Integrand size = 32, antiderivative size = 498 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^5} \, dx=\frac {2 B^2 d^3 (c+d x)}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B^2 d^2 (c+d x)^2}{4 (b c-a d)^4 g^5 (a+b x)^2}+\frac {2 b^2 B^2 d (c+d x)^3}{9 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B^2 (c+d x)^4}{32 (b c-a d)^4 g^5 (a+b x)^4}-\frac {B^2 d^4 \log ^2\left (\frac {c+d x}{a+b x}\right )}{4 b (b c-a d)^4 g^5}-\frac {2 B d^3 (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^4 g^5 (a+b x)}+\frac {3 b B d^2 (c+d x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 (b c-a d)^4 g^5 (a+b x)^2}-\frac {2 b^2 B d (c+d x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 (b c-a d)^4 g^5 (a+b x)^3}+\frac {b^3 B (c+d x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{8 (b c-a d)^4 g^5 (a+b x)^4}+\frac {B d^4 \log \left (\frac {c+d x}{a+b x}\right ) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4} \]
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)^4-1/4*B^2*d^4*ln((d*x+c)/(b*x +a))^2/b/(-a*d+b*c)^4/g^5-2*B*d^3*(d*x+c)*(A+B*ln(e*(d*x+c)/(b*x+a)))/(-a* d+b*c)^4/g^5/(b*x+a)+3/2*b*B*d^2*(d*x+c)^2*(A+B*ln(e*(d*x+c)/(b*x+a)))/(-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*(d*x+c)/(b*x+a)))/ (-a*d+b*c)^4/g^5/(b*x+a)^3+1/8*b^3*B*(d*x+c)^4*(A+B*ln(e*(d*x+c)/(b*x+a))) /(-a*d+b*c)^4/g^5/(b*x+a)^4+1/2*B*d^4*ln((d*x+c)/(b*x+a))*(A+B*ln(e*(d*x+c )/(b*x+a)))/b/(-a*d+b*c)^4/g^5-1/4*(A+B*ln(e*(d*x+c)/(b*x+a)))^2/b/g^5/(b* x+a)^4
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.50 (sec) , antiderivative size = 666, normalized size of antiderivative = 1.34 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^5} \, dx=\frac {-72 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2+\frac {B \left (36 A (b c-a d)^4-9 B (b c-a d)^4+28 B d (b c-a d)^3 (a+b x)+48 A d (-b c+a d)^3 (a+b x)+72 A d^2 (b c-a d)^2 (a+b x)^2-78 B d^2 (b c-a d)^2 (a+b x)^2+300 B d^3 (b c-a d) (a+b x)^3+144 A d^3 (-b c+a d) (a+b x)^3-144 A d^4 (a+b x)^4 \log (a+b x)+300 B d^4 (a+b x)^4 \log (a+b x)-72 B d^4 (a+b x)^4 \log ^2(a+b x)+144 A d^4 (a+b x)^4 \log (c+d x)-300 B d^4 (a+b x)^4 \log (c+d x)+144 B d^4 (a+b x)^4 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)-72 B d^4 (a+b x)^4 \log ^2(c+d x)+144 B d^4 (a+b x)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+36 B (b c-a d)^4 \log \left (\frac {e (c+d x)}{a+b x}\right )+48 B d (-b c+a d)^3 (a+b x) \log \left (\frac {e (c+d x)}{a+b x}\right )+72 B d^2 (b c-a d)^2 (a+b x)^2 \log \left (\frac {e (c+d x)}{a+b x}\right )+144 B d^3 (-b c+a d) (a+b x)^3 \log \left (\frac {e (c+d x)}{a+b x}\right )-144 B d^4 (a+b x)^4 \log (a+b x) \log \left (\frac {e (c+d x)}{a+b x}\right )+144 B d^4 (a+b x)^4 \log (c+d x) \log \left (\frac {e (c+d x)}{a+b x}\right )+144 B d^4 (a+b x)^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+144 B d^4 (a+b x)^4 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{(b c-a d)^4}}{288 b g^5 (a+b x)^4} \]
(-72*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2 + (B*(36*A*(b*c - a*d)^4 - 9*B *(b*c - a*d)^4 + 28*B*d*(b*c - a*d)^3*(a + b*x) + 48*A*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 + 300*B*d^3*(b*c - a*d)*(a + b*x)^3 + 144*A*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*Lo g[a + b*x] - 72*B*d^4*(a + b*x)^4*Log[a + b*x]^2 + 144*A*d^4*(a + b*x)^4*L og[c + d*x] - 300*B*d^4*(a + b*x)^4*Log[c + d*x] + 144*B*d^4*(a + b*x)^4*L og[(d*(a + b*x))/(-(b*c) + a*d)]*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)] + 36*B*(b*c - a*d)^4*Log[(e*(c + d*x))/(a + b*x)] + 48*B*d*(-(b*c) + a *d)^3*(a + b*x)*Log[(e*(c + d*x))/(a + b*x)] + 72*B*d^2*(b*c - a*d)^2*(a + b*x)^2*Log[(e*(c + d*x))/(a + b*x)] + 144*B*d^3*(-(b*c) + a*d)*(a + b*x)^ 3*Log[(e*(c + d*x))/(a + b*x)] - 144*B*d^4*(a + b*x)^4*Log[a + b*x]*Log[(e *(c + d*x))/(a + b*x)] + 144*B*d^4*(a + b*x)^4*Log[c + d*x]*Log[(e*(c + d* x))/(a + b*x)] + 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)/(288*b*g^5*(a + b*x)^4)
Time = 0.47 (sec) , antiderivative size = 379, normalized size of antiderivative = 0.76, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2952, 2756, 2772, 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.
