Integrand size = 45, antiderivative size = 1172 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {5 B^2 (b c-a d)^6 g^3 i^3 n^2 x}{84 b^3 d^3}+\frac {B^2 (b c-a d)^3 g^3 i^3 n^2 (a+b x)^4}{140 b^4}-\frac {29 B^2 (b c-a d)^5 g^3 i^3 n^2 (c+d x)^2}{840 b^2 d^4}+\frac {47 B^2 (b c-a d)^4 g^3 i^3 n^2 (c+d x)^3}{1260 b d^4}-\frac {13 B^2 (b c-a d)^3 g^3 i^3 n^2 (c+d x)^4}{420 d^4}+\frac {b B^2 (b c-a d)^2 g^3 i^3 n^2 (c+d x)^5}{105 d^4}-\frac {B (b c-a d)^4 g^3 i^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{210 b^4 d}-\frac {3 B (b c-a d)^3 g^3 i^3 n (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{140 b^4}-\frac {B (b c-a d)^2 g^3 i^3 n (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 b^3}+\frac {2 B (b c-a d)^4 g^3 i^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{21 b d^4}-\frac {3 B (b c-a d)^3 g^3 i^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{14 d^4}+\frac {6 b B (b c-a d)^2 g^3 i^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{35 d^4}-\frac {b^2 B (b c-a d) g^3 i^3 n (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{21 d^4}+\frac {(b c-a d)^3 g^3 i^3 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{140 b^4}+\frac {(b c-a d)^2 g^3 i^3 (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{35 b^3}+\frac {(b c-a d) g^3 i^3 (a+b x)^4 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{14 b^2}+\frac {g^3 i^3 (a+b x)^4 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{7 b}+\frac {B (b c-a d)^5 g^3 i^3 n (a+b x)^2 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{420 b^4 d^2}-\frac {B (b c-a d)^6 g^3 i^3 n (a+b x) \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{420 b^4 d^3}-\frac {B (b c-a d)^7 g^3 i^3 n \left (6 A+11 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{420 b^4 d^4}-\frac {B^2 (b c-a d)^7 g^3 i^3 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{210 b^4 d^4}-\frac {11 B^2 (b c-a d)^7 g^3 i^3 n^2 \log (c+d x)}{420 b^4 d^4}-\frac {B^2 (b c-a d)^7 g^3 i^3 n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{70 b^4 d^4} \]
5/84*B^2*(-a*d+b*c)^6*g^3*i^3*n^2*x/b^3/d^3+1/140*B^2*(-a*d+b*c)^3*g^3*i^3 *n^2*(b*x+a)^4/b^4-29/840*B^2*(-a*d+b*c)^5*g^3*i^3*n^2*(d*x+c)^2/b^2/d^4+4 7/1260*B^2*(-a*d+b*c)^4*g^3*i^3*n^2*(d*x+c)^3/b/d^4-13/420*B^2*(-a*d+b*c)^ 3*g^3*i^3*n^2*(d*x+c)^4/d^4+1/105*b*B^2*(-a*d+b*c)^2*g^3*i^3*n^2*(d*x+c)^5 /d^4-1/210*B*(-a*d+b*c)^4*g^3*i^3*n*(b*x+a)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^ n))/b^4/d-3/140*B*(-a*d+b*c)^3*g^3*i^3*n*(b*x+a)^4*(A+B*ln(e*((b*x+a)/(d*x +c))^n))/b^4-1/35*B*(-a*d+b*c)^2*g^3*i^3*n*(b*x+a)^4*(d*x+c)*(A+B*ln(e*((b *x+a)/(d*x+c))^n))/b^3+2/21*B*(-a*d+b*c)^4*g^3*i^3*n*(d*x+c)^3*(A+B*ln(e*( (b*x+a)/(d*x+c))^n))/b/d^4-3/14*B*(-a*d+b*c)^3*g^3*i^3*n*(d*x+c)^4*(A+B*ln (e*((b*x+a)/(d*x+c))^n))/d^4+6/35*b*B*(-a*d+b*c)^2*g^3*i^3*n*(d*x+c)^5*(A+ B*ln(e*((b*x+a)/(d*x+c))^n))/d^4-1/21*b^2*B*(-a*d+b*c)*g^3*i^3*n*(d*x+c)^6 *(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^4+1/140*(-a*d+b*c)^3*g^3*i^3*(b*x+a)^4* (A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^4+1/35*(-a*d+b*c)^2*g^3*i^3*(b*x+a)^4* (d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^3+1/14*(-a*d+b*c)*g^3*i^3*(b*x +a)^4*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^2+1/7*g^3*i^3*(b*x+a)^ 4*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b+1/420*B*(-a*d+b*c)^5*g^3*i ^3*n*(b*x+a)^2*(3*A+B*n+3*B*ln(e*((b*x+a)/(d*x+c))^n))/b^4/d^2-1/420*B*(-a *d+b*c)^6*g^3*i^3*n*(b*x+a)*(6*A+5*B*n+6*B*ln(e*((b*x+a)/(d*x+c))^n))/b^4/ d^3-1/420*B*(-a*d+b*c)^7*g^3*i^3*n*(6*A+11*B*n+6*B*ln(e*((b*x+a)/(d*x+c))^ n))*ln((-a*d+b*c)/b/(d*x+c))/b^4/d^4-1/210*B^2*(-a*d+b*c)^7*g^3*i^3*n^2...
