\(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(f+g x)^5} \, dx\) [75]

Optimal result

Integrand size = 32, antiderivative size = 1208 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^5} \, dx=-\frac {B^2 (b c-a d)^2 g^3 n^2 (c+d x)^2}{12 (b f-a g)^2 (d f-c g)^4 (f+g x)^2}-\frac {B^2 (b c-a d)^3 g^3 n^2 (c+d x)}{6 (b f-a g)^3 (d f-c g)^4 (f+g x)}+\frac {B^2 (b c-a d)^2 g^2 (4 b d f-b c g-3 a d g) n^2 (c+d x)}{4 (b f-a g)^3 (d f-c g)^4 (f+g x)}+\frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 (b f-a g) (d f-c g)^4 (f+g x)^3}-\frac {B (b c-a d) g^2 (4 b d f-b c g-3 a d g) n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 (b f-a g)^2 (d f-c g)^4 (f+g x)^2}+\frac {B (b c-a d) g \left (3 a^2 d^2 g^2-2 a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b f-a g)^4 (d f-c g)^3 (f+g x)}+\frac {b^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (b f-a g)^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}-\frac {B^2 (b c-a d)^4 g^3 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{6 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^3 g^2 (4 b d f-b c g-3 a d g) n^2 \log \left (\frac {a+b x}{c+d x}\right )}{4 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^4 g^3 n^2 \log \left (\frac {f+g x}{c+d x}\right )}{6 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d)^3 g^2 (4 b d f-b c g-3 a d g) n^2 \log \left (\frac {f+g x}{c+d x}\right )}{4 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^2 g \left (3 a^2 d^2 g^2-2 a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) n^2 \log \left (\frac {f+g x}{c+d x}\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n^2 \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{2 (b f-a g)^4 (d f-c g)^4} \]

[Out]

-1/12*B^2*(-a*d+b*c)^2*g^3*n^2*(d*x+c)^2/(-a*g+b*f)^2/(-c*g+d*f)^4/(g*x+f)^2-1/6*B^2*(-a*d+b*c)^3*g^3*n^2*(d*x
+c)/(-a*g+b*f)^3/(-c*g+d*f)^4/(g*x+f)+1/4*B^2*(-a*d+b*c)^2*g^2*(-3*a*d*g-b*c*g+4*b*d*f)*n^2*(d*x+c)/(-a*g+b*f)
^3/(-c*g+d*f)^4/(g*x+f)+1/6*B*(-a*d+b*c)*g^3*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*g+b*f)/(-c*g+d*f)
^4/(g*x+f)^3-1/4*B*(-a*d+b*c)*g^2*(-3*a*d*g-b*c*g+4*b*d*f)*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*g+b
*f)^2/(-c*g+d*f)^4/(g*x+f)^2+1/2*B*(-a*d+b*c)*g*(3*a^2*d^2*g^2-2*a*b*d*g*(-c*g+4*d*f)+b^2*(c^2*g^2-4*c*d*f*g+6
*d^2*f^2))*n*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*g+b*f)^4/(-c*g+d*f)^3/(g*x+f)+1/4*b^4*(A+B*ln(e*((b*x
+a)/(d*x+c))^n))^2/g/(-a*g+b*f)^4-1/4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/g/(g*x+f)^4-1/6*B^2*(-a*d+b*c)^4*g^3*n
^2*ln((b*x+a)/(d*x+c))/(-a*g+b*f)^4/(-c*g+d*f)^4+1/4*B^2*(-a*d+b*c)^3*g^2*(-3*a*d*g-b*c*g+4*b*d*f)*n^2*ln((b*x
+a)/(d*x+c))/(-a*g+b*f)^4/(-c*g+d*f)^4+1/6*B^2*(-a*d+b*c)^4*g^3*n^2*ln((g*x+f)/(d*x+c))/(-a*g+b*f)^4/(-c*g+d*f
)^4-1/4*B^2*(-a*d+b*c)^3*g^2*(-3*a*d*g-b*c*g+4*b*d*f)*n^2*ln((g*x+f)/(d*x+c))/(-a*g+b*f)^4/(-c*g+d*f)^4+1/2*B^
2*(-a*d+b*c)^2*g*(3*a^2*d^2*g^2-2*a*b*d*g*(-c*g+4*d*f)+b^2*(c^2*g^2-4*c*d*f*g+6*d^2*f^2))*n^2*ln((g*x+f)/(d*x+
c))/(-a*g+b*f)^4/(-c*g+d*f)^4-1/2*B*(-a*d+b*c)*(-a*d*g-b*c*g+2*b*d*f)*(2*a*b*d^2*f*g-a^2*d^2*g^2-b^2*(c^2*g^2-
2*c*d*f*g+2*d^2*f^2))*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln(1-(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/(-a*g+b*f)
^4/(-c*g+d*f)^4-1/2*B^2*(-a*d+b*c)*(-a*d*g-b*c*g+2*b*d*f)*(2*a*b*d^2*f*g-a^2*d^2*g^2-b^2*(c^2*g^2-2*c*d*f*g+2*
d^2*f^2))*n^2*polylog(2,(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/(-a*g+b*f)^4/(-c*g+d*f)^4

