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

Optimal. Leaf size=544 \[ -\frac{2 B^2 g^4 n^2 (b c-a d)^5 \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{5 b^5 d}+\frac{2 B g^4 n (b c-a d)^5 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 b^5 d}-\frac{2 B g^4 n (a+b x) (b c-a d)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 b^5}-\frac{B g^4 n (c+d x)^2 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 b^3 d}-\frac{2 B g^4 n (c+d x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{15 b^2 d}-\frac{B g^4 n (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{10 b d}+\frac{g^4 (c+d x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 d}+\frac{13 B^2 g^4 n^2 x (b c-a d)^4}{30 b^4}+\frac{7 B^2 g^4 n^2 (c+d x)^2 (b c-a d)^3}{60 b^3 d}+\frac{B^2 g^4 n^2 (c+d x)^3 (b c-a d)^2}{30 b^2 d}+\frac{13 B^2 g^4 n^2 (b c-a d)^5 \log \left (\frac{a+b x}{c+d x}\right )}{30 b^5 d}+\frac{5 B^2 g^4 n^2 (b c-a d)^5 \log (c+d x)}{6 b^5 d} \]

[Out]

(13*B^2*(b*c - a*d)^4*g^4*n^2*x)/(30*b^4) + (7*B^2*(b*c - a*d)^3*g^4*n^2*(c + d*x)^2)/(60*b^3*d) + (B^2*(b*c -
a*d)^2*g^4*n^2*(c + d*x)^3)/(30*b^2*d) - (2*B*(b*c - a*d)^4*g^4*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x)
)^n]))/(5*b^5) - (B*(b*c - a*d)^3*g^4*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*b^3*d) - (2*B*(
b*c - a*d)^2*g^4*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(15*b^2*d) - (B*(b*c - a*d)*g^4*n*(c +
d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(10*b*d) + (g^4*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^
n])^2)/(5*d) + (13*B^2*(b*c - a*d)^5*g^4*n^2*Log[(a + b*x)/(c + d*x)])/(30*b^5*d) + (5*B^2*(b*c - a*d)^5*g^4*n
^2*Log[c + d*x])/(6*b^5*d) + (2*B*(b*c - a*d)^5*g^4*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[1 - (b*(c + d
*x))/(d*(a + b*x))])/(5*b^5*d) - (2*B^2*(b*c - a*d)^5*g^4*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(5*b^5*
d)

________________________________________________________________________________________

Rubi [A]  time = 0.880468, antiderivative size = 634, normalized size of antiderivative = 1.17, number of steps used = 27, number of rules used = 13, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.371, Rules used = {2525, 12, 2528, 2486, 31, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 43} \[ -\frac{2 B^2 g^4 n^2 (b c-a d)^5 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{5 b^5 d}-\frac{2 B g^4 n (b c-a d)^5 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 b^5 d}-\frac{B g^4 n (c+d x)^2 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{5 b^3 d}-\frac{2 B g^4 n (c+d x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{15 b^2 d}-\frac{2 A B g^4 n x (b c-a d)^4}{5 b^4}-\frac{B g^4 n (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{10 b d}+\frac{g^4 (c+d x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 d}-\frac{2 B^2 g^4 n (a+b x) (b c-a d)^4 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{5 b^5}+\frac{13 B^2 g^4 n^2 x (b c-a d)^4}{30 b^4}+\frac{7 B^2 g^4 n^2 (c+d x)^2 (b c-a d)^3}{60 b^3 d}+\frac{B^2 g^4 n^2 (c+d x)^3 (b c-a d)^2}{30 b^2 d}+\frac{B^2 g^4 n^2 (b c-a d)^5 \log ^2(a+b x)}{5 b^5 d}+\frac{13 B^2 g^4 n^2 (b c-a d)^5 \log (a+b x)}{30 b^5 d}+\frac{2 B^2 g^4 n^2 (b c-a d)^5 \log (c+d x)}{5 b^5 d}-\frac{2 B^2 g^4 n^2 (b c-a d)^5 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{5 b^5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*B*(b*c - a*d)^4*g^4*n*x)/(5*b^4) + (13*B^2*(b*c - a*d)^4*g^4*n^2*x)/(30*b^4) + (7*B^2*(b*c - a*d)^3*g^4*
n^2*(c + d*x)^2)/(60*b^3*d) + (B^2*(b*c - a*d)^2*g^4*n^2*(c + d*x)^3)/(30*b^2*d) + (13*B^2*(b*c - a*d)^5*g^4*n
^2*Log[a + b*x])/(30*b^5*d) + (B^2*(b*c - a*d)^5*g^4*n^2*Log[a + b*x]^2)/(5*b^5*d) - (2*B^2*(b*c - a*d)^4*g^4*
n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(5*b^5) - (B*(b*c - a*d)^3*g^4*n*(c + d*x)^2*(A + B*Log[e*((a + b*
x)/(c + d*x))^n]))/(5*b^3*d) - (2*B*(b*c - a*d)^2*g^4*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(1
5*b^2*d) - (B*(b*c - a*d)*g^4*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(10*b*d) - (2*B*(b*c - a*d
)^5*g^4*n*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(5*b^5*d) + (g^4*(c + d*x)^5*(A + B*Log[e*((a +
b*x)/(c + d*x))^n])^2)/(5*d) + (2*B^2*(b*c - a*d)^5*g^4*n^2*Log[c + d*x])/(5*b^5*d) - (2*B^2*(b*c - a*d)^5*g^
4*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(5*b^5*d) - (2*B^2*(b*c - a*d)^5*g^4*n^2*PolyLog[2, -((d*(a
+ b*x))/(b*c - a*d))])/(5*b^5*d)

