\(\int \frac {(A+B \log (\frac {e (a+b x)^2}{(c+d x)^2}))^2}{(f+g x)^4} \, dx\) [279]

Optimal result

Integrand size = 31, antiderivative size = 724 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(f+g x)^4} \, dx=\frac {4 B^2 (b c-a d)^2 g^2 (c+d x)}{3 (b f-a g)^2 (d f-c g)^3 (f+g x)}-\frac {2 B (b c-a d) g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 (b f-a g) (d f-c g)^3 (f+g x)^2}+\frac {4 B (b c-a d) g (3 b d f-b c g-2 a d g) (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 (b f-a g)^3 (d f-c g)^2 (f+g x)}+\frac {b^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 g (b f-a g)^3}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 g (f+g x)^3}+\frac {4 B^2 (b c-a d)^3 g^2 \log \left (\frac {a+b x}{c+d x}\right )}{3 (b f-a g)^3 (d f-c g)^3}-\frac {4 B^2 (b c-a d)^3 g^2 \log \left (\frac {f+g x}{c+d x}\right )}{3 (b f-a g)^3 (d f-c g)^3}+\frac {8 B^2 (b c-a d)^2 g (3 b d f-b c g-2 a d g) \log \left (\frac {f+g x}{c+d x}\right )}{3 (b f-a g)^3 (d f-c g)^3}+\frac {4 B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{3 (b f-a g)^3 (d f-c g)^3}+\frac {8 B^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (3 d^2 f^2-3 c d f g+c^2 g^2\right )\right ) \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{3 (b f-a g)^3 (d f-c g)^3} \]

[Out]

4/3*B^2*(-a*d+b*c)^2*g^2*(d*x+c)/(-a*g+b*f)^2/(-c*g+d*f)^3/(g*x+f)-2/3*B*(-a*d+b*c)*g^2*(d*x+c)^2*(A+B*ln(e*(b
*x+a)^2/(d*x+c)^2))/(-a*g+b*f)/(-c*g+d*f)^3/(g*x+f)^2+4/3*B*(-a*d+b*c)*g*(-2*a*d*g-b*c*g+3*b*d*f)*(b*x+a)*(A+B
*ln(e*(b*x+a)^2/(d*x+c)^2))/(-a*g+b*f)^3/(-c*g+d*f)^2/(g*x+f)+1/3*b^3*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/g/(-a*
g+b*f)^3-1/3*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/g/(g*x+f)^3+4/3*B^2*(-a*d+b*c)^3*g^2*ln((b*x+a)/(d*x+c))/(-a*g+
b*f)^3/(-c*g+d*f)^3-4/3*B^2*(-a*d+b*c)^3*g^2*ln((g*x+f)/(d*x+c))/(-a*g+b*f)^3/(-c*g+d*f)^3+8/3*B^2*(-a*d+b*c)^
2*g*(-2*a*d*g-b*c*g+3*b*d*f)*ln((g*x+f)/(d*x+c))/(-a*g+b*f)^3/(-c*g+d*f)^3+4/3*B*(-a*d+b*c)*(a^2*d^2*g^2-a*b*d
*g*(-c*g+3*d*f)+b^2*(c^2*g^2-3*c*d*f*g+3*d^2*f^2))*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))*ln(1-(-c*g+d*f)*(b*x+a)/(-a
*g+b*f)/(d*x+c))/(-a*g+b*f)^3/(-c*g+d*f)^3+8/3*B^2*(-a*d+b*c)*(a^2*d^2*g^2-a*b*d*g*(-c*g+3*d*f)+b^2*(c^2*g^2-3
*c*d*f*g+3*d^2*f^2))*polylog(2,(-c*g+d*f)*(b*x+a)/(-a*g+b*f)/(d*x+c))/(-a*g+b*f)^3/(-c*g+d*f)^3

Rubi [A] (verified)

