Integrand size = 45, antiderivative size = 766 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {3 B^2 (b c-a d)^5 g^3 i^2 n^2 x}{20 b^2 d^3}+\frac {B^2 (b c-a d)^2 g^3 i^2 n^2 (a+b x)^4}{60 b^3}-\frac {3 B^2 (b c-a d)^4 g^3 i^2 n^2 (c+d x)^2}{40 b d^4}+\frac {B^2 (b c-a d)^3 g^3 i^2 n^2 (c+d x)^3}{60 d^4}-\frac {B (b c-a d)^3 g^3 i^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{90 b^3 d}-\frac {B (b c-a d)^2 g^3 i^2 n (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{20 b^3}-\frac {B (b c-a d) g^3 i^2 n (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 b^2}+\frac {(b c-a d)^2 g^3 i^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{60 b^3}+\frac {(b c-a d) g^3 i^2 (a+b x)^4 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{15 b^2}+\frac {g^3 i^2 (a+b x)^4 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 b}+\frac {B (b c-a d)^4 g^3 i^2 n (a+b x)^2 \left (3 A+B n+3 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{180 b^3 d^2}-\frac {B (b c-a d)^5 g^3 i^2 n (a+b x) \left (6 A+5 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{180 b^3 d^3}-\frac {B (b c-a d)^6 g^3 i^2 n \left (6 A+11 B n+6 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{180 b^3 d^4}-\frac {B^2 (b c-a d)^6 g^3 i^2 n^2 \log (c+d x)}{20 b^3 d^4}-\frac {B^2 (b c-a d)^6 g^3 i^2 n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{30 b^3 d^4} \] Output:
3/20*B^2*(-a*d+b*c)^5*g^3*i^2*n^2*x/b^2/d^3+1/60*B^2*(-a*d+b*c)^2*g^3*i^2* n^2*(b*x+a)^4/b^3-3/40*B^2*(-a*d+b*c)^4*g^3*i^2*n^2*(d*x+c)^2/b/d^4+1/60*B ^2*(-a*d+b*c)^3*g^3*i^2*n^2*(d*x+c)^3/d^4-1/90*B*(-a*d+b*c)^3*g^3*i^2*n*(b *x+a)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^3/d-1/20*B*(-a*d+b*c)^2*g^3*i^2* n*(b*x+a)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^3-1/15*B*(-a*d+b*c)*g^3*i^2* n*(b*x+a)^4*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^2+1/60*(-a*d+b*c)^2* g^3*i^2*(b*x+a)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^3+1/15*(-a*d+b*c)*g^ 3*i^2*(b*x+a)^4*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^2+1/6*g^3*i^2* (b*x+a)^4*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b+1/180*B*(-a*d+b*c) ^4*g^3*i^2*n*(b*x+a)^2*(3*A+B*n+3*B*ln(e*((b*x+a)/(d*x+c))^n))/b^3/d^2-1/1 80*B*(-a*d+b*c)^5*g^3*i^2*n*(b*x+a)*(6*A+5*B*n+6*B*ln(e*((b*x+a)/(d*x+c))^ n))/b^3/d^3-1/180*B*(-a*d+b*c)^6*g^3*i^2*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^3/d^4-1/20*B^2*(-a*d+b*c)^6*g^3*i^2 *n^2*ln(d*x+c)/b^3/d^4-1/30*B^2*(-a*d+b*c)^6*g^3*i^2*n^2*polylog(2,d*(b*x+ a)/b/(d*x+c))/b^3/d^4
Leaf count is larger than twice the leaf count of optimal. \(1634\) vs. \(2(766)=1532\).
Time = 1.58 (sec) , antiderivative size = 1634, normalized size of antiderivative = 2.13 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx =\text {Too large to display} \] Input:
Integrate[(a*g + b*g*x)^3*(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x ))^n])^2,x]
Output:
(g^3*i^2*(15*(b*c - a*d)^2*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^ n])^2 + 24*d*(b*c - a*d)*(a + b*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n] )^2 + 10*d^2*(a + b*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - (5*B*( b*c - a*d)^3*n*(6*A*b*d*(b*c - a*d)^2*x + 6*B*d*(b*c - a*d)^2*(a + b*x)*Lo g[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*Lo g[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2 , (b*(c + d*x))/(b*c - a*d)])))/d^4 + (2*B*(b*c - a*d)^2*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 - a*d)^2* x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*c - a*d...
