\(\int \frac {(c i+d i x) (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^5} \, dx\) [63]

Optimal result

Integrand size = 40, antiderivative size = 445 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {B^2 d^2 i (c+d x)^2}{4 (b c-a d)^3 g^5 (a+b x)^2}+\frac {4 b B^2 d i (c+d x)^3}{27 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 B^2 i (c+d x)^4}{32 (b c-a d)^3 g^5 (a+b x)^4}-\frac {B d^2 i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {4 b B d i (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 B i (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 (b c-a d)^3 g^5 (a+b x)^4}-\frac {d^2 i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {2 b d i (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 i (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 (b c-a d)^3 g^5 (a+b x)^4} \]

[Out]

-1/4*B^2*d^2*i*(d*x+c)^2/(-a*d+b*c)^3/g^5/(b*x+a)^2+4/27*b*B^2*d*i*(d*x+c)^3/(-a*d+b*c)^3/g^5/(b*x+a)^3-1/32*b
^2*B^2*i*(d*x+c)^4/(-a*d+b*c)^3/g^5/(b*x+a)^4-1/2*B*d^2*i*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g
^5/(b*x+a)^2+4/9*b*B*d*i*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^5/(b*x+a)^3-1/8*b^2*B*i*(d*x+c)^
4*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^5/(b*x+a)^4-1/2*d^2*i*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a
*d+b*c)^3/g^5/(b*x+a)^2+2/3*b*d*i*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^3/g^5/(b*x+a)^3-1/4*b^2*i
*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^3/g^5/(b*x+a)^4

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 445, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2562, 2395, 2342, 2341} \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {b^2 i (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{4 g^5 (a+b x)^4 (b c-a d)^3}-\frac {b^2 B i (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{8 g^5 (a+b x)^4 (b c-a d)^3}-\frac {d^2 i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^5 (a+b x)^2 (b c-a d)^3}-\frac {B d^2 i (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^5 (a+b x)^2 (b c-a d)^3}+\frac {2 b d i (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 g^5 (a+b x)^3 (b c-a d)^3}+\frac {4 b B d i (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{9 g^5 (a+b x)^3 (b c-a d)^3}-\frac {b^2 B^2 i (c+d x)^4}{32 g^5 (a+b x)^4 (b c-a d)^3}-\frac {B^2 d^2 i (c+d x)^2}{4 g^5 (a+b x)^2 (b c-a d)^3}+\frac {4 b B^2 d i (c+d x)^3}{27 g^5 (a+b x)^3 (b c-a d)^3} \]

[In]

Int[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a*g + b*g*x)^5,x]

[Out]

-1/4*(B^2*d^2*i*(c + d*x)^2)/((b*c - a*d)^3*g^5*(a + b*x)^2) + (4*b*B^2*d*i*(c + d*x)^3)/(27*(b*c - a*d)^3*g^5
*(a + b*x)^3) - (b^2*B^2*i*(c + d*x)^4)/(32*(b*c - a*d)^3*g^5*(a + b*x)^4) - (B*d^2*i*(c + d*x)^2*(A + B*Log[(
e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^3*g^5*(a + b*x)^2) + (4*b*B*d*i*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/
(c + d*x)]))/(9*(b*c - a*d)^3*g^5*(a + b*x)^3) - (b^2*B*i*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(8
*(b*c - a*d)^3*g^5*(a + b*x)^4) - (d^2*i*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^3*
g^5*(a + b*x)^2) + (2*b*d*i*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(b*c - a*d)^3*g^5*(a + b*x)
^3) - (b^2*i*(c + d*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(4*(b*c - a*d)^3*g^5*(a + b*x)^4)

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo
g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2395

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
]))

Rule 2562

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[x^m*(
(A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h,
i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i
, 0] && IntegersQ[m, q]

