\(\int (c+d x)^3 \text {log$\Gamma $}(a+b x) \, dx\) [203]

Optimal result

Integrand size = 14, antiderivative size = 74 \[ \int (c+d x)^3 \text {log$\Gamma $}(a+b x) \, dx=-\frac {6 d^3 \psi ^{(-5)}(a+b x)}{b^4}+\frac {6 d^2 (c+d x) \psi ^{(-4)}(a+b x)}{b^3}-\frac {3 d (c+d x)^2 \psi ^{(-3)}(a+b x)}{b^2}+\frac {(c+d x)^3 \psi ^{(-2)}(a+b x)}{b} \] Output:

-6*d^3*Psi(-5,b*x+a)/b^4+6*d^2*(d*x+c)*Psi(-4,b*x+a)/b^3-3*d*(d*x+c)^2*Psi 
(-3,b*x+a)/b^2+(d*x+c)^3*Psi(-2,b*x+a)/b
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.93 \[ \int (c+d x)^3 \text {log$\Gamma $}(a+b x) \, dx=\frac {-6 d^3 \psi ^{(-5)}(a+b x)+b (c+d x) \left (6 d^2 \psi ^{(-4)}(a+b x)+b (c+d x) (-3 d \psi ^{(-3)}(a+b x)+b (c+d x) \psi ^{(-2)}(a+b x))\right )}{b^4} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {7122, 7125, 7125, 7124}

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.

\(\displaystyle \int (c+d x)^3 \text {log$\Gamma $}(a+b x) \, dx\)

\(\Big \downarrow \) 7122

\(\displaystyle \frac {(c+d x)^3 \psi ^{(-2)}(a+b x)}{b}-\frac {3 d \int (c+d x)^2 \psi ^{(-2)}(a+b x)dx}{b}\)

\(\Big \downarrow \) 7125

\(\displaystyle \frac {(c+d x)^3 \psi ^{(-2)}(a+b x)}{b}-\frac {3 d \left (\frac {(c+d x)^2 \psi ^{(-3)}(a+b x)}{b}-\frac {2 d \int (c+d x) \psi ^{(-3)}(a+b x)dx}{b}\right )}{b}\)

\(\Big \downarrow \) 7125

\(\displaystyle \frac {(c+d x)^3 \psi ^{(-2)}(a+b x)}{b}-\frac {3 d \left (\frac {(c+d x)^2 \psi ^{(-3)}(a+b x)}{b}-\frac {2 d \left (\frac {(c+d x) \psi ^{(-4)}(a+b x)}{b}-\frac {d \int \psi ^{(-4)}(a+b x)dx}{b}\right )}{b}\right )}{b}\)

\(\Big \downarrow \) 7124

\(\displaystyle \frac {(c+d x)^3 \psi ^{(-2)}(a+b x)}{b}-\frac {3 d \left (\frac {(c+d x)^2 \psi ^{(-3)}(a+b x)}{b}-\frac {2 d \left (\frac {(c+d x) \psi ^{(-4)}(a+b x)}{b}-\frac {d \psi ^{(-5)}(a+b x)}{b^2}\right )}{b}\right )}{b}\)

Input:

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

Output:

(-3*d*((-2*d*(-((d*PolyGamma[-5, a + b*x])/b^2) + ((c + d*x)*PolyGamma[-4, 
 a + b*x])/b))/b + ((c + d*x)^2*PolyGamma[-3, a + b*x])/b))/b + ((c + d*x) 
^3*PolyGamma[-2, a + b*x])/b
 

Defintions of rubi rules used

rule 7122

Int[LogGamma[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> S 
imp[(c + d*x)^m*(PolyGamma[-2, a + b*x]/b), x] - Simp[d*(m/b)   Int[(c + d* 
x)^(m - 1)*PolyGamma[-2, a + b*x], x], x] /; FreeQ[{a, b, c, d}, x] && IGtQ 
[m, 0]
 

rule 7124

Int[PolyGamma[n_, (a_.) + (b_.)*(x_)], x_Symbol] :> Simp[PolyGamma[n - 1, a 
 + b*x]/b, x] /; FreeQ[{a, b, n}, 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 = 0.40 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.82

method	result	size
derivativedivides	\(\frac {\Psi \left (-2, b x +a \right ) \left (d x +c \right )^{3}-\frac {3 d \left (\Psi \left (-3, b x +a \right ) \left (\frac {d \left (b x +a \right )}{b}-\frac {a d}{b}+c \right )^{2}-\frac {2 d \left (\Psi \left (-4, b x +a \right ) \left (\frac {d \left (b x +a \right )}{b}-\frac {a d}{b}+c \right )-\frac {d \Psi \left (-5, b x +a \right )}{b}\right )}{b}\right )}{b}+\frac {\ln \left (2 \pi \right ) \left (\frac {d \left (b x +a \right )}{b}-\frac {a d}{b}+c \right )^{4} b}{8 d}}{b}\)	\(135\)
default	\(\frac {\Psi \left (-2, b x +a \right ) \left (d x +c \right )^{3}-\frac {3 d \left (\Psi \left (-3, b x +a \right ) \left (\frac {d \left (b x +a \right )}{b}-\frac {a d}{b}+c \right )^{2}-\frac {2 d \left (\Psi \left (-4, b x +a \right ) \left (\frac {d \left (b x +a \right )}{b}-\frac {a d}{b}+c \right )-\frac {d \Psi \left (-5, b x +a \right )}{b}\right )}{b}\right )}{b}+\frac {\ln \left (2 \pi \right ) \left (\frac {d \left (b x +a \right )}{b}-\frac {a d}{b}+c \right )^{4} b}{8 d}}{b}\)	\(135\)
parts	\(\frac {d^{3} \operatorname {lnGAMMA}\left (b x +a \right ) x^{4}}{4}+d^{2} \operatorname {lnGAMMA}\left (b x +a \right ) c \,x^{3}+\frac {3 d \operatorname {lnGAMMA}\left (b x +a \right ) c^{2} x^{2}}{2}+\operatorname {lnGAMMA}\left (b x +a \right ) c^{3} x +\frac {\operatorname {lnGAMMA}\left (b x +a \right ) c^{4}}{4 d}-\frac {\Psi \left (-1, b x +a \right ) \left (d x +c \right )^{4}-\frac {4 d \left (\Psi \left (-2, b x +a \right ) \left (\frac {d \left (b x +a \right )}{b}-\frac {a d}{b}+c \right )^{3}-\frac {3 d \left (\Psi \left (-3, b x +a \right ) \left (\frac {d \left (b x +a \right )}{b}-\frac {a d}{b}+c \right )^{2}-\frac {2 d \left (\Psi \left (-4, b x +a \right ) \left (\frac {d \left (b x +a \right )}{b}-\frac {a d}{b}+c \right )-\frac {d \Psi \left (-5, b x +a \right )}{b}\right )}{b}\right )}{b}\right )}{b}}{4 d}\)	\(210\)

Input:

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

Output:

1/b*(Psi(-2,b*x+a)*(d*x+c)^3-3*d/b*(Psi(-3,b*x+a)*(d/b*(b*x+a)-a*d/b+c)^2- 
2*d/b*(Psi(-4,b*x+a)*(d/b*(b*x+a)-a*d/b+c)-d/b*Psi(-5,b*x+a)))+1/8*ln(2*Pi 
)*(d/b*(b*x+a)-a*d/b+c)^4/d*b)
 

Fricas [F]

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

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

Output:

integral((d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*lngamma(b*x + a), x)
 

Sympy [F]

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

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

Output:

Integral((c + d*x)**3*lnGAMMA(a + b*x), x)
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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