3.17 \(\int \frac{(A+B \log (e (\frac{a+b x}{c+d x})^n))^2}{(a g+b g x)^4} \, dx\)

Optimal. Leaf size=448 \[ -\frac{b^2 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 g^4 (a+b x)^3 (b c-a d)^3}-\frac{2 b^2 B n (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{9 g^4 (a+b x)^3 (b c-a d)^3}-\frac{d^2 (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^4 (a+b x) (b c-a d)^3}-\frac{2 B d^2 n (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 (a+b x) (b c-a d)^3}+\frac{b d (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^4 (a+b x)^2 (b c-a d)^3}+\frac{b B d n (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 (a+b x)^2 (b c-a d)^3}-\frac{2 b^2 B^2 n^2 (c+d x)^3}{27 g^4 (a+b x)^3 (b c-a d)^3}-\frac{2 B^2 d^2 n^2 (c+d x)}{g^4 (a+b x) (b c-a d)^3}+\frac{b B^2 d n^2 (c+d x)^2}{2 g^4 (a+b x)^2 (b c-a d)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 1.08656, antiderivative size = 736, normalized size of antiderivative = 1.64, number of steps used = 32, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314, Rules used = {2525, 12, 2528, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{2 B^2 d^3 n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{3 b g^4 (b c-a d)^3}-\frac{2 B^2 d^3 n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{3 b g^4 (b c-a d)^3}-\frac{2 B d^3 n \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b g^4 (b c-a d)^3}+\frac{2 B d^3 n \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b g^4 (b c-a d)^3}-\frac{2 B d^2 n \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b g^4 (a+b x) (b c-a d)^2}+\frac{B d n \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b g^4 (a+b x)^2 (b c-a d)}-\frac{\left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b g^4 (a+b x)^3}-\frac{2 B n \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{9 b g^4 (a+b x)^3}-\frac{11 B^2 d^2 n^2}{9 b g^4 (a+b x) (b c-a d)^2}+\frac{B^2 d^3 n^2 \log ^2(a+b x)}{3 b g^4 (b c-a d)^3}+\frac{B^2 d^3 n^2 \log ^2(c+d x)}{3 b g^4 (b c-a d)^3}-\frac{11 B^2 d^3 n^2 \log (a+b x)}{9 b g^4 (b c-a d)^3}+\frac{11 B^2 d^3 n^2 \log (c+d x)}{9 b g^4 (b c-a d)^3}-\frac{2 B^2 d^3 n^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{3 b g^4 (b c-a d)^3}-\frac{2 B^2 d^3 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b g^4 (b c-a d)^3}+\frac{5 B^2 d n^2}{18 b g^4 (a+b x)^2 (b c-a d)}-\frac{2 B^2 n^2}{27 b g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*B^2*n^2)/(27*b*g^4*(a + b*x)^3) + (5*B^2*d*n^2)/(18*b*(b*c - a*d)*g^4*(a + b*x)^2) - (11*B^2*d^2*n^2)/(9*b
*(b*c - a*d)^2*g^4*(a + b*x)) - (11*B^2*d^3*n^2*Log[a + b*x])/(9*b*(b*c - a*d)^3*g^4) + (B^2*d^3*n^2*Log[a + b
*x]^2)/(3*b*(b*c - a*d)^3*g^4) - (2*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*b*g^4*(a + b*x)^3) + (B*d*n
*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b*(b*c - a*d)*g^4*(a + b*x)^2) - (2*B*d^2*n*(A + B*Log[e*((a + b*x
)/(c + d*x))^n]))/(3*b*(b*c - a*d)^2*g^4*(a + b*x)) - (2*B*d^3*n*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x
))^n]))/(3*b*(b*c - a*d)^3*g^4) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(3*b*g^4*(a + b*x)^3) + (11*B^2*d^3
*n^2*Log[c + d*x])/(9*b*(b*c - a*d)^3*g^4) - (2*B^2*d^3*n^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(3
*b*(b*c - a*d)^3*g^4) + (2*B*d^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x])/(3*b*(b*c - a*d)^3*g^4
) + (B^2*d^3*n^2*Log[c + d*x]^2)/(3*b*(b*c - a*d)^3*g^4) - (2*B^2*d^3*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c
- a*d)])/(3*b*(b*c - a*d)^3*g^4) - (2*B^2*d^3*n^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(3*b*(b*c - a*d)^3
*g^4) - (2*B^2*d^3*n^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(3*b*(b*c - a*d)^3*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 44

