\(\int x^2 \psi ^{(1)}(a+b x) \, dx\) [229]

Optimal result

Integrand size = 11, antiderivative size = 39 \[ \int x^2 \psi ^{(1)}(a+b x) \, dx=-\frac {2 x \text {log$\Gamma $}(a+b x)}{b^2}+\frac {2 \psi ^{(-2)}(a+b x)}{b^3}+\frac {x^2 \psi ^{(0)}(a+b x)}{b} \] Output:

-2*x*lnGAMMA(b*x+a)/b^2+2*Psi(-2,b*x+a)/b^3+x^2*Psi(b*x+a)/b
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.03 \[ \int x^2 \psi ^{(1)}(a+b x) \, dx=\frac {-\frac {2 x \text {log$\Gamma $}(a+b x)}{b}+\frac {2 \psi ^{(-2)}(a+b x)}{b^2}+x^2 \psi ^{(0)}(a+b x)}{b} \] Input:

Integrate[x^2*PolyGamma[1, a + b*x],x]
 

Output:

((-2*x*LogGamma[a + b*x])/b + (2*PolyGamma[-2, a + b*x])/b^2 + x^2*PolyGam 
ma[0, a + b*x])/b
 

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {7125, 7125, 7121}

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.

\(\displaystyle \int x^2 \psi ^{(1)}(a+b x) \, dx\)

\(\Big \downarrow \) 7125

\(\displaystyle \frac {x^2 \psi ^{(0)}(a+b x)}{b}-\frac {2 \int x \psi ^{(0)}(a+b x)dx}{b}\)

\(\Big \downarrow \) 7125

\(\displaystyle \frac {x^2 \psi ^{(0)}(a+b x)}{b}-\frac {2 \left (\frac {x \text {log$\Gamma $}(a+b x)}{b}-\frac {\int \text {log$\Gamma $}(a+b x)dx}{b}\right )}{b}\)

\(\Big \downarrow \) 7121

\(\displaystyle \frac {x^2 \psi ^{(0)}(a+b x)}{b}-\frac {2 \left (\frac {x \text {log$\Gamma $}(a+b x)}{b}-\frac {\psi ^{(-2)}(a+b x)}{b^2}\right )}{b}\)

Input:

Int[x^2*PolyGamma[1, a + b*x],x]
 

Output:

(-2*((x*LogGamma[a + b*x])/b - PolyGamma[-2, a + b*x]/b^2))/b + (x^2*PolyG 
amma[0, a + b*x])/b
 

Defintions of rubi rules used

rule 7121

Int[LogGamma[(a_.) + (b_.)*(x_)], x_Symbol] :> Simp[PolyGamma[-2, a + b*x]/ 
b, x] /; FreeQ[{a, b}, x]
 

rule 7125

Int[((c_.) + (d_.)*(x_))^(m_.)*PolyGamma[n_, (a_.) + (b_.)*(x_)], x_Symbol] 
 :> Simp[(c + d*x)^m*(PolyGamma[n - 1, a + b*x]/b), x] - Simp[d*(m/b)   Int 
[(c + d*x)^(m - 1)*PolyGamma[n - 1, a + b*x], x], x] /; FreeQ[{a, b, c, d, 
n}, x] && GtQ[m, 0]
 

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {\Psi \left (b x +a \right ) b^{2} x^{2}-2 \Psi \left (-1, b x +a \right ) b x +2 \Psi \left (-2, b x +a \right )}{b^{3}}\)	\(39\)
default	\(\frac {\Psi \left (b x +a \right ) b^{2} x^{2}-2 \Psi \left (-1, b x +a \right ) b x +2 \Psi \left (-2, b x +a \right )}{b^{3}}\)	\(39\)

Input:

int(x^2*Psi(1,b*x+a),x,method=_RETURNVERBOSE)
 

Output:

1/b^3*(Psi(b*x+a)*b^2*x^2-2*Psi(-1,b*x+a)*b*x+2*Psi(-2,b*x+a))
 

Fricas [F]

\[ \int x^2 \psi ^{(1)}(a+b x) \, dx=\int { x^{2} \Psi \left (1, b x + a\right ) \,d x } \] Input:

integrate(x^2*Psi(1,b*x+a),x, algorithm="fricas")
 

Output:

integral(x^2*Psi(1, b*x + a), x)
 

Sympy [F]

\[ \int x^2 \psi ^{(1)}(a+b x) \, dx=\int x^{2} \operatorname {diEulerGamma}{\left (1,a + b x \right )}\, dx \] Input:

integrate(x**2*diEulerGamma(1,b*x+a),x)
 

Output:

Integral(x**2*diEulerGamma(1, a + b*x), x)
 

Maxima [F]

\[ \int x^2 \psi ^{(1)}(a+b x) \, dx=\int { x^{2} \Psi \left (1, b x + a\right ) \,d x } \] Input:

integrate(x^2*Psi(1,b*x+a),x, algorithm="maxima")
 

Output:

integrate(x^2*Psi(1, b*x + a), x)
 

Giac [F]

\[ \int x^2 \psi ^{(1)}(a+b x) \, dx=\int { x^{2} \Psi \left (1, b x + a\right ) \,d x } \] Input:

integrate(x^2*Psi(1,b*x+a),x, algorithm="giac")
 

Output:

integrate(x^2*Psi(1, b*x + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^2 \psi ^{(1)}(a+b x) \, dx=\int x^2\,\psi ^{\left (a+b\,x\right )}\left (1\right ) \,d x \] Input:

int(x^2*psi(a + b*x, 1),x)
 

Output:

int(x^2*psi(a + b*x, 1), x)
 

Reduce [F]

\[ \int x^2 \psi ^{(1)}(a+b x) \, dx=\int \mathit {polygamma} \left (1, b x +a \right ) x^{2}d x \] Input:

int(x^2*Psi(1,b*x+a),x)
 

Output:

int(polygamma(1,a + b*x)*x**2,x)