\(\int (g+h x)^2 \log (e (f (a+b x)^p (c+d x)^q)^r) \, dx\) [27]

Optimal result

Integrand size = 29, antiderivative size = 218 \[ \int (g+h x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {(b g-a h)^2 p r x}{3 b^2}-\frac {(d g-c h)^2 q r x}{3 d^2}-\frac {(b g-a h) p r (g+h x)^2}{6 b h}-\frac {(d g-c h) q r (g+h x)^2}{6 d h}-\frac {p r (g+h x)^3}{9 h}-\frac {q r (g+h x)^3}{9 h}-\frac {(b g-a h)^3 p r \log (a+b x)}{3 b^3 h}-\frac {(d g-c h)^3 q r \log (c+d x)}{3 d^3 h}+\frac {(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h} \] Output:

-1/3*(-a*h+b*g)^2*p*r*x/b^2-1/3*(-c*h+d*g)^2*q*r*x/d^2-1/6*(-a*h+b*g)*p*r* 
(h*x+g)^2/b/h-1/6*(-c*h+d*g)*q*r*(h*x+g)^2/d/h-1/9*p*r*(h*x+g)^3/h-1/9*q*r 
*(h*x+g)^3/h-1/3*(-a*h+b*g)^3*p*r*ln(b*x+a)/b^3/h-1/3*(-c*h+d*g)^3*q*r*ln( 
d*x+c)/d^3/h+1/3*(h*x+g)^3*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/h
 

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 209, normalized size of antiderivative = 0.96 \[ \int (g+h x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {-\frac {r \left (6 d^3 (b g-a h)^3 p \log (a+b x)+b \left (6 a^2 d^3 h^3 p x-3 a b d^3 h p \left (g^2+6 g h x+h^2 x^2\right )+b^2 d \left (6 c^2 h^3 q x-3 c d h q \left (g^2+6 g h x+h^2 x^2\right )+d^2 (p+q) \left (5 g^3+18 g^2 h x+9 g h^2 x^2+2 h^3 x^3\right )\right )+6 b^2 (d g-c h)^3 q \log (c+d x)\right )\right )}{6 b^3 d^3}+(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h} \] Input:

Integrate[(g + h*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r],x]
 

Output:

(-1/6*(r*(6*d^3*(b*g - a*h)^3*p*Log[a + b*x] + b*(6*a^2*d^3*h^3*p*x - 3*a* 
b*d^3*h*p*(g^2 + 6*g*h*x + h^2*x^2) + b^2*d*(6*c^2*h^3*q*x - 3*c*d*h*q*(g^ 
2 + 6*g*h*x + h^2*x^2) + d^2*(p + q)*(5*g^3 + 18*g^2*h*x + 9*g*h^2*x^2 + 2 
*h^3*x^3)) + 6*b^2*(d*g - c*h)^3*q*Log[c + d*x])))/(b^3*d^3) + (g + h*x)^3 
*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(3*h)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.93, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {2981, 49, 2009}

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.

\(\displaystyle \int (g+h x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx\)

\(\Big \downarrow \) 2981

\(\displaystyle -\frac {b p r \int \frac {(g+h x)^3}{a+b x}dx}{3 h}-\frac {d q r \int \frac {(g+h x)^3}{c+d x}dx}{3 h}+\frac {(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}\)

\(\Big \downarrow \) 49

\(\displaystyle -\frac {b p r \int \left (\frac {(b g-a h)^3}{b^3 (a+b x)}+\frac {h (b g-a h)^2}{b^3}+\frac {h (g+h x) (b g-a h)}{b^2}+\frac {h (g+h x)^2}{b}\right )dx}{3 h}-\frac {d q r \int \left (\frac {(d g-c h)^3}{d^3 (c+d x)}+\frac {h (d g-c h)^2}{d^3}+\frac {h (g+h x) (d g-c h)}{d^2}+\frac {h (g+h x)^2}{d}\right )dx}{3 h}+\frac {(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {b p r \left (\frac {(b g-a h)^3 \log (a+b x)}{b^4}+\frac {h x (b g-a h)^2}{b^3}+\frac {(g+h x)^2 (b g-a h)}{2 b^2}+\frac {(g+h x)^3}{3 b}\right )}{3 h}+\frac {(g+h x)^3 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{3 h}-\frac {d q r \left (\frac {(d g-c h)^3 \log (c+d x)}{d^4}+\frac {h x (d g-c h)^2}{d^3}+\frac {(g+h x)^2 (d g-c h)}{2 d^2}+\frac {(g+h x)^3}{3 d}\right )}{3 h}\)