Time = 0.91 (sec) , antiderivative size = 724, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {2554, 2398, 2404, 2338, 2356, 46, 2351, 31, 2354, 2438} \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(f+g x)^4} \, dx=\frac {4 B (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right ) \log \left (1-\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right ) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 (b f-a g)^3 (d f-c g)^3}+\frac {8 B^2 (b c-a d) \left (a^2 d^2 g^2-a b d g (3 d f-c g)+b^2 \left (c^2 g^2-3 c d f g+3 d^2 f^2\right )\right ) \operatorname {PolyLog}\left (2,\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{3 (b f-a g)^3 (d f-c g)^3}+\frac {b^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{3 g (b f-a g)^3}-\frac {2 B g^2 (c+d x)^2 (b c-a d) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 (f+g x)^2 (b f-a g) (d f-c g)^3}-\frac {\left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{3 g (f+g x)^3}+\frac {4 B g (a+b x) (b c-a d) (-2 a d g-b c g+3 b d f) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 (f+g x) (b f-a g)^3 (d f-c g)^2}+\frac {4 B^2 g^2 (c+d x) (b c-a d)^2}{3 (f+g x) (b f-a g)^2 (d f-c g)^3}+\frac {4 B^2 g^2 (b c-a d)^3 \log \left (\frac {a+b x}{c+d x}\right )}{3 (b f-a g)^3 (d f-c g)^3}-\frac {4 B^2 g^2 (b c-a d)^3 \log \left (\frac {f+g x}{c+d x}\right )}{3 (b f-a g)^3 (d f-c g)^3}+\frac {8 B^2 g (b c-a d)^2 (-2 a d g-b c g+3 b d f) \log \left (\frac {f+g x}{c+d x}\right )}{3 (b f-a g)^3 (d f-c g)^3} \]

[In]

Int[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(f + g*x)^4,x]

[Out]

(4*B^2*(b*c - a*d)^2*g^2*(c + d*x))/(3*(b*f - a*g)^2*(d*f - c*g)^3*(f + g*x)) - (2*B*(b*c - a*d)*g^2*(c + d*x)
^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*(b*f - a*g)*(d*f - c*g)^3*(f + g*x)^2) + (4*B*(b*c - a*d)*g*(3
*b*d*f - b*c*g - 2*a*d*g)*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]))/(3*(b*f - a*g)^3*(d*f - c*g)^2*(
f + g*x)) + (b^3*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2)/(3*g*(b*f - a*g)^3) - (A + B*Log[(e*(a + b*x)^2)/
(c + d*x)^2])^2/(3*g*(f + g*x)^3) + (4*B^2*(b*c - a*d)^3*g^2*Log[(a + b*x)/(c + d*x)])/(3*(b*f - a*g)^3*(d*f -
c*g)^3) - (4*B^2*(b*c - a*d)^3*g^2*Log[(f + g*x)/(c + d*x)])/(3*(b*f - a*g)^3*(d*f - c*g)^3) + (8*B^2*(b*c -
a*d)^2*g*(3*b*d*f - b*c*g - 2*a*d*g)*Log[(f + g*x)/(c + d*x)])/(3*(b*f - a*g)^3*(d*f - c*g)^3) + (4*B*(b*c - a
*d)*(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))*(A + B*Log[(e*(a + b*x)^2)/(
c + d*x)^2])*Log[1 - ((d*f - c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(3*(b*f - a*g)^3*(d*f - c*g)^3) + (8*B^
2*(b*c - a*d)*(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))*PolyLog[2, ((d*f -
c*g)*(a + b*x))/((b*f - a*g)*(c + d*x))])/(3*(b*f - a*g)^3*(d*f - c*g)^3)

