Integrand size = 45, antiderivative size = 762 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx=\frac {B^2 d (b c-a d)^2 i^3 n^2 x}{3 b^3 g}-\frac {5 B d (b c-a d)^2 i^3 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g}-\frac {B (b c-a d) i^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 g}+\frac {d (b c-a d)^2 i^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^4 g}+\frac {(b c-a d) i^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 b^2 g}+\frac {i^3 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b g}+\frac {2 B (b c-a d)^3 i^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{b^4 g}+\frac {B^2 (b c-a d)^3 i^3 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{3 b^4 g}+\frac {2 B^2 (b c-a d)^3 i^3 n^2 \log (c+d x)}{b^4 g}+\frac {5 B (b c-a d)^3 i^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{3 b^4 g}-\frac {(b c-a d)^3 i^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g}+\frac {2 B^2 (b c-a d)^3 i^3 n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{b^4 g}-\frac {5 B^2 (b c-a d)^3 i^3 n^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{3 b^4 g}+\frac {2 B (b c-a d)^3 i^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g}+\frac {2 B^2 (b c-a d)^3 i^3 n^2 \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )}{b^4 g} \]
1/3*B^2*d*(-a*d+b*c)^2*i^3*n^2*x/b^3/g-5/3*B*d*(-a*d+b*c)^2*i^3*n*(b*x+a)* (A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^4/g-1/3*B*(-a*d+b*c)*i^3*n*(d*x+c)^2*(A+ B*ln(e*((b*x+a)/(d*x+c))^n))/b^2/g+d*(-a*d+b*c)^2*i^3*(b*x+a)*(A+B*ln(e*(( b*x+a)/(d*x+c))^n))^2/b^4/g+1/2*(-a*d+b*c)*i^3*(d*x+c)^2*(A+B*ln(e*((b*x+a )/(d*x+c))^n))^2/b^2/g+1/3*i^3*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2 /b/g+2*B*(-a*d+b*c)^3*i^3*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln((-a*d+b*c)/ b/(d*x+c))/b^4/g+1/3*B^2*(-a*d+b*c)^3*i^3*n^2*ln((b*x+a)/(d*x+c))/b^4/g+2* B^2*(-a*d+b*c)^3*i^3*n^2*ln(d*x+c)/b^4/g+5/3*B*(-a*d+b*c)^3*i^3*n*(A+B*ln( e*((b*x+a)/(d*x+c))^n))*ln(1-b*(d*x+c)/d/(b*x+a))/b^4/g-(-a*d+b*c)^3*i^3*( A+B*ln(e*((b*x+a)/(d*x+c))^n))^2*ln(1-b*(d*x+c)/d/(b*x+a))/b^4/g+2*B^2*(-a *d+b*c)^3*i^3*n^2*polylog(2,d*(b*x+a)/b/(d*x+c))/b^4/g-5/3*B^2*(-a*d+b*c)^ 3*i^3*n^2*polylog(2,b*(d*x+c)/d/(b*x+a))/b^4/g+2*B*(-a*d+b*c)^3*i^3*n*(A+B *ln(e*((b*x+a)/(d*x+c))^n))*polylog(2,b*(d*x+c)/d/(b*x+a))/b^4/g+2*B^2*(-a *d+b*c)^3*i^3*n^2*polylog(3,b*(d*x+c)/d/(b*x+a))/b^4/g
Leaf count is larger than twice the leaf count of optimal. \(4969\) vs. \(2(762)=1524\).