Input:

Int[(g + h*x)^2*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r],x]
 

Output:

-1/3*(b*p*r*((h*(b*g - a*h)^2*x)/b^3 + ((b*g - a*h)*(g + h*x)^2)/(2*b^2) + 
 (g + h*x)^3/(3*b) + ((b*g - a*h)^3*Log[a + b*x])/b^4))/h - (d*q*r*((h*(d* 
g - c*h)^2*x)/d^3 + ((d*g - c*h)*(g + h*x)^2)/(2*d^2) + (g + h*x)^3/(3*d) 
+ ((d*g - c*h)^3*Log[c + d*x])/d^4))/(3*h) + ((g + h*x)^3*Log[e*(f*(a + b* 
x)^p*(c + d*x)^q)^r])/(3*h)
 

Defintions of rubi rules used

rule 49

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int 
[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] 
&& IGtQ[m, 0] && IGtQ[m + n + 2, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2981

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) 
^(r_.)]*((g_.) + (h_.)*(x_))^(m_.), x_Symbol] :> Simp[(g + h*x)^(m + 1)*(Lo 
g[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(h*(m + 1))), x] + (-Simp[b*p*(r/(h*(m + 
 1)))   Int[(g + h*x)^(m + 1)/(a + b*x), x], x] - Simp[d*q*(r/(h*(m + 1))) 
  Int[(g + h*x)^(m + 1)/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h 
, m, p, q, r}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(754\) vs. \(2(200)=400\).

Time = 137.89 (sec) , antiderivative size = 755, normalized size of antiderivative = 3.46

method	result	size
parallelrisch	\(\frac {18 x a \,b^{2} d^{3} g h p r +18 x \,b^{3} c \,d^{2} g h q r -18 \ln \left (b x +a \right ) a^{2} b \,d^{3} g h p r -18 \ln \left (d x +c \right ) b^{3} c^{2} d g h q r -6 x \,a^{2} b \,d^{3} h^{2} p r -6 x \,b^{3} c^{2} d \,h^{2} q r -18 \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) a \,b^{2} c \,d^{2} g h +36 \ln \left (b x +a \right ) a \,b^{2} d^{3} g^{2} p r +18 \ln \left (b x +a \right ) b^{3} c \,d^{2} g^{2} p r +18 \ln \left (d x +c \right ) a \,b^{2} d^{3} g^{2} q r -9 a c \,b^{2} d^{2} g h p r -9 a c \,b^{2} d^{2} g h q r +6 x^{3} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) b^{3} d^{3} h^{2}+18 x \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) b^{3} d^{3} g^{2}-18 \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) a \,b^{2} d^{3} g^{2}-18 \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) b^{3} c \,d^{2} g^{2}+36 \ln \left (d x +c \right ) b^{3} c \,d^{2} g^{2} q r +3 x^{2} a \,b^{2} d^{3} h^{2} p r +3 x^{2} b^{3} c \,d^{2} h^{2} q r -9 x^{2} b^{3} d^{3} g h p r -9 x^{2} b^{3} d^{3} g h q r +18 \ln \left (b x +a \right ) a \,b^{2} c \,d^{2} g h p r +18 \ln \left (d x +c \right ) a \,b^{2} c \,d^{2} g h q r +6 a^{3} h^{2} p r \,d^{3}+6 b^{3} c^{3} h^{2} q r +3 a^{2} c b \,d^{2} h^{2} p r +3 a \,c^{2} b^{2} d \,h^{2} q r -18 a^{2} g h p r \,d^{3} b -18 b^{3} c^{2} d g h q r +18 a \,b^{2} d^{3} g^{2} q r +18 b^{3} c \,d^{2} g^{2} p r +18 a \,b^{2} d^{3} g^{2} p r +18 b^{3} c \,d^{2} g^{2} q r -2 x^{3} b^{3} d^{3} h^{2} p r -2 x^{3} b^{3} d^{3} h^{2} q r +18 x^{2} \ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right ) b^{3} d^{3} g h -18 x \,b^{3} d^{3} g^{2} p r -18 x \,b^{3} d^{3} g^{2} q r +6 \ln \left (b x +a \right ) a^{3} d^{3} h^{2} p r +6 \ln \left (d x +c \right ) b^{3} c^{3} h^{2} q r}{18 b^{3} d^{3}}\)	\(755\)