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 2554

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(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] && EqQ[n + mn, 0] && IGt
Q[n, 0] && NeQ[b*c - a*d, 0] && IntegerQ[m] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = (b c-a d) \text {Subst}\left (\int \frac {(b-d x)^2 \left (A+B \log \left (e x^2\right )\right )^2}{(b f-a g-(d f-c g) x)^4} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = -\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 g (f+g x)^3}+\frac {(4 B) \text {Subst}\left (\int \frac {(b-d x)^3 \left (A+B \log \left (e x^2\right )\right )}{x (b f-a g+(-d f+c g) x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 g} \\ & = -\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 g (f+g x)^3}+\frac {(4 B) \text {Subst}\left (\int \left (\frac {b^3 \left (A+B \log \left (e x^2\right )\right )}{(b f-a g)^3 x}+\frac {(-b c+a d)^3 g^3 \left (A+B \log \left (e x^2\right )\right )}{(b f-a g) (d f-c g)^2 (b f-a g-(d f-c g) x)^3}+\frac {(b c-a d)^2 g^2 (3 b d f-b c g-2 a d g) \left (A+B \log \left (e x^2\right )\right )}{(b f-a g)^2 (d f-c g)^2 (b f-a g-(d f-c g) x)^2}+\frac {(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 ) \left (A+B \log \left (e x^2\right )\right )}{(b f-a g)^3 (d f-c g)^2 (b f-a g-(d f-c g) x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{3 g} \\ & = -\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 g (f+g x)^3}+\frac {\left (4 b^3 B\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^2\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 g (b f-a g)^3}-\frac {\left (4 B (b c-a d)^3 g^2\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^2\right )}{(b f-a g+(-d f+c g) x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 (b f-a g) (d f-c g)^2}+\frac {\left (4 B (b c-a d)^2 g (3 b d f-b c g-2 a d g)\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^2\right )}{(b f-a g+(-d f+c g) x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 (b f-a g)^2 (d f-c g)^2}-\frac {\left (4 B (b c-a d) \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 )\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^2\right )}{b f-a g+(-d f+c g) x} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 (b f-a g)^3 (d f-c g)^2} \\ & = -\frac {2 B (b c-a d) g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 (b f-a g) (d f-c g)^3 (f+g x)^2}+\frac {4 B (b c-a d) g (3 b d f-b c g-2 a d g) (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 (b f-a g)^3 (d f-c g)^2 (f+g x)}+\frac {b^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 g (b f-a g)^3}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 g (f+g x)^3}+\frac {4 B (b c-a d) \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 ) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{3 (b f-a g)^3 (d f-c g)^3}+\frac {\left (4 B^2 (b c-a d)^3 g^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 )}{3 (b f-a g) (d f-c g)^3}-\frac {\left (8 B^2 (b c-a d)^2 g (3 b d f-b c g-2 a d g)\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 )}{3 (b f-a g)^3 (d f-c g)^2}-\frac {\left (8 B^2 (b c-a d) \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 )\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 )}{3 (b f-a g)^3 (d f-c g)^3} \\ & = -\frac {2 B (b c-a d) g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 (b f-a g) (d f-c g)^3 (f+g x)^2}+\frac {4 B (b c-a d) g (3 b d f-b c g-2 a d g) (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 (b f-a g)^3 (d f-c g)^2 (f+g x)}+\frac {b^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 g (b f-a g)^3}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 g (f+g x)^3}+\frac {8 B^2 (b c-a d)^2 g (3 b d f-b c g-2 a d g) \log \left (\frac {f+g x}{c+d x}\right )}{3 (b f-a g)^3 (d f-c g)^3}+\frac {4 B (b c-a d) \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 ) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{3 (b f-a g)^3 (d f-c g)^3}+\frac {8 B^2 (b c-a d) \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 ) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{3 (b f-a g)^3 (d f-c g)^3}+\frac {\left (4 B^2 (b c-a d)^3 g^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 )}{3 (b f-a g) (d f-c g)^3} \\ & = \frac {4 B^2 (b c-a d)^2 g^2 (c+d x)}{3 (b f-a g)^2 (d f-c g)^3 (f+g x)}-\frac {2 B (b c-a d) g^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 (b f-a g) (d f-c g)^3 (f+g x)^2}+\frac {4 B (b c-a d) g (3 b d f-b c g-2 a d g) (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 (b f-a g)^3 (d f-c g)^2 (f+g x)}+\frac {b^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 g (b f-a g)^3}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{3 g (f+g x)^3}+\frac {4 B^2 (b c-a d)^3 g^2 \log \left (\frac {a+b x}{c+d x}\right )}{3 (b f-a g)^3 (d f-c g)^3}-\frac {4 B^2 (b c-a d)^3 g^2 \log \left (\frac {f+g x}{c+d x}\right )}{3 (b f-a g)^3 (d f-c g)^3}+\frac {8 B^2 (b c-a d)^2 g (3 b d f-b c g-2 a d g) \log \left (\frac {f+g x}{c+d x}\right )}{3 (b f-a g)^3 (d f-c g)^3}+\frac {4 B (b c-a d) \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 ) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{3 (b f-a g)^3 (d f-c g)^3}+\frac {8 B^2 (b c-a d) \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 ) \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{3 (b f-a g)^3 (d f-c g)^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.54 (sec) , antiderivative size = 909, normalized size of antiderivative = 1.26 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(f+g x)^4} \, dx=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2+\frac {2 B (f+g x) \left ((b c-a d) g (b f-a g)^2 (d f-c g)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+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) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )-2 b^3 (d f-c g)^3 (f+g x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+2 d^3 (b f-a g)^3 (f+g x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \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 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (f+g x)-4 B (b c-a d) g (2 b d f-b c g-a d g) (f+g x)^2 (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))+2 B (b c-a d) g (f+g x) \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 )+2 b^3 B (d f-c g)^3 (f+g x)^2 \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 )-2 B d^3 (b f-a g)^3 (f+g x)^2 \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 )+4 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 ) (f+g x)^2 \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)^3 (d f-c g)^3}}{3 g (f+g x)^3} \]

