Integrand size = 34, antiderivative size = 587 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=\frac {8 B^2 d^3 (c+d x)}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B^2 d^2 (c+d x)^2}{(b c-a d)^4 g^5 (a+b x)^2}+\frac {8 b^2 B^2 d (c+d x)^3}{9 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B^2 (c+d x)^4}{8 (b c-a d)^4 g^5 (a+b x)^4}+\frac {4 B d^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B d^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^4 g^5 (a+b x)^2}+\frac {4 b^2 B d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 (b c-a d)^4 g^5 (a+b x)^4}+\frac {d^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b d^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 (b c-a d)^4 g^5 (a+b x)^2}+\frac {b^2 d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 (b c-a d)^4 g^5 (a+b x)^4} \]
8*B^2*d^3*(d*x+c)/(-a*d+b*c)^4/g^5/(b*x+a)-3*b*B^2*d^2*(d*x+c)^2/(-a*d+b*c )^4/g^5/(b*x+a)^2+8/9*b^2*B^2*d*(d*x+c)^3/(-a*d+b*c)^4/g^5/(b*x+a)^3-1/8*b ^3*B^2*(d*x+c)^4/(-a*d+b*c)^4/g^5/(b*x+a)^4+4*B*d^3*(d*x+c)*(A+B*ln(e*(b*x +a)^2/(d*x+c)^2))/(-a*d+b*c)^4/g^5/(b*x+a)-3*b*B*d^2*(d*x+c)^2*(A+B*ln(e*( b*x+a)^2/(d*x+c)^2))/(-a*d+b*c)^4/g^5/(b*x+a)^2+4/3*b^2*B*d*(d*x+c)^3*(A+B *ln(e*(b*x+a)^2/(d*x+c)^2))/(-a*d+b*c)^4/g^5/(b*x+a)^3-1/4*b^3*B*(d*x+c)^4 *(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(-a*d+b*c)^4/g^5/(b*x+a)^4+d^3*(d*x+c)*(A +B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(-a*d+b*c)^4/g^5/(b*x+a)-3/2*b*d^2*(d*x+c) ^2*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(-a*d+b*c)^4/g^5/(b*x+a)^2+b^2*d*(d*x +c)^3*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(-a*d+b*c)^4/g^5/(b*x+a)^3-1/4*b^3 *(d*x+c)^4*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(-a*d+b*c)^4/g^5/(b*x+a)^4
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.58 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.16 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {18 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2+\frac {B \left (18 A (b c-a d)^4+9 B (b c-a d)^4+24 A d (-b c+a d)^3 (a+b x)+28 B d (-b c+a d)^3 (a+b x)+36 A d^2 (b c-a d)^2 (a+b x)^2+78 B d^2 (b c-a d)^2 (a+b x)^2+72 A d^3 (-b c+a d) (a+b x)^3+300 B d^3 (-b c+a d) (a+b x)^3-72 A d^4 (a+b x)^4 \log (a+b x)-300 B d^4 (a+b x)^4 \log (a+b x)+72 B d^4 (a+b x)^4 \log ^2(a+b x)+18 B (b c-a d)^4 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+24 B d (-b c+a d)^3 (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+36 B d^2 (b c-a d)^2 (a+b x)^2 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+72 B d^3 (-b c+a d) (a+b x)^3 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )-72 B d^4 (a+b x)^4 \log (a+b x) \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+72 A d^4 (a+b x)^4 \log (c+d x)+300 B d^4 (a+b x)^4 \log (c+d x)-144 B d^4 (a+b x)^4 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+72 B d^4 (a+b x)^4 \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right ) \log (c+d x)+72 B d^4 (a+b x)^4 \log ^2(c+d x)-144 B d^4 (a+b x)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )-144 B d^4 (a+b x)^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )-144 B d^4 (a+b x)^4 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^4}}{72 b g^5 (a+b x)^4} \]
-1/72*(18*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2 + (B*(18*A*(b*c - a*d )^4 + 9*B*(b*c - a*d)^4 + 24*A*d*(-(b*c) + a*d)^3*(a + b*x) + 28*B*d*(-(b* c) + a*d)^3*(a + b*x) + 36*A*d^2*(b*c - a*d)^2*(a + b*x)^2 + 78*B*d^2*(b*c - a*d)^2*(a + b*x)^2 + 72*A*d^3*(-(b*c) + a*d)*(a + b*x)^3 + 300*B*d^3*(- (b*c) + a*d)*(a + b*x)^3 - 72*A*d^4*(a + b*x)^4*Log[a + b*x] - 300*B*d^4*( a + b*x)^4*Log[a + b*x] + 72*B*d^4*(a + b*x)^4*Log[a + b*x]^2 + 18*B*(b*c - a*d)^4*Log[(e*(a + b*x)^2)/(c + d*x)^2] + 24*B*d*(-(b*c) + a*d)^3*(a + b *x)*Log[(e*(a + b*x)^2)/(c + d*x)^2] + 36*B*d^2*(b*c - a*d)^2*(a + b*x)^2* Log[(e*(a + b*x)^2)/(c + d*x)^2] + 