\(\int (c+d x)^2 \log (\operatorname {Gamma}(a+b x)) \, dx\) [212]

Optimal result

Integrand size = 15, antiderivative size = 93 \[ \int (c+d x)^2 \log (\operatorname {Gamma}(a+b x)) \, dx=\frac {(c+d x)^3 \log (\operatorname {Gamma}(a+b x))}{3 d}-\frac {(c+d x)^3 \text {log$\Gamma $}(a+b x)}{3 d}+\frac {2 d^2 \psi ^{(-4)}(a+b x)}{b^3}-\frac {2 d (c+d x) \psi ^{(-3)}(a+b x)}{b^2}+\frac {(c+d x)^2 \psi ^{(-2)}(a+b x)}{b} \] Output:

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

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.24 \[ \int (c+d x)^2 \log (\operatorname {Gamma}(a+b x)) \, dx=\frac {-b^3 x \left (3 c^2+3 c d x+d^2 x^2\right ) \text {log$\Gamma $}(a+b x)+6 d^2 \psi ^{(-4)}(a+b x)+b \left (-6 d (c+d x) \psi ^{(-3)}(a+b x)+b \left (b x \left (3 c^2+3 c d x+d^2 x^2\right ) \log (\operatorname {Gamma}(a+b x))+3 (c+d x)^2 \psi ^{(-2)}(a+b x)\right )\right )}{3 b^3} \] Input:

Integrate[(c + d*x)^2*Log[Gamma[a + b*x]],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.17, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3031, 27, 7125, 7122, 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.

\(\displaystyle \int (c+d x)^2 \log (\operatorname {Gamma}(a+b x)) \, dx\)

\(\Big \downarrow \) 3031

\(\displaystyle \frac {(c+d x)^3 \log (\operatorname {Gamma}(a+b x))}{3 d}-\frac {\int b (c+d x)^3 \psi ^{(0)}(a+b x)dx}{3 d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {(c+d x)^3 \log (\operatorname {Gamma}(a+b x))}{3 d}-\frac {b \int (c+d x)^3 \psi ^{(0)}(a+b x)dx}{3 d}\)

\(\Big \downarrow \) 7125

\(\displaystyle \frac {(c+d x)^3 \log (\operatorname {Gamma}(a+b x))}{3 d}-\frac {b \left (\frac {(c+d x)^3 \text {log$\Gamma $}(a+b x)}{b}-\frac {3 d \int (c+d x)^2 \text {log$\Gamma $}(a+b x)dx}{b}\right )}{3 d}\)

\(\Big \downarrow \) 7122

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

\(\Big \downarrow \) 7125

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

\(\Big \downarrow \) 7124

\(\displaystyle \frac {(c+d x)^3 \log (\operatorname {Gamma}(a+b x))}{3 d}-\frac {b \left (\frac {(c+d x)^3 \text {log$\Gamma $}(a+b x)}{b}-\frac {3 d \left (\frac {(c+d x)^2 \psi ^{(-2)}(a+b x)}{b}-\frac {2 d \left (\frac {(c+d x) \psi ^{(-3)}(a+b x)}{b}-\frac {d \psi ^{(-4)}(a+b x)}{b^2}\right )}{b}\right )}{b}\right )}{3 d}\)

Input:

Int[(c + d*x)^2*Log[Gamma[a + b*x]],x]
 

Output:

((c + d*x)^3*Log[Gamma[a + b*x]])/(3*d) - (b*(((c + d*x)^3*LogGamma[a + b* 
x])/b - (3*d*((-2*d*(-((d*PolyGamma[-4, a + b*x])/b^2) + ((c + d*x)*PolyGa 
mma[-3, a + b*x])/b))/b + ((c + d*x)^2*PolyGamma[-2, a + b*x])/b))/b))/(3* 
d)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3031

Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[(a + b*x)^(m + 1) 
*(Log[u]/(b*(m + 1))), x] - Simp[1/(b*(m + 1))   Int[SimplifyIntegrand[(a + 
 b*x)^(m + 1)*(D[u, x]/u), x], x], x] /; FreeQ[{a, b, m}, x] && InverseFunc 
tionFreeQ[u, x] && NeQ[m, -1]
 

