\right )}{4 b^4} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.59 (sec) , antiderivative size = 147, normalized size of antiderivative = 1.02, 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.

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

\(\Big \downarrow \) 7119

\(\displaystyle \frac {b \int e^{-a-b x} (a+b x)^{n-1} (c+d x)^4dx}{4 d}+\frac {(c+d x)^4 \Gamma (n,a+b x)}{4 d}\)

\(\Big \downarrow \) 2629

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

((c + d*x)^4*Gamma[n, a + b*x])/(4*d) + (b*(-(((b*c - a*d)^4*Gamma[n, a + 
b*x])/b^5) - (4*d*(b*c - a*d)^3*Gamma[1 + n, a + b*x])/b^5 - (6*d^2*(b*c - 
 a*d)^2*Gamma[2 + n, a + b*x])/b^5 - (4*d^3*(b*c - a*d)*Gamma[3 + n, a + b 
*x])/b^5 - (d^4*Gamma[4 + n, a + b*x])/b^5))/(4*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. \(1928\) vs. \(2(136)=272\).

Time = 5.84 (sec) , antiderivative size = 1929, normalized size of antiderivative = 13.40

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1929\)

Input:

int((d*x+c)^3*GAMMA(n,b*x+a),x,method=_RETURNVERBOSE)
 

Output:

-1/4*(8*x^2*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^3*c*d^2+5*x^2*exp(-b*x-a)*(b*x+ 
a)^(-1+n)*a*b^2*d^3*n+4*x^3*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^4*c*d^2+x^3*exp 
(-b*x-a)*(b*x+a)^(-1+n)*a*b^3*d^3*n-x^2*exp(-b*x-a)*(b*x+a)^(-1+n)*a^2*b^2 
*d^3*n+6*x^2*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^4*c^2*d+x^2*exp(-b*x-a)*(b*x+a 
)^(-1+n)*a*b^2*d^3*n^2+x*exp(-b*x-a)*(b*x+a)^(-1+n)*a^3*b*d^3*n-2*x*exp(-b 
*x-a)*(b*x+a)^(-1+n)*a^2*b*d^3*n^2+x*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b*d^3*n^ 
3+4*x*exp(-b*x-a)*(b*x+a)^(-1+n)*a^2*b^2*c*d^2-2*x*exp(-b*x-a)*(b*x+a)^(-1 
+n)*a^2*b*d^3*n+6*x*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^3*c^2*d+6*x*exp(-b*x-a) 
*(b*x+a)^(-1+n)*a*b*d^3*n^2-8*exp(-b*x-a)*(b*x+a)^(-1+n)*a^3*b*c*d^2*n+6*e 
xp(-b*x-a)*(b*x+a)^(-1+n)*a^2*b^2*c^2*d*n+4*exp(-b*x-a)*(b*x+a)^(-1+n)*a^2 
*b*c*d^2*n^2+8*x*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^2*c*d^2+11*x*exp(-b*x-a)*( 
b*x+a)^(-1+n)*a*b*d^3*n+12*exp(-b*x-a)*(b*x+a)^(-1+n)*a^2*b*c*d^2*n-x^4*GA 
MMA(n,b*x+a)*a*b^4*d^3-4*x*GAMMA(n,b*x+a)*a*b^4*c^3+3*exp(-b*x-a)*(b*x+a)^ 
(-1+n)*a^4*d^3*n-3*exp(-b*x-a)*(b*x+a)^(-1+n)*a^3*d^3*n^2+4*exp(-b*x-a)*(b 
*x+a)^(-1+n)*a^2*b^3*c^3+exp(-b*x-a)*(b*x+a)^(-1+n)*a^2*d^3*n^3+4*x^2*exp( 
-b*x-a)*(b*x+a)^(-1+n)*a*b^3*c*d^2*n-4*x*exp(-b*x-a)*(b*x+a)^(-1+n)*a^2*b^ 
2*c*d^2*n+6*x*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^3*c^2*d*n+4*x*exp(-b*x-a)*(b* 
x+a)^(-1+n)*a*b^2*c*d^2*n^2+12*x*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^2*c*d^2*n+ 
x^2*exp(-b*x-a)*(b*x+a)^(-1+n)*a^2*b^2*d^3-6*x^2*GAMMA(n,b*x+a)*a*b^4*c^2* 
d+4*x*exp(-b*x-a)*(b*x+a)^(-1+n)*a*b^4*c^3+6*x^2*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. 618 vs. \(2 (136) = 272\).