Leaf count is larger than twice the leaf count of optimal. \(2448\) vs. \(2(1172)=2344\).
Time = 1.85 (sec) , antiderivative size = 2448, normalized size of antiderivative = 2.09 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Result too large to show} \]
(g^3*i^3*(35*(b*c - a*d)^3*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^ n])^2 + 84*d*(b*c - a*d)^2*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^ n])^2 + 70*d^2*(b*c - a*d)*(a + b*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^ n])^2 + 20*d^3*(a + b*x)^7*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - (35* B*(b*c - a*d)^4*n*(6*A*b*d*(b*c - a*d)^2*x + 6*B*d*(b*c - a*d)^2*(a + b*x) *Log[e*((a + b*x)/(c + d*x))^n] + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2*(A + B* Log[e*((a + b*x)/(c + d*x))^n]) + 2*d^3*(a + b*x)^3*(A + B*Log[e*((a + b*x )/(c + d*x))^n]) - 6*B*(b*c - a*d)^3*n*Log[c + d*x] - 6*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + B*(b*c - a*d)*n*(2*b*d*( b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]) + 3*B*(b*c - a*d)^2*n*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 3*B*(b*c - a*d)^3*n*((2 *Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLo g[2, (b*(c + d*x))/(b*c - a*d)])))/(3*d^4) + (7*B*(b*c - a*d)^3*n*(24*A*b* d*(b*c - a*d)^3*x + 24*B*d*(b*c - a*d)^3*(a + b*x)*Log[e*((a + b*x)/(c + d *x))^n] - 12*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d* x))^n]) + 8*d^3*(b*c - a*d)*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x)) ^n]) - 6*d^4*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 24*B*(b* c - a*d)^4*n*Log[c + d*x] - 24*(b*c - a*d)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 4*B*(b*c - a*d)^2*n*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c + d*x]) + B*(b*c - a*d)*n*(6*b*d*(b*c...
Time = 3.52 (sec) , antiderivative size = 1455, normalized size of antiderivative = 1.24, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.489, Rules used = {2961, 2783, 2782, 27, 2123, 2009, 2783, 2782, 27, 87, 49, 2009, 2783, 2773, 49, 2009, 2781, 2784, 2784, 2784, 2754, 2838}
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.