Time = 2.37 (sec) , antiderivative size = 1003, normalized size of antiderivative = 1.31, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.378, Rules used = {2961, 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.
| \(\displaystyle \int (a g+b g x)^3 (c i+d i x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2 \, dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle g^3 i^2 (b c-a d)^6 \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^7}d\frac {a+b x}{c+d x}\) |
| \(\Big \downarrow \) 2783 |
| \(\displaystyle g^3 i^2 (b c-a d)^6 \left (-\frac {B n \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{3 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{3 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\) |
| \(\Big \downarrow \) 2782 |
| \(\displaystyle g^3 i^2 (b c-a d)^6 \left (-\frac {B n \left (-B n \int \frac {(a+b x)^3 \left (5 b-\frac {d (a+b x)}{c+d x}\right )}{20 b^2 (c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )}{3 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{3 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle g^3 i^2 (b c-a d)^6 \left (-\frac {B n \left (-\frac {B n \int \frac {(a+b x)^3 \left (5 b-\frac {d (a+b x)}{c+d x}\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{20 b^2}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )}{3 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{3 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle g^3 i^2 (b c-a d)^6 \left (-\frac {B n \left (-\frac {B n \left (\int \frac {(a+b x)^3}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}+\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{20 b^2}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )}{3 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{3 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle g^3 i^2 (b c-a d)^6 \left (-\frac {B n \left (-\frac {B n \left (\int \left (\frac {b^3}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3 b^2}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {3 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {1}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )d\frac {a+b x}{c+d x}+\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{20 b^2}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}\right )}{3 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{3 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle g^3 i^2 (b c-a d)^6 \left (\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}d\frac {a+b x}{c+d x}}{3 b}-\frac {B n \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{20 b^2}\right )}{3 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\) |
| \(\Big \downarrow \) 2783 |
| \(\displaystyle g^3 i^2 (b c-a d)^6 \left (\frac {-\frac {2 B n \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}}{3 b}-\frac {B n \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{20 b^2}\right )}{3 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\) |
| \(\Big \downarrow \) 2773 |
| \(\displaystyle g^3 i^2 (b c-a d)^6 \left (\frac {-\frac {2 B n \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \int \frac {(a+b x)^3}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{4 b}\right )}{5 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}}{3 b}-\frac {B n \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{20 b^2}\right )}{3 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\) |
| \(\Big \downarrow \) 49 |
| \(\displaystyle g^3 i^2 (b c-a d)^6 \left (\frac {-\frac {2 B n \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \int \left (\frac {b^3}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {3 b^2}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}+\frac {3 b}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {1}{d^3 \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )d\frac {a+b x}{c+d x}}{4 b}\right )}{5 b}+\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}}{3 b}-\frac {B n \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{20 b^2}\right )}{3 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle g^3 i^2 (b c-a d)^6 \left (\frac {\frac {\int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}d\frac {a+b x}{c+d x}}{5 b}-\frac {2 B n \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}\right )}{4 b}\right )}{5 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}}{3 b}-\frac {B n \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{20 b^2}\right )}{3 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\) |
| \(\Big \downarrow \) 2781 |
| \(\displaystyle g^3 i^2 (b c-a d)^6 \left (\frac {\frac {\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \int \frac {(a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(c+d x)^3 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{2 b}}{5 b}-\frac {2 B n \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}-\frac {B n \left (\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}\right )}{4 b}\right )}{5 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}}{3 b}-\frac {B n \left (\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^2 (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^5}-\frac {B n \left (\frac {b^3}{3 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {3 b^2}{2 d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {3 b}{d^4 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{d^4}+\frac {(a+b x)^4}{(c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^4}\right )}{20 b^2}\right )}{3 b}+\frac {(a+b x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b (c+d x)^4 \left (b-\frac {d (a+b x)}{c+d x}\right )^6}\right )\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle (b c-a d)^6 g^3 i^2 \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 )\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle (b c-a d)^6 g^3 i^2 \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 )\) |
| \(\Big \downarrow \) 2784 |
| \(\displaystyle (b c-a d)^6 g^3 i^2 \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 )\) |
| \(\Big \downarrow \) 2754 |
| \(\displaystyle (b c-a d)^6 g^3 i^2 \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 )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle (b c-a d)^6 g^3 i^2 \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 )\) |
Input:
Int[(a*g + b*g*x)^3*(c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) ^2,x]
Output:
(b*c - a*d)^6*g^3*i^2*(((a + b*x)^4*(A + B*Log[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]))/(2 0*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 + d*x))^5) - (2*B*n*(((a + b*x)^ 4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*b*(c + d*x)^4*(b - (d*(a + b* x))/(c + d*x))^4) - (B*n*(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))/(4*b)))/(5*b) + ( ((a + b*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*b*(c + d*x)^4*(b - (d*(a + b*x))/(c + d*x))^4) - (B*n*(((a + b*x)^3*(A + B*Log[e*((a + b*x )/(c + d*x))^n]))/(3*d*(c + d*x)^3*(b - (d*(a + b*x))/(c + d*x))^3) - (((a + b*x)^2*(3*A + B*n + 3*B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*d*(c + d*x) ^2*(b - (d*(a + b*x))/(c + d*x))^2) - (((a + b*x)*(6*A + 5*B*n + 6*B*Log[e *((a + b*x)/(c + d*x))^n]))/(d*(c + d*x)*(b - (d*(a + b*x))/(c + d*x)))...