72*B*d^3*(-(b*c) + a*d)*(a + b*x)^3*Log [(e*(a + b*x)^2)/(c + d*x)^2] - 72*B*d^4*(a + b*x)^4*Log[a + b*x]*Log[(e*( a + b*x)^2)/(c + d*x)^2] + 72*A*d^4*(a + b*x)^4*Log[c + d*x] + 300*B*d^4*( a + b*x)^4*Log[c + d*x] - 144*B*d^4*(a + b*x)^4*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] + 72*B*d^4*(a + b*x)^4*Log[(e*(a + b*x)^2)/(c + d*x)^ 2]*Log[c + d*x] + 72*B*d^4*(a + b*x)^4*Log[c + d*x]^2 - 144*B*d^4*(a + b*x )^4*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] - 144*B*d^4*(a + b*x)^4*Po lyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] - 144*B*d^4*(a + b*x)^4*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]))/(b*c - a*d)^4)/(b*g^5*(a + b*x)^4)
Time = 0.55 (sec) , antiderivative size = 445, normalized size of antiderivative = 0.76, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.088, Rules used = {2950, 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.
| \(\displaystyle \int \frac {\left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{(a g+b g x)^5} \, dx\) |
| \(\Big \downarrow \) 2950 |
| \(\displaystyle \frac {\int \frac {(c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a+b x)^5}d\frac {a+b x}{c+d x}}{g^5 (b c-a d)^4}\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \frac {\int \left (\frac {b^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 (c+d x)^5}{(a+b x)^5}-\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 (c+d x)^4}{(a+b x)^4}+\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 (c+d x)^3}{(a+b x)^3}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2 (c+d x)^2}{(a+b x)^2}\right )d\frac {a+b x}{c+d x}}{g^5 (b c-a d)^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b^3 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{4 (a+b x)^4}-\frac {b^3 B (c+d x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{4 (a+b x)^4}+\frac {b^2 d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{(a+b x)^3}+\frac {4 b^2 B d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 (a+b x)^3}+\frac {d^3 (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{a+b x}+\frac {4 B d^3 (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{a+b x}-\frac {3 b d^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{2 (a+b x)^2}-\frac {3 b B d^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{(a+b x)^2}-\frac {b^3 B^2 (c+d x)^4}{8 (a+b x)^4}+\frac {8 b^2 B^2 d (c+d x)^3}{9 (a+b x)^3}+\frac {8 B^2 d^3 (c+d x)}{a+b x}-\frac {3 b B^2 d^2 (c+d x)^2}{(a+b x)^2}}{g^5 (b c-a d)^4}\) |
((8*B^2*d^3*(c + d*x))/(a + b*x) - (3*b*B^2*d^2*(c + d*x)^2)/(a + b*x)^2 + (8*b^2*B^2*d*(c + d*x)^3)/(9*(a + b*x)^3) - (b^3*B^2*(c + d*x)^4)/(8*(a + b*x)^4) + (4*B*d^3*(c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(a + b*x) - (3*b*B*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])) /(a + b*x)^2 + (4*b^2*B*d*(c + d*x)^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x) ^2]))/(3*(a + b*x)^3) - (b^3*B*(c + d*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(4*(a + b*x)^4) + (d^3*(c + d*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(a + b*x) - (3*b*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x)^2)/ (c + d*x)^2])^2)/(2*(a + b*x)^2) + (b^2*d*(c + d*x)^3*(A + B*Log[(e*(a + b *x)^2)/(c + d*x)^2])^2)/(a + b*x)^3 - (b^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(4*(a + b*x)^4))/((b*c - a*d)^4*g^5)
3.2.36.3.1 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_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_ )]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*c - a*d)^( m + 1)*(g/b)^m Subst[Int[x^m*((A + B*Log[e*x^n])^p/(b - d*x)^(m + 2)), x] , x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] & & EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && IntegersQ[m, p] && E qQ[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
Leaf count of result is larger than twice the leaf count of optimal. \(1485\) vs. \(2(575)=1150\).