| \(\displaystyle \int \frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{(a g+b g x)^5} \, dx\) |
| \(\Big \downarrow \) 2952 |
| \(\displaystyle \frac {\int \left (d-\frac {b (c+d x)}{a+b x}\right )^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2d\frac {c+d x}{a+b x}}{g^5 (b c-a d)^4}\) |
| \(\Big \downarrow \) 2756 |
| \(\displaystyle \frac {\frac {B \int \frac {(a+b x) \left (d-\frac {b (c+d x)}{a+b x}\right )^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{c+d x}d\frac {c+d x}{a+b x}}{2 b}-\frac {\left (d-\frac {b (c+d x)}{a+b x}\right )^4 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{4 b}}{g^5 (b c-a d)^4}\) |
| \(\Big \downarrow \) 2772 |
| \(\displaystyle \frac {\frac {B \left (-B \int \left (\frac {(c+d x)^3 b^4}{4 (a+b x)^3}-\frac {4 d (c+d x)^2 b^3}{3 (a+b x)^2}+\frac {3 d^2 (c+d x) b^2}{a+b x}-4 d^3 b+\frac {d^4 (a+b x) \log \left (\frac {c+d x}{a+b x}\right )}{c+d x}\right )d\frac {c+d x}{a+b x}+\frac {b^4 (c+d x)^4 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{4 (a+b x)^4}-\frac {4 b^3 d (c+d x)^3 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{3 (a+b x)^3}+\frac {3 b^2 d^2 (c+d x)^2 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{(a+b x)^2}+d^4 \log \left (\frac {c+d x}{a+b x}\right ) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )-\frac {4 b d^3 (c+d x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{a+b x}\right )}{2 b}-\frac {\left (d-\frac {b (c+d x)}{a+b x}\right )^4 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{4 b}}{g^5 (b c-a d)^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {B \left (\frac {b^4 (c+d x)^4 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{4 (a+b x)^4}-\frac {4 b^3 d (c+d x)^3 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{3 (a+b x)^3}+\frac {3 b^2 d^2 (c+d x)^2 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{(a+b x)^2}+d^4 \log \left (\frac {c+d x}{a+b x}\right ) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )-\frac {4 b d^3 (c+d x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{a+b x}-B \left (\frac {b^4 (c+d x)^4}{16 (a+b x)^4}-\frac {4 b^3 d (c+d x)^3}{9 (a+b x)^3}+\frac {3 b^2 d^2 (c+d x)^2}{2 (a+b x)^2}+\frac {1}{2} d^4 \log ^2\left (\frac {c+d x}{a+b x}\right )-\frac {4 b d^3 (c+d x)}{a+b x}\right )\right )}{2 b}-\frac {\left (d-\frac {b (c+d x)}{a+b x}\right )^4 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{4 b}}{g^5 (b c-a d)^4}\) |
(-1/4*((d - (b*(c + d*x))/(a + b*x))^4*(A + B*Log[(e*(c + d*x))/(a + b*x)] )^2)/b + (B*(-(B*((-4*b*d^3*(c + d*x))/(a + b*x) + (3*b^2*d^2*(c + d*x)^2) /(2*(a + b*x)^2) - (4*b^3*d*(c + d*x)^3)/(9*(a + b*x)^3) + (b^4*(c + d*x)^ 4)/(16*(a + b*x)^4) + (d^4*Log[(c + d*x)/(a + b*x)]^2)/2)) - (4*b*d^3*(c + d*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(a + b*x) + (3*b^2*d^2*(c + d* x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(a + b*x)^2 - (4*b^3*d*(c + d*x )^3*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(3*(a + b*x)^3) + (b^4*(c + d*x) ^4*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(4*(a + b*x)^4) + d^4*Log[(c + d* x)/(a + b*x)]*(A + B*Log[(e*(c + d*x))/(a + b*x)])))/(2*b))/((b*c - a*d)^4 *g^5)
3.2.90.3.1 Defintions of rubi rules used
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_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_ .))^(q_.), x_Symbol] :> With[{u = IntHide[x^m*(d + e*x^r)^q, x]}, Simp[(a + b*Log[c*x^n]) u, x] - Simp[b*n Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] && !(EqQ[q , 1] && EqQ[m, -1])
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/d)^m Subst[Int[(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] && EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])
Leaf count of result is larger than twice the leaf count of optimal. \(1111\) vs. \(2(480)=960\).