Rubi steps \begin{align*} \text {integral}& = \frac {i \text {Subst}\left (\int \frac {(b-d x)^2 (A+B \log (e x))^2}{x^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^5} \\ & = \frac {i \text {Subst}\left (\int \left (\frac {b^2 (A+B \log (e x))^2}{x^5}-\frac {2 b d (A+B \log (e x))^2}{x^4}+\frac {d^2 (A+B \log (e x))^2}{x^3}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^5} \\ & = \frac {\left (b^2 i\right ) \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{x^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^5}-\frac {(2 b d i) \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{x^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^5}+\frac {\left (d^2 i\right ) \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{x^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^5} \\ & = -\frac {d^2 i (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)^3 g^5 (a+b x)^2}+\frac {2 b d i (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 i (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)^3 g^5 (a+b x)^4}+\frac {\left (b^2 B i\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{x^5} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^3 g^5}-\frac {(4 b B d i) \text {Subst}\left (\int \frac {A+B \log (e x)}{x^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 (b c-a d)^3 g^5}+\frac {\left (B d^2 i\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{x^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^5} \\ & = -\frac {B^2 d^2 i (c+d x)^2}{4 (b c-a d)^3 g^5 (a+b x)^2}+\frac {4 b B^2 d i (c+d x)^3}{27 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 B^2 i (c+d x)^4}{32 (b c-a d)^3 g^5 (a+b x)^4}-\frac {B d^2 i (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^5 (a+b x)^2}+\frac {4 b B d i (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{9 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 B i (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 (b c-a d)^3 g^5 (a+b x)^4}-\frac {d^2 i (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)^3 g^5 (a+b x)^2}+\frac {2 b d i (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 (b c-a d)^3 g^5 (a+b x)^3}-\frac {b^2 i (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)^3 g^5 (a+b x)^4} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.69 (sec) , antiderivative size = 1255, normalized size of antiderivative = 2.82 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {i \left (216 (b c-a d)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2-288 d (-b c+a d)^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+16 B d (a+b x) \left (12 A (b c-a d)^3+4 B (b c-a d)^3-18 A d (b c-a d)^2 (a+b x)-15 B d (b c-a d)^2 (a+b x)+36 A d^2 (b c-a d) (a+b x)^2+66 B d^2 (b c-a d) (a+b x)^2+36 A d^3 (a+b x)^3 \log (a+b x)+66 B d^3 (a+b x)^3 \log (a+b x)-18 B d^3 (a+b x)^3 \log ^2(a+b x)+12 B (b c-a d)^3 \log \left (\frac {e (a+b x)}{c+d x}\right )-18 B d (b c-a d)^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+36 B d^2 (b c-a d) (a+b x)^2 \log \left (\frac {e (a+b x)}{c+d x}\right )+36 B d^3 (a+b x)^3 \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )-36 A d^3 (a+b x)^3 \log (c+d x)-66 B d^3 (a+b x)^3 \log (c+d x)+36 B d^3 (a+b x)^3 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)-36 B d^3 (a+b x)^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)-18 B d^3 (a+b x)^3 \log ^2(c+d x)+36 B d^3 (a+b x)^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )+36 B d^3 (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )+36 B d^3 (a+b x)^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )+3 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 )\right )}{864 b^2 (b c-a d)^3 g^5 (a+b x)^4} \]

[In]

Integrate[((c*i + d*i*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(a*g + b*g*x)^5,x]

[Out]

-1/864*(i*(216*(b*c - a*d)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 - 288*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*
Log[(e*(a + b*x))/(c + d*x)])^2 + 16*B*d*(a + b*x)*(12*A*(b*c - a*d)^3 + 4*B*(b*c - a*d)^3 - 18*A*d*(b*c - a*d
)^2*(a + b*x) - 15*B*d*(b*c - a*d)^2*(a + b*x) + 36*A*d^2*(b*c - a*d)*(a + b*x)^2 + 66*B*d^2*(b*c - a*d)*(a +
b*x)^2 + 36*A*d^3*(a + b*x)^3*Log[a + b*x] + 66*B*d^3*(a + b*x)^3*Log[a + b*x] - 18*B*d^3*(a + b*x)^3*Log[a +
b*x]^2 + 12*B*(b*c - a*d)^3*Log[(e*(a + b*x))/(c + d*x)] - 18*B*d*(b*c - a*d)^2*(a + b*x)*Log[(e*(a + b*x))/(c
+ d*x)] + 36*B*d^2*(b*c - a*d)*(a + b*x)^2*Log[(e*(a + b*x))/(c + d*x)] + 36*B*d^3*(a + b*x)^3*Log[a + b*x]*L
og[(e*(a + b*x))/(c + d*x)] - 36*A*d^3*(a + b*x)^3*Log[c + d*x] - 66*B*d^3*(a + b*x)^3*Log[c + d*x] + 36*B*d^3
*(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*Log[(e*(a + b*x))/(c + d*x)
]*Log[c + d*x] - 18*B*d^3*(a + b*x)^3*Log[c + d*x]^2 + 36*B*d^3*(a + b*x)^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*
c - a*d)] + 36*B*d^3*(a + b*x)^3*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 36*B*d^3*(a + b*x)^3*PolyLog[2, (b
*(c + d*x))/(b*c - a*d)]) + 3*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)*Log[(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)] - 1
44*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^2*(b*c - a*d)^3*g^5*(a + b*x)^4)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(925\) vs. \(2(427)=854\).