Input:

int((h*x+g)^2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x,method=_RETURNVERBOSE)
 

Output:

1/18*(18*x*a*b^2*d^3*g*h*p*r+18*x*b^3*c*d^2*g*h*q*r-18*ln(b*x+a)*a^2*b*d^3 
*g*h*p*r-18*ln(d*x+c)*b^3*c^2*d*g*h*q*r-6*x*a^2*b*d^3*h^2*p*r-6*x*b^3*c^2* 
d*h^2*q*r-18*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*a*b^2*c*d^2*g*h+36*ln(b*x+a)* 
a*b^2*d^3*g^2*p*r+18*ln(b*x+a)*b^3*c*d^2*g^2*p*r+18*ln(d*x+c)*a*b^2*d^3*g^ 
2*q*r-9*a*c*b^2*d^2*g*h*p*r-9*a*c*b^2*d^2*g*h*q*r+6*x^3*ln(e*(f*(b*x+a)^p* 
(d*x+c)^q)^r)*b^3*d^3*h^2+18*x*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*b^3*d^3*g^2 
-18*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*a*b^2*d^3*g^2-18*ln(e*(f*(b*x+a)^p*(d* 
x+c)^q)^r)*b^3*c*d^2*g^2+36*ln(d*x+c)*b^3*c*d^2*g^2*q*r+3*x^2*a*b^2*d^3*h^ 
2*p*r+3*x^2*b^3*c*d^2*h^2*q*r-9*x^2*b^3*d^3*g*h*p*r-9*x^2*b^3*d^3*g*h*q*r+ 
18*ln(b*x+a)*a*b^2*c*d^2*g*h*p*r+18*ln(d*x+c)*a*b^2*c*d^2*g*h*q*r+6*a^3*h^ 
2*p*r*d^3+6*b^3*c^3*h^2*q*r+3*a^2*c*b*d^2*h^2*p*r+3*a*c^2*b^2*d*h^2*q*r-18 
*a^2*g*h*p*r*d^3*b-18*b^3*c^2*d*g*h*q*r+18*a*b^2*d^3*g^2*q*r+18*b^3*c*d^2* 
g^2*p*r+18*a*b^2*d^3*g^2*p*r+18*b^3*c*d^2*g^2*q*r-2*x^3*b^3*d^3*h^2*p*r-2* 
x^3*b^3*d^3*h^2*q*r+18*x^2*ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)*b^3*d^3*g*h-18* 
x*b^3*d^3*g^2*p*r-18*x*b^3*d^3*g^2*q*r+6*ln(b*x+a)*a^3*d^3*h^2*p*r+6*ln(d* 
x+c)*b^3*c^3*h^2*q*r)/b^3/d^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (200) = 400\).