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] && L
tQ[m + n + 2, 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]

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}{(a g+b g x)^4} \, dx &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{(2 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 )}{g^3 (a+b x)^4 (c+d x)} \, dx}{3 b g}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{(2 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)^4 (c+d x)} \, dx}{3 b g^4}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{(2 B (b c-a d) n) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^4}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)}+\frac{d^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b g^4}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{(2 B n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{3 g^4}-\frac{\left (2 B d^3 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{3 (b c-a d)^3 g^4}+\frac{\left (2 B d^4 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}+\frac{\left (2 B d^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{3 (b c-a d)^2 g^4}-\frac{(2 B d n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{3 (b c-a d) g^4}\\ &=-\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b g^4 (a+b x)^3}+\frac{B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac{2 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac{2 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{2 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac{\left (2 B^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{9 b g^4}+\frac{\left (2 B^2 d^3 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}{3 b (b c-a d)^3 g^4}-\frac{\left (2 B^2 d^3 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}{3 b (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3 b (b c-a d)^2 g^4}-\frac{\left (B^2 d n^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{3 b (b c-a d) g^4}\\ &=-\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b g^4 (a+b x)^3}+\frac{B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac{2 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac{2 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{2 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}-\frac{\left (B^2 d n^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{3 b g^4}+\frac{\left (2 B^2 d^3 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}{3 b (b c-a d)^3 g^4}-\frac{\left (2 B^2 d^3 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}{3 b (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^2 n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{3 b (b c-a d) g^4}+\frac{\left (2 B^2 (b c-a d) n^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{9 b g^4}\\ &=-\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b g^4 (a+b x)^3}+\frac{B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac{2 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac{2 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{2 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}-\frac{\left (B^2 d n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{3 b g^4}+\frac{\left (2 B^2 d^3 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{3 (b c-a d)^3 g^4}-\frac{\left (2 B^2 d^3 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{3 (b c-a d)^3 g^4}-\frac{\left (2 B^2 d^4 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^4 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^2 n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3 b (b c-a d) g^4}+\frac{\left (2 B^2 (b c-a d) n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{9 b g^4}\\ &=-\frac{2 B^2 n^2}{27 b g^4 (a+b x)^3}+\frac{5 B^2 d n^2}{18 b (b c-a d) g^4 (a+b x)^2}-\frac{11 B^2 d^2 n^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac{11 B^2 d^3 n^2 \log (a+b x)}{9 b (b c-a d)^3 g^4}-\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b g^4 (a+b x)^3}+\frac{B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac{2 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac{2 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{11 B^2 d^3 n^2 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac{2 B^2 d^3 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac{2 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}-\frac{2 B^2 d^3 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^3 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{3 b (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^3 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{3 b (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^4 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}\\ &=-\frac{2 B^2 n^2}{27 b g^4 (a+b x)^3}+\frac{5 B^2 d n^2}{18 b (b c-a d) g^4 (a+b x)^2}-\frac{11 B^2 d^2 n^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac{11 B^2 d^3 n^2 \log (a+b x)}{9 b (b c-a d)^3 g^4}+\frac{B^2 d^3 n^2 \log ^2(a+b x)}{3 b (b c-a d)^3 g^4}-\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b g^4 (a+b x)^3}+\frac{B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac{2 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac{2 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{11 B^2 d^3 n^2 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac{2 B^2 d^3 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac{2 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac{B^2 d^3 n^2 \log ^2(c+d x)}{3 b (b c-a d)^3 g^4}-\frac{2 B^2 d^3 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^3 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 )}{3 b (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^3 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 )}{3 b (b c-a d)^3 g^4}\\ &=-\frac{2 B^2 n^2}{27 b g^4 (a+b x)^3}+\frac{5 B^2 d n^2}{18 b (b c-a d) g^4 (a+b x)^2}-\frac{11 B^2 d^2 n^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac{11 B^2 d^3 n^2 \log (a+b x)}{9 b (b c-a d)^3 g^4}+\frac{B^2 d^3 n^2 \log ^2(a+b x)}{3 b (b c-a d)^3 g^4}-\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{9 b g^4 (a+b x)^3}+\frac{B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac{2 B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac{2 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{11 B^2 d^3 n^2 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac{2 B^2 d^3 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac{2 B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac{B^2 d^3 n^2 \log ^2(c+d x)}{3 b (b c-a d)^3 g^4}-\frac{2 B^2 d^3 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}-\frac{2 B^2 d^3 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}-\frac{2 B^2 d^3 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}\\ \end{align*}