Time = 3.50 (sec) , antiderivative size = 1486, normalized size of antiderivative = 2.53
| method | result | size |
| derivativedivides | \(\text {Expression too large to display}\) | \(1486\) |
| default | \(\text {Expression too large to display}\) | \(1486\) |
| norman | \(\text {Expression too large to display}\) | \(1816\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2110\) |
| risch | \(\text {Expression too large to display}\) | \(2235\) |
| parts | \(\text {Expression too large to display}\) | \(2235\) |
-1/d*(d^5/g^5*A^2*(-1/(a*d-b*c)^4/(a*d/(d*x+c)-b*c/(d*x+c)+b)+1/4*b^3/(a*d -b*c)^4/(a*d/(d*x+c)-b*c/(d*x+c)+b)^4-b^2/(a*d-b*c)^4/(a*d/(d*x+c)-b*c/(d* x+c)+b)^3+3/2*b/(a*d-b*c)^4/(a*d/(d*x+c)-b*c/(d*x+c)+b)^2)+(415/72*B^2/b*d ^5/g/(d*x+c)^4-25/12*b^3*B^2*d^5/g/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^ 2-4*a*b^3*c^3*d+b^4*c^4)*ln(e*(a*d/(d*x+c)-b*c/(d*x+c)+b)^2/d^2)+25/6*B^2* b^2*d^5/g/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(d*x+c)+163/12*B^2 *b*d^5/g/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)^2+271/18*B^2*d^5/g/(a*d-b*c)/ (d*x+c)^3-4*B^2*d^5/g/(a*d-b*c)/(d*x+c)^3*ln(e*(a*d/(d*x+c)-b*c/(d*x+c)+b) ^2/d^2)-1/4*B^2*b^3*d^5/g/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3 *c^3*d+b^4*c^4)*ln(e*(a*d/(d*x+c)-b*c/(d*x+c)+b)^2/d^2)^2-B^2*d^5/g/(a*d-b *c)/(d*x+c)^3*ln(e*(a*d/(d*x+c)-b*c/(d*x+c)+b)^2/d^2)^2-9*B^2*d^5*b/g/(a^2 *d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)^2*ln(e*(a*d/(d*x+c)-b*c/(d*x+c)+b)^2/d^2)- 22/3*B^2*d^5*b^2/g/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(d*x+c)*l n(e*(a*d/(d*x+c)-b*c/(d*x+c)+b)^2/d^2)-3/2*B^2*b*d^5/g/(a^2*d^2-2*a*b*c*d+ b^2*c^2)/(d*x+c)^2*ln(e*(a*d/(d*x+c)-b*c/(d*x+c)+b)^2/d^2)^2-B^2*b^2*d^5/g /(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(d*x+c)*ln(e*(a*d/(d*x+c)-b *c/(d*x+c)+b)^2/d^2)^2)/(a*d/(d*x+c)-b*c/(d*x+c)+b)^4/g^4+(A*B*b^2*d^5/g/( a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(d*x+c)+25/12*A*B/b*d^5/g/(d* x+c)^4-1/2*b^3*A*B*d^5/g/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3* c^3*d+b^4*c^4)*ln(e*(a*d/(d*x+c)-b*c/(d*x+c)+b)^2/d^2)+7/2*A*B*b*d^5/g/...
