Integrand size = 45, antiderivative size = 319 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\frac {2 B^2 d i^2 n^2 (c+d x)^3}{27 (b c-a d)^2 g^5 (a+b x)^3}-\frac {b B^2 i^2 n^2 (c+d x)^4}{32 (b c-a d)^2 g^5 (a+b x)^4}+\frac {2 B d i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 (b c-a d)^2 g^5 (a+b x)^3}-\frac {b B i^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 (b c-a d)^2 g^5 (a+b x)^4}+\frac {d i^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 (b c-a d)^2 g^5 (a+b x)^3}-\frac {b i^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 (b c-a d)^2 g^5 (a+b x)^4} \]
2/27*B^2*d*i^2*n^2*(d*x+c)^3/(-a*d+b*c)^2/g^5/(b*x+a)^3-1/32*b*B^2*i^2*n^2 *(d*x+c)^4/(-a*d+b*c)^2/g^5/(b*x+a)^4+2/9*B*d*i^2*n*(d*x+c)^3*(A+B*ln(e*(( b*x+a)/(d*x+c))^n))/(-a*d+b*c)^2/g^5/(b*x+a)^3-1/8*b*B*i^2*n*(d*x+c)^4*(A+ B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^2/g^5/(b*x+a)^4+1/3*d*i^2*(d*x+c)^ 3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^2/g^5/(b*x+a)^3-1/4*b*i^2*( d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^2/g^5/(b*x+a)^4
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 1.63 (sec) , antiderivative size = 1787, normalized size of antiderivative = 5.60 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx =\text {Too large to display} \]
-1/864*(i^2*(216*(b*c - a*d)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - 576*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 432*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) ^2 + 32*B*d*n*(a + b*x)*(12*A*(b*c - a*d)^3 + 4*B*(b*c - a*d)^3*n - 18*A*d *(b*c - a*d)^2*(a + b*x) - 15*B*d*(b*c - a*d)^2*n*(a + b*x) + 36*A*d^2*(b* c - a*d)*(a + b*x)^2 + 66*B*d^2*(b*c - a*d)*n*(a + b*x)^2 + 36*A*d^3*(a + b*x)^3*Log[a + b*x] + 66*B*d^3*n*(a + b*x)^3*Log[a + b*x] - 18*B*d^3*n*(a + b*x)^3*Log[a + b*x]^2 + 12*B*(b*c - a*d)^3*Log[e*((a + b*x)/(c + d*x))^n ] - 18*B*d*(b*c - a*d)^2*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + 36*B*d ^2*(b*c - a*d)*(a + b*x)^2*Log[e*((a + b*x)/(c + d*x))^n] + 36*B*d^3*(a + b*x)^3*Log[a + b*x]*Log[e*((a + b*x)/(c + d*x))^n] - 36*A*d^3*(a + b*x)^3* Log[c + d*x] - 66*B*d^3*n*(a + b*x)^3*Log[c + d*x] + 36*B*d^3*n*(a + b*x)^ 3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] - 36*B*d^3*(a + b*x)^3*Lo g[e*((a + b*x)/(c + d*x))^n]*Log[c + d*x] - 18*B*d^3*n*(a + b*x)^3*Log[c + d*x]^2 + 36*B*d^3*n*(a + b*x)^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d )] + 36*B*d^3*n*(a + b*x)^3*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 36* B*d^3*n*(a + b*x)^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) + 3*B*n*(36*A*( b*c - a*d)^4 + 9*B*(b*c - a*d)^4*n + 48*A*d*(-(b*c) + a*d)^3*(a + b*x) + 2 8*B*d*(-(b*c) + a*d)^3*n*(a + b*x) + 72*A*d^2*(b*c - a*d)^2*(a + b*x)^2 + 78*B*d^2*(b*c - a*d)^2*n*(a + b*x)^2 + 144*A*d^3*(-(b*c) + a*d)*(a + b*...
