Integrand size = 32, antiderivative size = 575 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\frac {2 B^2 d^3 (c+d x)}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B^2 d^2 (c+d x)^2}{4 (b c-a d)^4 g^5 (a+b x)^2}+\frac {2 b^2 B^2 d (c+d x)^3}{9 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B^2 (c+d x)^4}{32 (b c-a d)^4 g^5 (a+b x)^4}+\frac {2 B d^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B d^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^4 g^5 (a+b x)^2}+\frac {2 b^2 B d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 (b c-a d)^4 g^5 (a+b x)^4}+\frac {d^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b d^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^4 g^5 (a+b x)^2}+\frac {b^2 d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 (b c-a d)^4 g^5 (a+b x)^4} \]
2*B^2*d^3*(d*x+c)/(-a*d+b*c)^4/g^5/(b*x+a)-3/4*b*B^2*d^2*(d*x+c)^2/(-a*d+b *c)^4/g^5/(b*x+a)^2+2/9*b^2*B^2*d*(d*x+c)^3/(-a*d+b*c)^4/g^5/(b*x+a)^3-1/3 2*b^3*B^2*(d*x+c)^4/(-a*d+b*c)^4/g^5/(b*x+a)^4+2*B*d^3*(d*x+c)*(A+B*ln(e*( b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^5/(b*x+a)-3/2*b*B*d^2*(d*x+c)^2*(A+B*ln(e* (b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^5/(b*x+a)^2+2/3*b^2*B*d*(d*x+c)^3*(A+B*ln (e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^5/(b*x+a)^3-1/8*b^3*B*(d*x+c)^4*(A+B*l n(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^5/(b*x+a)^4+d^3*(d*x+c)*(A+B*ln(e*(b* x+a)/(d*x+c)))^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)/(d*x+c)))^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)/(d*x+c)))^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)/(d*x+c)))^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 = 0.59 (sec) , antiderivative size = 666, normalized size of antiderivative = 1.16 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {72 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {B \left (36 A (b c-a d)^4+9 B (b c-a d)^4+48 A d (-b c+a d)^3 (a+b x)+28 B d (-b c+a d)^3 (a+b x)+72 A d^2 (b c-a d)^2 (a+b x)^2+78 B d^2 (b c-a d)^2 (a+b x)^2+144 A d^3 (-b c+a d) (a+b x)^3+300 B d^3 (-b c+a d) (a+b x)^3-144 A d^4 (a+b x)^4 \log (a+b x)-300 B d^4 (a+b x)^4 \log (a+b x)+72 B d^4 (a+b x)^4 \log ^2(a+b x)+36 B (b c-a d)^4 \log \left (\frac {e (a+b x)}{c+d x}\right )+48 B d (-b c+a d)^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+72 B d^2 (b c-a d)^2 (a+b x)^2 \log \left (\frac {e (a+b x)}{c+d x}\right )+144 B d^3 (-b c+a d) (a+b x)^3 \log \left (\frac {e (a+b x)}{c+d x}\right )-144 B d^4 (a+b x)^4 \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+144 A d^4 (a+b x)^4 \log (c+d x)+300 B d^4 (a+b x)^4 \log (c+d x)-144 B d^4 (a+b x)^4 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+144 B d^4 (a+b x)^4 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)+72 B d^4 (a+b x)^4 \log ^2(c+d x)-144 B d^4 (a+b x)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )-144 B d^4 (a+b x)^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )-144 B d^4 (a+b x)^4 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^4}}{288 b g^5 (a+b x)^4} \]
-1/288*(72*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + (B*(36*A*(b*c - a*d)^4 + 9*B*(b*c - a*d)^4 + 48*A*d*(-(b*c) + a*d)^3*(a + b*x) + 28*B*d*(-(b*c) + a*d)^3*(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*(a + b*x)^2 + 144*A*d^3*(-(b*c) + a*d)*(a + b*x)^3 + 300*B*d^3*(-(b *c) + a*d)*(a + b*x)^3 - 144*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 + 36*B*(b*c - a*d)^4*Log[(e*(a + b*x))/(c + d*x)] + 48*B*d*(-(b*c) + a*d)^3*(a + b*x)*L og[(e*(a + b*x))/(c + d*x)] + 72*B*d^2*(b*c - a*d)^2*(a + b*x)^2*Log[(e*(a + b*x))/(c + d*x)] + 144*B*d^3*(-(b*c) + a*d)*(a + b*x)^3*Log[(e*(a + b*x ))/(c + d*x)] - 144*B*d^4*(a + b*x)^4*Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)] + 144*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] + 144*B*d^4*(a + b*x)^4*Log[(e*(a + b*x))/(c + d*x)]*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*PolyLog[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 = 0.56 (sec) , antiderivative size = 433, normalized size of antiderivative = 0.75, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, 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.