| \(\displaystyle \int (a g+b g x)^3 (c i+d i x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2 \, dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle g^3 i^3 (b c-a d)^7 \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^8}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2783 |
| \(\displaystyle g^3 i^3 (b c-a d)^7 \left (-\frac {2 B n \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}d\frac {a+b x}{c+d x}}{7 b}+\frac {3 \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}d\frac {a+b x}{c+d x}}{7 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}\right )\) |
| \(\Big \downarrow \) 2782 |
| \(\displaystyle g^3 i^3 (b c-a d)^7 \left (-\frac {2 B n \left (-B n \int -\frac {(c+d x) \left (b^3-\frac {6 d (a+b x) b^2}{c+d x}+\frac {15 d^2 (a+b x)^2 b}{(c+d x)^2}-\frac {20 d^3 (a+b x)^3}{(c+d x)^3}\right )}{60 d^4 (a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}+\frac {b^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{7 b}+\frac {3 \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}d\frac {a+b x}{c+d x}}{7 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle g^3 i^3 (b c-a d)^7 \left (-\frac {2 B n \left (\frac {B n \int \frac {(c+d x) \left (b^3-\frac {6 d (a+b x) b^2}{c+d x}+\frac {15 d^2 (a+b x)^2 b}{(c+d x)^2}-\frac {20 d^3 (a+b x)^3}{(c+d x)^3}\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{60 d^4}+\frac {b^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{7 b}+\frac {3 \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}d\frac {a+b x}{c+d x}}{7 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}\right )\) |
| \(\Big \downarrow \) 2123 |
| \(\displaystyle g^3 i^3 (b c-a d)^7 \left (-\frac {2 B n \left (\frac {B n \int \left (-\frac {10 d b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^6}+\frac {26 d b}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {19 d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3 b}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2 b^2}+\frac {c+d x}{(a+b x) b^3}+\frac {d}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^3}\right )d\frac {a+b x}{c+d x}}{60 d^4}+\frac {b^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{7 b}+\frac {3 \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}d\frac {a+b x}{c+d x}}{7 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle g^3 i^3 (b c-a d)^7 \left (\frac {3 \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}d\frac {a+b x}{c+d x}}{7 b}-\frac {2 B n \left (\frac {b^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{60 d^4}\right )}{7 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}\right )\) |
| \(\Big \downarrow \) 2783 |
| \(\displaystyle g^3 i^3 (b c-a d)^7 \left (\frac {3 \left (-\frac {B n \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{3 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{3 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )}{7 b}-\frac {2 B n \left (\frac {b^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 b \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {1}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}\right )}{60 d^4}\right )}{7 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}\right )\) |
| \(\Big \downarrow \) 2782 |
| \(\displaystyle (b c-a d)^7 g^3 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^3}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}\right )}{60 d^4}\right )}{7 b}+\frac {3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-B n \int \frac {(a+b x)^3 \left (5 b-\frac {d (a+b x)}{c+d x}\right )}{20 b^2 (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}\right )}{3 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{3 b}\right )}{7 b}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle (b c-a d)^7 g^3 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^3}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}\right )}{60 d^4}\right )}{7 b}+\frac {3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \int \frac {(a+b x)^3 \left (5 b-\frac {d (a+b x)}{c+d x}\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{20 b^2}\right )}{3 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{3 b}\right )}{7 b}\right )\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle (b c-a d)^7 g^3 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^3}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}\right )}{60 d^4}\right )}{7 b}+\frac {3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\int \frac {(a+b x)^3}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}\right )}{20 b^2}\right )}{3 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{3 b}\right )}{7 b}\right )\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle (b c-a d)^7 g^3 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^3}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}\right )}{60 d^4}\right )}{7 b}+\frac {3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\int \left (\frac {b^3}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3 b^2}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {3 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {1}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )d\frac {a+b x}{c+d x}\right )}{20 b^2}\right )}{3 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{3 b}\right )}{7 b}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle (b c-a d)^7 g^3 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^3}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}\right )}{60 d^4}\right )}{7 b}+\frac {3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{20 b^2}\right )}{3 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{3 b}\right )}{7 b}\right )\) |