Time = 0.30 (sec) , antiderivative size = 1084, normalized size of antiderivative = 1.85 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {9 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{4} c^{4} - 8 \, {\left (9 \, A^{2} + 12 \, A B + 8 \, B^{2}\right )} a b^{3} c^{3} d + 108 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a^{2} b^{2} c^{2} d^{2} - 72 \, {\left (A^{2} + 4 \, A B + 8 \, B^{2}\right )} a^{3} b c d^{3} + {\left (18 \, A^{2} + 150 \, A B + 415 \, B^{2}\right )} a^{4} d^{4} - 12 \, {\left ({\left (6 \, A B + 25 \, B^{2}\right )} b^{4} c d^{3} - {\left (6 \, A B + 25 \, B^{2}\right )} a b^{3} d^{4}\right )} x^{3} + 6 \, {\left ({\left (6 \, A B + 13 \, B^{2}\right )} b^{4} c^{2} d^{2} - 16 \, {\left (3 \, A B + 11 \, B^{2}\right )} a b^{3} c d^{3} + {\left (42 \, A B + 163 \, B^{2}\right )} a^{2} b^{2} d^{4}\right )} x^{2} - 18 \, {\left (B^{2} b^{4} d^{4} x^{4} + 4 \, B^{2} a b^{3} d^{4} x^{3} + 6 \, B^{2} a^{2} b^{2} d^{4} x^{2} + 4 \, B^{2} a^{3} b d^{4} x - B^{2} b^{4} c^{4} + 4 \, B^{2} a b^{3} c^{3} d - 6 \, B^{2} a^{2} b^{2} c^{2} d^{2} + 4 \, B^{2} a^{3} b c d^{3}\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} - 4 \, {\left ({\left (6 \, A B + 7 \, B^{2}\right )} b^{4} c^{3} d - 12 \, {\left (3 \, A B + 5 \, B^{2}\right )} a b^{3} c^{2} d^{2} + 108 \, {\left (A B + 3 \, B^{2}\right )} a^{2} b^{2} c d^{3} - {\left (78 \, A B + 271 \, B^{2}\right )} a^{3} b d^{4}\right )} x - 6 \, {\left ({\left (6 \, A B + 25 \, B^{2}\right )} b^{4} d^{4} x^{4} - 3 \, {\left (2 \, A B + B^{2}\right )} b^{4} c^{4} + 8 \, {\left (3 \, A B + 2 \, B^{2}\right )} a b^{3} c^{3} d - 36 \, {\left (A B + B^{2}\right )} a^{2} b^{2} c^{2} d^{2} + 24 \, {\left (A B + 2 \, B^{2}\right )} a^{3} b c d^{3} + 4 \, {\left (3 \, B^{2} b^{4} c d^{3} + 2 \, {\left (3 \, A B + 11 \, B^{2}\right )} a b^{3} d^{4}\right )} x^{3} - 6 \, {\left (B^{2} b^{4} c^{2} d^{2} - 8 \, B^{2} a b^{3} c d^{3} - 6 \, {\left (A B + 3 \, B^{2}\right )} a^{2} b^{2} d^{4}\right )} x^{2} + 4 \, {\left (B^{2} b^{4} c^{3} d - 6 \, B^{2} a b^{3} c^{2} d^{2} + 18 \, B^{2} a^{2} b^{2} c d^{3} + 6 \, {\left (A B + 2 \, B^{2}\right )} a^{3} b d^{4}\right )} x\right )} \log \left (\frac {b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{72 \, {\left ({\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} g^{5} x + {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} g^{5}\right )}} \]
