Integrand size = 13, antiderivative size = 71 \[ \int (c+d x) \log (\operatorname {Gamma}(a+b x)) \, dx=\frac {(c+d x)^2 \log (\operatorname {Gamma}(a+b x))}{2 d}-\frac {(c+d x)^2 \text {log$\Gamma $}(a+b x)}{2 d}-\frac {d \psi ^{(-3)}(a+b x)}{b^2}+\frac {(c+d x) \psi ^{(-2)}(a+b x)}{b} \] Output:
1/2*(d*x+c)^2*ln(GAMMA(b*x+a))/d-1/2*(d*x+c)^2*lnGAMMA(b*x+a)/d-d*Psi(-3,b *x+a)/b^2+(d*x+c)*Psi(-2,b*x+a)/b
Time = 0.06 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.46 \[ \int (c+d x) \log (\operatorname {Gamma}(a+b x)) \, dx=\frac {1}{2} d x^2 \log (\operatorname {Gamma}(a+b x))+\frac {c (a+b x) \log (\operatorname {Gamma}(a+b x))}{b}-\frac {c (a+b x) \text {log$\Gamma $}(a+b x)}{b}+\frac {c \psi ^{(-2)}(a+b x)}{b}+\frac {1}{2} d \left (-x^2 \text {log$\Gamma $}(a+b x)-\frac {2 \psi ^{(-3)}(a+b x)}{b^2}+\frac {2 x \psi ^{(-2)}(a+b x)}{b}\right ) \] Input:
Integrate[(c + d*x)*Log[Gamma[a + b*x]],x]
Output:
(d*x^2*Log[Gamma[a + b*x]])/2 + (c*(a + b*x)*Log[Gamma[a + b*x]])/b - (c*( a + b*x)*LogGamma[a + b*x])/b + (c*PolyGamma[-2, a + b*x])/b + (d*(-(x^2*L ogGamma[a + b*x]) - (2*PolyGamma[-3, a + b*x])/b^2 + (2*x*PolyGamma[-2, a + b*x])/b))/2
Time = 0.45 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.18, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3031, 27, 7125, 7122, 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.
\(\displaystyle \int (c+d x) \log (\operatorname {Gamma}(a+b x)) \, dx\) |
\(\Big \downarrow \) 3031 |
\(\displaystyle \frac {(c+d x)^2 \log (\operatorname {Gamma}(a+b x))}{2 d}-\frac {\int b (c+d x)^2 \psi ^{(0)}(a+b x)dx}{2 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {(c+d x)^2 \log (\operatorname {Gamma}(a+b x))}{2 d}-\frac {b \int (c+d x)^2 \psi ^{(0)}(a+b x)dx}{2 d}\) |
\(\Big \downarrow \) 7125 |
\(\displaystyle \frac {(c+d x)^2 \log (\operatorname {Gamma}(a+b x))}{2 d}-\frac {b \left (\frac {(c+d x)^2 \text {log$\Gamma $}(a+b x)}{b}-\frac {2 d \int (c+d x) \text {log$\Gamma $}(a+b x)dx}{b}\right )}{2 d}\) |
\(\Big \downarrow \) 7122 |
\(\displaystyle \frac {(c+d x)^2 \log (\operatorname {Gamma}(a+b x))}{2 d}-\frac {b \left (\frac {(c+d x)^2 \text {log$\Gamma $}(a+b x)}{b}-\frac {2 d \left (\frac {(c+d x) \psi ^{(-2)}(a+b x)}{b}-\frac {d \int \psi ^{(-2)}(a+b x)dx}{b}\right )}{b}\right )}{2 d}\) |
\(\Big \downarrow \) 7124 |
\(\displaystyle \frac {(c+d x)^2 \log (\operatorname {Gamma}(a+b x))}{2 d}-\frac {b \left (\frac {(c+d x)^2 \text {log$\Gamma $}(a+b x)}{b}-\frac {2 d \left (\frac {(c+d x) \psi ^{(-2)}(a+b x)}{b}-\frac {d \psi ^{(-3)}(a+b x)}{b^2}\right )}{b}\right )}{2 d}\) |