Time = 7.27 (sec) , antiderivative size = 4969, normalized size of antiderivative = 6.52 \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx=\text {Result too large to show} \]
(i^3*(36*b*d*(3*b^2*c^2 - 3*a*b*c*d + a^2*d^2)*x*(A + B*Log[e*((a + b*x)/( c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])^2 + 18*b^2*d^2*(3*b*c - a*d)* x^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])^ 2 + 12*b^3*d^3*x^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b* x)/(c + d*x)])^2 + 36*(b*c - a*d)^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/( c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])^2 - 108*b^2*B*c^2*n*(A + B*Lo g[e*((a + b*x)/(c + d*x))^n] - B*n*Log[(a + b*x)/(c + d*x)])*(a*d*Log[a/b + x]^2 - 2*a*d*Log[a/b + x]*(1 + Log[a + b*x]) + 2*(-(b*c) + a*d + Log[c/d + x]*(b*c + a*d*Log[a + b*x] - a*d*Log[(d*(a + b*x))/(-(b*c) + a*d)]) + ( -(b*d*x) + a*d*Log[a + b*x])*Log[(a + b*x)/(c + d*x)]) - 2*a*d*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) - 12*B*n*(-A - B*Log[e*((a + b*x)/(c + d*x))^n ] + B*n*Log[(a + b*x)/(c + d*x)])*(6*a^2*b*c*d^2 - 6*a^3*d^3 + 2*b^3*c^2*d *x + 3*a*b^2*c*d^2*x - 5*a^2*b*d^3*x - b^3*c*d^2*x^2 + a*b^2*d^3*x^2 - 3*a ^3*d^3*Log[a/b + x]^2 - 6*a^2*b*c*d^2*Log[c/d + x] + 5*a^3*d^3*Log[a + b*x ] - 6*a^3*d^3*Log[c/d + x]*Log[a + b*x] + 6*a^3*d^3*Log[a/b + x]*(1 + Log[ a + b*x]) + 6*a^3*d^3*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + 6*a ^2*b*d^3*x*Log[(a + b*x)/(c + d*x)] - 3*a*b^2*d^3*x^2*Log[(a + b*x)/(c + d *x)] + 2*b^3*d^3*x^3*Log[(a + b*x)/(c + d*x)] - 6*a^3*d^3*Log[a + b*x]*Log [(a + b*x)/(c + d*x)] - 2*b^3*c^3*Log[c + d*x] - 3*a*b^2*c^2*d*Log[c + d*x ] + 6*a^3*d^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) + 36*b^3*B*c^3*n*(...
Time = 3.08 (sec) , antiderivative size = 901, normalized size of antiderivative = 1.18, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.356, Rules used = {2961, 2789, 2756, 2789, 2756, 54, 2009, 2789, 2751, 16, 2755, 2754, 2779, 2821, 2838, 7143}
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.
\(\displaystyle \int \frac {(c i+d i x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{a g+b g x} \, dx\) |
\(\Big \downarrow \) 2961 |
\(\displaystyle \frac {i^3 (b c-a d)^3 \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{g}\) |
\(\Big \downarrow \) 2789 |
\(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {d \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^4}d\frac {a+b x}{c+d x}}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{b}\right )}{g}\) |
\(\Big \downarrow \) 2756 |
\(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {d \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{3 d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{b}\right )}{g}\) |
\(\Big \downarrow \) 2789 |
\(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {d \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {d \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}\right )}{3 d}\right )}{b}+\frac {\frac {d \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^3}d\frac {a+b x}{c+d x}}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}}{b}\right )}{g}\) |
\(\Big \downarrow \) 2756 |
\(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {d \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \frac {c+d x}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{2 d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}\right )}{3 d}\right )}{b}+\frac {\frac {d \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}}{b}\right )}{g}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {d \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \left (\frac {d}{b^2 \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {d}{b \left (b-\frac {d (a+b x)}{c+d x}\right )^2}+\frac {c+d x}{b^2 (a+b x)}\right )d\frac {a+b x}{c+d x}}{2 d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}\right )}{3 d}\right )}{b}+\frac {\frac {d \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}}{b}\right )}{g}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {d \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}\right )}{3 d}\right )}{b}+\frac {\frac {d \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}}{b}\right )}{g}\) |
\(\Big \downarrow \) 2789 |
\(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {d \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {\frac {d \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}\right )}{3 d}\right )}{b}+\frac {\frac {d \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {d \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}\right )}{d}\right )}{b}+\frac {\frac {d \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}}{b}}{b}\right )}{g}\) |