Time = 0.08 (sec) , antiderivative size = 441, normalized size of antiderivative = 2.02 \[ \int (g+h x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {2 \, {\left (b^{3} d^{3} h^{2} p + b^{3} d^{3} h^{2} q\right )} r x^{3} + 3 \, {\left ({\left (3 \, b^{3} d^{3} g h - a b^{2} d^{3} h^{2}\right )} p + {\left (3 \, b^{3} d^{3} g h - b^{3} c d^{2} h^{2}\right )} q\right )} r x^{2} + 6 \, {\left ({\left (3 \, b^{3} d^{3} g^{2} - 3 \, a b^{2} d^{3} g h + a^{2} b d^{3} h^{2}\right )} p + {\left (3 \, b^{3} d^{3} g^{2} - 3 \, b^{3} c d^{2} g h + b^{3} c^{2} d h^{2}\right )} q\right )} r x - 6 \, {\left (b^{3} d^{3} h^{2} p r x^{3} + 3 \, b^{3} d^{3} g h p r x^{2} + 3 \, b^{3} d^{3} g^{2} p r x + {\left (3 \, a b^{2} d^{3} g^{2} - 3 \, a^{2} b d^{3} g h + a^{3} d^{3} h^{2}\right )} p r\right )} \log \left (b x + a\right ) - 6 \, {\left (b^{3} d^{3} h^{2} q r x^{3} + 3 \, b^{3} d^{3} g h q r x^{2} + 3 \, b^{3} d^{3} g^{2} q r x + {\left (3 \, b^{3} c d^{2} g^{2} - 3 \, b^{3} c^{2} d g h + b^{3} c^{3} h^{2}\right )} q r\right )} \log \left (d x + c\right ) - 6 \, {\left (b^{3} d^{3} h^{2} x^{3} + 3 \, b^{3} d^{3} g h x^{2} + 3 \, b^{3} d^{3} g^{2} x\right )} \log \left (e\right ) - 6 \, {\left (b^{3} d^{3} h^{2} r x^{3} + 3 \, b^{3} d^{3} g h r x^{2} + 3 \, b^{3} d^{3} g^{2} r x\right )} \log \left (f\right )}{18 \, b^{3} d^{3}} \] Input:

integrate((h*x+g)^2*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="fricas" 
)
 

Output:

-1/18*(2*(b^3*d^3*h^2*p + b^3*d^3*h^2*q)*r*x^3 + 3*((3*b^3*d^3*g*h - a*b^2 
*d^3*h^2)*p + (3*b^3*d^3*g*h - b^3*c*d^2*h^2)*q)*r*x^2 + 6*((3*b^3*d^3*g^2 
 - 3*a*b^2*d^3*g*h + a^2*b*d^3*h^2)*p + (3*b^3*d^3*g^2 - 3*b^3*c*d^2*g*h + 
 b^3*c^2*d*h^2)*q)*r*x - 6*(b^3*d^3*h^2*p*r*x^3 + 3*b^3*d^3*g*h*p*r*x^2 + 
3*b^3*d^3*g^2*p*r*x + (3*a*b^2*d^3*g^2 - 3*a^2*b*d^3*g*h + a^3*d^3*h^2)*p* 
r)*log(b*x + a) - 6*(b^3*d^3*h^2*q*r*x^3 + 3*b^3*d^3*g*h*q*r*x^2 + 3*b^3*d 
^3*g^2*q*r*x + (3*b^3*c*d^2*g^2 - 3*b^3*c^2*d*g*h + b^3*c^3*h^2)*q*r)*log( 
d*x + c) - 6*(b^3*d^3*h^2*x^3 + 3*b^3*d^3*g*h*x^2 + 3*b^3*d^3*g^2*x)*log(e 
) - 6*(b^3*d^3*h^2*r*x^3 + 3*b^3*d^3*g*h*r*x^2 + 3*b^3*d^3*g^2*r*x)*log(f) 
)/(b^3*d^3)
 

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 930 vs. \(2 (192) = 384\).

Time = 94.10 (sec) , antiderivative size = 930, normalized size of antiderivative = 4.27 \[ \int (g+h x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\text {Too large to display} \] Input:

integrate((h*x+g)**2*ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r),x)
 

Output:

Piecewise(((g**2*x + g*h*x**2 + h**2*x**3/3)*log(e*(a**p*c**q*f)**r), Eq(b 
, 0) & Eq(d, 0)), (c**3*h**2*log(e*(a**p*f*(c + d*x)**q)**r)/(3*d**3) - c* 
*2*g*h*log(e*(a**p*f*(c + d*x)**q)**r)/d**2 - c**2*h**2*q*r*x/(3*d**2) + c 
*g**2*log(e*(a**p*f*(c + d*x)**q)**r)/d + c*g*h*q*r*x/d + c*h**2*q*r*x**2/ 
(6*d) - g**2*q*r*x + g**2*x*log(e*(a**p*f*(c + d*x)**q)**r) - g*h*q*r*x**2 
/2 + g*h*x**2*log(e*(a**p*f*(c + d*x)**q)**r) - h**2*q*r*x**3/9 + h**2*x** 
3*log(e*(a**p*f*(c + d*x)**q)**r)/3, Eq(b, 0)), (a**3*h**2*log(e*(c**q*f*( 
a + b*x)**p)**r)/(3*b**3) - a**2*g*h*log(e*(c**q*f*(a + b*x)**p)**r)/b**2 
- a**2*h**2*p*r*x/(3*b**2) + a*g**2*log(e*(c**q*f*(a + b*x)**p)**r)/b + a* 
g*h*p*r*x/b + a*h**2*p*r*x**2/(6*b) - g**2*p*r*x + g**2*x*log(e*(c**q*f*(a 
 + b*x)**p)**r) - g*h*p*r*x**2/2 + g*h*x**2*log(e*(c**q*f*(a + b*x)**p)**r 
) - h**2*p*r*x**3/9 + h**2*x**3*log(e*(c**q*f*(a + b*x)**p)**r)/3, Eq(d, 0 
)), (-a**3*h**2*q*r*log(c/d + x)/(3*b**3) + a**3*h**2*log(e*(f*(a + b*x)** 
p*(c + d*x)**q)**r)/(3*b**3) + a**2*g*h*q*r*log(c/d + x)/b**2 - a**2*g*h*l 
og(e*(f*(a + b*x)**p*(c + d*x)**q)**r)/b**2 - a**2*h**2*p*r*x/(3*b**2) - a 
*g**2*q*r*log(c/d + x)/b + a*g**2*log(e*(f*(a + b*x)**p*(c + d*x)**q)**r)/ 
b + a*g*h*p*r*x/b + a*h**2*p*r*x**2/(6*b) + c**3*h**2*q*r*log(c/d + x)/(3* 
d**3) - c**2*g*h*q*r*log(c/d + x)/d**2 - c**2*h**2*q*r*x/(3*d**2) + c*g**2 
*q*r*log(c/d + x)/d + c*g*h*q*r*x/d + c*h**2*q*r*x**2/(6*d) - g**2*p*r*x - 
 g**2*q*r*x + g**2*x*log(e*(f*(a + b*x)**p*(c + d*x)**q)**r) - g*h*p*r*...
 

Maxima [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.23 \[ \int (g+h x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {1}{3} \, {\left (h^{2} x^{3} + 3 \, g h x^{2} + 3 \, g^{2} x\right )} \log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right ) + \frac {r {\left (\frac {6 \, {\left (3 \, a b^{2} f g^{2} p - 3 \, a^{2} b f g h p + a^{3} f h^{2} p\right )} \log \left (b x + a\right )}{b^{3}} + \frac {6 \, {\left (3 \, c d^{2} f g^{2} q - 3 \, c^{2} d f g h q + c^{3} f h^{2} q\right )} \log \left (d x + c\right )}{d^{3}} - \frac {2 \, b^{2} d^{2} f h^{2} {\left (p + q\right )} x^{3} - 3 \, {\left (a b d^{2} f h^{2} p - {\left (3 \, d^{2} f g h {\left (p + q\right )} - c d f h^{2} q\right )} b^{2}\right )} x^{2} - 6 \, {\left (3 \, a b d^{2} f g h p - a^{2} d^{2} f h^{2} p - {\left (3 \, d^{2} f g^{2} {\left (p + q\right )} - 3 \, c d f g h q + c^{2} f h^{2} q\right )} b^{2}\right )} x}{b^{2} d^{2}}\right )}}{18 \, f} \] Input:

integrate((h*x+g)^2*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="maxima" 
)
 

Output:

1/3*(h^2*x^3 + 3*g*h*x^2 + 3*g^2*x)*log(((b*x + a)^p*(d*x + c)^q*f)^r*e) + 
 1/18*r*(6*(3*a*b^2*f*g^2*p - 3*a^2*b*f*g*h*p + a^3*f*h^2*p)*log(b*x + a)/ 
b^3 + 6*(3*c*d^2*f*g^2*q - 3*c^2*d*f*g*h*q + c^3*f*h^2*q)*log(d*x + c)/d^3 
 - (2*b^2*d^2*f*h^2*(p + q)*x^3 - 3*(a*b*d^2*f*h^2*p - (3*d^2*f*g*h*(p + q 
) - c*d*f*h^2*q)*b^2)*x^2 - 6*(3*a*b*d^2*f*g*h*p - a^2*d^2*f*h^2*p - (3*d^ 
2*f*g^2*(p + q) - 3*c*d*f*g*h*q + c^2*f*h^2*q)*b^2)*x)/(b^2*d^2))/f
 

Giac [A] (verification not implemented)

Time = 41.68 (sec) , antiderivative size = 363, normalized size of antiderivative = 1.67 \[ \int (g+h x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=-\frac {1}{9} \, {\left (h^{2} p r + h^{2} q r - 3 \, h^{2} r \log \left (f\right ) - 3 \, h^{2} \log \left (e\right )\right )} x^{3} + \frac {1}{3} \, {\left (h^{2} p r x^{3} + 3 \, g h p r x^{2} + 3 \, g^{2} p r x\right )} \log \left (b x + a\right ) + \frac {1}{3} \, {\left (h^{2} q r x^{3} + 3 \, g h q r x^{2} + 3 \, g^{2} q r x\right )} \log \left (d x + c\right ) - \frac {{\left (3 \, b d g h p r - a d h^{2} p r + 3 \, b d g h q r - b c h^{2} q r - 6 \, b d g h r \log \left (f\right ) - 6 \, b d g h \log \left (e\right )\right )} x^{2}}{6 \, b d} + \frac {{\left (3 \, a b^{2} g^{2} p r - 3 \, a^{2} b g h p r + a^{3} h^{2} p r\right )} \log \left (b x + a\right )}{3 \, b^{3}} + \frac {{\left (3 \, c d^{2} g^{2} q r - 3 \, c^{2} d g h q r + c^{3} h^{2} q r\right )} \log \left (-d x - c\right )}{3 \, d^{3}} - \frac {{\left (3 \, b^{2} d^{2} g^{2} p r - 3 \, a b d^{2} g h p r + a^{2} d^{2} h^{2} p r + 3 \, b^{2} d^{2} g^{2} q r - 3 \, b^{2} c d g h q r + b^{2} c^{2} h^{2} q r - 3 \, b^{2} d^{2} g^{2} r \log \left (f\right ) - 3 \, b^{2} d^{2} g^{2} \log \left (e\right )\right )} x}{3 \, b^{2} d^{2}} \] Input:

integrate((h*x+g)^2*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x, algorithm="giac")
 

Output:

-1/9*(h^2*p*r + h^2*q*r - 3*h^2*r*log(f) - 3*h^2*log(e))*x^3 + 1/3*(h^2*p* 
r*x^3 + 3*g*h*p*r*x^2 + 3*g^2*p*r*x)*log(b*x + a) + 1/3*(h^2*q*r*x^3 + 3*g 
*h*q*r*x^2 + 3*g^2*q*r*x)*log(d*x + c) - 1/6*(3*b*d*g*h*p*r - a*d*h^2*p*r 
+ 3*b*d*g*h*q*r - b*c*h^2*q*r - 6*b*d*g*h*r*log(f) - 6*b*d*g*h*log(e))*x^2 
/(b*d) + 1/3*(3*a*b^2*g^2*p*r - 3*a^2*b*g*h*p*r + a^3*h^2*p*r)*log(b*x + a 
)/b^3 + 1/3*(3*c*d^2*g^2*q*r - 3*c^2*d*g*h*q*r + c^3*h^2*q*r)*log(-d*x - c 
)/d^3 - 1/3*(3*b^2*d^2*g^2*p*r - 3*a*b*d^2*g*h*p*r + a^2*d^2*h^2*p*r + 3*b 
^2*d^2*g^2*q*r - 3*b^2*c*d*g*h*q*r + b^2*c^2*h^2*q*r - 3*b^2*d^2*g^2*r*log 
(f) - 3*b^2*d^2*g^2*log(e))*x/(b^2*d^2)
 