Input:
Int[(c + d*x)*Log[Gamma[a + b*x]],x]
Output:
((c + d*x)^2*Log[Gamma[a + b*x]])/(2*d) - (b*(((c + d*x)^2*LogGamma[a + b* x])/b - (2*d*(-((d*PolyGamma[-3, a + b*x])/b^2) + ((c + d*x)*PolyGamma[-2, a + b*x])/b))/b))/(2*d)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
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]
Int[PolyGamma[n_, (a_.) + (b_.)*(x_)], x_Symbol] :> Simp[PolyGamma[n - 1, a + b*x]/b, x] /; FreeQ[{a, b, n}, x]
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]
\[\int \left (d x +c \right ) \ln \left (\Gamma \left (b x +a \right )\right )d x\]
Input:
int((d*x+c)*ln(GAMMA(b*x+a)),x)
Output:
int((d*x+c)*ln(GAMMA(b*x+a)),x)
Time = 0.09 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.59 \[ \int (c+d x) \log (\operatorname {Gamma}(a+b x)) \, dx=\frac {1}{2} \, {\left (d x^{2} + 2 \, c x\right )} \log \left (\Gamma \left (b x + a\right )\right ) - \frac {1}{6} \, {\left (b d x^{3} + 3 \, b c x^{2}\right )} \psi \left (b x + a\right ) \] Input:
integrate((d*x+c)*log(gamma(b*x+a)),x, algorithm="fricas")
Output:
1/2*(d*x^2 + 2*c*x)*log(gamma(b*x + a)) - 1/6*(b*d*x^3 + 3*b*c*x^2)*psi(b* x + a)
\[ \int (c+d x) \log (\operatorname {Gamma}(a+b x)) \, dx=\int \left (c + d x\right ) \log {\left (\Gamma \left (a + b x\right ) \right )}\, dx \] Input:
integrate((d*x+c)*ln(gamma(b*x+a)),x)
Output:
Integral((c + d*x)*log(gamma(a + b*x)), x)
\[ \int (c+d x) \log (\operatorname {Gamma}(a+b x)) \, dx=\int { {\left (d x + c\right )} \log \left (\Gamma \left (b x + a\right )\right ) \,d x } \] Input:
integrate((d*x+c)*log(gamma(b*x+a)),x, algorithm="maxima")
Output:
integrate((d*x + c)*log(gamma(b*x + a)), x)
\[ \int (c+d x) \log (\operatorname {Gamma}(a+b x)) \, dx=\int { {\left (d x + c\right )} \log \left (\Gamma \left (b x + a\right )\right ) \,d x } \] Input:
integrate((d*x+c)*log(gamma(b*x+a)),x, algorithm="giac")
Output:
integrate((d*x + c)*log(gamma(b*x + a)), x)
Timed out. \[ \int (c+d x) \log (\operatorname {Gamma}(a+b x)) \, dx=\int \ln \left (\Gamma \left (a+b\,x\right )\right )\,\left (c+d\,x\right ) \,d x \] Input:
int(log(gamma(a + b*x))*(c + d*x),x)
Output:
int(log(gamma(a + b*x))*(c + d*x), x)
\[ \int (c+d x) \log (\operatorname {Gamma}(a+b x)) \, dx=\left (\int \mathrm {log}\left (\gamma \left (b x +a \right )\right )d x \right ) c +\left (\int \mathrm {log}\left (\gamma \left (b x +a \right )\right ) x d x \right ) d \] Input:
int((d*x+c)*log(GAMMA(b*x+a)),x)
Output:
int(log(gamma(a + b*x)),x)*c + int(log(gamma(a + b*x))*x,x)*d