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[((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]
\[\int \left (b g x +a g \right )^{3} \left (d i x +c i \right )^{2} {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}d x\]
Input:
int((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,x)
Output:
int((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,x)
\[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{3} {\left (d i x + c i\right )}^{2} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} \,d x } \] Input:
integrate((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x , algorithm="fricas")
Output:
integral(A^2*b^3*d^2*g^3*i^2*x^5 + A^2*a^3*c^2*g^3*i^2 + (2*A^2*b^3*c*d + 3*A^2*a*b^2*d^2)*g^3*i^2*x^4 + (A^2*b^3*c^2 + 6*A^2*a*b^2*c*d + 3*A^2*a^2* b*d^2)*g^3*i^2*x^3 + (3*A^2*a*b^2*c^2 + 6*A^2*a^2*b*c*d + A^2*a^3*d^2)*g^3 *i^2*x^2 + (3*A^2*a^2*b*c^2 + 2*A^2*a^3*c*d)*g^3*i^2*x + (B^2*b^3*d^2*g^3* i^2*x^5 + B^2*a^3*c^2*g^3*i^2 + (2*B^2*b^3*c*d + 3*B^2*a*b^2*d^2)*g^3*i^2* x^4 + (B^2*b^3*c^2 + 6*B^2*a*b^2*c*d + 3*B^2*a^2*b*d^2)*g^3*i^2*x^3 + (3*B ^2*a*b^2*c^2 + 6*B^2*a^2*b*c*d + B^2*a^3*d^2)*g^3*i^2*x^2 + (3*B^2*a^2*b*c ^2 + 2*B^2*a^3*c*d)*g^3*i^2*x)*log(e*((b*x + a)/(d*x + c))^n)^2 + 2*(A*B*b ^3*d^2*g^3*i^2*x^5 + A*B*a^3*c^2*g^3*i^2 + (2*A*B*b^3*c*d + 3*A*B*a*b^2*d^ 2)*g^3*i^2*x^4 + (A*B*b^3*c^2 + 6*A*B*a*b^2*c*d + 3*A*B*a^2*b*d^2)*g^3*i^2 *x^3 + (3*A*B*a*b^2*c^2 + 6*A*B*a^2*b*c*d + A*B*a^3*d^2)*g^3*i^2*x^2 + (3* A*B*a^2*b*c^2 + 2*A*B*a^3*c*d)*g^3*i^2*x)*log(e*((b*x + a)/(d*x + c))^n), x)
Timed out. \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Timed out} \] Input:
integrate((b*g*x+a*g)**3*(d*i*x+c*i)**2*(A+B*ln(e*((b*x+a)/(d*x+c))**n))** 2,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 5952 vs. \(2 (735) = 1470\).
