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

Optimal. Leaf size=1208 \[ \text{result too large to display} \]

[Out]

-(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) - (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*Lo
g[(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)

________________________________________________________________________________________

Rubi [A]  time = 3.54545, antiderivative size = 1968, normalized size of antiderivative = 1.63, number of steps used = 41, number of rules used = 11, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.344, Rules used = {2525, 12, 2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 72} \[ \text{result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(B^2*(b*c - a*d)^2*g*n^2)/(12*(b*f - a*g)^2*(d*f - c*g)^2*(f + g*x)^2) - (5*B^2*(b*c - a*d)^2*g*(2*b*d*f - b*
c*g - a*d*g)*n^2)/(12*(b*f - a*g)^3*(d*f - c*g)^3*(f + g*x)) + (b^3*B^2*(b*c - a*d)*n^2*Log[a + b*x])/(6*(b*f
- a*g)^4*(d*f - c*g)) + (b^2*B^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*g)*n^2*Log[a + b*x])/(4*(b*f - a*g)^4*(d*f
- c*g)^2) + (b*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))*
n^2*Log[a + b*x])/(2*(b*f - a*g)^4*(d*f - c*g)^3) - (b^4*B^2*n^2*Log[a + b*x]^2)/(4*g*(b*f - a*g)^4) - (B*(b*c
- a*d)*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*(b*f - a*g)*(d*f - c*g)*(f + g*x)^3) - (B*(b*c - a*d)*(2*
b*d*f - b*c*g - a*d*g)*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*(b*f - a*g)^2*(d*f - c*g)^2*(f + g*x)^2) -
(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))*n*(A + B*Log[e*(
(a + b*x)/(c + d*x))^n]))/(2*(b*f - a*g)^3*(d*f - c*g)^3*(f + g*x)) + (b^4*B*n*Log[a + b*x]*(A + B*Log[e*((a +
b*x)/(c + d*x))^n]))/(2*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*
d^3*(b*c - a*d)*n^2*Log[c + d*x])/(6*(b*f - a*g)*(d*f - c*g)^4) - (B^2*d^2*(b*c - a*d)*(2*b*d*f - b*c*g - a*d*
g)*n^2*Log[c + d*x])/(4*(b*f - a*g)^2*(d*f - c*g)^4) - (B^2*d*(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))*n^2*Log[c + d*x])/(2*(b*f - a*g)^3*(d*f - c*g)^4) + (B^2*d^4*n^2*Log
[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(2*g*(d*f - c*g)^4) - (B*d^4*n*(A + B*Log[e*((a + b*x)/(c + d*x))
^n])*Log[c + d*x])/(2*g*(d*f - c*g)^4) - (B^2*d^4*n^2*Log[c + d*x]^2)/(4*g*(d*f - c*g)^4) + (b^4*B^2*n^2*Log[a
+ b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(2*g*(b*f - a*g)^4) + (B^2*(b*c - a*d)^2*g*(2*b*d*f - b*c*g - a*d*g)^2
*n^2*Log[f + g*x])/(4*(b*f - a*g)^4*(d*f - c*g)^4) + (2*B^2*(b*c - a*d)^2*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^2*Log[f + g*x])/(3*(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*Log[-((
g*(a + b*x))/(b*f - a*g))]*Log[f + g*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[f + g*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*Log[-((g*(c + d*x))/(d*f - c*g))]*Log[f + g*x])/
(2*(b*f - a*g)^4*(d*f - c*g)^4) + (b^4*B^2*n^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(2*g*(b*f - a*g)^4) +
(B^2*d^4*n^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(2*g*(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, (b*(f + g*x))/(b*
f - a*g)])/(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 + g*x))/(d*f - c*g)])/(2*(b*f - a*g)^4*(d
*f - c*g)^4)

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2528

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

Rule 2524

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2390

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

Rule 2301

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 2394

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

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

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 72

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rubi steps

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

Mathematica [A]  time = 7.32275, size = 1476, normalized size = 1.22 \[ \frac{B (b c-a d) n \left (\frac{\log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) b^4}{(b c-a d) (b f-a g)^4}-\frac{B n \left (\log ^2(a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right ) \log (a+b x)-2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )\right ) b^4}{2 (b c-a d) (b f-a g)^4}-\frac{g \left (\left (3 d^2 f^2-3 c d g f+c^2 g^2\right ) b^2-a d g (3 d f-c g) b+a^2 d^2 g^2\right ) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b f-a g)^3 (d f-c g)^3 (f+g x)}-\frac{g (2 b d f-b c g-a d g) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 (b f-a g)^2 (d f-c g)^2 (f+g x)^2}-\frac{g \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 (b f-a g) (d f-c g) (f+g x)^3}-\frac{d^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{(b c-a d) (d f-c g)^4}+\frac{g (2 b d f-b c g-a d g) \left (2 d^2 f^2 b^2+c^2 g^2 b^2-2 c d f g b^2-2 a d^2 f g b+a^2 d^2 g^2\right ) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{(b f-a g)^4 (d f-c g)^4}+\frac{B (b c-a d) g \left (\left (3 d^2 f^2-3 c d g f+c^2 g^2\right ) b^2-a d g (3 d f-c g) b+a^2 d^2 g^2\right ) n \left (\frac{b \log (a+b x)}{(b c-a d) (b f-a g)}-\frac{d \log (c+d x)}{(b c-a d) (d f-c g)}+\frac{g \log (f+g x)}{(b f-a g) (d f-c g)}\right )}{(b f-a g)^3 (d f-c g)^3}-\frac{B (b c-a d) g (2 b d f-b c g-a d g) n \left (-\frac{\log (a+b x) b^2}{(b c-a d) (b f-a g)^2}+\frac{d^2 \log (c+d x)}{(b c-a d) (d f-c g)^2}-\frac{g (2 b d f-b c g-a d g) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{g}{(b f-a g) (d f-c g) (f+g x)}\right )}{2 (b f-a g)^2 (d f-c g)^2}-\frac{B (b c-a d) g n \left (-\frac{2 \log (a+b x) b^3}{(b c-a d) (b f-a g)^3}+\frac{2 d^3 \log (c+d x)}{(b c-a d) (d f-c g)^3}-\frac{2 g \left (\left (3 d^2 f^2-3 c d g f+c^2 g^2\right ) b^2-a d g (3 d f-c g) b+a^2 d^2 g^2\right ) \log (f+g x)}{(b f-a g)^3 (d f-c g)^3}+\frac{2 g (2 b d f-b c g-a d g)}{(b f-a g)^2 (d f-c g)^2 (f+g x)}+\frac{g}{(b f-a g) (d f-c g) (f+g x)^2}\right )}{6 (b f-a g) (d f-c g)}+\frac{B d^4 n \left (-\log ^2(c+d x)+2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)+2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right )}{2 (b c-a d) (d f-c g)^4}-\frac{B g (2 b d f-b c g-a d g) \left (2 d^2 f^2 b^2+c^2 g^2 b^2-2 c d f g b^2-2 a d^2 f g b+a^2 d^2 g^2\right ) n \left (\log \left (-\frac{g (a+b x)}{b f-a g}\right ) \log (f+g x)-\log \left (-\frac{g (c+d x)}{d f-c g}\right ) \log (f+g x)+\text{PolyLog}\left (2,\frac{b (f+g x)}{b f-a g}\right )-\text{PolyLog}\left (2,\frac{d (f+g x)}{d f-c g}\right )\right )}{(b f-a g)^4 (d f-c g)^4}\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} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.52, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) ^{5}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{B^{2} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, A B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + 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\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)