\(\int (a g+b g x)^{-2-m} (c i+d i x)^m (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [219]

Optimal result

Integrand size = 49, antiderivative size = 223 \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=-\frac {2 B^2 n^2 (a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m}}{(b c-a d) i^2 (1+m)^3 (c+d x)}-\frac {2 B n (a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m} \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) i^2 (1+m)^2 (c+d x)}-\frac {(a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m} \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d) i^2 (1+m) (c+d x)} \] Output:

-2*B^2*n^2*(b*x+a)*(g*(b*x+a))^(-2-m)*(i*(d*x+c))^(2+m)/(-a*d+b*c)/i^2/(1+ 
m)^3/(d*x+c)-2*B*n*(b*x+a)*(g*(b*x+a))^(-2-m)*(i*(d*x+c))^(2+m)*(A+B*ln(e* 
((b*x+a)/(d*x+c))^n))/(-a*d+b*c)/i^2/(1+m)^2/(d*x+c)-(b*x+a)*(g*(b*x+a))^( 
-2-m)*(i*(d*x+c))^(2+m)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)/i^2/( 
1+m)/(d*x+c)
 

Mathematica [A] (verified)

Time = 3.00 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.60 \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=-\frac {(g (a+b x))^{-1-m} (c+d x) (i (c+d x))^m \left (A^2 (1+m)^2+2 A B (1+m) n+2 B^2 n^2+2 B (1+m) (A+A m+B n) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B^2 (1+m)^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) g (1+m)^3} \] Input:

Integrate[(a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m*(A + B*Log[e*((a + b*x)/( 
c + d*x))^n])^2,x]
 

Output:

-(((g*(a + b*x))^(-1 - m)*(c + d*x)*(i*(c + d*x))^m*(A^2*(1 + m)^2 + 2*A*B 
*(1 + m)*n + 2*B^2*n^2 + 2*B*(1 + m)*(A + A*m + B*n)*Log[e*((a + b*x)/(c + 
 d*x))^n] + B^2*(1 + m)^2*Log[e*((a + b*x)/(c + d*x))^n]^2))/((b*c - a*d)* 
g*(1 + m)^3))
 

Rubi [A] (verified)

Time = 0.54 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.86, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2963, 2742, 2741}

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.

\(\displaystyle \int (a g+b g x)^{-m-2} (c i+d i x)^m \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2 \, dx\)

\(\Big \downarrow \) 2963

\(\displaystyle \frac {(g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (\frac {a+b x}{c+d x}\right )^{m+2} \int \left (\frac {a+b x}{c+d x}\right )^{-m-2} \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2d\frac {a+b x}{c+d x}}{i^2 (b c-a d)}\)

\(\Big \downarrow \) 2742

\(\displaystyle \frac {(g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (\frac {a+b x}{c+d x}\right )^{m+2} \left (\frac {2 B n \int \left (\frac {a+b x}{c+d x}\right )^{-m-2} \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )d\frac {a+b x}{c+d x}}{m+1}-\frac {\left (\frac {a+b x}{c+d x}\right )^{-m-1} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{m+1}\right )}{i^2 (b c-a d)}\)

\(\Big \downarrow \) 2741

\(\displaystyle \frac {(g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (\frac {a+b x}{c+d x}\right )^{m+2} \left (\frac {2 B n \left (-\frac {\left (\frac {a+b x}{c+d x}\right )^{-m-1} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{m+1}-\frac {B n \left (\frac {a+b x}{c+d x}\right )^{-m-1}}{(m+1)^2}\right )}{m+1}-\frac {\left (\frac {a+b x}{c+d x}\right )^{-m-1} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{m+1}\right )}{i^2 (b c-a d)}\)

Input:

Int[(a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m*(A + B*Log[e*((a + b*x)/(c + d* 
x))^n])^2,x]
 

Output:

((g*(a + b*x))^(-2 - m)*((a + b*x)/(c + d*x))^(2 + m)*(i*(c + d*x))^(2 + m 
)*(-((((a + b*x)/(c + d*x))^(-1 - m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n] 
)^2)/(1 + m)) + (2*B*n*(-((B*n*((a + b*x)/(c + d*x))^(-1 - m))/(1 + m)^2) 
- (((a + b*x)/(c + d*x))^(-1 - m)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/ 
(1 + m)))/(1 + m)))/((b*c - a*d)*i^2)
 

Defintions of rubi rules used

rule 2741

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> 
Simp[(d*x)^(m + 1)*((a + b*Log[c*x^n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^( 
m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1]
 

rule 2742

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

rule 2963

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( 
B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol 
] :> Simp[d^2*((g*((a + b*x)/b))^m/(i^2*(b*c - a*d)*(i*((c + d*x)/d))^m*((a 
 + b*x)/(c + d*x))^m))   Subst[Int[x^m*(A + B*Log[e*x^n])^p, x], x, (a + b* 
x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, A, B, m, n, p, q}, x 
] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && EqQ[m + 
 q + 2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(2618\) vs. \(2(223)=446\).