Time = 0.65 (sec) , antiderivative size = 5952, normalized size of antiderivative = 7.77 \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \] Input:
integrate((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x , algorithm="maxima")
Output:
1/3*A*B*b^3*d^2*g^3*i^2*x^6*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/6*A ^2*b^3*d^2*g^3*i^2*x^6 + 4/5*A*B*b^3*c*d*g^3*i^2*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 6/5*A*B*a*b^2*d^2*g^3*i^2*x^5*log(e*(b*x/(d*x + c) + a /(d*x + c))^n) + 2/5*A^2*b^3*c*d*g^3*i^2*x^5 + 3/5*A^2*a*b^2*d^2*g^3*i^2*x ^5 + 1/2*A*B*b^3*c^2*g^3*i^2*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3*A*B*a*b^2*c*d*g^3*i^2*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*A *B*a^2*b*d^2*g^3*i^2*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A^2* b^3*c^2*g^3*i^2*x^4 + 3/2*A^2*a*b^2*c*d*g^3*i^2*x^4 + 3/4*A^2*a^2*b*d^2*g^ 3*i^2*x^4 + 2*A*B*a*b^2*c^2*g^3*i^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c) )^n) + 4*A*B*a^2*b*c*d*g^3*i^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2/3*A*B*a^3*d^2*g^3*i^2*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2 *a*b^2*c^2*g^3*i^2*x^3 + 2*A^2*a^2*b*c*d*g^3*i^2*x^3 + 1/3*A^2*a^3*d^2*g^3 *i^2*x^3 + 3*A*B*a^2*b*c^2*g^3*i^2*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c)) ^n) + 2*A*B*a^3*c*d*g^3*i^2*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3 /2*A^2*a^2*b*c^2*g^3*i^2*x^2 + A^2*a^3*c*d*g^3*i^2*x^2 - 1/180*A*B*b^3*d^2 *g^3*i^2*n*(60*a^6*log(b*x + a)/b^6 - 60*c^6*log(d*x + c)/d^6 + (12*(b^5*c *d^4 - a*b^4*d^5)*x^5 - 15*(b^5*c^2*d^3 - a^2*b^3*d^5)*x^4 + 20*(b^5*c^3*d ^2 - a^3*b^2*d^5)*x^3 - 30*(b^5*c^4*d - a^4*b*d^5)*x^2 + 60*(b^5*c^5 - a^5 *d^5)*x)/(b^5*d^5)) + 1/15*A*B*b^3*c*d*g^3*i^2*n*(12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2...
Timed out. \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Timed out} \] Input:
integrate((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x , algorithm="giac")
Output:
Timed out
Timed out. \[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int {\left (a\,g+b\,g\,x\right )}^3\,{\left (c\,i+d\,i\,x\right )}^2\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \] Input:
int((a*g + b*g*x)^3*(c*i + d*i*x)^2*(A + B*log(e*((a + b*x)/(c + d*x))^n)) ^2,x)
Output:
int((a*g + b*g*x)^3*(c*i + d*i*x)^2*(A + B*log(e*((a + b*x)/(c + d*x))^n)) ^2, x)
\[ \int (a g+b g x)^3 (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {too large to display} \] Input:
int((b*g*x+a*g)^3*(d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x)
Output:
(g**3*( - 12*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c *x + b*d*x**2),x)*a**6*b**2*d**7*n + 72*int((log(((a + b*x)**n*e)/(c + d*x )**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**5*b**3*c*d**6*n - 180*int( (log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x) *a**4*b**4*c**2*d**5*n + 240*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a *c + a*d*x + b*c*x + b*d*x**2),x)*a**3*b**5*c**3*d**4*n - 180*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*a**2*b* *6*c**4*d**3*n + 72*int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(a*c + a*d* x + b*c*x + b*d*x**2),x)*a*b**7*c**5*d**2*n - 12*int((log(((a + b*x)**n*e) /(c + d*x)**n)*x)/(a*c + a*d*x + b*c*x + b*d*x**2),x)*b**8*c**6*d*n - 12*l og(c + d*x)*a**7*d**6*n + 72*log(c + d*x)*a**6*b*c*d**5*n - 4*log(c + d*x) *a**6*b*d**6*n**2 - 180*log(c + d*x)*a**5*b**2*c**2*d**4*n + 24*log(c + d* x)*a**5*b**2*c*d**5*n**2 + 240*log(c + d*x)*a**4*b**3*c**3*d**3*n - 60*log (c + d*x)*a**4*b**3*c**2*d**4*n**2 - 180*log(c + d*x)*a**3*b**4*c**4*d**2* n + 80*log(c + d*x)*a**3*b**4*c**3*d**3*n**2 + 72*log(c + d*x)*a**2*b**5*c **5*d*n - 60*log(c + d*x)*a**2*b**5*c**4*d**2*n**2 - 12*log(c + d*x)*a*b** 6*c**6*n + 24*log(c + d*x)*a*b**6*c**5*d*n**2 - 4*log(c + d*x)*b**7*c**6*n **2 + 6*log(((a + b*x)**n*e)/(c + d*x)**n)**2*a**5*b**2*c*d**5 - 30*log((( a + b*x)**n*e)/(c + d*x)**n)**2*a**4*b**3*c**2*d**4 - 60*log(((a + b*x)**n *e)/(c + d*x)**n)**2*a**3*b**4*c**3*d**3 - 360*log(((a + b*x)**n*e)/(c ...