Time = 1.61 (sec) , antiderivative size = 926, normalized size of antiderivative = 2.08

method	result	size
parts	\(\frac {i \,A^{2} \left (-\frac {d}{3 b^{2} \left (b x +a \right )^{3}}-\frac {-a d +c b}{4 b^{2} \left (b x +a \right )^{4}}\right )}{g^{5}}-\frac {i \,B^{2} \left (a d -c b \right )^{2} e^{2} \left (\frac {d^{5} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{5}}-\frac {2 d^{4} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2 \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {2}{27 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{5}}+\frac {d^{3} e^{2} b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{8 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{32 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (a d -c b \right )^{5}}\right )}{g^{5} d^{3}}-\frac {2 i B A \left (a d -c b \right )^{2} e^{2} \left (\frac {d^{5} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{5}}-\frac {2 d^{4} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{5}}+\frac {d^{3} e^{2} b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (a d -c b \right )^{5}}\right )}{g^{5} d^{3}}\)	\(926\)
derivativedivides	\(\text {Expression too large to display}\)	\(1058\)
default	\(\text {Expression too large to display}\)	\(1058\)
norman	\(\text {Expression too large to display}\)	\(1577\)
parallelrisch	\(\text {Expression too large to display}\)	\(1926\)
risch	\(\text {Expression too large to display}\)	\(3657\)

[In]

int((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x,method=_RETURNVERBOSE)

[Out]

i*A^2/g^5*(-1/3*d/b^2/(b*x+a)^3-1/4*(-a*d+b*c)/b^2/(b*x+a)^4)-i*B^2/g^5/d^3*(a*d-b*c)^2*e^2*(d^5/(a*d-b*c)^5*(
-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)-2*d^4/(a*d-b*c)^5*b*e*(-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^3/(a*d-b*c)^5*e^2*b^2*(-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*i*B*A/g^5/d^3*(a*d-b*c)^2*e^2*(d^5/(a*d-b*c)^5*(-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)-2*d^4/(a*d-b*c)^5*b*e*(-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^3/(a*d-b*c)^5*
e^2*b^2*(-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))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 985 vs. \(2 (427) = 854\).