-1/72*(9*(2*A^2 + 2*A*B + B^2)*b^4*c^4 - 8*(9*A^2 + 12*A*B + 8*B^2)*a*b^3* c^3*d + 108*(A^2 + 2*A*B + 2*B^2)*a^2*b^2*c^2*d^2 - 72*(A^2 + 4*A*B + 8*B^ 2)*a^3*b*c*d^3 + (18*A^2 + 150*A*B + 415*B^2)*a^4*d^4 - 12*((6*A*B + 25*B^ 2)*b^4*c*d^3 - (6*A*B + 25*B^2)*a*b^3*d^4)*x^3 + 6*((6*A*B + 13*B^2)*b^4*c ^2*d^2 - 16*(3*A*B + 11*B^2)*a*b^3*c*d^3 + (42*A*B + 163*B^2)*a^2*b^2*d^4) *x^2 - 18*(B^2*b^4*d^4*x^4 + 4*B^2*a*b^3*d^4*x^3 + 6*B^2*a^2*b^2*d^4*x^2 + 4*B^2*a^3*b*d^4*x - B^2*b^4*c^4 + 4*B^2*a*b^3*c^3*d - 6*B^2*a^2*b^2*c^2*d ^2 + 4*B^2*a^3*b*c*d^3)*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c *d*x + c^2))^2 - 4*((6*A*B + 7*B^2)*b^4*c^3*d - 12*(3*A*B + 5*B^2)*a*b^3*c ^2*d^2 + 108*(A*B + 3*B^2)*a^2*b^2*c*d^3 - (78*A*B + 271*B^2)*a^3*b*d^4)*x - 6*((6*A*B + 25*B^2)*b^4*d^4*x^4 - 3*(2*A*B + B^2)*b^4*c^4 + 8*(3*A*B + 2*B^2)*a*b^3*c^3*d - 36*(A*B + B^2)*a^2*b^2*c^2*d^2 + 24*(A*B + 2*B^2)*a^3 *b*c*d^3 + 4*(3*B^2*b^4*c*d^3 + 2*(3*A*B + 11*B^2)*a*b^3*d^4)*x^3 - 6*(B^2 *b^4*c^2*d^2 - 8*B^2*a*b^3*c*d^3 - 6*(A*B + 3*B^2)*a^2*b^2*d^4)*x^2 + 4*(B ^2*b^4*c^3*d - 6*B^2*a*b^3*c^2*d^2 + 18*B^2*a^2*b^2*c*d^3 + 6*(A*B + 2*B^2 )*a^3*b*d^4)*x)*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c ^2)))/((b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^ 4*b^5*d^4)*g^5*x^4 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*g^5*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a^5*b^4*c*d^3 + a^6*b^3*d^4)*g^5*x^2 + 4*(a^3*b...
Timed out. \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2279 vs. \(2 (575) = 1150\).
Time = 0.37 (sec) , antiderivative size = 2279, normalized size of antiderivative = 3.88 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
1/72*(6*((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 + 2 5*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))*log (b^2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) - (9*b^4*c^4 - 64*a*b^3*c^3*d + 216*a^ 2*b^2*c^2*d^2 - 576*a^3*b*c*d^3 + 415*a^4*d^4 - 300*(b^4*c*d^3 - a*b^3*d^4 )*x^3 + 6*(13*b^4*c^2*d^2 - 176*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x^2 + 72*(b ^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4 )*log(b*x + a)^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(d*x + c)^2 - 4*(7*b^4*c^3*d - 60*a*b^3*c^2*d ^2 + 324*a^2*b^2*c*d^3 - 271*a^3*b*d^4)*x - 300*(b^4*d^4*x^4 + 4*a*b^3*d^4 *x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a) + 12*(25* b^4*d^4*x^4 + 100*a*b^3*d^4*x^3 + 150*a^2*b^2*d^4*x^2 + 100*a^3*b*d^4*x...