\(\Big \downarrow \) 2751 |
\(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {d \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {\frac {d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {B n \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{b}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}\right )}{3 d}\right )}{b}+\frac {\frac {d \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {B n \int \frac {1}{b-\frac {d (a+b x)}{c+d x}}d\frac {a+b x}{c+d x}}{b}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}\right )}{d}\right )}{b}+\frac {\frac {d \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}}{b}}{b}\right )}{g}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {i^3 (b c-a d)^3 \left (\frac {d \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}+\frac {d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}}{b}+\frac {d \left (\frac {B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}\right )}{3 d}\right )}{b}+\frac {\frac {d \left (\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}+\frac {d \left (\frac {(a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}\right )}{d}\right )}{b}+\frac {\frac {d \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{\left (b-\frac {d (a+b x)}{c+d x}\right )^2}d\frac {a+b x}{c+d x}}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}}{b}}{b}\right )}{g}\) |
\(\Big \downarrow \) 2755 |
\(\displaystyle \frac {(b c-a d)^3 i^3 \left (\frac {d \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {d \left (\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}+\frac {\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}}{b}\right )}{3 d}\right )}{b}+\frac {\frac {d \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}\right )}{d}\right )}{b}+\frac {\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {2 B n \int \frac {A+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}}{b}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}}{b}}{b}\right )}{g}\) |
\(\Big \downarrow \) 2754 |
\(\displaystyle \frac {(b c-a d)^3 i^3 \left (\frac {d \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {d \left (\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}+\frac {\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}}{b}\right )}{3 d}\right )}{b}+\frac {\frac {d \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}+\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}\right )}{d}\right )}{b}+\frac {\frac {\int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x) \left (b-\frac {d (a+b x)}{c+d x}\right )}d\frac {a+b x}{c+d x}}{b}+\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {2 B n \left (\frac {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 (A+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}\right )}{b}\right )}{b}}{b}}{b}\right )}{g}\) |
\(\Big \downarrow \) 2779 |
\(\displaystyle \frac {(b c-a d)^3 i^3 \left (\frac {d \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {d \left (\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}+\frac {\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}+\frac {\frac {B n \int \frac {(c+d x) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{b}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b}}{b}}{b}\right )}{3 d}\right )}{b}+\frac {\frac {d \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}+\frac {\frac {B n \int \frac {(c+d x) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{b}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b}}{b}\right )}{d}\right )}{b}+\frac {\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {2 B n \left (\frac {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 (A+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}\right )}{b}\right )}{b}+\frac {\frac {2 B n \int \frac {(c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{b}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b}}{b}}{b}}{b}\right )}{g}\) |
\(\Big \downarrow \) 2821 |