Time = 0.31 (sec) , antiderivative size = 985, normalized size of antiderivative = 2.21 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {12 \, {\left ({\left (12 \, A B + 13 \, B^{2}\right )} b^{4} c d^{3} - {\left (12 \, A B + 13 \, B^{2}\right )} a b^{3} d^{4}\right )} i x^{3} - 6 \, {\left ({\left (12 \, A B + B^{2}\right )} b^{4} c^{2} d^{2} - 16 \, {\left (6 \, A B + 5 \, B^{2}\right )} a b^{3} c d^{3} + {\left (84 \, A B + 79 \, B^{2}\right )} a^{2} b^{2} d^{4}\right )} i x^{2} + 4 \, {\left ({\left (72 \, A^{2} + 12 \, A B - 5 \, B^{2}\right )} b^{4} c^{3} d - 12 \, {\left (18 \, A^{2} + 6 \, A B - B^{2}\right )} a b^{3} c^{2} d^{2} + 108 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a^{2} b^{2} c d^{3} - {\left (72 \, A^{2} + 156 \, A B + 115 \, B^{2}\right )} a^{3} b d^{4}\right )} i x + 72 \, {\left (B^{2} b^{4} d^{4} i x^{4} + 4 \, B^{2} a b^{3} d^{4} i x^{3} + 6 \, B^{2} a^{2} b^{2} d^{4} i x^{2} + 4 \, {\left (B^{2} b^{4} c^{3} d - 3 \, B^{2} a b^{3} c^{2} d^{2} + 3 \, B^{2} a^{2} b^{2} c d^{3}\right )} i x + {\left (3 \, B^{2} b^{4} c^{4} - 8 \, B^{2} a b^{3} c^{3} d + 6 \, B^{2} a^{2} b^{2} c^{2} d^{2}\right )} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + {\left (27 \, {\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} b^{4} c^{4} - 64 \, {\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} - {\left (72 \, A^{2} + 156 \, A B + 115 \, B^{2}\right )} a^{4} d^{4}\right )} i + 12 \, {\left ({\left (12 \, A B + 13 \, B^{2}\right )} b^{4} d^{4} i x^{4} + 4 \, {\left (3 \, B^{2} b^{4} c d^{3} + 2 \, {\left (6 \, A B + 5 \, B^{2}\right )} a b^{3} d^{4}\right )} i 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 + B^{2}\right )} a^{2} b^{2} d^{4}\right )} i x^{2} + 4 \, {\left ({\left (12 \, A B + B^{2}\right )} b^{4} c^{3} d - 6 \, {\left (6 \, A B + B^{2}\right )} a b^{3} c^{2} d^{2} + 18 \, {\left (2 \, A B + B^{2}\right )} a^{2} b^{2} c d^{3}\right )} i x + {\left (9 \, {\left (4 \, A B + B^{2}\right )} b^{4} c^{4} - 32 \, {\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}\right )} i\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{864 \, {\left ({\left (b^{9} c^{3} - 3 \, a b^{8} c^{2} d + 3 \, a^{2} b^{7} c d^{2} - a^{3} b^{6} d^{3}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{3} - 3 \, a^{2} b^{7} c^{2} d + 3 \, a^{3} b^{6} c d^{2} - a^{4} b^{5} d^{3}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{3} - 3 \, a^{3} b^{6} c^{2} d + 3 \, a^{4} b^{5} c d^{2} - a^{5} b^{4} d^{3}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{3} - 3 \, a^{4} b^{5} c^{2} d + 3 \, a^{5} b^{4} c d^{2} - a^{6} b^{3} d^{3}\right )} g^{5} x + {\left (a^{4} b^{5} c^{3} - 3 \, a^{5} b^{4} c^{2} d + 3 \, a^{6} b^{3} c d^{2} - a^{7} b^{2} d^{3}\right )} g^{5}\right )}} \]

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x, algorithm="fricas")

[Out]

-1/864*(12*((12*A*B + 13*B^2)*b^4*c*d^3 - (12*A*B + 13*B^2)*a*b^3*d^4)*i*x^3 - 6*((12*A*B + B^2)*b^4*c^2*d^2 -
16*(6*A*B + 5*B^2)*a*b^3*c*d^3 + (84*A*B + 79*B^2)*a^2*b^2*d^4)*i*x^2 + 4*((72*A^2 + 12*A*B - 5*B^2)*b^4*c^3*
d - 12*(18*A^2 + 6*A*B - B^2)*a*b^3*c^2*d^2 + 108*(2*A^2 + 2*A*B + B^2)*a^2*b^2*c*d^3 - (72*A^2 + 156*A*B + 11
5*B^2)*a^3*b*d^4)*i*x + 72*(B^2*b^4*d^4*i*x^4 + 4*B^2*a*b^3*d^4*i*x^3 + 6*B^2*a^2*b^2*d^4*i*x^2 + 4*(B^2*b^4*c
^3*d - 3*B^2*a*b^3*c^2*d^2 + 3*B^2*a^2*b^2*c*d^3)*i*x + (3*B^2*b^4*c^4 - 8*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2
*d^2)*i)*log((b*e*x + a*e)/(d*x + c))^2 + (27*(8*A^2 + 4*A*B + B^2)*b^4*c^4 - 64*(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 - (72*A^2 + 156*A*B + 115*B^2)*a^4*d^4)*i + 12*((12*A*B + 1
3*B^2)*b^4*d^4*i*x^4 + 4*(3*B^2*b^4*c*d^3 + 2*(6*A*B + 5*B^2)*a*b^3*d^4)*i*x^3 - 6*(B^2*b^4*c^2*d^2 - 8*B^2*a*
b^3*c*d^3 - 6*(2*A*B + B^2)*a^2*b^2*d^4)*i*x^2 + 4*((12*A*B + B^2)*b^4*c^3*d - 6*(6*A*B + B^2)*a*b^3*c^2*d^2 +
18*(2*A*B + B^2)*a^2*b^2*c*d^3)*i*x + (9*(4*A*B + B^2)*b^4*c^4 - 32*(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)*i)*log((b*e*x + a*e)/(d*x + c)))/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^
3)*g^5*x^4 + 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*
b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*
b^3*d^3)*g^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*g^5)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Timed out} \]

[In]

integrate((d*i*x+c*i)*(A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**5,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4808 vs. \(2 (427) = 854\).