Rubi [A] (verified)

Time = 1.64 (sec) , antiderivative size = 1208, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2553, 2398, 2404, 2338, 2356, 46, 2351, 31, 2354, 2438} \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^5} \, dx=\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 b^4}{4 g (b f-a g)^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}+\frac {B (b c-a d) g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-2 a d g (4 d f-c g) b+3 a^2 d^2 g^2\right ) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b f-a g)^4 (d f-c g)^3 (f+g x)}-\frac {B (b c-a d) g^2 (4 b d f-b c g-3 a d g) n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 (b f-a g)^2 (d f-c g)^4 (f+g x)^2}+\frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 (b f-a g) (d f-c g)^4 (f+g x)^3}+\frac {B^2 (b c-a d)^3 g^2 (4 b d f-b c g-3 a d g) n^2 \log \left (\frac {a+b x}{c+d x}\right )}{4 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d)^4 g^3 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{6 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d)^3 g^2 (4 b d f-b c g-3 a d g) n^2 \log \left (\frac {f+g x}{c+d x}\right )}{4 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^2 g \left (\left (6 d^2 f^2-4 c d g f+c^2 g^2\right ) b^2-2 a d g (4 d f-c g) b+3 a^2 d^2 g^2\right ) n^2 \log \left (\frac {f+g x}{c+d x}\right )}{2 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^4 g^3 n^2 \log \left (\frac {f+g x}{c+d x}\right )}{6 (b f-a g)^4 (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (-\left (\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2\right )+2 a d^2 f g b-a^2 d^2 g^2\right ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (-\left (\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2\right )+2 a d^2 f g b-a^2 d^2 g^2\right ) n^2 \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{2 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^2 g^2 (4 b d f-b c g-3 a d g) n^2 (c+d x)}{4 (b f-a g)^3 (d f-c g)^4 (f+g x)}-\frac {B^2 (b c-a d)^3 g^3 n^2 (c+d x)}{6 (b f-a g)^3 (d f-c g)^4 (f+g x)}-\frac {B^2 (b c-a d)^2 g^3 n^2 (c+d x)^2}{12 (b f-a g)^2 (d f-c g)^4 (f+g x)^2} \]

[In]

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

[Out]