Time = 148.90 (sec) , antiderivative size = 2619, normalized size of antiderivative = 11.74

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(2619\)

Input:

int((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,x,m 
ethod=_RETURNVERBOSE)
 

Output:

(2*A*B*x*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*ln(e*((b*x+a)/(d*x+c))^n)*b^2*c* 
d*n+2*A*B*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*a*b*c*d*m*n^2+2*A*B*(i*(d*x+c)) 
^m*(g*(b*x+a))^(-2-m)*ln(e*((b*x+a)/(d*x+c))^n)*a*b*c*d*n+2*A*B*x^2*(i*(d* 
x+c))^m*(g*(b*x+a))^(-2-m)*ln(e*((b*x+a)/(d*x+c))^n)*b^2*d^2*m^2*n+B^2*x*( 
i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*ln(e*((b*x+a)/(d*x+c))^n)^2*a*b*d^2*m^2*n+ 
2*B^2*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*ln(e*((b*x+a)/(d*x+c))^n)^2*a*b*c*d 
*m*n+2*B^2*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*ln(e*((b*x+a)/(d*x+c))^n)*a*b* 
c*d*m*n^2+2*A*B*x*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*ln(e*((b*x+a)/(d*x+c))^ 
n)*a*b*d^2*n+2*A*B*x*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*b^2*c*d*m*n^2+B^2*(i 
*(d*x+c))^m*(g*(b*x+a))^(-2-m)*ln(e*((b*x+a)/(d*x+c))^n)^2*a*b*c*d*m^2*n+2 
*A*B*x*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*a*b*d^2*m*n^2+2*A^2*(i*(d*x+c))^m* 
(g*(b*x+a))^(-2-m)*a*b*c*d*m*n+2*A*B*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*a*b* 
c*d*n^2+4*A*B*x*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*ln(e*((b*x+a)/(d*x+c))^n) 
*b^2*c*d*m*n+4*A*B*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*ln(e*((b*x+a)/(d*x+c)) 
^n)*a*b*c*d*m*n+2*A*B*x*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*ln(e*((b*x+a)/(d* 
x+c))^n)*a*b*d^2*m^2*n+2*A*B*x*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*ln(e*((b*x 
+a)/(d*x+c))^n)*b^2*c*d*m^2*n+4*A*B*x*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*ln( 
e*((b*x+a)/(d*x+c))^n)*a*b*d^2*m*n+A^2*x^2*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m 
)*b^2*d^2*m^2*n+2*A*B*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*ln(e*((b*x+a)/(d*x+ 
c))^n)*a*b*c*d*m^2*n+B^2*x*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*ln(e*((b*x+...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 983 vs. \(2 (223) = 446\).

Time = 0.13 (sec) , antiderivative size = 983, normalized size of antiderivative = 4.41 \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:

integrate((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m*(A+B*log(e*((b*x+a)/(d*x+c))^n) 
)^2,x, algorithm="fricas")
 

Output:

-(A^2*a*c*m^2 + 2*B^2*a*c*n^2 + 2*A^2*a*c*m + A^2*a*c + (A^2*b*d*m^2 + 2*B 
^2*b*d*n^2 + 2*A^2*b*d*m + A^2*b*d + 2*(A*B*b*d*m + A*B*b*d)*n)*x^2 + (B^2 
*a*c*m^2 + 2*B^2*a*c*m + B^2*a*c + (B^2*b*d*m^2 + 2*B^2*b*d*m + B^2*b*d)*x 
^2 + (B^2*b*c + B^2*a*d + (B^2*b*c + B^2*a*d)*m^2 + 2*(B^2*b*c + B^2*a*d)* 
m)*x)*log(e)^2 + ((B^2*b*d*m^2 + 2*B^2*b*d*m + B^2*b*d)*n^2*x^2 + (B^2*b*c 
 + B^2*a*d + (B^2*b*c + B^2*a*d)*m^2 + 2*(B^2*b*c + B^2*a*d)*m)*n^2*x + (B 
^2*a*c*m^2 + 2*B^2*a*c*m + B^2*a*c)*n^2)*log((b*x + a)/(d*x + c))^2 + 2*(A 
*B*a*c*m + A*B*a*c)*n + (A^2*b*c + A^2*a*d + (A^2*b*c + A^2*a*d)*m^2 + 2*( 
B^2*b*c + B^2*a*d)*n^2 + 2*(A^2*b*c + A^2*a*d)*m + 2*(A*B*b*c + A*B*a*d + 
(A*B*b*c + A*B*a*d)*m)*n)*x + 2*(A*B*a*c*m^2 + 2*A*B*a*c*m + A*B*a*c + (A* 
B*b*d*m^2 + 2*A*B*b*d*m + A*B*b*d + (B^2*b*d*m + B^2*b*d)*n)*x^2 + (B^2*a* 
c*m + B^2*a*c)*n + (A*B*b*c + A*B*a*d + (A*B*b*c + A*B*a*d)*m^2 + 2*(A*B*b 
*c + A*B*a*d)*m + (B^2*b*c + B^2*a*d + (B^2*b*c + B^2*a*d)*m)*n)*x + ((B^2 
*b*d*m^2 + 2*B^2*b*d*m + B^2*b*d)*n*x^2 + (B^2*b*c + B^2*a*d + (B^2*b*c + 
B^2*a*d)*m^2 + 2*(B^2*b*c + B^2*a*d)*m)*n*x + (B^2*a*c*m^2 + 2*B^2*a*c*m + 
 B^2*a*c)*n)*log((b*x + a)/(d*x + c)))*log(e) + 2*((B^2*a*c*m + B^2*a*c)*n 
^2 + ((B^2*b*d*m + B^2*b*d)*n^2 + (A*B*b*d*m^2 + 2*A*B*b*d*m + A*B*b*d)*n) 
*x^2 + (A*B*a*c*m^2 + 2*A*B*a*c*m + A*B*a*c)*n + ((B^2*b*c + B^2*a*d + (B^ 
2*b*c + B^2*a*d)*m)*n^2 + (A*B*b*c + A*B*a*d + (A*B*b*c + A*B*a*d)*m^2 + 2 
*(A*B*b*c + A*B*a*d)*m)*n)*x)*log((b*x + a)/(d*x + c)))*(b*g*x + a*g)^(...
 