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 = 0.43 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.73

method	result	size
default	\(\frac {\ln \left (\Gamma \left (b x +a \right )\right ) d^{2} x^{3}}{3}+\ln \left (\Gamma \left (b x +a \right )\right ) d c \,x^{2}+\ln \left (\Gamma \left (b x +a \right )\right ) c^{2} x +\frac {\ln \left (\Gamma \left (b x +a \right )\right ) c^{3}}{3 d}-\frac {\Psi \left (-1, b x +a \right ) \left (d x +c \right )^{3}-\frac {3 d \left (\Psi \left (-2, b x +a \right ) \left (\frac {d \left (b x +a \right )}{b}-\frac {a d}{b}+c \right )^{2}-\frac {2 d \left (\Psi \left (-3, b x +a \right ) \left (\frac {d \left (b x +a \right )}{b}-\frac {a d}{b}+c \right )-\frac {d \Psi \left (-4, b x +a \right )}{b}\right )}{b}\right )}{b}}{3 d}\)	\(161\)
parts	\(\frac {\ln \left (\Gamma \left (b x +a \right )\right ) d^{2} x^{3}}{3}+\ln \left (\Gamma \left (b x +a \right )\right ) d c \,x^{2}+\ln \left (\Gamma \left (b x +a \right )\right ) c^{2} x +\frac {\ln \left (\Gamma \left (b x +a \right )\right ) c^{3}}{3 d}-\frac {\Psi \left (-1, b x +a \right ) \left (d x +c \right )^{3}-\frac {3 d \left (\Psi \left (-2, b x +a \right ) \left (\frac {d \left (b x +a \right )}{b}-\frac {a d}{b}+c \right )^{2}-\frac {2 d \left (\Psi \left (-3, b x +a \right ) \left (\frac {d \left (b x +a \right )}{b}-\frac {a d}{b}+c \right )-\frac {d \Psi \left (-4, b x +a \right )}{b}\right )}{b}\right )}{b}}{3 d}\)	\(161\)

Input:

int((d*x+c)^2*ln(GAMMA(b*x+a)),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.70 \[ \int (c+d x)^2 \log (\operatorname {Gamma}(a+b x)) \, dx=\frac {1}{3} \, {\left (d^{2} x^{3} + 3 \, c d x^{2} + 3 \, c^{2} x\right )} \log \left (\Gamma \left (b x + a\right )\right ) - \frac {1}{12} \, {\left (b d^{2} x^{4} + 4 \, b c d x^{3} + 6 \, b c^{2} x^{2}\right )} \psi \left (b x + a\right ) \] Input:

integrate((d*x+c)^2*log(gamma(b*x+a)),x, algorithm="fricas")
 

Output:

1/3*(d^2*x^3 + 3*c*d*x^2 + 3*c^2*x)*log(gamma(b*x + a)) - 1/12*(b*d^2*x^4 
+ 4*b*c*d*x^3 + 6*b*c^2*x^2)*psi(b*x + a)
 

Sympy [F]

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

integrate((d*x+c)**2*ln(gamma(b*x+a)),x)
 

Output:

Integral((c + d*x)**2*log(gamma(a + b*x)), x)
 

Maxima [F]

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

integrate((d*x+c)^2*log(gamma(b*x+a)),x, algorithm="maxima")
 

Output:

integrate((d*x + c)^2*log(gamma(b*x + a)), x)
 

Giac [F]

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

integrate((d*x+c)^2*log(gamma(b*x+a)),x, algorithm="giac")
 

Output:

integrate((d*x + c)^2*log(gamma(b*x + a)), x)
 

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^2 \log (\operatorname {Gamma}(a+b x)) \, dx=\int \ln \left (\Gamma \left (a+b\,x\right )\right )\,{\left (c+d\,x\right )}^2 \,d x \] Input:

int(log(gamma(a + b*x))*(c + d*x)^2,x)
 

Output:

int(log(gamma(a + b*x))*(c + d*x)^2, x)
 

Reduce [F]

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

int((d*x+c)^2*log(GAMMA(b*x+a)),x)
 

Output:

int(log(gamma(a + b*x)),x)*c**2 + int(log(gamma(a + b*x))*x**2,x)*d**2 + 2 
*int(log(gamma(a + b*x))*x,x)*c*d