-1/12*(B^2*(b*c - a*d)^2*g^3*n^2*(c + d*x)^2)/((b*f - a*g)^2*(d*f - c*g)^4*(f + g*x)^2) - (B^2*(b*c - a*d)^3*g
^3*n^2*(c + d*x))/(6*(b*f - a*g)^3*(d*f - c*g)^4*(f + g*x)) + (B^2*(b*c - a*d)^2*g^2*(4*b*d*f - b*c*g - 3*a*d*
g)*n^2*(c + d*x))/(4*(b*f - a*g)^3*(d*f - c*g)^4*(f + g*x)) + (B*(b*c - a*d)*g^3*n*(c + d*x)^3*(A + B*Log[e*((
a + b*x)/(c + d*x))^n]))/(6*(b*f - a*g)*(d*f - c*g)^4*(f + g*x)^3) - (B*(b*c - a*d)*g^2*(4*b*d*f - b*c*g - 3*a
*d*g)*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*(b*f - a*g)^2*(d*f - c*g)^4*(f + g*x)^2) + (B*(
b*c - a*d)*g*(3*a^2*d^2*g^2 - 2*a*b*d*g*(4*d*f - c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*n*(a + b*x)*(A
+ B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*f - a*g)^4*(d*f - c*g)^3*(f + g*x)) + (b^4*(A + B*Log[e*((a + b*x)/
(c + d*x))^n])^2)/(4*g*(b*f - a*g)^4) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(4*g*(f + g*x)^4) - (B^2*(b*c
- a*d)^4*g^3*n^2*Log[(a + b*x)/(c + d*x)])/(6*(b*f - a*g)^4*(d*f - c*g)^4) + (B^2*(b*c - a*d)^3*g^2*(4*b*d*f
- b*c*g - 3*a*d*g)*n^2*Log[(a + b*x)/(c + d*x)])/(4*(b*f - a*g)^4*(d*f - c*g)^4) + (B^2*(b*c - a*d)^4*g^3*n^2*
Log[(f + g*x)/(c + d*x)])/(6*(b*f - a*g)^4*(d*f - c*g)^4) - (B^2*(b*c - a*d)^3*g^2*(4*b*d*f - b*c*g - 3*a*d*g)
*n^2*Log[(f + g*x)/(c + d*x)])/(4*(b*f - a*g)^4*(d*f - c*g)^4) + (B^2*(b*c - a*d)^2*g*(3*a^2*d^2*g^2 - 2*a*b*d
*g*(4*d*f - c*g) + b^2*(6*d^2*f^2 - 4*c*d*f*g + c^2*g^2))*n^2*Log[(f + g*x)/(c + d*x)])/(2*(b*f - a*g)^4*(d*f
- c*g)^4) - (B*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 - b^2*(2*d^2*f^2 - 2*c*d*f*g
+ c^2*g^2))*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))]
)/(2*(b*f - a*g)^4*(d*f - c*g)^4) - (B^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*(2*a*b*d^2*f*g - a^2*d^2*g^2 -
b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*n^2*PolyLog[2, ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(2*(b*
f - a*g)^4*(d*f - c*g)^4)

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_) + (e_.)*(x_)^(r_.))^(q_), x_Symbol] :> Simp[x*(d + e*x^r)^(q +
1)*((a + b*Log[c*x^n])/d), x] - Dist[b*(n/d), Int[(d + e*x^r)^(q + 1), x], x] /; FreeQ[{a, b, c, d, e, n, q,
r}, x] && EqQ[r*(q + 1) + 1, 0]

Rule 2354

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

Rule 2356

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] - Dist[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] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2398

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol]
:> Simp[(f + g*x)^(m + 1)*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/((q + 1)*(e*f - d*g))), x] - Dist[b*n*(p/((q
+ 1)*(e*f - d*g))), Int[(f + g*x)^(m + 1)*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[{
a, b, c, d, e, f, g, m, n, q}, x] && NeQ[e*f - d*g, 0] && EqQ[m + q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 2404