| \(\displaystyle \int \frac {\left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(a g+b g x)^5} \, dx\) |
| \(\Big \downarrow \) 2950 |
| \(\displaystyle \frac {\int \frac {(c+d x)^5 \left (b-\frac {d (a+b x)}{c+d x}\right )^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^5}d\frac {a+b x}{c+d x}}{g^5 (b c-a d)^4}\) |
| \(\Big \downarrow \) 2795 |
| \(\displaystyle \frac {\int \left (\frac {b^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (c+d x)^5}{(a+b x)^5}-\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (c+d x)^4}{(a+b x)^4}+\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (c+d x)^3}{(a+b x)^3}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 (c+d x)^2}{(a+b x)^2}\right )d\frac {a+b x}{c+d x}}{g^5 (b c-a d)^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {b^3 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 (a+b x)^4}-\frac {b^3 B (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{8 (a+b x)^4}+\frac {b^2 d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{(a+b x)^3}+\frac {2 b^2 B d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 (a+b x)^3}+\frac {d^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{a+b x}+\frac {2 B d^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac {3 b d^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 (a+b x)^2}-\frac {3 b B d^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 (a+b x)^2}-\frac {b^3 B^2 (c+d x)^4}{32 (a+b x)^4}+\frac {2 b^2 B^2 d (c+d x)^3}{9 (a+b x)^3}+\frac {2 B^2 d^3 (c+d x)}{a+b x}-\frac {3 b B^2 d^2 (c+d x)^2}{4 (a+b x)^2}}{g^5 (b c-a d)^4}\) |
((2*B^2*d^3*(c + d*x))/(a + b*x) - (3*b*B^2*d^2*(c + d*x)^2)/(4*(a + b*x)^ 2) + (2*b^2*B^2*d*(c + d*x)^3)/(9*(a + b*x)^3) - (b^3*B^2*(c + d*x)^4)/(32 *(a + b*x)^4) + (2*B*d^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/( a + b*x) - (3*b*B*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2 *(a + b*x)^2) + (2*b^2*B*d*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)] ))/(3*(a + b*x)^3) - (b^3*B*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x) ]))/(8*(a + b*x)^4) + (d^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^ 2)/(a + b*x) - (3*b*d^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 )/(2*(a + b*x)^2) + (b^2*d*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)] )^2)/(a + b*x)^3 - (b^3*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 )/(4*(a + b*x)^4))/((b*c - a*d)^4*g^5)
3.2.5.3.1 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. \(1178\) vs. \(2(559)=1118\).
Time = 2.32 (sec) , antiderivative size = 1179, normalized size of antiderivative = 2.05
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1179\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1393\) |
| default | \(\text {Expression too large to display}\) | \(1393\) |
| norman | \(\text {Expression too large to display}\) | \(1796\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2035\) |
| risch | \(\text {Expression too large to display}\) | \(3080\) |
-1/4*A^2/g^5/(b*x+a)^4/b-B^2/g^5/d^2*(a*d-b*c)*e*(d^5/(a*d-b*c)^5*(-1/(b*e /d+(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-2/(b*e/d+(a*d- b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-2/(b*e/d+(a*d-b*c)*e/d/( d*x+c)))-3*d^4/(a*d-b*c)^5*b*e*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b* e/d+(a*d-b*c)*e/d/(d*x+c))^2-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+ (a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2)+3*d^3/(a*d-b*c )^5*b^2*e^2*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/( d*x+c))^2-2/9/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+ c))-2/27/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3)-d^2/(a*d-b*c)^5*b^3*e^3*(-1/4/(b *e/d+(a*d-b*c)*e/d/(d*x+c))^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-1/8/(b*e/d +(a*d-b*c)*e/d/(d*x+c))^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/32/(b*e/d+(a*d -b*c)*e/d/(d*x+c))^4))-2*B*A/g^5/d^2*(a*d-b*c)*e*(d^5/(a*d-b*c)^5*(-1/(b*e /d+(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/(b*e/d+(a*d-b* c)*e/d/(d*x+c)))-3*d^4/(a*d-b*c)^5*b*e*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c)) ^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2)+3* d^3/(a*d-b*c)^5*b^2*e^2*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a* d-b*c)*e/d/(d*x+c))-1/9/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3)-d^2/(a*d-b*c)^5*b ^3*e^3*(-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c ))-1/16/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4))