Time = 0.14 (sec) , antiderivative size = 618, normalized size of antiderivative = 4.29 \[ \int (c+d x)^3 \Gamma (n,a+b x) \, dx=-\frac {{\left (b^{4} d^{3} x^{4} + 4 \, a b^{3} c^{3} + a d^{3} n^{3} - 6 \, {\left (a^{2} - a\right )} b^{2} c^{2} d + 4 \, {\left (a^{3} - a^{2} + 2 \, a\right )} b c d^{2} - {\left (a^{4} - a^{3} + 2 \, a^{2} - 6 \, a\right )} d^{3} + {\left (4 \, b^{4} c d^{2} + b^{3} d^{3} n + 3 \, b^{3} d^{3}\right )} x^{3} + {\left (4 \, a b c d^{2} - 3 \, {\left (a^{2} - 2 \, a\right )} d^{3}\right )} n^{2} + {\left (6 \, b^{4} c^{2} d + b^{2} d^{3} n^{2} + 8 \, b^{3} c d^{2} + {\left (a + 6\right )} b^{2} d^{3} + {\left (4 \, b^{3} c d^{2} - {\left (a - 5\right )} b^{2} d^{3}\right )} n\right )} x^{2} + {\left (6 \, a b^{2} c^{2} d - 4 \, {\left (2 \, a^{2} - 3 \, a\right )} b c d^{2} + {\left (3 \, a^{3} - 7 \, a^{2} + 11 \, a\right )} d^{3}\right )} n + {\left (4 \, b^{4} c^{3} + b d^{3} n^{3} + 6 \, b^{3} c^{2} d + 4 \, {\left (a + 2\right )} b^{2} c d^{2} - {\left (a^{2} - 4 \, a - 6\right )} b d^{3} + 2 \, {\left (2 \, b^{2} c d^{2} - {\left (a - 3\right )} b d^{3}\right )} n^{2} + {\left (6 \, b^{3} c^{2} d - 4 \, {\left (a - 3\right )} b^{2} c d^{2} + {\left (a^{2} - 2 \, a + 11\right )} b d^{3}\right )} n\right )} x\right )} {\left (b x + a\right )}^{n - 1} e^{\left (-b x - a\right )} - {\left (b^{4} d^{3} x^{4} + 4 \, b^{4} c d^{2} x^{3} + 6 \, b^{4} c^{2} d x^{2} + 4 \, b^{4} c^{3} x + 4 \, a b^{3} c^{3} - 6 \, a^{2} b^{2} c^{2} d + 4 \, a^{3} b c d^{2} - a^{4} d^{3} - d^{3} n^{4} - 2 \, {\left (2 \, b c d^{2} - {\left (2 \, a - 3\right )} d^{3}\right )} n^{3} - {\left (6 \, b^{2} c^{2} d - 12 \, {\left (a - 1\right )} b c d^{2} + {\left (6 \, a^{2} - 12 \, a + 11\right )} d^{3}\right )} n^{2} - 2 \, {\left (2 \, b^{3} c^{3} - 3 \, {\left (2 \, a - 1\right )} b^{2} c^{2} d + 2 \, {\left (3 \, a^{2} - 3 \, a + 2\right )} b c d^{2} - {\left (2 \, a^{3} - 3 \, a^{2} + 4 \, a - 3\right )} d^{3}\right )} n\right )} \Gamma \left (n, b x + a\right )}{4 \, b^{4}} \] Input:

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

Output:

-1/4*((b^4*d^3*x^4 + 4*a*b^3*c^3 + a*d^3*n^3 - 6*(a^2 - a)*b^2*c^2*d + 4*( 
a^3 - a^2 + 2*a)*b*c*d^2 - (a^4 - a^3 + 2*a^2 - 6*a)*d^3 + (4*b^4*c*d^2 + 
b^3*d^3*n + 3*b^3*d^3)*x^3 + (4*a*b*c*d^2 - 3*(a^2 - 2*a)*d^3)*n^2 + (6*b^ 
4*c^2*d + b^2*d^3*n^2 + 8*b^3*c*d^2 + (a + 6)*b^2*d^3 + (4*b^3*c*d^2 - (a 
- 5)*b^2*d^3)*n)*x^2 + (6*a*b^2*c^2*d - 4*(2*a^2 - 3*a)*b*c*d^2 + (3*a^3 - 
 7*a^2 + 11*a)*d^3)*n + (4*b^4*c^3 + b*d^3*n^3 + 6*b^3*c^2*d + 4*(a + 2)*b 
^2*c*d^2 - (a^2 - 4*a - 6)*b*d^3 + 2*(2*b^2*c*d^2 - (a - 3)*b*d^3)*n^2 + ( 
6*b^3*c^2*d - 4*(a - 3)*b^2*c*d^2 + (a^2 - 2*a + 11)*b*d^3)*n)*x)*(b*x + a 
)^(n - 1)*e^(-b*x - a) - (b^4*d^3*x^4 + 4*b^4*c*d^2*x^3 + 6*b^4*c^2*d*x^2 
+ 4*b^4*c^3*x + 4*a*b^3*c^3 - 6*a^2*b^2*c^2*d + 4*a^3*b*c*d^2 - a^4*d^3 - 
d^3*n^4 - 2*(2*b*c*d^2 - (2*a - 3)*d^3)*n^3 - (6*b^2*c^2*d - 12*(a - 1)*b* 
c*d^2 + (6*a^2 - 12*a + 11)*d^3)*n^2 - 2*(2*b^3*c^3 - 3*(2*a - 1)*b^2*c^2* 
d + 2*(3*a^2 - 3*a + 2)*b*c*d^2 - (2*a^3 - 3*a^2 + 4*a - 3)*d^3)*n)*gamma( 
n, b*x + a))/b^4
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

int(igamma(n, a + b*x)*(c + d*x)^3,x)
 

Output:

int(igamma(n, a + b*x)*(c + d*x)^3, x)
 

Reduce [F]

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

int((d*x+c)^3*GAMMA(n,b*x+a),x)
                                                                                    
                                                                                    
 

Output:

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