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2553

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.), x_Symbol] :> Dist[b*c - a*d, Subst[Int[(b*f - a*g - (d*f - c*g)*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] && NeQ[b*c - a*d, 0] && Inte
gerQ[m] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = (b c-a d) \text {Subst}\left (\int \frac {(b-d x)^3 \left (A+B \log \left (e x^n\right )\right )^2}{(b f-a g-(d f-c g) x)^5} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = -\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}+\frac {(B n) \text {Subst}\left (\int \frac {(b-d x)^4 \left (A+B \log \left (e x^n\right )\right )}{x (b f-a g+(-d f+c g) x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 g} \\ & = -\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}+\frac {(B n) \text {Subst}\left (\int \left (\frac {b^4 \left (A+B \log \left (e x^n\right )\right )}{(b f-a g)^4 x}+\frac {(b c-a d)^4 g^4 \left (A+B \log \left (e x^n\right )\right )}{(b f-a g) (d f-c g)^3 (b f-a g-(d f-c g) x)^4}+\frac {(b c-a d)^3 g^3 (-4 b d f+b c g+3 a d g) \left (A+B \log \left (e x^n\right )\right )}{(b f-a g)^2 (d f-c g)^3 (b f-a g-(d f-c g) x)^3}+\frac {(b c-a d)^2 g^2 \left (3 a^2 d^2 g^2-2 a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (e x^n\right )\right )}{(b f-a g)^3 (d f-c g)^3 (b f-a g-(d f-c g) x)^2}+\frac {(b c-a d) g (2 b d f-b c g-a d g) \left (-2 b^2 d^2 f^2+2 b^2 c d f g+2 a b d^2 f g-b^2 c^2 g^2-a^2 d^2 g^2\right ) \left (A+B \log \left (e x^n\right )\right )}{(b f-a g)^4 (d f-c g)^3 (b f-a g-(d f-c g) x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{2 g} \\ & = -\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}+\frac {\left (b^4 B n\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 g (b f-a g)^4}+\frac {\left (B (b c-a d)^4 g^3 n\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{(b f-a g+(-d f+c g) x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 (b f-a g) (d f-c g)^3}-\frac {\left (B (b c-a d)^3 g^2 (4 b d f-b c g-3 a d g) n\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{(b f-a g+(-d f+c g) x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 (b f-a g)^2 (d f-c g)^3}+\frac {\left (B (b c-a d)^2 g \left (3 a^2 d^2 g^2-2 a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) n\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{(b f-a g+(-d f+c g) x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 (b f-a g)^3 (d f-c g)^3}+\frac {\left (B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{b f-a g+(-d f+c g) x} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 (b f-a g)^4 (d f-c g)^3} \\ & = \frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 (b f-a g) (d f-c g)^4 (f+g x)^3}-\frac {B (b c-a d) g^2 (4 b d f-b c g-3 a d g) n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 (b f-a g)^2 (d f-c g)^4 (f+g x)^2}+\frac {B (b c-a d) g \left (3 a^2 d^2 g^2-2 a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b f-a g)^4 (d f-c g)^3 (f+g x)}+\frac {b^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (b f-a g)^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {\left (B^2 (b c-a d)^4 g^3 n^2\right ) \text {Subst}\left (\int \frac {1}{x (b f-a g+(-d f+c g) x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{6 (b f-a g) (d f-c g)^4}+\frac {\left (B^2 (b c-a d)^3 g^2 (4 b d f-b c g-3 a d g) n^2\right ) \text {Subst}\left (\int \frac {1}{x (b f-a g+(-d f+c g) x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{4 (b f-a g)^2 (d f-c g)^4}-\frac {\left (B^2 (b c-a d)^2 g \left (3 a^2 d^2 g^2-2 a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) n^2\right ) \text {Subst}\left (\int \frac {1}{b f-a g+(-d f+c g) x} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 (b f-a g)^4 (d f-c g)^3}+\frac {\left (B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {(-d f+c g) x}{b f-a g}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 (b f-a g)^4 (d f-c g)^4} \\ & = \frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 (b f-a g) (d f-c g)^4 (f+g x)^3}-\frac {B (b c-a d) g^2 (4 b d f-b c g-3 a d g) n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 (b f-a g)^2 (d f-c g)^4 (f+g x)^2}+\frac {B (b c-a d) g \left (3 a^2 d^2 g^2-2 a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b f-a g)^4 (d f-c g)^3 (f+g x)}+\frac {b^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (b f-a g)^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}+\frac {B^2 (b c-a d)^2 g \left (3 a^2 d^2 g^2-2 a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) n^2 \log \left (\frac {f+g x}{c+d x}\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n^2 \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {\left (B^2 (b c-a d)^4 g^3 n^2\right ) \text {Subst}\left (\int \left (\frac {1}{(b f-a g)^3 x}+\frac {d f-c g}{(b f-a g) (b f-a g-(d f-c g) x)^3}+\frac {d f-c g}{(b f-a g)^2 (b f-a g-(d f-c g) x)^2}+\frac {d f-c g}{(b f-a g)^3 (b f-a g-(d f-c g) x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{6 (b f-a g) (d f-c g)^4}+\frac {\left (B^2 (b c-a d)^3 g^2 (4 b d f-b c g-3 a d g) n^2\right ) \text {Subst}\left (\int \left (\frac {1}{(b f-a g)^2 x}+\frac {d f-c g}{(b f-a g) (b f-a g-(d f-c g) x)^2}+\frac {d f-c g}{(b f-a g)^2 (b f-a g-(d f-c g) x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{4 (b f-a g)^2 (d f-c g)^4} \\ & = -\frac {B^2 (b c-a d)^2 g^3 n^2 (c+d x)^2}{12 (b f-a g)^2 (d f-c g)^4 (f+g x)^2}-\frac {B^2 (b c-a d)^3 g^3 n^2 (c+d x)}{6 (b f-a g)^3 (d f-c g)^4 (f+g x)}+\frac {B^2 (b c-a d)^2 g^2 (4 b d f-b c g-3 a d g) n^2 (c+d x)}{4 (b f-a g)^3 (d f-c g)^4 (f+g x)}+\frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 (b f-a g) (d f-c g)^4 (f+g x)^3}-\frac {B (b c-a d) g^2 (4 b d f-b c g-3 a d g) n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 (b f-a g)^2 (d f-c g)^4 (f+g x)^2}+\frac {B (b c-a d) g \left (3 a^2 d^2 g^2-2 a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b f-a g)^4 (d f-c g)^3 (f+g x)}+\frac {b^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (b f-a g)^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g (f+g x)^4}-\frac {B^2 (b c-a d)^4 g^3 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{6 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^3 g^2 (4 b d f-b c g-3 a d g) n^2 \log \left (\frac {a+b x}{c+d x}\right )}{4 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^4 g^3 n^2 \log \left (\frac {f+g x}{c+d x}\right )}{6 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d)^3 g^2 (4 b d f-b c g-3 a d g) n^2 \log \left (\frac {f+g x}{c+d x}\right )}{4 (b f-a g)^4 (d f-c g)^4}+\frac {B^2 (b c-a d)^2 g \left (3 a^2 d^2 g^2-2 a b d g (4 d f-c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) n^2 \log \left (\frac {f+g x}{c+d x}\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{2 (b f-a g)^4 (d f-c g)^4}-\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n^2 \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{2 (b f-a g)^4 (d f-c g)^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.89 (sec) , antiderivative size = 1329, normalized size of antiderivative = 1.10 \[ \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(f+g x)^5} \, dx=-\frac {3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {B n (f+g x) \left (2 (b c-a d) g (b f-a g)^3 (d f-c g)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-3 (b c-a d) g (b f-a g)^2 (d f-c g)^2 (-2 b d f+b c g+a d g) (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+6 (b c-a d) g (b f-a g) (d f-c g) \left (a^2 d^2 g^2+a b d g (-3 d f+c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) (f+g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 b^4 (d f-c g)^4 (f+g x)^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+6 d^4 (b f-a g)^4 (f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+6 (b c-a d) g (-2 b d f+b c g+a d g) \left (-2 a b d^2 f g+a^2 d^2 g^2+b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) (f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)-6 B (b c-a d) g \left (a^2 d^2 g^2+a b d g (-3 d f+c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) n (f+g x)^3 (b (d f-c g) \log (a+b x)+(-b d f+a d g) \log (c+d x)+(b c-a d) g \log (f+g x))+3 B (b c-a d) g (2 b d f-b c g-a d g) n (f+g x)^2 \left ((b c-a d) g (b f-a g) (d f-c g)-b^2 (d f-c g)^2 (f+g x) \log (a+b x)+d^2 (b f-a g)^2 (f+g x) \log (c+d x)+(b c-a d) g (-2 b d f+b c g+a d g) (f+g x) \log (f+g x)\right )+B (b c-a d) g n (f+g x) \left ((b c-a d) g (b f-a g)^2 (d f-c g)^2+2 (b c-a d) g (b f-a g) (-d f+c g) (-2 b d f+b c g+a d g) (f+g x)-2 b^3 (d f-c g)^3 (f+g x)^2 \log (a+b x)+2 d^3 (b f-a g)^3 (f+g x)^2 \log (c+d x)-2 (b c-a d) g \left (a^2 d^2 g^2+a b d g (-3 d f+c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) (f+g x)^2 \log (f+g x)\right )+3 b^4 B (d f-c g)^4 n (f+g x)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )-3 B d^4 (b f-a g)^4 n (f+g x)^3 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )-6 B (b c-a d) g (-2 b d f+b c g+a d g) \left (-2 a b d^2 f g+a^2 d^2 g^2+b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n (f+g x)^3 \left (\left (\log \left (\frac {g (a+b x)}{-b f+a g}\right )-\log \left (\frac {g (c+d x)}{-d f+c g}\right )\right ) \log (f+g x)+\operatorname {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )-\operatorname {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )\right )\right )}{(b f-a g)^4 (d f-c g)^4}}{12 g (f+g x)^4} \]