Time = 2.38 (sec) , antiderivative size = 1112, normalized size of antiderivative = 2.23
| method | result | size |
| parts | \(\text {Expression too large to display}\) | \(1112\) |
| derivativedivides | \(\text {Expression too large to display}\) | \(1422\) |
| default | \(\text {Expression too large to display}\) | \(1422\) |
| norman | \(\text {Expression too large to display}\) | \(1796\) |
| parallelrisch | \(\text {Expression too large to display}\) | \(2035\) |
| risch | \(\text {Expression too large to display}\) | \(2601\) |
-1/4*A^2/g^5/(b*x+a)^4/b-B^2/g^5*b^3/e^4/(a*d-b*c)^4*(1/4*(d*e/b-e*(a*d-b* c)/b/(b*x+a))^4*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))^2-1/8*(d*e/b-e*(a*d-b*c)/b /(b*x+a))^4*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))+1/32*(d*e/b-e*(a*d-b*c)/b/(b*x +a))^4-3*d*e/b*(1/3*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^3*ln(d*e/b-e*(a*d-b*c)/b /(b*x+a))^2-2/9*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^3*ln(d*e/b-e*(a*d-b*c)/b/(b* x+a))+2/27*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^3)+3/b^2*d^2*e^2*(1/2*(d*e/b-e*(a *d-b*c)/b/(b*x+a))^2*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))^2-1/2*(d*e/b-e*(a*d-b *c)/b/(b*x+a))^2*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))+1/4*(d*e/b-e*(a*d-b*c)/b/ (b*x+a))^2)-1/b^3*d^3*e^3*((d*e/b-e*(a*d-b*c)/b/(b*x+a))*ln(d*e/b-e*(a*d-b *c)/b/(b*x+a))^2-2*(d*e/b-e*(a*d-b*c)/b/(b*x+a))*ln(d*e/b-e*(a*d-b*c)/b/(b *x+a))-2*e*(a*d-b*c)/b/(b*x+a)+2*d*e/b))-2*B*A/g^5*b^3/e^4/(a*d-b*c)^4*(1/ 4*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^4*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))-1/16*(d* e/b-e*(a*d-b*c)/b/(b*x+a))^4-3*d*e/b*(1/3*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^3* ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))-1/9*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^3)+3*d^2 *e^2/b^2*(1/2*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^2*ln(d*e/b-e*(a*d-b*c)/b/(b*x+ a))-1/4*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^2)-d^3*e^3/b^3*((d*e/b-e*(a*d-b*c)/b /(b*x+a))*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))+e*(a*d-b*c)/b/(b*x+a)-d*e/b))
Leaf count of result is larger than twice the leaf count of optimal. 1045 vs. \(2 (480) = 960\).
Time = 0.29 (sec) , antiderivative size = 1045, normalized size of antiderivative = 2.10 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b 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 {d e x + c e}{b x + a}\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 {d e x + c e}{b x + a}\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((d*e*x + c*e)/(b*x + a))^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((d* e*x + c*e)/(b*x + a)))/((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. \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 2122 vs. \(2 (480) = 960\).
Time = 0.36 (sec) , antiderivative size = 2122, normalized size of antiderivative = 4.26 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
-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))* log(d*e*x/(b*x + a) + c*e/(b*x + a)) + (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...
