\(\int x^{100} \Gamma (n,a x) \, dx\) [86]

Optimal result

Integrand size = 9, antiderivative size = 27 \[ \int x^{100} \Gamma (n,a x) \, dx=\frac {1}{101} x^{101} \Gamma (n,a x)-\frac {\Gamma (101+n,a x)}{101 a^{101}} \] Output:

1/101*x^101*GAMMA(n,a*x)-1/101*GAMMA(101+n,a*x)/a^101
 

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int x^{100} \Gamma (n,a x) \, dx=\frac {1}{101} x^{101} \Gamma (n,a x)-\frac {\Gamma (101+n,a x)}{101 a^{101}} \] Input:

Integrate[x^100*Gamma[n, a*x],x]
 

Output:

(x^101*Gamma[n, a*x])/101 - Gamma[101 + n, a*x]/(101*a^101)
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {7116}

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.

\(\displaystyle \int x^{100} \Gamma (n,a x) \, dx\)

\(\Big \downarrow \) 7116

\(\displaystyle \frac {1}{101} x^{101} \Gamma (n,a x)-\frac {\Gamma (n+101,a x)}{101 a^{101}}\)

Input:

Int[x^100*Gamma[n, a*x],x]
 

Output:

(x^101*Gamma[n, a*x])/101 - Gamma[101 + n, a*x]/(101*a^101)
 

Defintions of rubi rules used

rule 7116

Int[Gamma[n_, (b_.)*(x_)]*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 
1)*(Gamma[n, b*x]/(d*(m + 1))), x] - Simp[(d*x)^m*(Gamma[m + n + 1, b*x]/(b 
*(m + 1)*(b*x)^m)), x] /; FreeQ[{b, d, m, n}, x] && NeQ[m, -1]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4.

Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.37

\[\frac {\frac {\pi \csc \left (\pi n \right ) x^{101} a^{101}}{101 \Gamma \left (-n +1\right )}-\frac {x^{101+n} a^{101+n} \operatorname {hypergeom}\left (\left [n , 101+n \right ], \left [1+n , 102+n \right ], -x a \right )}{n \left (101+n \right )}}{a^{101}}\]

Input:

int(x^100*GAMMA(n,x*a),x)
 

Output:

1/a^101*(1/101*Pi*csc(Pi*n)/GAMMA(-n+1)*x^101*a^101-1/n/(101+n)*x^(101+n)* 
a^(101+n)*hypergeom([n,101+n],[1+n,102+n],-x*a))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 41441 vs. \(2 (23) = 46\).

Time = 3.03 (sec) , antiderivative size = 41441, normalized size of antiderivative = 1534.85 \[ \int x^{100} \Gamma (n,a x) \, dx=\text {Too large to display} \] Input:

integrate(x^100*gamma(n,a*x),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int x^{100} \Gamma (n,a x) \, dx=\text {Timed out} \] Input:

integrate(x**100*uppergamma(n,a*x),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^{100} \Gamma (n,a x) \, dx=\int { x^{100} \Gamma \left (n, a x\right ) \,d x } \] Input:

integrate(x^100*gamma(n,a*x),x, algorithm="maxima")
 

Output:

integrate(x^100*gamma(n, a*x), x)
 

Giac [F]

\[ \int x^{100} \Gamma (n,a x) \, dx=\int { x^{100} \Gamma \left (n, a x\right ) \,d x } \] Input:

integrate(x^100*gamma(n,a*x),x, algorithm="giac")
 

Output:

integrate(x^100*gamma(n, a*x), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^{100} \Gamma (n,a x) \, dx=\int x^{100}\,\Gamma \left (n,a\,x\right ) \,d x \] Input:

int(x^100*igamma(n, a*x),x)
 

Output:

int(x^100*igamma(n, a*x), x)
 

Reduce [F]

\[ \int x^{100} \Gamma (n,a x) \, dx=\int \gamma \left (n , a x \right ) x^{100}d x \] Input:

int(x^100*GAMMA(n,a*x),x)
 

Output:

int(gamma(n,a*x)*x**100,x)