[In]

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

[Out]

-1/12*(3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + (B*n*(f + g*x)*(2*(b*c - a*d)*g*(b*f - a*g)^3*(d*f - c*g)^
3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 3*(b*c - a*d)*g*(b*f - a*g)^2*(d*f - c*g)^2*(-2*b*d*f + b*c*g + a*d
*g)*(f + g*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 6*(b*c - a*d)*g*(b*f - a*g)*(d*f - c*g)*(a^2*d^2*g^2 +
a*b*d*g*(-3*d*f + c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*(f + g*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))
^n]) - 6*b^4*(d*f - c*g)^4*(f + g*x)^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 6*d^4*(b*f - a*g)
^4*(f + g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 6*(b*c - a*d)*g*(-2*b*d*f + b*c*g + a*d*g
)*(-2*a*b*d^2*f*g + a^2*d^2*g^2 + b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*(f + g*x)^3*(A + B*Log[e*((a + b*x)/(
c + d*x))^n])*Log[f + g*x] - 6*B*(b*c - a*d)*g*(a^2*d^2*g^2 + a*b*d*g*(-3*d*f + c*g) + b^2*(3*d^2*f^2 - 3*c*d*
f*g + c^2*g^2))*n*(f + g*x)^3*(b*(d*f - c*g)*Log[a + b*x] + (-(b*d*f) + a*d*g)*Log[c + d*x] + (b*c - a*d)*g*Lo
g[f + g*x]) + 3*B*(b*c - a*d)*g*(2*b*d*f - b*c*g - a*d*g)*n*(f + g*x)^2*((b*c - a*d)*g*(b*f - a*g)*(d*f - c*g)
- b^2*(d*f - c*g)^2*(f + g*x)*Log[a + b*x] + d^2*(b*f - a*g)^2*(f + g*x)*Log[c + d*x] + (b*c - a*d)*g*(-2*b*d
*f + b*c*g + a*d*g)*(f + g*x)*Log[f + g*x]) + B*(b*c - a*d)*g*n*(f + g*x)*((b*c - a*d)*g*(b*f - a*g)^2*(d*f -
c*g)^2 + 2*(b*c - a*d)*g*(b*f - a*g)*(-(d*f) + c*g)*(-2*b*d*f + b*c*g + a*d*g)*(f + g*x) - 2*b^3*(d*f - c*g)^3
*(f + g*x)^2*Log[a + b*x] + 2*d^3*(b*f - a*g)^3*(f + g*x)^2*Log[c + d*x] - 2*(b*c - a*d)*g*(a^2*d^2*g^2 + a*b*
d*g*(-3*d*f + c*g) + b^2*(3*d^2*f^2 - 3*c*d*f*g + c^2*g^2))*(f + g*x)^2*Log[f + g*x]) + 3*b^4*B*(d*f - c*g)^4*
n*(f + g*x)^3*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(
b*c) + a*d)]) - 3*B*d^4*(b*f - a*g)^4*n*(f + g*x)^3*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[
c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]) - 6*B*(b*c - a*d)*g*(-2*b*d*f + b*c*g + a*d*g)*(-2*a*b*d^2
*f*g + a^2*d^2*g^2 + b^2*(2*d^2*f^2 - 2*c*d*f*g + c^2*g^2))*n*(f + g*x)^3*((Log[(g*(a + b*x))/(-(b*f) + a*g)]
- Log[(g*(c + d*x))/(-(d*f) + c*g)])*Log[f + g*x] + PolyLog[2, (b*(f + g*x))/(b*f - a*g)] - PolyLog[2, (d*(f +
g*x))/(d*f - c*g)])))/((b*f - a*g)^4*(d*f - c*g)^4))/(g*(f + g*x)^4)