Time = 0.57 (sec) , antiderivative size = 4808, normalized size of antiderivative = 10.80 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x, algorithm="maxima")

[Out]

-1/12*(4*b*x + a)*B^2*d*i*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^
5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) + 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(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 + 1
00*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*a^3*b*d^4*x + a^4*d^4)
*log(b*x + a))*log(d*x + c))/(a^4*b^5*c^4*g^5 - 4*a^5*b^4*c^3*d*g^5 + 6*a^6*b^3*c^2*d^2*g^5 - 4*a^7*b^2*c*d^3*
g^5 + a^8*b*d^4*g^5 + (b^9*c^4*g^5 - 4*a*b^8*c^3*d*g^5 + 6*a^2*b^7*c^2*d^2*g^5 - 4*a^3*b^6*c*d^3*g^5 + a^4*b^5
*d^4*g^5)*x^4 + 4*(a*b^8*c^4*g^5 - 4*a^2*b^7*c^3*d*g^5 + 6*a^3*b^6*c^2*d^2*g^5 - 4*a^4*b^5*c*d^3*g^5 + a^5*b^4
*d^4*g^5)*x^3 + 6*(a^2*b^7*c^4*g^5 - 4*a^3*b^6*c^3*d*g^5 + 6*a^4*b^5*c^2*d^2*g^5 - 4*a^5*b^4*c*d^3*g^5 + a^6*b
^3*d^4*g^5)*x^2 + 4*(a^3*b^6*c^4*g^5 - 4*a^4*b^5*c^3*d*g^5 + 6*a^5*b^4*c^2*d^2*g^5 - 4*a^6*b^3*c*d^3*g^5 + a^7
*b^2*d^4*g^5)*x))*B^2*c*i - 1/864*(12*((7*a*b^3*c^3 - 33*a^2*b^2*c^2*d + 75*a^3*b*c*d^2 - 13*a^4*d^3 + 12*(4*b
^4*c*d^2 - a*b^3*d^3)*x^3 - 6*(4*b^4*c^2*d - 29*a*b^3*c*d^2 + 7*a^2*b^2*d^3)*x^2 + 4*(4*b^4*c^3 - 21*a*b^3*c^2
*d + 57*a^2*b^2*c*d^2 - 13*a^3*b*d^3)*x)/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 +
4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3
*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g^5*
x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*g^5) + 12*(4*b*c*d^3 - a*d^4)*log(b*x + a)
/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5) - 12*(4*b*c*d^3 - a*d^4)*
log(d*x + c)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5))*log(b*e*x/(d
*x + c) + a*e/(d*x + c)) + (37*a*b^4*c^4 - 304*a^2*b^3*c^3*d + 1512*a^3*b^2*c^2*d^2 - 1360*a^4*b*c*d^3 + 115*a
^5*d^4 + 12*(88*b^5*c^2*d^2 - 101*a*b^4*c*d^3 + 13*a^2*b^3*d^4)*x^3 - 6*(40*b^5*c^3*d - 609*a*b^4*c^2*d^2 + 64
8*a^2*b^3*c*d^3 - 79*a^3*b^2*d^4)*x^2 - 72*(4*a^4*b*c*d^3 - a^5*d^4 + (4*b^5*c*d^3 - a*b^4*d^4)*x^4 + 4*(4*a*b
^4*c*d^3 - a^2*b^3*d^4)*x^3 + 6*(4*a^2*b^3*c*d^3 - a^3*b^2*d^4)*x^2 + 4*(4*a^3*b^2*c*d^3 - a^4*b*d^4)*x)*log(b
*x + a)^2 - 72*(4*a^4*b*c*d^3 - a^5*d^4 + (4*b^5*c*d^3 - a*b^4*d^4)*x^4 + 4*(4*a*b^4*c*d^3 - a^2*b^3*d^4)*x^3
+ 6*(4*a^2*b^3*c*d^3 - a^3*b^2*d^4)*x^2 + 4*(4*a^3*b^2*c*d^3 - a^4*b*d^4)*x)*log(d*x + c)^2 + 4*(16*b^5*c^4 -
163*a*b^4*c^3*d + 1068*a^2*b^3*c^2*d^2 - 1036*a^3*b^2*c*d^3 + 115*a^4*b*d^4)*x + 12*(88*a^4*b*c*d^3 - 13*a^5*d