Mupad [B] (verification not implemented)

Time = 26.73 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.50 \[ \int (g+h x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=x\,\left (\frac {\left (\frac {h\,r\,\left (b\,c\,h\,p+3\,b\,d\,g\,p+a\,d\,h\,q+3\,b\,d\,g\,q\right )}{3\,b\,d}-\frac {h^2\,r\,\left (p+q\right )\,\left (3\,a\,d+3\,b\,c\right )}{9\,b\,d}\right )\,\left (3\,a\,d+3\,b\,c\right )}{3\,b\,d}-\frac {g\,r\,\left (b\,c\,h\,p+b\,d\,g\,p+a\,d\,h\,q+b\,d\,g\,q\right )}{b\,d}+\frac {a\,c\,h^2\,r\,\left (p+q\right )}{3\,b\,d}\right )-x^2\,\left (\frac {h\,r\,\left (b\,c\,h\,p+3\,b\,d\,g\,p+a\,d\,h\,q+3\,b\,d\,g\,q\right )}{6\,b\,d}-\frac {h^2\,r\,\left (p+q\right )\,\left (3\,a\,d+3\,b\,c\right )}{18\,b\,d}\right )+\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\left (g^2\,x+g\,h\,x^2+\frac {h^2\,x^3}{3}\right )+\frac {\ln \left (a+b\,x\right )\,\left (p\,r\,a^3\,h^2-3\,p\,r\,a^2\,b\,g\,h+3\,p\,r\,a\,b^2\,g^2\right )}{3\,b^3}+\frac {\ln \left (c+d\,x\right )\,\left (q\,r\,c^3\,h^2-3\,q\,r\,c^2\,d\,g\,h+3\,q\,r\,c\,d^2\,g^2\right )}{3\,d^3}-\frac {h^2\,r\,x^3\,\left (p+q\right )}{9} \] Input:

int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)*(g + h*x)^2,x)
 

Output:

x*((((h*r*(b*c*h*p + 3*b*d*g*p + a*d*h*q + 3*b*d*g*q))/(3*b*d) - (h^2*r*(p 
 + q)*(3*a*d + 3*b*c))/(9*b*d))*(3*a*d + 3*b*c))/(3*b*d) - (g*r*(b*c*h*p + 
 b*d*g*p + a*d*h*q + b*d*g*q))/(b*d) + (a*c*h^2*r*(p + q))/(3*b*d)) - x^2* 
((h*r*(b*c*h*p + 3*b*d*g*p + a*d*h*q + 3*b*d*g*q))/(6*b*d) - (h^2*r*(p + q 
)*(3*a*d + 3*b*c))/(18*b*d)) + log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)*(g^2*x 
 + (h^2*x^3)/3 + g*h*x^2) + (log(a + b*x)*(a^3*h^2*p*r + 3*a*b^2*g^2*p*r - 
 3*a^2*b*g*h*p*r))/(3*b^3) + (log(c + d*x)*(c^3*h^2*q*r + 3*c*d^2*g^2*q*r 
- 3*c^2*d*g*h*q*r))/(3*d^3) - (h^2*r*x^3*(p + q))/9
 