Maple [F]

\[\int \frac {{\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}}{\left (g x +f \right )^{5}}d x\]

[In]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(g*x+f)^5,x)

[Out]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(g*x+f)^5,x)

Fricas [F]

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

[In]

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

[Out]

integral((B^2*log(e*((b*x + a)/(d*x + c))^n)^2 + 2*A*B*log(e*((b*x + a)/(d*x + c))^n) + A^2)/(g^5*x^5 + 5*f*g^
4*x^4 + 10*f^2*g^3*x^3 + 10*f^3*g^2*x^2 + 5*f^4*g*x + f^5), x)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

1/12*(6*b^4*log(b*x + a)/(b^4*f^4*g - 4*a*b^3*f^3*g^2 + 6*a^2*b^2*f^2*g^3 - 4*a^3*b*f*g^4 + a^4*g^5) - 6*d^4*l
og(d*x + c)/(d^4*f^4*g - 4*c*d^3*f^3*g^2 + 6*c^2*d^2*f^2*g^3 - 4*c^3*d*f*g^4 + c^4*g^5) + 6*(4*(b^4*c*d^3 - a*
b^3*d^4)*f^3 - 6*(b^4*c^2*d^2 - a^2*b^2*d^4)*f^2*g + 4*(b^4*c^3*d - a^3*b*d^4)*f*g^2 - (b^4*c^4 - a^4*d^4)*g^3
)*log(g*x + f)/(b^4*d^4*f^8 + a^4*c^4*g^8 - 4*(b^4*c*d^3 + a*b^3*d^4)*f^7*g + 2*(3*b^4*c^2*d^2 + 8*a*b^3*c*d^3
+ 3*a^2*b^2*d^4)*f^6*g^2 - 4*(b^4*c^3*d + 6*a*b^3*c^2*d^2 + 6*a^2*b^2*c*d^3 + a^3*b*d^4)*f^5*g^3 + (b^4*c^4 +
16*a*b^3*c^3*d + 36*a^2*b^2*c^2*d^2 + 16*a^3*b*c*d^3 + a^4*d^4)*f^4*g^4 - 4*(a*b^3*c^4 + 6*a^2*b^2*c^3*d + 6*
a^3*b*c^2*d^2 + a^4*c*d^3)*f^3*g^5 + 2*(3*a^2*b^2*c^4 + 8*a^3*b*c^3*d + 3*a^4*c^2*d^2)*f^2*g^6 - 4*(a^3*b*c^4
+ a^4*c^3*d)*f*g^7) - (26*(b^3*c*d^2 - a*b^2*d^3)*f^4 - 31*(b^3*c^2*d - a^2*b*d^3)*f^3*g + (11*b^3*c^3 + 15*a*
b^2*c^2*d - 15*a^2*b*c*d^2 - 11*a^3*d^3)*f^2*g^2 - 7*(a*b^2*c^3 - a^3*c*d^2)*f*g^3 + 2*(a^2*b*c^3 - a^3*c^2*d)
*g^4 + 6*(3*(b^3*c*d^2 - a*b^2*d^3)*f^2*g^2 - 3*(b^3*c^2*d - a^2*b*d^3)*f*g^3 + (b^3*c^3 - a^3*d^3)*g^4)*x^2 +
3*(14*(b^3*c*d^2 - a*b^2*d^3)*f^3*g - 15*(b^3*c^2*d - a^2*b*d^3)*f^2*g^2 + (5*b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2
*b*c*d^2 - 5*a^3*d^3)*f*g^3 - (a*b^2*c^3 - a^3*c*d^2)*g^4)*x)/(b^3*d^3*f^9 + a^3*c^3*f^3*g^6 - 3*(b^3*c*d^2 +
a*b^2*d^3)*f^8*g + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*f^7*g^2 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^