\(\displaystyle \frac {(b c-a d)^3 i^3 \left (\frac {d \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {d \left (\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}+\frac {\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}+\frac {\frac {B n \int \frac {(c+d x) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{b}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b}}{b}}{b}\right )}{3 d}\right )}{b}+\frac {\frac {d \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}+\frac {\frac {B n \int \frac {(c+d x) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{a+b x}d\frac {a+b x}{c+d x}}{b}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b}}{b}\right )}{d}\right )}{b}+\frac {\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {2 B n \left (\frac {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 (A+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}\right )}{b}\right )}{b}+\frac {\frac {2 B n \left (\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )-B n \int \frac {(c+d x) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{a+b x}d\frac {a+b x}{c+d x}\right )}{b}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b}}{b}}{b}}{b}\right )}{g}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle \frac {(b c-a d)^3 i^3 \left (\frac {d \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {d \left (\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}+\frac {\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}+\frac {\frac {B n \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b}}{b}}{b}\right )}{3 d}\right )}{b}+\frac {\frac {d \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}+\frac {\frac {B n \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b}}{b}\right )}{d}\right )}{b}+\frac {\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {2 B n \left (-\frac {\left (A+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 {B n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}\right )}{b}\right )}{b}+\frac {\frac {2 B n \left (\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )-B n \int \frac {(c+d x) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{a+b x}d\frac {a+b x}{c+d x}\right )}{b}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b}}{b}}{b}}{b}\right )}{g}\) |
\(\Big \downarrow \) 7143 |
\(\displaystyle \frac {(b c-a d)^3 i^3 \left (\frac {d \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d \left (b-\frac {d (a+b x)}{c+d x}\right )^3}-\frac {2 B n \left (\frac {d \left (\frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {\log \left (\frac {a+b x}{c+d x}\right )}{b^2}-\frac {\log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b^2}+\frac {1}{b \left (b-\frac {d (a+b x)}{c+d x}\right )}\right )}{2 d}\right )}{b}+\frac {\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}+\frac {\frac {B n \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b}}{b}}{b}\right )}{3 d}\right )}{b}+\frac {\frac {d \left (\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 d \left (b-\frac {d (a+b x)}{c+d x}\right )^2}-\frac {B n \left (\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}+\frac {B n \log \left (b-\frac {d (a+b x)}{c+d x}\right )}{b d}\right )}{b}+\frac {\frac {B n \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{b}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b}}{b}\right )}{d}\right )}{b}+\frac {\frac {d \left (\frac {(a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b (c+d x) \left (b-\frac {d (a+b x)}{c+d x}\right )}-\frac {2 B n \left (-\frac {\left (A+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 {B n \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d}\right )}{b}\right )}{b}+\frac {\frac {2 B n \left (\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )+B n \operatorname {PolyLog}\left (3,\frac {b (c+d x)}{d (a+b x)}\right )\right )}{b}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{b}}{b}}{b}}{b}\right )}{g}\) |
((b*c - a*d)^3*i^3*((d*((A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(3*d*(b - (d*(a + b*x))/(c + d*x))^3) - (2*B*n*((d*((A + B*Log[e*((a + b*x)/(c + d* x))^n])/(2*d*(b - (d*(a + b*x))/(c + d*x))^2) - (B*n*(1/(b*(b - (d*(a + b* x))/(c + d*x))) + Log[(a + b*x)/(c + d*x)]/b^2 - Log[b - (d*(a + b*x))/(c + d*x)]/b^2))/(2*d)))/b + ((d*(((a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x ))^n]))/(b*(c + d*x)*(b - (d*(a + b*x))/(c + d*x))) + (B*n*Log[b - (d*(a + b*x))/(c + d*x)])/(b*d)))/b + (-(((A + B*Log[e*((a + b*x)/(c + d*x))^n])* Log[1 - (b*(c + d*x))/(d*(a + b*x))])/b) + (B*n*PolyLog[2, (b*(c + d*x))/( d*(a + b*x))])/b)/b)/b))/(3*d)))/b + ((d*((A + B*Log[e*((a + b*x)/(c + d*x ))^n])^2/(2*d*(b - (d*(a + b*x))/(c + d*x))^2) - (B*n*((d*(((a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b*(c + d*x)*(b - (d*(a + b*x))/(c + d *x))) + (B*n*Log[b - (d*(a + b*x))/(c + d*x)])/(b*d)))/b + (-(((A + B*Log[ e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d*x))/(d*(a + b*x))])/b) + (B* n*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/b)/b))/d))/b + ((d*(((a + b*x)* (A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b*(c + d*x)*(b - (d*(a + b*x))/ (c + d*x))) - (2*B*n*(-(((A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (d *(a + b*x))/(b*(c + d*x))])/d) - (B*n*PolyLog[2, (d*(a + b*x))/(b*(c + d*x ))])/d))/b))/b + (-(((A + B*Log[e*((a + b*x)/(c + d*x))^n])^2*Log[1 - (b*( c + d*x))/(d*(a + b*x))])/b) + (2*B*n*((A + B*Log[e*((a + b*x)/(c + d*x))^ n])*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))] + B*n*PolyLog[3, (b*(c + d*...