Time = 0.28 (sec) , antiderivative size = 1035, normalized size of antiderivative = 1.80 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {9 \, {\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} b^{4} c^{4} - 32 \, {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a b^{3} c^{3} d + 216 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a^{2} b^{2} c^{2} d^{2} - 288 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a^{3} b c d^{3} + {\left (72 \, A^{2} + 300 \, A B + 415 \, B^{2}\right )} a^{4} d^{4} - 12 \, {\left ({\left (12 \, A B + 25 \, B^{2}\right )} b^{4} c d^{3} - {\left (12 \, A B + 25 \, B^{2}\right )} a b^{3} d^{4}\right )} x^{3} + 6 \, {\left ({\left (12 \, A B + 13 \, B^{2}\right )} b^{4} c^{2} d^{2} - 16 \, {\left (6 \, A B + 11 \, B^{2}\right )} a b^{3} c d^{3} + {\left (84 \, A B + 163 \, B^{2}\right )} a^{2} b^{2} d^{4}\right )} x^{2} - 72 \, {\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 e x + a e}{d x + c}\right )^{2} - 4 \, {\left ({\left (12 \, A B + 7 \, B^{2}\right )} b^{4} c^{3} d - 12 \, {\left (6 \, A B + 5 \, B^{2}\right )} a b^{3} c^{2} d^{2} + 108 \, {\left (2 \, A B + 3 \, B^{2}\right )} a^{2} b^{2} c d^{3} - {\left (156 \, A B + 271 \, B^{2}\right )} a^{3} b d^{4}\right )} x - 12 \, {\left ({\left (12 \, A B + 25 \, B^{2}\right )} b^{4} d^{4} x^{4} - 3 \, {\left (4 \, A B + B^{2}\right )} b^{4} c^{4} + 16 \, {\left (3 \, A B + B^{2}\right )} a b^{3} c^{3} d - 36 \, {\left (2 \, A B + B^{2}\right )} a^{2} b^{2} c^{2} d^{2} + 48 \, {\left (A B + B^{2}\right )} a^{3} b c d^{3} + 4 \, {\left (3 \, B^{2} b^{4} c d^{3} + 2 \, {\left (6 \, 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 (2 \, 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} + 12 \, {\left (A B + B^{2}\right )} a^{3} b d^{4}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{288 \, {\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 )}} \]
-1/288*(9*(8*A^2 + 4*A*B + B^2)*b^4*c^4 - 32*(9*A^2 + 6*A*B + 2*B^2)*a*b^3 *c^3*d + 216*(2*A^2 + 2*A*B + B^2)*a^2*b^2*c^2*d^2 - 288*(A^2 + 2*A*B + 2* B^2)*a^3*b*c*d^3 + (72*A^2 + 300*A*B + 415*B^2)*a^4*d^4 - 12*((12*A*B + 25 *B^2)*b^4*c*d^3 - (12*A*B + 25*B^2)*a*b^3*d^4)*x^3 + 6*((12*A*B + 13*B^2)* b^4*c^2*d^2 - 16*(6*A*B + 11*B^2)*a*b^3*c*d^3 + (84*A*B + 163*B^2)*a^2*b^2 *d^4)*x^2 - 72*(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*e*x + a*e)/(d*x + c))^2 - 4*((12*A*B + 7*B^2)*b^4*c^3*d - 12*(6*A*B + 5*B^2)*a*b^3*c^2*d^2 + 108*(2*A*B + 3*B^2) *a^2*b^2*c*d^3 - (156*A*B + 271*B^2)*a^3*b*d^4)*x - 12*((12*A*B + 25*B^2)* b^4*d^4*x^4 - 3*(4*A*B + B^2)*b^4*c^4 + 16*(3*A*B + B^2)*a*b^3*c^3*d - 36* (2*A*B + B^2)*a^2*b^2*c^2*d^2 + 48*(A*B + B^2)*a^3*b*c*d^3 + 4*(3*B^2*b^4* c*d^3 + 2*(6*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*(2*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 + 12*(A*B + B^2)*a^3*b*d^4)*x)*log((b* e*x + a*e)/(d*x + c)))/((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^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*...
Timed out. \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2123 vs. \(2 (559) = 1118\).
Time = 0.36 (sec) , antiderivative size = 2123, normalized size of antiderivative = 3.69 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
1/288*(12*((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))*l og(b*e*x/(d*x + c) + a*e/(d*x + c)) - (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 + 25*a^4*d^4 - 12*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*...