| \(\Big \downarrow \) 2783 |
| \(\displaystyle (b c-a d)^7 g^3 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^3}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}\right )}{60 d^4}\right )}{7 b}+\frac {3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{20 b^2}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}}{3 b}\right )}{7 b}\right )\) |
| \(\Big \downarrow \) 2773 |
| \(\displaystyle (b c-a d)^7 g^3 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^3}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}\right )}{60 d^4}\right )}{7 b}+\frac {3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{20 b^2}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {(a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \int \frac {(a+b x)^3}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{4 b}\right )}{5 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}}{3 b}\right )}{7 b}\right )\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle (b c-a d)^7 g^3 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^3}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}\right )}{60 d^4}\right )}{7 b}+\frac {3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{20 b^2}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {(a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \int \left (\frac {b^3}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3 b^2}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {3 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {1}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )d\frac {a+b x}{c+d x}}{4 b}\right )}{5 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}}{3 b}\right )}{7 b}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle (b c-a d)^7 g^3 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^3}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}\right )}{60 d^4}\right )}{7 b}+\frac {3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{20 b^2}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {(a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}\right )}{4 b}\right )}{5 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}}{3 b}\right )}{7 b}\right )\) |
| \(\Big \downarrow \) 2781 |
| \(\displaystyle (b c-a d)^7 g^3 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^3}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}\right )}{60 d^4}\right )}{7 b}+\frac {3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{20 b^2}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {(a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}\right )}{4 b}\right )}{5 b}+\frac {\frac {(a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}}{5 b}}{3 b}\right )}{7 b}\right )\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle (b c-a d)^7 g^3 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^3}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}\right )}{60 d^4}\right )}{7 b}+\frac {3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{20 b^2}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {(a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}\right )}{4 b}\right )}{5 b}+\frac {\frac {(a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\int \frac {(a+b x)^2 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 d}\right )}{2 b}}{5 b}}{3 b}\right )}{7 b}\right )\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle (b c-a d)^7 g^3 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^3}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}\right )}{60 d^4}\right )}{7 b}+\frac {3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{20 b^2}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {(a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}\right )}{4 b}\right )}{5 b}+\frac {\frac {(a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\int \frac {(a+b x) \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 d}}{3 d}\right )}{2 b}}{5 b}}{3 b}\right )}{7 b}\right )\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle (b c-a d)^7 g^3 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^3}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}\right )}{60 d^4}\right )}{7 b}+\frac {3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{20 b^2}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {(a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}\right )}{4 b}\right )}{5 b}+\frac {\frac {(a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\int \frac {6 A+11 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{d}}{2 d}}{3 d}\right )}{2 b}}{5 b}}{3 b}\right )}{7 b}\right )\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle (b c-a d)^7 g^3 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^3}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}\right )}{60 d^4}\right )}{7 b}+\frac {3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{20 b^2}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {(a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}\right )}{4 b}\right )}{5 b}+\frac {\frac {(a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {\frac {6 B n \int \frac {(c+d x) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{d}-\frac {\left (6 A+11 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}}{2 d}}{3 d}\right )}{2 b}}{5 b}}{3 b}\right )}{7 b}\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle (b c-a d)^7 g^3 i^3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{7 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}-\frac {2 B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^3}{6 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^2}{5 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b}{4 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {B n \left (-\frac {2 b^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^5}+\frac {13 b}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {19}{3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {1}{2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2 b}+\frac {1}{\left (b-\frac {d (a+b x)}{c+d x}\right ) b^2}+\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^3}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^3}\right )}{60 d^4}\right )}{7 b}+\frac {3 \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}-\frac {B n \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}\right )}{20 b^2}\right )}{3 b}+\frac {\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (a+b x)^4}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {2 B n \left (\frac {(a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}\right )}{4 b}\right )}{5 b}+\frac {\frac {(a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {\frac {(a+b x)^2 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 d (c+d x)^2 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {\frac {(a+b x) \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{d (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {-\frac {\left (6 A+11 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{d}-\frac {6 B n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}}{d}}{2 d}}{3 d}\right )}{2 b}}{5 b}}{3 b}\right )}{7 b}\right )\) |
(b*c - a*d)^7*g^3*i^3*(((a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) ^2)/(7*b*(c + d*x)^4*(b - (d*(a + b*x))/(c + d*x))^7) - (2*B*n*((b^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*d^4*(b - (d*(a + b*x))/(c + d*x))^6) - (3*b^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*d^4*(b - (d*(a + b*x) )/(c + d*x))^5) + (3*b*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*d^4*(b - (d*(a + b*x))/(c + d*x))^4) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(3*d ^4*(b - (d*(a + b*x))/(c + d*x))^3) + (B*n*((-2*b^2)/(b - (d*(a + b*x))/(c + d*x))^5 + (13*b)/(2*(b - (d*(a + b*x))/(c + d*x))^4) - 19/(3*(b - (d*(a + b*x))/(c + d*x))^3) + 1/(2*b*(b - (d*(a + b*x))/(c + d*x))^2) + 1/(b^2* (b - (d*(a + b*x))/(c + d*x))) + Log[(a + b*x)/(c + d*x)]/b^3 - Log[b - (d *(a + b*x))/(c + d*x)]/b^3))/(60*d^4)))/(7*b) + (3*(((a + b*x)^4*(A + B*Lo g[e*((a + b*x)/(c + d*x))^n])^2)/(6*b*(c + d*x)^4*(b - (d*(a + b*x))/(c + d*x))^6) - (B*n*(((a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*b *(c + d*x)^4*(b - (d*(a + b*x))/(c + d*x))^5) + ((a + b*x)^4*(A + B*Log[e* ((a + b*x)/(c + d*x))^n]))/(20*b^2*(c + d*x)^4*(b - (d*(a + b*x))/(c + d*x ))^4) - (B*n*((a + b*x)^4/((c + d*x)^4*(b - (d*(a + b*x))/(c + d*x))^4) + b^3/(3*d^4*(b - (d*(a + b*x))/(c + d*x))^3) - (3*b^2)/(2*d^4*(b - (d*(a + b*x))/(c + d*x))^2) + (3*b)/(d^4*(b - (d*(a + b*x))/(c + d*x))) + Log[b - (d*(a + b*x))/(c + d*x)]/d^4))/(20*b^2)))/(3*b) + (((a + b*x)^4*(A + B*Log [e*((a + b*x)/(c + d*x))^n])^2)/(5*b*(c + d*x)^4*(b - (d*(a + b*x))/(c ...
3.2.78.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c , d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2])