3.2.82.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x _Symbol] :> Simp[x*(d + e*x^r)^(q + 1)*((a + b*Log[c*x^n])/d), x] - Simp[b* (n/d) Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q, r}, x] && EqQ[r*(q + 1) + 1, 0]
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_.))^(p_.)/((d_) + (e_.)*(x_))^2, x_Sy mbol] :> Simp[x*((a + b*Log[c*x^n])^p/(d*(d + e*x))), x] - Simp[b*n*(p/d) Int[(a + b*Log[c*x^n])^(p - 1)/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && GtQ[p, 0]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Simp[b*n*(p/(e*(q + 1))) Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (IntegersQ[2*p, 2*q] && !IGtQ[q, 0]) || (EqQ[p, 2] & & NeQ[q, 1]))
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_)^(r _.))), x_Symbol] :> Simp[(-Log[1 + d/(e*x^r)])*((a + b*Log[c*x^n])^p/(d*r)) , x] + Simp[b*n*(p/(d*r)) Int[Log[1 + d/(e*x^r)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[p, 0]
Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/ (x_), x_Symbol] :> Simp[1/d Int[(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x ), x], x] - Simp[e/d Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; Free Q[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b _.))^(p_.))/(x_), x_Symbol] :> Simp[(-PolyLog[2, (-d)*f*x^m])*((a + b*Log[c *x^n])^p/m), x] + Simp[b*n*(p/m) Int[PolyLog[2, (-d)*f*x^m]*((a + b*Log[c *x^n])^(p - 1)/x), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[p, 0] && EqQ[d*e, 1]
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[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
\[\int \frac {\left (d i x +c i \right )^{3} {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}}{b g x +a g}d x\]
\[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx=\int { \frac {{\left (d i x + c i\right )}^{3} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{b g x + a g} \,d x } \]
integral((A^2*d^3*i^3*x^3 + 3*A^2*c*d^2*i^3*x^2 + 3*A^2*c^2*d*i^3*x + A^2* c^3*i^3 + (B^2*d^3*i^3*x^3 + 3*B^2*c*d^2*i^3*x^2 + 3*B^2*c^2*d*i^3*x + B^2 *c^3*i^3)*log(e*((b*x + a)/(d*x + c))^n)^2 + 2*(A*B*d^3*i^3*x^3 + 3*A*B*c* d^2*i^3*x^2 + 3*A*B*c^2*d*i^3*x + A*B*c^3*i^3)*log(e*((b*x + a)/(d*x + c)) ^n))/(b*g*x + a*g), x)
Timed out. \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx=\text {Timed out} \]
\[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx=\int { \frac {{\left (d i x + c i\right )}^{3} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{b g x + a g} \,d x } \]
3*A^2*c^2*d*i^3*(x/(b*g) - a*log(b*x + a)/(b^2*g)) - 1/6*A^2*d^3*i^3*(6*a^ 3*log(b*x + a)/(b^4*g) - (2*b^2*x^3 - 3*a*b*x^2 + 6*a^2*x)/(b^3*g)) + 3/2* A^2*c*d^2*i^3*(2*a^2*log(b*x + a)/(b^3*g) + (b*x^2 - 2*a*x)/(b^2*g)) + A^2 *c^3*i^3*log(b*g*x + a*g)/(b*g) + 1/6*(2*B^2*b^3*d^3*i^3*x^3 + 3*(3*b^3*c* d^2*i^3 - a*b^2*d^3*i^3)*B^2*x^2 + 6*(3*b^3*c^2*d*i^3 - 3*a*b^2*c*d^2*i^3 + a^2*b*d^3*i^3)*B^2*x + 6*(b^3*c^3*i^3 - 3*a*b^2*c^2*d*i^3 + 3*a^2*b*c*d^ 2*i^3 - a^3*d^3*i^3)*B^2*log(b*x + a))*log((d*x + c)^n)^2/(b^4*g) - integr ate(-1/3*(3*B^2*b^4*c^4*i^3*log(e)^2 + 6*A*B*b^4*c^4*i^3*log(e) + 3*(B^2*b ^4*d^4*i^3*log(e)^2 + 2*A*B*b^4*d^4*i^3*log(e))*x^4 + 12*(B^2*b^4*c*d^3*i^ 3*log(e)^2 + 2*A*B*b^4*c*d^3*i^3*log(e))*x^3 + 18*(B^2*b^4*c^2*d^2*i^3*log (e)^2 + 2*A*B*b^4*c^2*d^2*i^3*log(e))*x^2 + 3*(B^2*b^4*d^4*i^3*x^4 + 4*B^2 *b^4*c*d^3*i^3*x^3 + 6*B^2*b^4*c^2*d^2*i^3*x^2 + 4*B^2*b^4*c^3*d*i^3*x + B ^2*b^4*c^4*i^3)*log((b*x + a)^n)^2 + 12*(B^2*b^4*c^3*d*i^3*log(e)^2 + 2*A* B*b^4*c^3*d*i^3*log(e))*x + 6*(B^2*b^4*c^4*i^3*log(e) + A*B*b^4*c^4*i^3 + (B^2*b^4*d^4*i^3*log(e) + A*B*b^4*d^4*i^3)*x^4 + 4*(B^2*b^4*c*d^3*i^3*log( e) + A*B*b^4*c*d^3*i^3)*x^3 + 6*(B^2*b^4*c^2*d^2*i^3*log(e) + A*B*b^4*c^2* d^2*i^3)*x^2 + 4*(B^2*b^4*c^3*d*i^3*log(e) + A*B*b^4*c^3*d*i^3)*x)*log((b* x + a)^n) - (6*B^2*b^4*c^4*i^3*log(e) + 6*A*B*b^4*c^4*i^3 + 2*(3*A*B*b^4*d ^4*i^3 + (i^3*n + 3*i^3*log(e))*B^2*b^4*d^4)*x^4 + (24*A*B*b^4*c*d^3*i^3 - (a*b^3*d^4*i^3*n - 3*(3*i^3*n + 8*i^3*log(e))*b^4*c*d^3)*B^2)*x^3 + 3*...
\[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx=\int { \frac {{\left (d i x + c i\right )}^{3} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{b g x + a g} \,d x } \]
Timed out. \[ \int \frac {(c i+d i x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{a g+b g x} \, dx=\int \frac {{\left (c\,i+d\,i\,x\right )}^3\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{a\,g+b\,g\,x} \,d x \]