^4 + (88*b^5*c*d^3 - 13*a*b^4*d^4)*x^4 + 4*(88*a*b^4*c*d^3 - 13*a^2*b^3*d^4)*x^3 + 6*(88*a^2*b^3*c*d^3 - 13*a^
3*b^2*d^4)*x^2 + 4*(88*a^3*b^2*c*d^3 - 13*a^4*b*d^4)*x)*log(b*x + a) - 12*(88*a^4*b*c*d^3 - 13*a^5*d^4 + (88*b
^5*c*d^3 - 13*a*b^4*d^4)*x^4 + 4*(88*a*b^4*c*d^3 - 13*a^2*b^3*d^4)*x^3 + 6*(88*a^2*b^3*c*d^3 - 13*a^3*b^2*d^4)
*x^2 + 4*(88*a^3*b^2*c*d^3 - 13*a^4*b*d^4)*x - 12*(4*a^4*b*c*d^3 - a^5*d^4 + (4*b^5*c*d^3 - a*b^4*d^4)*x^4 + 4
*(4*a*b^4*c*d^3 - a^2*b^3*d^4)*x^3 + 6*(4*a^2*b^3*c*d^3 - a^3*b^2*d^4)*x^2 + 4*(4*a^3*b^2*c*d^3 - a^4*b*d^4)*x
)*log(b*x + a))*log(d*x + c))/(a^4*b^6*c^4*g^5 - 4*a^5*b^5*c^3*d*g^5 + 6*a^6*b^4*c^2*d^2*g^5 - 4*a^7*b^3*c*d^3
*g^5 + a^8*b^2*d^4*g^5 + (b^10*c^4*g^5 - 4*a*b^9*c^3*d*g^5 + 6*a^2*b^8*c^2*d^2*g^5 - 4*a^3*b^7*c*d^3*g^5 + a^4
*b^6*d^4*g^5)*x^4 + 4*(a*b^9*c^4*g^5 - 4*a^2*b^8*c^3*d*g^5 + 6*a^3*b^7*c^2*d^2*g^5 - 4*a^4*b^6*c*d^3*g^5 + a^5
*b^5*d^4*g^5)*x^3 + 6*(a^2*b^8*c^4*g^5 - 4*a^3*b^7*c^3*d*g^5 + 6*a^4*b^6*c^2*d^2*g^5 - 4*a^5*b^5*c*d^3*g^5 + a
^6*b^4*d^4*g^5)*x^2 + 4*(a^3*b^7*c^4*g^5 - 4*a^4*b^6*c^3*d*g^5 + 6*a^5*b^5*c^2*d^2*g^5 - 4*a^6*b^4*c*d^3*g^5 +
a^7*b^3*d^4*g^5)*x))*B^2*d*i - 1/72*A*B*d*i*(12*(4*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^6*g^5*x^4
+ 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) + (7*a*b^3*c^3 - 33*a^2*b^2*c^2*d + 75
*a^3*b*c*d^2 - 13*a^4*d^3 + 12*(4*b^4*c*d^2 - a*b^3*d^3)*x^3 - 6*(4*b^4*c^2*d - 29*a*b^3*c*d^2 + 7*a^2*b^2*d^3
)*x^2 + 4*(4*b^4*c^3 - 21*a*b^3*c^2*d + 57*a^2*b^2*c*d^2 - 13*a^3*b*d^3)*x)/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*
b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 + 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6
*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d +
3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*g^5) +
12*(4*b*c*d^3 - a*d^4)*log(b*x + a)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*
d^4)*g^5) - 12*(4*b*c*d^3 - a*d^4)*log(d*x + c)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^
3 + a^4*b^2*d^4)*g^5)) + 1/24*A*B*c*i*((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*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b
^2*g^5*x + a^4*b*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)) - 1/4*B^2*c*i*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5
*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) - 1/12*(4*b*x + a)*A^2*d*i/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*
x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) - 1/4*A^2*c*i/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*
b^2*g^5*x + a^4*b*g^5)