Mathematica [C]  time = 0.803217, size = 609, normalized size = 1.36 \[ -\frac{\frac{B n \left (-18 B d^3 n (a+b x)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+18 B d^3 n (a+b x)^3 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+36 d^2 (a+b x)^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+36 d^3 (a+b x)^3 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-36 d^3 (a+b x)^3 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+12 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-18 d (a+b x) (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+36 B d^2 n (a+b x)^2 (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)-9 B d n (a+b x) \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )+2 B n \left (6 d^2 (a+b x)^2 (b c-a d)-6 d^3 (a+b x)^3 \log (c+d x)-3 d (a+b x) (b c-a d)^2+2 (b c-a d)^3+6 d^3 (a+b x)^3 \log (a+b x)\right )\right )}{(b c-a d)^3}+18 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{54 b g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(18*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + (B*n*(12*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])
- 18*d*(b*c - a*d)^2*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 36*d^2*(b*c - a*d)*(a + b*x)^2*(A + B*
Log[e*((a + b*x)/(c + d*x))^n]) + 36*d^3*(a + b*x)^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 36*
d^3*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 36*B*d^2*n*(a + b*x)^2*(b*c - a*d + d*(a
+ b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) - 9*B*d*n*(a + b*x)*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a +
b*x) - 2*d^2*(a + b*x)^2*Log[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d*x]) + 2*B*n*(2*(b*c - a*d)^3 - 3*d*(b*c -
a*d)^2*(a + b*x) + 6*d^2*(b*c - a*d)*(a + b*x)^2 + 6*d^3*(a + b*x)^3*Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[c +
d*x]) - 18*B*d^3*n*(a + b*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)]) + 18*B*d^3*n*(a + b*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)])))/(b*c - a*d)^3)/(54*b*g^4*(a + b*x)^3)

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Maple [F]  time = 0.439, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{4}} \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/(b*g*x+a*g)^4,x)

[Out]

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

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Maxima [B]  time = 1.63949, size = 1933, normalized size = 4.31 \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/(b*g*x+a*g)^4,x, algorithm="maxima")

[Out]

-1/9*A*B*n*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b
^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b
^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 -
3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c
*d^2 - a^3*b*d^3)*g^4)) - 1/54*(6*n*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*
d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 +
3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*d^3*
log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*
a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + (4*b^3*c^3 - 27*a*b^
2*c^2*d + 108*a^2*b*c*d^2 - 85*a^3*d^3 + 66*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 +
3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a)^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(d*x
+ c)^2 - 3*(5*b^3*c^2*d - 54*a*b^2*c*d^2 + 49*a^2*b*d^3)*x + 66*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*
x + a^3*d^3)*log(b*x + a) - 6*(11*b^3*d^3*x^3 + 33*a*b^2*d^3*x^2 + 33*a^2*b*d^3*x + 11*a^3*d^3 - 6*(b^3*d^3*x^
3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a))*log(d*x + c))*n^2/(a^3*b^4*c^3*g^4 - 3*a^4*b^3*c^
2*d*g^4 + 3*a^5*b^2*c*d^2*g^4 - a^6*b*d^3*g^4 + (b^7*c^3*g^4 - 3*a*b^6*c^2*d*g^4 + 3*a^2*b^5*c*d^2*g^4 - a^3*b
^4*d^3*g^4)*x^3 + 3*(a*b^6*c^3*g^4 - 3*a^2*b^5*c^2*d*g^4 + 3*a^3*b^4*c*d^2*g^4 - a^4*b^3*d^3*g^4)*x^2 + 3*(a^2
*b^5*c^3*g^4 - 3*a^3*b^4*c^2*d*g^4 + 3*a^4*b^3*c*d^2*g^4 - a^5*b^2*d^3*g^4)*x))*B^2 - 1/3*B^2*log(e*(b*x/(d*x
+ c) + a/(d*x + c))^n)^2/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) - 2/3*A*B*log(e*(b*x/(d
*x + c) + a/(d*x + c))^n)/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) - 1/3*A^2/(b^4*g^4*x^3
+ 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4)