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symb ol] :> Simp[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^p/e), x] - Simp[b*n*(p/e) Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_)^(r_.))^(q_), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/(d*f*(m + 1))), x] - Simp[b*(n/(d*(m + 1))) Int[(f*x)^m*(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q, r}, x] && Eq Q[m + r*(q + 1) + 1, 0] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + Simp[b*n*(p/(d*(q + 1))) Int[(f*x) ^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q _), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x)^q, x]}, Simp[(a + b*Log[c* x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[ {a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + (Simp[(m + q + 2)/(d*(q + 1)) Int[ (f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p, x], x] + Simp[b*n*(p/(d*(q + 1))) Int[(f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[p, 0] && L tQ[q, -1] && GtQ[m, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)* (x_))^(q_.), x_Symbol] :> Simp[(f*x)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n] )/(e*(q + 1))), x] - Simp[f/(e*(q + 1)) Int[(f*x)^(m - 1)*(d + e*x)^(q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*( B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol ] :> Simp[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q Subst[Int[x^m*((A + B*L og[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; Fre eQ[{a, b, c, d, e, f, g, h, i, A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[ b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]
\[\int \left (b g x +a g \right )^{3} \left (d i x +c i \right )^{3} {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}d x\]
\[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{3} {\left (d i x + c i\right )}^{3} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} \,d x } \]
integrate((b*g*x+a*g)^3*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x , algorithm="fricas")
integral(A^2*b^3*d^3*g^3*i^3*x^6 + A^2*a^3*c^3*g^3*i^3 + 3*(A^2*b^3*c*d^2 + A^2*a*b^2*d^3)*g^3*i^3*x^5 + 3*(A^2*b^3*c^2*d + 3*A^2*a*b^2*c*d^2 + A^2* a^2*b*d^3)*g^3*i^3*x^4 + (A^2*b^3*c^3 + 9*A^2*a*b^2*c^2*d + 9*A^2*a^2*b*c* d^2 + A^2*a^3*d^3)*g^3*i^3*x^3 + 3*(A^2*a*b^2*c^3 + 3*A^2*a^2*b*c^2*d + A^ 2*a^3*c*d^2)*g^3*i^3*x^2 + 3*(A^2*a^2*b*c^3 + A^2*a^3*c^2*d)*g^3*i^3*x + ( B^2*b^3*d^3*g^3*i^3*x^6 + B^2*a^3*c^3*g^3*i^3 + 3*(B^2*b^3*c*d^2 + B^2*a*b ^2*d^3)*g^3*i^3*x^5 + 3*(B^2*b^3*c^2*d + 3*B^2*a*b^2*c*d^2 + B^2*a^2*b*d^3 )*g^3*i^3*x^4 + (B^2*b^3*c^3 + 9*B^2*a*b^2*c^2*d + 9*B^2*a^2*b*c*d^2 + B^2 *a^3*d^3)*g^3*i^3*x^3 + 3*(B^2*a*b^2*c^3 + 3*B^2*a^2*b*c^2*d + B^2*a^3*c*d ^2)*g^3*i^3*x^2 + 3*(B^2*a^2*b*c^3 + B^2*a^3*c^2*d)*g^3*i^3*x)*log(e*((b*x + a)/(d*x + c))^n)^2 + 2*(A*B*b^3*d^3*g^3*i^3*x^6 + A*B*a^3*c^3*g^3*i^3 + 3*(A*B*b^3*c*d^2 + A*B*a*b^2*d^3)*g^3*i^3*x^5 + 3*(A*B*b^3*c^2*d + 3*A*B* a*b^2*c*d^2 + A*B*a^2*b*d^3)*g^3*i^3*x^4 + (A*B*b^3*c^3 + 9*A*B*a*b^2*c^2* d + 9*A*B*a^2*b*c*d^2 + A*B*a^3*d^3)*g^3*i^3*x^3 + 3*(A*B*a*b^2*c^3 + 3*A* B*a^2*b*c^2*d + A*B*a^3*c*d^2)*g^3*i^3*x^2 + 3*(A*B*a^2*b*c^3 + A*B*a^3*c^ 2*d)*g^3*i^3*x)*log(e*((b*x + a)/(d*x + c))^n), x)
Timed out. \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 7845 vs. \(2 (1125) = 2250\).
Time = 0.83 (sec) , antiderivative size = 7845, normalized size of antiderivative = 6.69 \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \]
integrate((b*g*x+a*g)^3*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x , algorithm="maxima")
2/7*A*B*b^3*d^3*g^3*i^3*x^7*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/7*A ^2*b^3*d^3*g^3*i^3*x^7 + A*B*b^3*c*d^2*g^3*i^3*x^6*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A*B*a*b^2*d^3*g^3*i^3*x^6*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/2*A^2*b^3*c*d^2*g^3*i^3*x^6 + 1/2*A^2*a*b^2*d^3*g^3*i^3*x^6 + 6/5*A*B*b^3*c^2*d*g^3*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 18 /5*A*B*a*b^2*c*d^2*g^3*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 6/ 5*A*B*a^2*b*d^3*g^3*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/5*A ^2*b^3*c^2*d*g^3*i^3*x^5 + 9/5*A^2*a*b^2*c*d^2*g^3*i^3*x^5 + 3/5*A^2*a^2*b *d^3*g^3*i^3*x^5 + 1/2*A*B*b^3*c^3*g^3*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d *x + c))^n) + 9/2*A*B*a*b^2*c^2*d*g^3*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d* x + c))^n) + 9/2*A*B*a^2*b*c*d^2*g^3*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/2*A*B*a^3*d^3*g^3*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c) )^n) + 1/4*A^2*b^3*c^3*g^3*i^3*x^4 + 9/4*A^2*a*b^2*c^2*d*g^3*i^3*x^4 + 9/4 *A^2*a^2*b*c*d^2*g^3*i^3*x^4 + 1/4*A^2*a^3*d^3*g^3*i^3*x^4 + 2*A*B*a*b^2*c ^3*g^3*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 6*A*B*a^2*b*c^2*d* g^3*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*A*B*a^3*c*d^2*g^3*i ^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2*a*b^2*c^3*g^3*i^3*x^3 + 3*A^2*a^2*b*c^2*d*g^3*i^3*x^3 + A^2*a^3*c*d^2*g^3*i^3*x^3 + 3*A*B*a^2*b* c^3*g^3*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3*A*B*a^3*c^2*d*g ^3*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*A^2*a^2*b*c^3*g...
Timed out. \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Timed out} \]
integrate((b*g*x+a*g)^3*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x , algorithm="giac")
Timed out. \[ \int (a g+b g x)^3 (c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int {\left (a\,g+b\,g\,x\right )}^3\,{\left (c\,i+d\,i\,x\right )}^3\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]