Giac [A] (verification not implemented)

none

Time = 0.56 (sec) , antiderivative size = 744, normalized size of antiderivative = 1.67 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=-\frac {1}{864} \, {\left (\frac {72 \, {\left (3 \, B^{2} b^{2} e^{5} i - \frac {8 \, {\left (b e x + a e\right )} B^{2} b d e^{4} i}{d x + c} + \frac {6 \, {\left (b e x + a e\right )}^{2} B^{2} d^{2} e^{3} i}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{\frac {{\left (b e x + a e\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b e x + a e\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b e x + a e\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {12 \, {\left (36 \, A B b^{2} e^{5} i + 9 \, B^{2} b^{2} e^{5} i - \frac {96 \, {\left (b e x + a e\right )} A B b d e^{4} i}{d x + c} - \frac {32 \, {\left (b e x + a e\right )} B^{2} b d e^{4} i}{d x + c} + \frac {72 \, {\left (b e x + a e\right )}^{2} A B d^{2} e^{3} i}{{\left (d x + c\right )}^{2}} + \frac {36 \, {\left (b e x + a e\right )}^{2} B^{2} d^{2} e^{3} i}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b e x + a e\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b e x + a e\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {216 \, A^{2} b^{2} e^{5} i + 108 \, A B b^{2} e^{5} i + 27 \, B^{2} b^{2} e^{5} i - \frac {576 \, {\left (b e x + a e\right )} A^{2} b d e^{4} i}{d x + c} - \frac {384 \, {\left (b e x + a e\right )} A B b d e^{4} i}{d x + c} - \frac {128 \, {\left (b e x + a e\right )} B^{2} b d e^{4} i}{d x + c} + \frac {432 \, {\left (b e x + a e\right )}^{2} A^{2} d^{2} e^{3} i}{{\left (d x + c\right )}^{2}} + \frac {432 \, {\left (b e x + a e\right )}^{2} A B d^{2} e^{3} i}{{\left (d x + c\right )}^{2}} + \frac {216 \, {\left (b e x + a e\right )}^{2} B^{2} d^{2} e^{3} i}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b e x + a e\right )}^{4} b^{2} c^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {2 \, {\left (b e x + a e\right )}^{4} a b c d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {{\left (b e x + a e\right )}^{4} a^{2} d^{2} g^{5}}{{\left (d x + c\right )}^{4}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]

[In]

integrate((d*i*x+c*i)*(A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^5,x, algorithm="giac")

[Out]

-1/864*(72*(3*B^2*b^2*e^5*i - 8*(b*e*x + a*e)*B^2*b*d*e^4*i/(d*x + c) + 6*(b*e*x + a*e)^2*B^2*d^2*e^3*i/(d*x +
c)^2)*log((b*e*x + a*e)/(d*x + c))^2/((b*e*x + a*e)^4*b^2*c^2*g^5/(d*x + c)^4 - 2*(b*e*x + a*e)^4*a*b*c*d*g^5
/(d*x + c)^4 + (b*e*x + a*e)^4*a^2*d^2*g^5/(d*x + c)^4) + 12*(36*A*B*b^2*e^5*i + 9*B^2*b^2*e^5*i - 96*(b*e*x +
a*e)*A*B*b*d*e^4*i/(d*x + c) - 32*(b*e*x + a*e)*B^2*b*d*e^4*i/(d*x + c) + 72*(b*e*x + a*e)^2*A*B*d^2*e^3*i/(d
*x + c)^2 + 36*(b*e*x + a*e)^2*B^2*d^2*e^3*i/(d*x + c)^2)*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^4*b^2*c^
2*g^5/(d*x + c)^4 - 2*(b*e*x + a*e)^4*a*b*c*d*g^5/(d*x + c)^4 + (b*e*x + a*e)^4*a^2*d^2*g^5/(d*x + c)^4) + (21
6*A^2*b^2*e^5*i + 108*A*B*b^2*e^5*i + 27*B^2*b^2*e^5*i - 576*(b*e*x + a*e)*A^2*b*d*e^4*i/(d*x + c) - 384*(b*e*
x + a*e)*A*B*b*d*e^4*i/(d*x + c) - 128*(b*e*x + a*e)*B^2*b*d*e^4*i/(d*x + c) + 432*(b*e*x + a*e)^2*A^2*d^2*e^3
*i/(d*x + c)^2 + 432*(b*e*x + a*e)^2*A*B*d^2*e^3*i/(d*x + c)^2 + 216*(b*e*x + a*e)^2*B^2*d^2*e^3*i/(d*x + c)^2
)/((b*e*x + a*e)^4*b^2*c^2*g^5/(d*x + c)^4 - 2*(b*e*x + a*e)^4*a*b*c*d*g^5/(d*x + c)^4 + (b*e*x + a*e)^4*a^2*d
^2*g^5/(d*x + c)^4))*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))