Time = 0.56 (sec) , antiderivative size = 1014, normalized size of antiderivative = 1.76 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
-1/288*(72*(B^2*b^3*e^5 - 4*(b*e*x + a*e)*B^2*b^2*d*e^4/(d*x + c) + 6*(b*e *x + a*e)^2*B^2*b*d^2*e^3/(d*x + c)^2 - 4*(b*e*x + a*e)^3*B^2*d^3*e^2/(d*x + c)^3)*log((b*e*x + a*e)/(d*x + c))^2/((b*e*x + a*e)^4*b^3*c^3*g^5/(d*x + c)^4 - 3*(b*e*x + a*e)^4*a*b^2*c^2*d*g^5/(d*x + c)^4 + 3*(b*e*x + a*e)^4 *a^2*b*c*d^2*g^5/(d*x + c)^4 - (b*e*x + a*e)^4*a^3*d^3*g^5/(d*x + c)^4) + 12*(12*A*B*b^3*e^5 + 3*B^2*b^3*e^5 - 48*(b*e*x + a*e)*A*B*b^2*d*e^4/(d*x + c) - 16*(b*e*x + a*e)*B^2*b^2*d*e^4/(d*x + c) + 72*(b*e*x + a*e)^2*A*B*b* d^2*e^3/(d*x + c)^2 + 36*(b*e*x + a*e)^2*B^2*b*d^2*e^3/(d*x + c)^2 - 48*(b *e*x + a*e)^3*A*B*d^3*e^2/(d*x + c)^3 - 48*(b*e*x + a*e)^3*B^2*d^3*e^2/(d* x + c)^3)*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^4*b^3*c^3*g^5/(d*x + c)^4 - 3*(b*e*x + a*e)^4*a*b^2*c^2*d*g^5/(d*x + c)^4 + 3*(b*e*x + a*e)^4* a^2*b*c*d^2*g^5/(d*x + c)^4 - (b*e*x + a*e)^4*a^3*d^3*g^5/(d*x + c)^4) + ( 72*A^2*b^3*e^5 + 36*A*B*b^3*e^5 + 9*B^2*b^3*e^5 - 288*(b*e*x + a*e)*A^2*b^ 2*d*e^4/(d*x + c) - 192*(b*e*x + a*e)*A*B*b^2*d*e^4/(d*x + c) - 64*(b*e*x + a*e)*B^2*b^2*d*e^4/(d*x + c) + 432*(b*e*x + a*e)^2*A^2*b*d^2*e^3/(d*x + c)^2 + 432*(b*e*x + a*e)^2*A*B*b*d^2*e^3/(d*x + c)^2 + 216*(b*e*x + a*e)^2 *B^2*b*d^2*e^3/(d*x + c)^2 - 288*(b*e*x + a*e)^3*A^2*d^3*e^2/(d*x + c)^3 - 576*(b*e*x + a*e)^3*A*B*d^3*e^2/(d*x + c)^3 - 576*(b*e*x + a*e)^3*B^2*d^3 *e^2/(d*x + c)^3)/((b*e*x + a*e)^4*b^3*c^3*g^5/(d*x + c)^4 - 3*(b*e*x + a* e)^4*a*b^2*c^2*d*g^5/(d*x + c)^4 + 3*(b*e*x + a*e)^4*a^2*b*c*d^2*g^5/(d...
Time = 7.61 (sec) , antiderivative size = 1881, normalized size of antiderivative = 3.27 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
(B*d^4*atan((B*d^4*(12*A + 25*B)*(24*b^5*c^4*g^5 - 24*a^4*b*d^4*g^5 - 48*a *b^4*c^3*d*g^5 + 48*a^3*b^2*c*d^3*g^5)*1i)/(24*b*g^5*(a*d - b*c)^4*(25*B^2 *d^4 + 12*A*B*d^4)) + (B*d^5*x*(12*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 + 12*A*B*d^4)))*(12*A + 25*B)*1i)/(12*b*g^5*(a*d - b*c)^4) - log((e*(a + b*x))/(c + d*x))^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))/(c + d*x))* ((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)/( 12*b*d^3) + (a*(a*d - b*c))/(4*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)/(12*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)/(4*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)/(12*b*d^3) + (a*(a*d - b*c))/(4*b* d^2)) + (4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(6*d^3) + (a*(a*d - b*c))/(2*d^2 )) - a*((b^2*c - a*b*d)/(4*d^2) - (b*(a*d - b*c))/(2*d^2)) + (b^3*c^2 + 4* a^2*b*d^2 - 5*a*b^2*c*d)/(4*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)/(4*d^2) - (b*(a*d - b*c))/(2*d^2)) + (b^3*c - a*b^2*d)/(4*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*...