Time = 0.49 (sec) , antiderivative size = 240, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2961, 2795, 2009}
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.
| \(\displaystyle \int \frac {(c i+d i x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{(a g+b g x)^5} \, dx\) |
| \(\Big \downarrow \) 2961 |
| \(\displaystyle \frac {i^2 \int \frac {(c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right ) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^5}d\frac {a+b x}{c+d x}}{g^5 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \frac {i^2 \int \left (\frac {b (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^5}-\frac {d (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^4}\right )d\frac {a+b x}{c+d x}}{g^5 (b c-a d)^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {i^2 \left (-\frac {b (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 (a+b x)^4}-\frac {b B n (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{8 (a+b x)^4}+\frac {d (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 (a+b x)^3}+\frac {2 B d n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 (a+b x)^3}-\frac {b B^2 n^2 (c+d x)^4}{32 (a+b x)^4}+\frac {2 B^2 d n^2 (c+d x)^3}{27 (a+b x)^3}\right )}{g^5 (b c-a d)^2}\) |
(i^2*((2*B^2*d*n^2*(c + d*x)^3)/(27*(a + b*x)^3) - (b*B^2*n^2*(c + d*x)^4) /(32*(a + b*x)^4) + (2*B*d*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x) )^n]))/(9*(a + b*x)^3) - (b*B*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d *x))^n]))/(8*(a + b*x)^4) + (d*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d* x))^n])^2)/(3*(a + b*x)^3) - (b*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d *x))^n])^2)/(4*(a + b*x)^4)))/((b*c - a*d)^2*g^5)
3.2.76.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[ c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b , c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0 ] && IntegerQ[m] && IntegerQ[r]))
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]
Leaf count of result is larger than twice the leaf count of optimal. \(1883\) vs. \(2(307)=614\).
Time = 27.47 (sec) , antiderivative size = 1884, normalized size of antiderivative = 5.91
1/864*(-1152*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b*c^4*d*i^2*n+144*A*B*x^4 *ln(e*((b*x+a)/(d*x+c))^n)*a^6*b*c*d^4*i^2*n-864*A*B*x^2*ln(e*((b*x+a)/(d* x+c))^n)*a^6*b*c^3*d^2*i^2*n-192*A*B*x^4*a^3*b^4*c^4*d*i^2*n^2+144*B^2*x^3 *ln(e*((b*x+a)/(d*x+c))^n)*a^6*b*c^2*d^3*i^2*n^2+576*A*B*x^3*ln(e*((b*x+a) /(d*x+c))^n)*a^7*c*d^4*i^2*n+144*A*B*x^3*a^6*b*c^2*d^3*i^2*n^2-768*A*B*x^3 *a^4*b^3*c^4*d*i^2*n^2-432*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)^2*a^6*b*c^3*d ^2*i^2*n-72*B^2*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b*c^3*d^2*i^2*n^2+1728*A *B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^7*c^2*d^3*i^2*n-72*A*B*x^2*a^6*b*c^3*d^ 2*i^2*n^2-1152*A*B*x^2*a^5*b^2*c^4*d*i^2*n^2-576*B^2*x*ln(e*((b*x+a)/(d*x+ c))^n)^2*a^6*b*c^4*d*i^2*n-240*B^2*x*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b*c^4*d *i^2*n^2+1728*A*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^7*c^3*d^2*i^2*n-1008*A*B*x *a^6*b*c^4*d*i^2*n^2+72*B^2*x^4*ln(e*((b*x+a)/(d*x+c))^n)^2*a^6*b*c*d^4*i^ 2*n+84*B^2*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a^6*b*c*d^4*i^2*n^2+84*A*B*x^4*a^ 6*b*c*d^4*i^2*n^2+108*A*B*x^4*a^2*b^5*c^5*i^2*n^2+288*B^2*x^3*ln(e*((b*x+a )/(d*x+c))^n)^2*a^7*c*d^4*i^2*n+192*B^2*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^7* c*d^4*i^2*n^2+84*B^2*x^3*a^6*b*c^2*d^3*i^2*n^3-256*B^2*x^3*a^4*b^3*c^4*d*i ^2*n^3+72*A^2*x^4*a^6*b*c*d^4*i^2*n-288*A^2*x^4*a^3*b^4*c^4*d*i^2*n+192*A* B*x^3*a^7*c*d^4*i^2*n^2+432*A*B*x^3*a^3*b^4*c^5*i^2*n^2+864*B^2*x^2*ln(e*( (b*x+a)/(d*x+c))^n)^2*a^7*c^2*d^3*i^2*n+576*B^2*x^2*ln(e*((b*x+a)/(d*x+c)) ^n)*a^7*c^2*d^3*i^2*n^2+30*B^2*x^2*a^6*b*c^3*d^2*i^2*n^3-384*B^2*x^2*a^...