Sympy [F(-2)]

Exception generated. \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((b*g*x+a*g)**(-2-m)*(d*i*x+c*i)**m*(A+B*ln(e*((b*x+a)/(d*x+c))** 
n))**2,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} {\left (b g x + a g\right )}^{-m - 2} {\left (d i x + c i\right )}^{m} \,d x } \] Input:

integrate((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m*(A+B*log(e*((b*x+a)/(d*x+c))^n) 
)^2,x, algorithm="maxima")
 

Output:

integrate((B*log(e*((b*x + a)/(d*x + c))^n) + A)^2*(b*g*x + a*g)^(-m - 2)* 
(d*i*x + c*i)^m, x)
 

Giac [F]

\[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} {\left (b g x + a g\right )}^{-m - 2} {\left (d i x + c i\right )}^{m} \,d x } \] Input:

integrate((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m*(A+B*log(e*((b*x+a)/(d*x+c))^n) 
)^2,x, algorithm="giac")
 

Output:

integrate((B*log(e*((b*x + a)/(d*x + c))^n) + A)^2*(b*g*x + a*g)^(-m - 2)* 
(d*i*x + c*i)^m, x)
 

Mupad [F(-1)]

Timed out. \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int \frac {{\left (c\,i+d\,i\,x\right )}^m\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{{\left (a\,g+b\,g\,x\right )}^{m+2}} \,d x \] Input:

int(((c*i + d*i*x)^m*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2)/(a*g + b*g* 
x)^(m + 2),x)
 

Output:

int(((c*i + d*i*x)^m*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2)/(a*g + b*g* 
x)^(m + 2), x)
 

Reduce [F]

\[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {\left (\int \frac {\left (d i x +c i \right )^{m}}{\left (b g x +a g \right )^{m} a^{2}+2 \left (b g x +a g \right )^{m} a b x +\left (b g x +a g \right )^{m} b^{2} x^{2}}d x \right ) a^{2}+\left (\int \frac {\left (d i x +c i \right )^{m} \mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )^{2}}{\left (b g x +a g \right )^{m} a^{2}+2 \left (b g x +a g \right )^{m} a b x +\left (b g x +a g \right )^{m} b^{2} x^{2}}d x \right ) b^{2}+2 \left (\int \frac {\left (d i x +c i \right )^{m} \mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right )}{\left (b g x +a g \right )^{m} a^{2}+2 \left (b g x +a g \right )^{m} a b x +\left (b g x +a g \right )^{m} b^{2} x^{2}}d x \right ) a b}{g^{2}} \] Input:

int((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x)
                                                                                    
                                                                                    
 

Output:

(int((c*i + d*i*x)**m/((a*g + b*g*x)**m*a**2 + 2*(a*g + b*g*x)**m*a*b*x + 
(a*g + b*g*x)**m*b**2*x**2),x)*a**2 + int(((c*i + d*i*x)**m*log(((a + b*x) 
**n*e)/(c + d*x)**n)**2)/((a*g + b*g*x)**m*a**2 + 2*(a*g + b*g*x)**m*a*b*x 
 + (a*g + b*g*x)**m*b**2*x**2),x)*b**2 + 2*int(((c*i + d*i*x)**m*log(((a + 
 b*x)**n*e)/(c + d*x)**n))/((a*g + b*g*x)**m*a**2 + 2*(a*g + b*g*x)**m*a*b 
*x + (a*g + b*g*x)**m*b**2*x**2),x)*a*b)/g**2