Mupad [B] (verification not implemented)

Time = 7.44 (sec) , antiderivative size = 1870, normalized size of antiderivative = 4.20 \[ \int \frac {(c i+d i x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx=\text {Too large to display} \]

[In]

int(((c*i + d*i*x)*(A + B*log((e*(a + b*x))/(c + d*x)))^2)/(a*g + b*g*x)^5,x)

[Out]

((72*A^2*a^3*d^3*i + 216*A^2*b^3*c^3*i + 115*B^2*a^3*d^3*i + 27*B^2*b^3*c^3*i + 156*A*B*a^3*d^3*i + 108*A*B*b^
3*c^3*i - 360*A^2*a*b^2*c^2*d*i + 72*A^2*a^2*b*c*d^2*i - 101*B^2*a*b^2*c^2*d*i + 115*B^2*a^2*b*c*d^2*i - 276*A
*B*a*b^2*c^2*d*i + 156*A*B*a^2*b*c*d^2*i)/(12*(a*d - b*c)) + (x^2*(79*B^2*a*b^2*d^3*i - B^2*b^3*c*d^2*i + 84*A
*B*a*b^2*d^3*i - 12*A*B*b^3*c*d^2*i))/(2*(a*d - b*c)) + (x*(72*A^2*a^2*b*d^3*i + 115*B^2*a^2*b*d^3*i + 72*A^2*
b^3*c^2*d*i - 5*B^2*b^3*c^2*d*i + 156*A*B*a^2*b*d^3*i + 12*A*B*b^3*c^2*d*i - 144*A^2*a*b^2*c*d^2*i + 7*B^2*a*b
^2*c*d^2*i - 60*A*B*a*b^2*c*d^2*i))/(3*(a*d - b*c)) + (d*x^3*(13*B^2*b^3*d^2*i + 12*A*B*b^3*d^2*i))/(a*d - b*c
))/(x*(288*a^3*b^4*c*g^5 - 288*a^4*b^3*d*g^5) - x^3*(288*a^2*b^5*d*g^5 - 288*a*b^6*c*g^5) + x^4*(72*b^7*c*g^5
- 72*a*b^6*d*g^5) + x^2*(432*a^2*b^5*c*g^5 - 432*a^3*b^4*d*g^5) + 72*a^4*b^3*c*g^5 - 72*a^5*b^2*d*g^5) - log((
e*(a + b*x))/(c + d*x))^2*(((B^2*c*i)/(4*b^2*g^5) + (B^2*a*d*i)/(12*b^3*g^5) + (B^2*d*i*x)/(3*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*i)/(12*b^2*g^5*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d
- 3*a^2*b*c*d^2))) - (log((e*(a + b*x))/(c + d*x))*(x*((2*A*B*i)/(3*b^2*g^5) + (B^2*d^4*i*(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)))/(6*b^2*g^5*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) + (A*B*a*i)/(6*b^3*g^5)
+ (B*i*(3*A*b*c - B*a*d + B*b*c))/(6*b^3*d*g^5) + (B^2*d^4*i*(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)))/(6*b^2*g^5*(a^3*d^3 - b^3*c^3
+ 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (B^2*d^4*i*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)))/(6*b^2*g^5*(a^3*d^3
- b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (B^2*d^4*i*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)))/(6*b^2*g^5*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))))/((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) - (B*d^4*i*atan((B*d^4*i*(12*A + 13*B)*(72
*b^5*c^3*g^5 + 72*a^3*b^2*d^3*g^5 - 72*a*b^4*c^2*d*g^5 - 72*a^2*b^3*c*d^2*g^5)*1i)/(72*b^2*g^5*(13*B^2*d^4*i +
12*A*B*d^4*i)*(a*d - b*c)^3) + (B*d^5*i*x*(12*A + 13*B)*(b^4*c^2*g^5 + a^2*b^2*d^2*g^5 - 2*a*b^3*c*d*g^5)*2i)
/(b*g^5*(13*B^2*d^4*i + 12*A*B*d^4*i)*(a*d - b*c)^3))*(12*A + 13*B)*1i)/(36*b^2*g^5*(a*d - b*c)^3)