Leaf count of result is larger than twice the leaf count of optimal. 1729 vs. \(2 (307) = 614\).
Time = 0.38 (sec) , antiderivative size = 1729, normalized size of antiderivative = 5.42 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^5,x , algorithm="fricas")
-1/864*((27*B^2*b^4*c^4 - 64*B^2*a*b^3*c^3*d + 37*B^2*a^4*d^4)*i^2*n^2 + 1 2*(9*A*B*b^4*c^4 - 16*A*B*a*b^3*c^3*d + 7*A*B*a^4*d^4)*i^2*n - 12*(7*(B^2* b^4*c*d^3 - B^2*a*b^3*d^4)*i^2*n^2 + 12*(A*B*b^4*c*d^3 - A*B*a*b^3*d^4)*i^ 2*n)*x^3 + 72*(3*A^2*b^4*c^4 - 4*A^2*a*b^3*c^3*d + A^2*a^4*d^4)*i^2 - 6*(( 5*B^2*b^4*c^2*d^2 + 32*B^2*a*b^3*c*d^3 - 37*B^2*a^2*b^2*d^4)*i^2*n^2 - 12* (A*B*b^4*c^2*d^2 - 8*A*B*a*b^3*c*d^3 + 7*A*B*a^2*b^2*d^4)*i^2*n - 72*(A^2* b^4*c^2*d^2 - 2*A^2*a*b^3*c*d^3 + A^2*a^2*b^2*d^4)*i^2)*x^2 + 72*(6*(B^2*b ^4*c^2*d^2 - 2*B^2*a*b^3*c*d^3 + B^2*a^2*b^2*d^4)*i^2*x^2 + 4*(2*B^2*b^4*c ^3*d - 3*B^2*a*b^3*c^2*d^2 + B^2*a^3*b*d^4)*i^2*x + (3*B^2*b^4*c^4 - 4*B^2 *a*b^3*c^3*d + B^2*a^4*d^4)*i^2)*log(e)^2 - 72*(B^2*b^4*d^4*i^2*n^2*x^4 + 4*B^2*a*b^3*d^4*i^2*n^2*x^3 - 6*(B^2*b^4*c^2*d^2 - 2*B^2*a*b^3*c*d^3)*i^2* n^2*x^2 - 4*(2*B^2*b^4*c^3*d - 3*B^2*a*b^3*c^2*d^2)*i^2*n^2*x - (3*B^2*b^4 *c^4 - 4*B^2*a*b^3*c^3*d)*i^2*n^2)*log((b*x + a)/(d*x + c))^2 + 4*((11*B^2 *b^4*c^3*d - 48*B^2*a*b^3*c^2*d^2 + 37*B^2*a^3*b*d^4)*i^2*n^2 + 12*(5*A*B* b^4*c^3*d - 12*A*B*a*b^3*c^2*d^2 + 7*A*B*a^3*b*d^4)*i^2*n + 72*(2*A^2*b^4* c^3*d - 3*A^2*a*b^3*c^2*d^2 + A^2*a^3*b*d^4)*i^2)*x - 12*(12*(B^2*b^4*c*d^ 3 - B^2*a*b^3*d^4)*i^2*n*x^3 - (9*B^2*b^4*c^4 - 16*B^2*a*b^3*c^3*d + 7*B^2 *a^4*d^4)*i^2*n - 12*(3*A*B*b^4*c^4 - 4*A*B*a*b^3*c^3*d + A*B*a^4*d^4)*i^2 - 6*((B^2*b^4*c^2*d^2 - 8*B^2*a*b^3*c*d^3 + 7*B^2*a^2*b^2*d^4)*i^2*n + 12 *(A*B*b^4*c^2*d^2 - 2*A*B*a*b^3*c*d^3 + A*B*a^2*b^2*d^4)*i^2)*x^2 - 4*(...
Timed out. \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 8087 vs. \(2 (307) = 614\).