Time = 1.07 (sec) , antiderivative size = 874, normalized size of antiderivative = 1.49 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=\frac {1}{4} \, {\left (\frac {B^{2} d^{4}}{b^{5} c^{4} g^{5} - 4 \, a b^{4} c^{3} d g^{5} + 6 \, a^{2} b^{3} c^{2} d^{2} g^{5} - 4 \, a^{3} b^{2} c d^{3} g^{5} + a^{4} b d^{4} g^{5}} - \frac {B^{2}}{{\left (b g x + a g\right )}^{4} b g}\right )} \log \left (\frac {b^{2} e}{\frac {b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac {2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {2 \, b c d g}{b g x + a g} - \frac {2 \, a d^{2} g}{b g x + a g} + d^{2}}\right )^{2} + \frac {1}{12} \, {\left (\frac {12 \, B^{2} d^{3}}{{\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )} {\left (b g x + a g\right )} b g} - \frac {6 \, B^{2} d^{2}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (b g x + a g\right )}^{2} b g^{2}} + \frac {4 \, B^{2} d}{{\left (b g x + a g\right )}^{3} {\left (b c - a d\right )} b g^{2}} - \frac {3 \, {\left (2 \, A B b^{3} g^{3} + B^{2} b^{3} g^{3}\right )}}{{\left (b g x + a g\right )}^{4} b^{4} g^{4}}\right )} \log \left (\frac {b^{2} e}{\frac {b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac {2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {2 \, b c d g}{b g x + a g} - \frac {2 \, a d^{2} g}{b g x + a g} + d^{2}}\right ) - \frac {{\left (6 \, A B d^{4} + 25 \, B^{2} d^{4}\right )} \log \left (-\frac {b c g}{b g x + a g} + \frac {a d g}{b g x + a g} - d\right )}{6 \, {\left (b^{5} c^{4} g^{5} - 4 \, a b^{4} c^{3} d g^{5} + 6 \, a^{2} b^{3} c^{2} d^{2} g^{5} - 4 \, a^{3} b^{2} c d^{3} g^{5} + a^{4} b d^{4} g^{5}\right )}} + \frac {6 \, A B d^{3} + 25 \, B^{2} d^{3}}{6 \, {\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )} {\left (b g x + a g\right )} b g} - \frac {6 \, A B b d^{2} + 13 \, B^{2} b d^{2}}{12 \, {\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (b g x + a g\right )}^{2} b^{2} g^{2}} + \frac {6 \, A B b^{2} d g + 7 \, B^{2} b^{2} d g}{18 \, {\left (b g x + a g\right )}^{3} {\left (b c - a d\right )} b^{3} g^{3}} - \frac {2 \, A^{2} b^{3} g^{3} + 2 \, A B b^{3} g^{3} + B^{2} b^{3} g^{3}}{8 \, {\left (b g x + a g\right )}^{4} b^{4} g^{4}} \]
1/4*(B^2*d^4/(b^5*c^4*g^5 - 4*a*b^4*c^3*d*g^5 + 6*a^2*b^3*c^2*d^2*g^5 - 4* a^3*b^2*c*d^3*g^5 + a^4*b*d^4*g^5) - B^2/((b*g*x + a*g)^4*b*g))*log(b^2*e/ (b^2*c^2*g^2/(b*g*x + a*g)^2 - 2*a*b*c*d*g^2/(b*g*x + a*g)^2 + a^2*d^2*g^2 /(b*g*x + a*g)^2 + 2*b*c*d*g/(b*g*x + a*g) - 2*a*d^2*g/(b*g*x + a*g) + d^2 ))^2 + 1/12*(12*B^2*d^3/((b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2* g^3 - a^3*d^3*g^3)*(b*g*x + a*g)*b*g) - 6*B^2*d^2/((b^2*c^2*g - 2*a*b*c*d* g + a^2*d^2*g)*(b*g*x + a*g)^2*b*g^2) + 4*B^2*d/((b*g*x + a*g)^3*(b*c - a* d)*b*g^2) - 3*(2*A*B*b^3*g^3 + B^2*b^3*g^3)/((b*g*x + a*g)^4*b^4*g^4))*log (b^2*e/(b^2*c^2*g^2/(b*g*x + a*g)^2 - 2*a*b*c*d*g^2/(b*g*x + a*g)^2 + a^2* d^2*g^2/(b*g*x + a*g)^2 + 2*b*c*d*g/(b*g*x + a*g) - 2*a*d^2*g/(b*g*x + a*g ) + d^2)) - 1/6*(6*A*B*d^4 + 25*B^2*d^4)*log(-b*c*g/(b*g*x + a*g) + a*d*g/ (b*g*x + a*g) - d)/(b^5*c^4*g^5 - 4*a*b^4*c^3*d*g^5 + 6*a^2*b^3*c^2*d^2*g^ 5 - 4*a^3*b^2*c*d^3*g^5 + a^4*b*d^4*g^5) + 1/6*(6*A*B*d^3 + 25*B^2*d^3)/(( b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^3)*(b*g*x + a*g)*b*g) - 1/12*(6*A*B*b*d^2 + 13*B^2*b*d^2)/((b^2*c^2*g - 2*a*b*c*d*g + a^2*d^2*g)*(b*g*x + a*g)^2*b^2*g^2) + 1/18*(6*A*B*b^2*d*g + 7*B^2*b^2*d* g)/((b*g*x + a*g)^3*(b*c - a*d)*b^3*g^3) - 1/8*(2*A^2*b^3*g^3 + 2*A*B*b^3* g^3 + B^2*b^3*g^3)/((b*g*x + a*g)^4*b^4*g^4)
Time = 7.76 (sec) , antiderivative size = 1883, normalized size of antiderivative = 3.21 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
(B*d^4*atan((B*d^4*(6*A + 25*B)*(6*b^5*c^4*g^5 - 6*a^4*b*d^4*g^5 - 12*a*b^ 4*c^3*d*g^5 + 12*a^3*b^2*c*d^3*g^5)*1i)/(6*b*g^5*(a*d - b*c)^4*(25*B^2*d^4 + 6*A*B*d^4)) + (B*d^5*x*(6*A + 25*B)*(b^4*c^3*g^5 - a^3*b*d^3*g^5 - 3*a* b^3*c^2*d*g^5 + 3*a^2*b^2*c*d^2*g^5)*2i)/(g^5*(a*d - b*c)^4*(25*B^2*d^4 + 6*A*B*d^4)))*(6*A + 25*B)*1i)/(3*b*g^5*(a*d - b*c)^4) - log((e*(a + b*x)^2 )/(c + d*x)^2)^2*(B^2/(4*b^2*g^5*(4*a^3*x + a^4/b + b^3*x^4 + 6*a^2*b*x^2 + 4*a*b^2*x^3)) - (B^2*d^4)/(4*b*g^5*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^ 2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))) - (log((e*(a + b*x)^2)/(c + d*x)^2)*( (A*B)/(2*b^2*d*g^5) + (B^2*d^4*(a*(a*((4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(6 *b*d^3) + (a*(a*d - b*c))/(2*b*d^2)) + (6*a^3*d^3 - b^3*c^3 + 5*a*b^2*c^2* d - 10*a^2*b*c*d^2)/(6*b*d^4)) + (4*a^4*d^4 + b^4*c^4 + 10*a^2*b^2*c^2*d^2 - 5*a*b^3*c^3*d - 10*a^3*b*c*d^3)/(2*b*d^5)))/(2*b*g^5*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (B^2*d^4*x^2*(b*( b*((4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(2*b*d^2) ) + (4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(3*d^3) + (a*(a*d - b*c))/d^2) - a*( (b^2*c - a*b*d)/(2*d^2) - (b*(a*d - b*c))/d^2) + (b^3*c^2 + 4*a^2*b*d^2 - 5*a*b^2*c*d)/(2*d^3)))/(2*b*g^5*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4 *a*b^3*c^3*d - 4*a^3*b*c*d^3)) - (B^2*d^4*x^3*(b*((b^2*c - a*b*d)/(2*d^2) - (b*(a*d - b*c))/d^2) + (b^3*c - a*b^2*d)/(2*d^2)))/(2*b*g^5*(a^4*d^4 + b ^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (B^2*d^4...