[In]

Integrate[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(f + g*x)^4,x]

[Out]

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

Maple [F]

\[\int \frac {{\left (A +B \ln \left (\frac {e \left (b x +a \right )^{2}}{\left (d x +c \right )^{2}}\right )\right )}^{2}}{\left (g x +f \right )^{4}}d x\]

[In]

int((A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(g*x+f)^4,x)

[Out]

int((A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(g*x+f)^4,x)

Fricas [F]

\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(f+g x)^4} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}}{{\left (g x + f\right )}^{4}} \,d x } \]

[In]

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2/(g*x+f)^4,x, algorithm="fricas")

[Out]

integral((B^2*log((b^2*e*x^2 + 2*a*b*e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2))^2 + 2*A*B*log((b^2*e*x^2 + 2*a*b*
e*x + a^2*e)/(d^2*x^2 + 2*c*d*x + c^2)) + A^2)/(g^4*x^4 + 4*f*g^3*x^3 + 6*f^2*g^2*x^2 + 4*f^3*g*x + f^4), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(f+g x)^4} \, dx=\text {Timed out} \]

[In]

integrate((A+B*ln(e*(b*x+a)**2/(d*x+c)**2))**2/(g*x+f)**4,x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(f+g x)^4} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}}{{\left (g x + f\right )}^{4}} \,d x } \]

[In]

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2/(g*x+f)^4,x, algorithm="maxima")

[Out]