Leaf count of result is larger than twice the leaf count of optimal. 1194 vs. \(2 (480) = 960\).
Time = 0.56 (sec) , antiderivative size = 1194, normalized size of antiderivative = 2.40 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
-1/288*(72*((d*e*x + c*e)^4*B^2*b^3/((b^3*c^3*e^3*g^5 - 3*a*b^2*c^2*d*e^3* g^5 + 3*a^2*b*c*d^2*e^3*g^5 - a^3*d^3*e^3*g^5)*(b*x + a)^4) - 4*(d*e*x + c *e)^3*B^2*b^2*d/((b^3*c^3*e^2*g^5 - 3*a*b^2*c^2*d*e^2*g^5 + 3*a^2*b*c*d^2* e^2*g^5 - a^3*d^3*e^2*g^5)*(b*x + a)^3) + 6*(d*e*x + c*e)^2*B^2*b*d^2/((b^ 3*c^3*e*g^5 - 3*a*b^2*c^2*d*e*g^5 + 3*a^2*b*c*d^2*e*g^5 - a^3*d^3*e*g^5)*( b*x + a)^2) - 4*(d*e*x + c*e)*B^2*d^3/((b^3*c^3*g^5 - 3*a*b^2*c^2*d*g^5 + 3*a^2*b*c*d^2*g^5 - a^3*d^3*g^5)*(b*x + a)))*log((d*e*x + c*e)/(b*x + a))^ 2 + 12*(3*(4*A*B*b^3 - B^2*b^3)*(d*e*x + c*e)^4/((b^3*c^3*e^3*g^5 - 3*a*b^ 2*c^2*d*e^3*g^5 + 3*a^2*b*c*d^2*e^3*g^5 - a^3*d^3*e^3*g^5)*(b*x + a)^4) - 16*(3*A*B*b^2*d - B^2*b^2*d)*(d*e*x + c*e)^3/((b^3*c^3*e^2*g^5 - 3*a*b^2*c ^2*d*e^2*g^5 + 3*a^2*b*c*d^2*e^2*g^5 - a^3*d^3*e^2*g^5)*(b*x + a)^3) + 36* (2*A*B*b*d^2 - B^2*b*d^2)*(d*e*x + c*e)^2/((b^3*c^3*e*g^5 - 3*a*b^2*c^2*d* e*g^5 + 3*a^2*b*c*d^2*e*g^5 - a^3*d^3*e*g^5)*(b*x + a)^2) - 48*(A*B*d^3 - B^2*d^3)*(d*e*x + c*e)/((b^3*c^3*g^5 - 3*a*b^2*c^2*d*g^5 + 3*a^2*b*c*d^2*g ^5 - a^3*d^3*g^5)*(b*x + a)))*log((d*e*x + c*e)/(b*x + a)) + 9*(8*A^2*b^3 - 4*A*B*b^3 + B^2*b^3)*(d*e*x + c*e)^4/((b^3*c^3*e^3*g^5 - 3*a*b^2*c^2*d*e ^3*g^5 + 3*a^2*b*c*d^2*e^3*g^5 - a^3*d^3*e^3*g^5)*(b*x + a)^4) - 32*(9*A^2 *b^2*d - 6*A*B*b^2*d + 2*B^2*b^2*d)*(d*e*x + c*e)^3/((b^3*c^3*e^2*g^5 - 3* a*b^2*c^2*d*e^2*g^5 + 3*a^2*b*c*d^2*e^2*g^5 - a^3*d^3*e^2*g^5)*(b*x + a)^3 ) + 216*(2*A^2*b*d^2 - 2*A*B*b*d^2 + B^2*b*d^2)*(d*e*x + c*e)^2/((b^3*c...
Time = 7.81 (sec) , antiderivative size = 1880, normalized size of antiderivative = 3.78 \[ \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]
(log((e*(c + d*x))/(a + b*x))*((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)) - (A*B)/ (2*b^2*d*g^5) + (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*d^3)) + (B^2*d^4*x*(b*(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)) + a*(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)) + (6*a^3*d^3 - b^3*c^ 3 + 5*a*b^2*c^2*d - 10*a^2*b*c*d^2)/(4*d^4)))/(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))))/((4*a^3*x)/d + a^4 /(b*d) + (b^3*x^4)/d + (6*a^2*b*x^2)/d + (4*a*b^2*x^3)/d) - log((e*(c +...