2 + a^3*d^3)*f^6*g^3 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*f^5*g^4 - 3*(a^2*b*c^3 + a^3*c^2*d)*f^4*g^5 +
(b^3*d^3*f^6*g^3 + a^3*c^3*g^9 - 3*(b^3*c*d^2 + a*b^2*d^3)*f^5*g^4 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3
)*f^4*g^5 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*f^3*g^6 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c
*d^2)*f^2*g^7 - 3*(a^2*b*c^3 + a^3*c^2*d)*f*g^8)*x^3 + 3*(b^3*d^3*f^7*g^2 + a^3*c^3*f*g^8 - 3*(b^3*c*d^2 + a*b
^2*d^3)*f^6*g^3 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*f^5*g^4 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2
+ a^3*d^3)*f^4*g^5 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*f^3*g^6 - 3*(a^2*b*c^3 + a^3*c^2*d)*f^2*g^7)*x
^2 + 3*(b^3*d^3*f^8*g + a^3*c^3*f^2*g^7 - 3*(b^3*c*d^2 + a*b^2*d^3)*f^7*g^2 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a
^2*b*d^3)*f^6*g^3 - (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*f^5*g^4 + 3*(a*b^2*c^3 + 3*a^2*b*c^2*d
+ a^3*c*d^2)*f^4*g^5 - 3*(a^2*b*c^3 + a^3*c^2*d)*f^3*g^6)*x))*A*B*n - 1/4*B^2*(log((d*x + c)^n)^2/(g^5*x^4 +
4*f*g^4*x^3 + 6*f^2*g^3*x^2 + 4*f^3*g^2*x + f^4*g) + 4*integrate(-1/2*(2*d*g*x*log(e)^2 + 2*c*g*log(e)^2 + 2*(
d*g*x + c*g)*log((b*x + a)^n)^2 + 4*(d*g*x*log(e) + c*g*log(e))*log((b*x + a)^n) + (d*f*n + (g*n - 4*g*log(e))
*d*x - 4*c*g*log(e) - 4*(d*g*x + c*g)*log((b*x + a)^n))*log((d*x + c)^n))/(d*g^6*x^6 + c*f^5*g + (5*d*f*g^5 +
c*g^6)*x^5 + 5*(2*d*f^2*g^4 + c*f*g^5)*x^4 + 10*(d*f^3*g^3 + c*f^2*g^4)*x^3 + 5*(d*f^4*g^2 + 2*c*f^3*g^3)*x^2
+ (d*f^5*g + 5*c*f^4*g^2)*x), x)) - 1/2*A*B*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(g^5*x^4 + 4*f*g^4*x^3 + 6*
f^2*g^3*x^2 + 4*f^3*g^2*x + f^4*g) - 1/4*A^2/(g^5*x^4 + 4*f*g^4*x^3 + 6*f^2*g^3*x^2 + 4*f^3*g^2*x + f^4*g)

Giac [F]

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

[In]

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

[Out]

integrate((B*log(e*((b*x + a)/(d*x + c))^n) + A)^2/(g*x + f)^5, x)

Mupad [F(-1)]

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

[In]

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

[Out]

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