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Fricas [B]  time = 1.06668, size = 2383, normalized size = 5.32 \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/(b*g*x+a*g)^4,x, algorithm="fricas")

[Out]

-1/54*(18*A^2*b^3*c^3 - 54*A^2*a*b^2*c^2*d + 54*A^2*a^2*b*c*d^2 - 18*A^2*a^3*d^3 + (4*B^2*b^3*c^3 - 27*B^2*a*b
^2*c^2*d + 108*B^2*a^2*b*c*d^2 - 85*B^2*a^3*d^3)*n^2 + 6*(11*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*n^2 + 6*(A*B*b^3*
c*d^2 - A*B*a*b^2*d^3)*n)*x^2 + 18*(B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2*a^2*b*c*d^2 - B^2*a^3*d^3)*log(e)^
2 + 18*(B^2*b^3*d^3*n^2*x^3 + 3*B^2*a*b^2*d^3*n^2*x^2 + 3*B^2*a^2*b*d^3*n^2*x + (B^2*b^3*c^3 - 3*B^2*a*b^2*c^2
*d + 3*B^2*a^2*b*c*d^2)*n^2)*log((b*x + a)/(d*x + c))^2 + 6*(2*A*B*b^3*c^3 - 9*A*B*a*b^2*c^2*d + 18*A*B*a^2*b*
c*d^2 - 11*A*B*a^3*d^3)*n - 3*((5*B^2*b^3*c^2*d - 54*B^2*a*b^2*c*d^2 + 49*B^2*a^2*b*d^3)*n^2 + 6*(A*B*b^3*c^2*
d - 6*A*B*a*b^2*c*d^2 + 5*A*B*a^2*b*d^3)*n)*x + 6*(6*A*B*b^3*c^3 - 18*A*B*a*b^2*c^2*d + 18*A*B*a^2*b*c*d^2 - 6
*A*B*a^3*d^3 + 6*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*n*x^2 - 3*(B^2*b^3*c^2*d - 6*B^2*a*b^2*c*d^2 + 5*B^2*a^2*b*d^
3)*n*x + (2*B^2*b^3*c^3 - 9*B^2*a*b^2*c^2*d + 18*B^2*a^2*b*c*d^2 - 11*B^2*a^3*d^3)*n + 6*(B^2*b^3*d^3*n*x^3 +
3*B^2*a*b^2*d^3*n*x^2 + 3*B^2*a^2*b*d^3*n*x + (B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2*a^2*b*c*d^2)*n)*log((b*
x + a)/(d*x + c)))*log(e) + 6*((11*B^2*b^3*d^3*n^2 + 6*A*B*b^3*d^3*n)*x^3 + (2*B^2*b^3*c^3 - 9*B^2*a*b^2*c^2*d
+ 18*B^2*a^2*b*c*d^2)*n^2 + 3*(6*A*B*a*b^2*d^3*n + (2*B^2*b^3*c*d^2 + 9*B^2*a*b^2*d^3)*n^2)*x^2 + 6*(A*B*b^3*
c^3 - 3*A*B*a*b^2*c^2*d + 3*A*B*a^2*b*c*d^2)*n + 3*(6*A*B*a^2*b*d^3*n - (B^2*b^3*c^2*d - 6*B^2*a*b^2*c*d^2 - 6
*B^2*a^2*b*d^3)*n^2)*x)*log((b*x + a)/(d*x + c)))/((b^7*c^3 - 3*a*b^6*c^2*d + 3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*g
^4*x^3 + 3*(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*g^4*x^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*
c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*g^4*x + (a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3*a^5*b^2*c*d^2 - a^6*b*d^3)*g
^4)

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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/(b*g*x+a*g)**4,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 (b g x + a g\right )}^{4}}\,{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/(b*g*x+a*g)^4,x, algorithm="giac")

[Out]

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