Integrand size = 49, antiderivative size = 223 \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=-\frac {2 B^2 n^2 (a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m}}{(b c-a d) i^2 (1+m)^3 (c+d x)}-\frac {2 B n (a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m} \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) i^2 (1+m)^2 (c+d x)}-\frac {(a+b x) (g (a+b x))^{-2-m} (i (c+d x))^{2+m} \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d) i^2 (1+m) (c+d x)} \]
-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)
Time = 1.78 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.60 \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=-\frac {(g (a+b x))^{-1-m} (c+d x) (i (c+d x))^m \left (A^2 (1+m)^2+2 A B (1+m) n+2 B^2 n^2+2 B (1+m) (A+A m+B n) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+B^2 (1+m)^2 \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) g (1+m)^3} \]
-(((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))
Time = 0.50 (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 |
| \(\displaystyle \frac {(g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (\frac {a+b x}{c+d x}\right )^{m+2} \left (\frac {2 B n \left (-\frac {\left (\frac {a+b x}{c+d x}\right )^{-m-1} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{m+1}-\frac {B n \left (\frac {a+b x}{c+d x}\right )^{-m-1}}{(m+1)^2}\right )}{m+1}-\frac {\left (\frac {a+b x}{c+d x}\right )^{-m-1} \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{m+1}\right )}{i^2 (b c-a d)}\) |
((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)
3.3.19.3.1 Defintions of rubi rules used
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]
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]
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]
Leaf count of result is larger than twice the leaf count of optimal. \(2618\) vs. \(2(223)=446\).
Time = 131.30 (sec) , antiderivative size = 2619, normalized size of antiderivative = 11.74
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)
(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)*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*B^2*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^2+2*B^2* 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^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+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+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+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+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+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*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 *b^2*c*d*n+2*B^2*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+2*B^2*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+2*A^2*x*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*a*b*d^2*m*n+ 2*A^2*x*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*b^2*c*d*m*n+A^2*(i*(d*x+c))^m*(g* (b*x+a))^(-2-m)*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*n^2+2*A*B*x*(i*(d*x+c))^m*(g*(b*x+a))^(-2-m)*b^2*c*d*n^2+B^2*(i*(d...
Leaf count of result is larger than twice the leaf count of optimal. 983 vs. \(2 (223) = 446\).
Time = 0.34 (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=-\frac {{\left (A^{2} a c m^{2} + 2 \, B^{2} a c n^{2} + 2 \, A^{2} a c m + A^{2} a c + {\left (A^{2} b d m^{2} + 2 \, B^{2} b d n^{2} + 2 \, A^{2} b d m + A^{2} b d + 2 \, {\left (A B b d m + A B b d\right )} n\right )} x^{2} + {\left (B^{2} a c m^{2} + 2 \, B^{2} a c m + B^{2} a c + {\left (B^{2} b d m^{2} + 2 \, B^{2} b d m + B^{2} b d\right )} x^{2} + {\left (B^{2} b c + B^{2} a d + {\left (B^{2} b c + B^{2} a d\right )} m^{2} + 2 \, {\left (B^{2} b c + B^{2} a d\right )} m\right )} x\right )} \log \left (e\right )^{2} + {\left ({\left (B^{2} b d m^{2} + 2 \, B^{2} b d m + B^{2} b d\right )} n^{2} x^{2} + {\left (B^{2} b c + B^{2} a d + {\left (B^{2} b c + B^{2} a d\right )} m^{2} + 2 \, {\left (B^{2} b c + B^{2} a d\right )} m\right )} n^{2} x + {\left (B^{2} a c m^{2} + 2 \, B^{2} a c m + B^{2} a c\right )} n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + 2 \, {\left (A B a c m + A B a c\right )} n + {\left (A^{2} b c + A^{2} a d + {\left (A^{2} b c + A^{2} a d\right )} m^{2} + 2 \, {\left (B^{2} b c + B^{2} a d\right )} n^{2} + 2 \, {\left (A^{2} b c + A^{2} a d\right )} m + 2 \, {\left (A B b c + A B a d + {\left (A B b c + A B a d\right )} m\right )} n\right )} x + 2 \, {\left (A B a c m^{2} + 2 \, A B a c m + A B a c + {\left (A B b d m^{2} + 2 \, A B b d m + A B b d + {\left (B^{2} b d m + B^{2} b d\right )} n\right )} x^{2} + {\left (B^{2} a c m + B^{2} a c\right )} n + {\left (A B b c + A B a d + {\left (A B b c + A B a d\right )} m^{2} + 2 \, {\left (A B b c + A B a d\right )} m + {\left (B^{2} b c + B^{2} a d + {\left (B^{2} b c + B^{2} a d\right )} m\right )} n\right )} x + {\left ({\left (B^{2} b d m^{2} + 2 \, B^{2} b d m + B^{2} b d\right )} n x^{2} + {\left (B^{2} b c + B^{2} a d + {\left (B^{2} b c + B^{2} a d\right )} m^{2} + 2 \, {\left (B^{2} b c + B^{2} a d\right )} m\right )} n x + {\left (B^{2} a c m^{2} + 2 \, B^{2} a c m + B^{2} a c\right )} n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \left (e\right ) + 2 \, {\left ({\left (B^{2} a c m + B^{2} a c\right )} n^{2} + {\left ({\left (B^{2} b d m + B^{2} b d\right )} n^{2} + {\left (A B b d m^{2} + 2 \, A B b d m + A B b d\right )} n\right )} x^{2} + {\left (A B a c m^{2} + 2 \, A B a c m + A B a c\right )} n + {\left ({\left (B^{2} b c + B^{2} a d + {\left (B^{2} b c + B^{2} a d\right )} m\right )} n^{2} + {\left (A B b c + A B a d + {\left (A B b c + A B a d\right )} m^{2} + 2 \, {\left (A B b c + A B a d\right )} m\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} {\left (b g x + a g\right )}^{-m - 2} e^{\left (m \log \left (b g x + a g\right ) - m \log \left (\frac {b x + a}{d x + c}\right ) + m \log \left (\frac {i}{g}\right )\right )}}{{\left (b c - a d\right )} m^{3} + 3 \, {\left (b c - a d\right )} m^{2} + b c - a d + 3 \, {\left (b c - a d\right )} m} \]
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")
-(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)^(...
Exception generated. \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Exception raised: HeuristicGCDFailed} \]
\[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} {\left (b g x + a g\right )}^{-m - 2} {\left (d i x + c i\right )}^{m} \,d x } \]
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")
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)
\[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} {\left (b g x + a g\right )}^{-m - 2} {\left (d i x + c i\right )}^{m} \,d x } \]
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")
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)
Timed out. \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int \frac {{\left (c\,i+d\,i\,x\right )}^m\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{{\left (a\,g+b\,g\,x\right )}^{m+2}} \,d x \]