Time = 0.73 (sec) , antiderivative size = 8087, normalized size of antiderivative = 25.35 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^5,x , algorithm="maxima")
1/24*A*B*c^2*i^2*n*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2* b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5* a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d ^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c *d^2 - a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d ^2 - a^7*b*d^3)*g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a ^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/( (b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4 )*g^5)) - 1/72*A*B*d^2*i^2*n*((13*a^2*b^3*c^3 - 75*a^3*b^2*c^2*d + 33*a^4* b*c*d^2 - 7*a^5*d^3 - 12*(6*b^5*c^2*d - 4*a*b^4*c*d^2 + a^2*b^3*d^3)*x^3 + 6*(6*b^5*c^3 - 46*a*b^4*c^2*d + 29*a^2*b^3*c*d^2 - 7*a^3*b^2*d^3)*x^2 + 4 *(10*a*b^4*c^3 - 63*a^2*b^3*c^2*d + 33*a^3*b^2*c*d^2 - 7*a^4*b*d^3)*x)/((b ^10*c^3 - 3*a*b^9*c^2*d + 3*a^2*b^8*c*d^2 - a^3*b^7*d^3)*g^5*x^4 + 4*(a*b^ 9*c^3 - 3*a^2*b^8*c^2*d + 3*a^3*b^7*c*d^2 - a^4*b^6*d^3)*g^5*x^3 + 6*(a^2* b^8*c^3 - 3*a^3*b^7*c^2*d + 3*a^4*b^6*c*d^2 - a^5*b^5*d^3)*g^5*x^2 + 4*(a^ 3*b^7*c^3 - 3*a^4*b^6*c^2*d + 3*a^5*b^5*c*d^2 - a^6*b^4*d^3)*g^5*x + (a^4* b^6*c^3 - 3*a^5*b^5*c^2*d + 3*a^6*b^4*c*d^2 - a^7*b^3*d^3)*g^5) - 12*(6*b^ 2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(b*x + a)/((b^7*c^4 - 4*a*b^6*c^3...
Time = 4.55 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.70 \[ \int \frac {(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}{(a g+b g x)^5} \, dx=-\frac {1}{864} \, {\left (\frac {72 \, {\left (3 \, B^{2} b i^{2} n^{2} - \frac {4 \, {\left (b x + a\right )} B^{2} d i^{2} n^{2}}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{\frac {{\left (b x + a\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x + a\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {12 \, {\left (9 \, B^{2} b i^{2} n^{2} - \frac {16 \, {\left (b x + a\right )} B^{2} d i^{2} n^{2}}{d x + c} + 36 \, B^{2} b i^{2} n \log \left (e\right ) - \frac {48 \, {\left (b x + a\right )} B^{2} d i^{2} n \log \left (e\right )}{d x + c} + 36 \, A B b i^{2} n - \frac {48 \, {\left (b x + a\right )} A B d i^{2} n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x + a\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {27 \, B^{2} b i^{2} n^{2} - \frac {64 \, {\left (b x + a\right )} B^{2} d i^{2} n^{2}}{d x + c} + 108 \, B^{2} b i^{2} n \log \left (e\right ) - \frac {192 \, {\left (b x + a\right )} B^{2} d i^{2} n \log \left (e\right )}{d x + c} + 216 \, B^{2} b i^{2} \log \left (e\right )^{2} - \frac {288 \, {\left (b x + a\right )} B^{2} d i^{2} \log \left (e\right )^{2}}{d x + c} + 108 \, A B b i^{2} n - \frac {192 \, {\left (b x + a\right )} A B d i^{2} n}{d x + c} + 432 \, A B b i^{2} \log \left (e\right ) - \frac {576 \, {\left (b x + a\right )} A B d i^{2} \log \left (e\right )}{d x + c} + 216 \, A^{2} b i^{2} - \frac {288 \, {\left (b x + a\right )} A^{2} d i^{2}}{d x + c}}{\frac {{\left (b x + a\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x + a\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]
integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^5,x , algorithm="giac")
-1/864*(72*(3*B^2*b*i^2*n^2 - 4*(b*x + a)*B^2*d*i^2*n^2/(d*x + c))*log((b* x + a)/(d*x + c))^2/((b*x + a)^4*b*c*g^5/(d*x + c)^4 - (b*x + a)^4*a*d*g^5 /(d*x + c)^4) + 12*(9*B^2*b*i^2*n^2 - 16*(b*x + a)*B^2*d*i^2*n^2/(d*x + c) + 36*B^2*b*i^2*n*log(e) - 48*(b*x + a)*B^2*d*i^2*n*log(e)/(d*x + c) + 36* A*B*b*i^2*n - 48*(b*x + a)*A*B*d*i^2*n/(d*x + c))*log((b*x + a)/(d*x + c)) /((b*x + a)^4*b*c*g^5/(d*x + c)^4 - (b*x + a)^4*a*d*g^5/(d*x + c)^4) + (27 *B^2*b*i^2*n^2 - 64*(b*x + a)*B^2*d*i^2*n^2/(d*x + c) + 108*B^2*b*i^2*n*lo g(e) - 192*(b*x + a)*B^2*d*i^2*n*log(e)/(d*x + c) + 216*B^2*b*i^2*log(e)^2 - 288*(b*x + a)*B^2*d*i^2*log(e)^2/(d*x + c) + 108*A*B*b*i^2*n - 192*(b*x + a)*A*B*d*i^2*n/(d*x + c) + 432*A*B*b*i^2*log(e) - 576*(b*x + a)*A*B*d*i ^2*log(e)/(d*x + c) + 216*A^2*b*i^2 - 288*(b*x + a)*A^2*d*i^2/(d*x + c))/( (b*x + a)^4*b*c*g^5/(d*x + c)^4 - (b*x + a)^4*a*d*g^5/(d*x + c)^4))*(b*c/( b*c - a*d)^2 - a*d/(b*c - a*d)^2)
Time = 5.31 (sec) , antiderivative size = 1934, normalized size of antiderivative = 6.06 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
- log(e*((a + b*x)/(c + d*x))^n)*((a*(A*B*a*d^2*i^2 - (B^2*a*d^2*i^2*n)/2 + (B^2*b*c*d*i^2*n)/2 + 2*A*B*b*c*d*i^2) + x*(b*(A*B*a*d^2*i^2 - (B^2*a*d^ 2*i^2*n)/2 + (B^2*b*c*d*i^2*n)/2 + 2*A*B*b*c*d*i^2) + 3*A*B*a*b*d^2*i^2 + 6*A*B*b^2*c*d*i^2 - (3*B^2*a*b*d^2*i^2*n)/2 + (3*B^2*b^2*c*d*i^2*n)/2) + 3 *A*B*b^2*c^2*i^2 - B^2*a^2*d^2*i^2*n + (B^2*b^2*c^2*i^2*n)/2 + 6*A*B*b^2*d ^2*i^2*x^2 + (B^2*a*b*c*d*i^2*n)/2)/(6*a^4*b^3*g^5 + 6*b^7*g^5*x^4 + 24*a^ 3*b^4*g^5*x + 24*a*b^6*g^5*x^3 + 36*a^2*b^5*g^5*x^2) + (B^2*d^4*i^2*(x^2*( b*(b*((3*a*b^3*g^5*n*(a*d - b*c))/(2*d) + (b^3*g^5*n*(a*d - b*c)*(4*a*d - b*c))/(2*d^2)) + (3*a*b^4*g^5*n*(a*d - b*c))/d + (b^4*g^5*n*(a*d - b*c)*(4 *a*d - b*c))/d^2) + (9*a*b^5*g^5*n*(a*d - b*c))/(2*d) + (3*b^5*g^5*n*(a*d - b*c)*(4*a*d - b*c))/(2*d^2)) + a*(a*((3*a*b^3*g^5*n*(a*d - b*c))/(2*d) + (b^3*g^5*n*(a*d - b*c)*(4*a*d - b*c))/(2*d^2)) + (b^3*g^5*n*(a*d - b*c)*( 6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(2*d^3)) + x*(a*(b*((3*a*b^3*g^5*n*(a*d - b*c))/(2*d) + (b^3*g^5*n*(a*d - b*c)*(4*a*d - b*c))/(2*d^2)) + (3*a*b^4* g^5*n*(a*d - b*c))/d + (b^4*g^5*n*(a*d - b*c)*(4*a*d - b*c))/d^2) + b*(a*( (3*a*b^3*g^5*n*(a*d - b*c))/(2*d) + (b^3*g^5*n*(a*d - b*c)*(4*a*d - b*c))/ (2*d^2)) + (b^3*g^5*n*(a*d - b*c)*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(2*d^ 3)) + (3*b^4*g^5*n*(a*d - b*c)*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(2*d^3)) + (3*b^3*g^5*n*(a*d - b*c)*(4*a^3*d^3 - b^3*c^3 + 4*a*b^2*c^2*d - 6*a^2*b *c*d^2))/(2*d^4) + (6*b^6*g^5*n*x^3*(a*d - b*c))/d))/(6*b^3*g^5*(a^2*d^...