2/3*(2*b^3*log(b*x + a)/(b^3*f^3*g - 3*a*b^2*f^2*g^2 + 3*a^2*b*f*g^3 - a^3*g^4) - 2*d^3*log(d*x + c)/(d^3*f^3*
g - 3*c*d^2*f^2*g^2 + 3*c^2*d*f*g^3 - c^3*g^4) + 2*(3*(b^3*c*d^2 - a*b^2*d^3)*f^2 - 3*(b^3*c^2*d - a^2*b*d^3)*
f*g + (b^3*c^3 - a^3*d^3)*g^2)*log(g*x + f)/(b^3*d^3*f^6 + a^3*c^3*g^6 - 3*(b^3*c*d^2 + a*b^2*d^3)*f^5*g + 3*(
b^3*c^2*d + 3*a*b^2*c*d^2 + a^2*b*d^3)*f^4*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^3*g^3 +
3*(a*b^2*c^3 + 3*a^2*b*c^2*d + a^3*c*d^2)*f^2*g^4 - 3*(a^2*b*c^3 + a^3*c^2*d)*f*g^5) - (5*(b^2*c*d - a*b*d^2)
*f^2 - 3*(b^2*c^2 - a^2*d^2)*f*g + (a*b*c^2 - a^2*c*d)*g^2 + 2*(2*(b^2*c*d - a*b*d^2)*f*g - (b^2*c^2 - a^2*d^2
)*g^2)*x)/(b^2*d^2*f^6 + a^2*c^2*f^2*g^4 - 2*(b^2*c*d + a*b*d^2)*f^5*g + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^4*g
^2 - 2*(a*b*c^2 + a^2*c*d)*f^3*g^3 + (b^2*d^2*f^4*g^2 + a^2*c^2*g^6 - 2*(b^2*c*d + a*b*d^2)*f^3*g^3 + (b^2*c^2
+ 4*a*b*c*d + a^2*d^2)*f^2*g^4 - 2*(a*b*c^2 + a^2*c*d)*f*g^5)*x^2 + 2*(b^2*d^2*f^5*g + a^2*c^2*f*g^5 - 2*(b^2
*c*d + a*b*d^2)*f^4*g^2 + (b^2*c^2 + 4*a*b*c*d + a^2*d^2)*f^3*g^3 - 2*(a*b*c^2 + a^2*c*d)*f^2*g^4)*x) - log(b^
2*e*x^2/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*e*x/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2))/(g^
4*x^3 + 3*f*g^3*x^2 + 3*f^2*g^2*x + f^3*g))*A*B - 1/3*B^2*(4*log(d*x + c)^2/(g^4*x^3 + 3*f*g^3*x^2 + 3*f^2*g^2
*x + f^3*g) + 3*integrate(-1/3*(3*d*g*x*log(e)^2 + 3*c*g*log(e)^2 + 12*(d*g*x + c*g)*log(b*x + a)^2 + 12*(d*g*
x*log(e) + c*g*log(e))*log(b*x + a) - 4*((3*g*log(e) - 2*g)*d*x + 3*c*g*log(e) - 2*d*f + 6*(d*g*x + c*g)*log(b
*x + a))*log(d*x + c))/(d*g^5*x^5 + c*f^4*g + (4*d*f*g^4 + c*g^5)*x^4 + 2*(3*d*f^2*g^3 + 2*c*f*g^4)*x^3 + 2*(2
*d*f^3*g^2 + 3*c*f^2*g^3)*x^2 + (d*f^4*g + 4*c*f^3*g^2)*x), x)) - 1/3*A^2/(g^4*x^3 + 3*f*g^3*x^2 + 3*f^2*g^2*x
+ f^3*g)

Giac [F]

\[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(f+g x)^4} \, dx=\int { \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{2} e}{{\left (d x + c\right )}^{2}}\right ) + A\right )}^{2}}{{\left (g x + f\right )}^{4}} \,d x } \]

[In]

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2/(g*x+f)^4,x, algorithm="giac")

[Out]

integrate((B*log((b*x + a)^2*e/(d*x + c)^2) + A)^2/(g*x + f)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(f+g x)^4} \, dx=\int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\right )}^2}{{\left (f+g\,x\right )}^4} \,d x \]

[In]

int((A + B*log((e*(a + b*x)^2)/(c + d*x)^2))^2/(f + g*x)^4,x)

[Out]

int((A + B*log((e*(a + b*x)^2)/(c + d*x)^2))^2/(f + g*x)^4, x)