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 524, normalized size of antiderivative = 2.40 \[ \int (g+h x)^2 \log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right ) \, dx=\frac {-6 \,\mathrm {log}\left (d x +c \right ) a^{3} d^{3} h^{2} q r +18 \,\mathrm {log}\left (d x +c \right ) a^{2} b \,d^{3} g h q r -18 \,\mathrm {log}\left (d x +c \right ) a \,b^{2} d^{3} g^{2} q r +6 \,\mathrm {log}\left (d x +c \right ) b^{3} c^{3} h^{2} q r -18 \,\mathrm {log}\left (d x +c \right ) b^{3} c^{2} d g h q r +18 \,\mathrm {log}\left (d x +c \right ) b^{3} c \,d^{2} g^{2} q r +6 \,\mathrm {log}\left (f^{r} \left (d x +c \right )^{q r} \left (b x +a \right )^{p r} e \right ) a^{3} d^{3} h^{2}-18 \,\mathrm {log}\left (f^{r} \left (d x +c \right )^{q r} \left (b x +a \right )^{p r} e \right ) a^{2} b \,d^{3} g h +18 \,\mathrm {log}\left (f^{r} \left (d x +c \right )^{q r} \left (b x +a \right )^{p r} e \right ) a \,b^{2} d^{3} g^{2}+18 \,\mathrm {log}\left (f^{r} \left (d x +c \right )^{q r} \left (b x +a \right )^{p r} e \right ) b^{3} d^{3} g^{2} x +18 \,\mathrm {log}\left (f^{r} \left (d x +c \right )^{q r} \left (b x +a \right )^{p r} e \right ) b^{3} d^{3} g h \,x^{2}+6 \,\mathrm {log}\left (f^{r} \left (d x +c \right )^{q r} \left (b x +a \right )^{p r} e \right ) b^{3} d^{3} h^{2} x^{3}-6 a^{2} b \,d^{3} h^{2} p r x +18 a \,b^{2} d^{3} g h p r x +3 a \,b^{2} d^{3} h^{2} p r \,x^{2}-6 b^{3} c^{2} d \,h^{2} q r x +18 b^{3} c \,d^{2} g h q r x +3 b^{3} c \,d^{2} h^{2} q r \,x^{2}-18 b^{3} d^{3} g^{2} p r x -18 b^{3} d^{3} g^{2} q r x -9 b^{3} d^{3} g h p r \,x^{2}-9 b^{3} d^{3} g h q r \,x^{2}-2 b^{3} d^{3} h^{2} p r \,x^{3}-2 b^{3} d^{3} h^{2} q r \,x^{3}}{18 b^{3} d^{3}} \] Input:

int((h*x+g)^2*log(e*(f*(b*x+a)^p*(d*x+c)^q)^r),x)
 

Output:

( - 6*log(c + d*x)*a**3*d**3*h**2*q*r + 18*log(c + d*x)*a**2*b*d**3*g*h*q* 
r - 18*log(c + d*x)*a*b**2*d**3*g**2*q*r + 6*log(c + d*x)*b**3*c**3*h**2*q 
*r - 18*log(c + d*x)*b**3*c**2*d*g*h*q*r + 18*log(c + d*x)*b**3*c*d**2*g** 
2*q*r + 6*log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)*a**3*d**3*h**2 - 1 
8*log(f**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)*a**2*b*d**3*g*h + 18*log(f 
**r*(c + d*x)**(q*r)*(a + b*x)**(p*r)*e)*a*b**2*d**3*g**2 + 18*log(f**r*(c 
 + d*x)**(q*r)*(a + b*x)**(p*r)*e)*b**3*d**3*g**2*x + 18*log(f**r*(c + d*x 
)**(q*r)*(a + b*x)**(p*r)*e)*b**3*d**3*g*h*x**2 + 6*log(f**r*(c + d*x)**(q 
*r)*(a + b*x)**(p*r)*e)*b**3*d**3*h**2*x**3 - 6*a**2*b*d**3*h**2*p*r*x + 1 
8*a*b**2*d**3*g*h*p*r*x + 3*a*b**2*d**3*h**2*p*r*x**2 - 6*b**3*c**2*d*h**2 
*q*r*x + 18*b**3*c*d**2*g*h*q*r*x + 3*b**3*c*d**2*h**2*q*r*x**2 - 18*b**3* 
d**3*g**2*p*r*x - 18*b**3*d**3*g**2*q*r*x - 9*b**3*d**3*g*h*p*r*x**2 - 9*b 
**3*d**3*g*h*q*r*x**2 - 2*b**3*d**3*h**2*p*r*x**3 - 2*b**3*d**3*h**2*q*r*x 
**3)/(18*b**3*d**3)