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 2486

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

Rule 31

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

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (c g+d g x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\frac{g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d}-\frac{(2 B n) \int \frac{(b c-a d) g^5 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{5 d g}\\ &=\frac{g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d}-\frac{\left (2 B (b c-a d) g^4 n\right ) \int \frac{(c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{5 d}\\ &=\frac{g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d}-\frac{\left (2 B (b c-a d) g^4 n\right ) \int \left (\frac{d (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4}+\frac{(b c-a d)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 (a+b x)}+\frac{d (b c-a d)^2 (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3}+\frac{d (b c-a d) (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac{d (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{5 d}\\ &=\frac{g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d}-\frac{\left (2 B (b c-a d) g^4 n\right ) \int (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b}-\frac{\left (2 B (b c-a d)^2 g^4 n\right ) \int (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b^2}-\frac{\left (2 B (b c-a d)^3 g^4 n\right ) \int (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b^3}-\frac{\left (2 B (b c-a d)^4 g^4 n\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b^4}-\frac{\left (2 B (b c-a d)^5 g^4 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{5 b^4 d}\\ &=-\frac{2 A B (b c-a d)^4 g^4 n x}{5 b^4}-\frac{B (b c-a d)^3 g^4 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac{2 B (b c-a d)^2 g^4 n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{15 b^2 d}-\frac{B (b c-a d) g^4 n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b d}-\frac{2 B (b c-a d)^5 g^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^5 d}+\frac{g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d}-\frac{\left (2 B^2 (b c-a d)^4 g^4 n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{5 b^4}+\frac{\left (B^2 (b c-a d) g^4 n^2\right ) \int \frac{(b c-a d) (c+d x)^3}{a+b x} \, dx}{10 b d}+\frac{\left (2 B^2 (b c-a d)^2 g^4 n^2\right ) \int \frac{(b c-a d) (c+d x)^2}{a+b x} \, dx}{15 b^2 d}+\frac{\left (B^2 (b c-a d)^3 g^4 n^2\right ) \int \frac{(b c-a d) (c+d x)}{a+b x} \, dx}{5 b^3 d}+\frac{\left (2 B^2 (b c-a d)^5 g^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 (a+b x)}{a+b x} \, dx}{5 b^5 d}\\ &=-\frac{2 A B (b c-a d)^4 g^4 n x}{5 b^4}-\frac{2 B^2 (b c-a d)^4 g^4 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{5 b^5}-\frac{B (b c-a d)^3 g^4 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac{2 B (b c-a d)^2 g^4 n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{15 b^2 d}-\frac{B (b c-a d) g^4 n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b d}-\frac{2 B (b c-a d)^5 g^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^5 d}+\frac{g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d}+\frac{\left (B^2 (b c-a d)^2 g^4 n^2\right ) \int \frac{(c+d x)^3}{a+b x} \, dx}{10 b d}+\frac{\left (2 B^2 (b c-a d)^3 g^4 n^2\right ) \int \frac{(c+d x)^2}{a+b x} \, dx}{15 b^2 d}+\frac{\left (B^2 (b c-a d)^4 g^4 n^2\right ) \int \frac{c+d x}{a+b x} \, dx}{5 b^3 d}+\frac{\left (2 B^2 (b c-a d)^5 g^4 n^2\right ) \int \frac{1}{c+d x} \, dx}{5 b^5}+\frac{\left (2 B^2 (b c-a d)^5 g^4 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}{5 b^5 d}\\ &=-\frac{2 A B (b c-a d)^4 g^4 n x}{5 b^4}-\frac{2 B^2 (b c-a d)^4 g^4 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{5 b^5}-\frac{B (b c-a d)^3 g^4 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac{2 B (b c-a d)^2 g^4 n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{15 b^2 d}-\frac{B (b c-a d) g^4 n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b d}-\frac{2 B (b c-a d)^5 g^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^5 d}+\frac{g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d}+\frac{2 B^2 (b c-a d)^5 g^4 n^2 \log (c+d x)}{5 b^5 d}+\frac{\left (B^2 (b c-a d)^2 g^4 n^2\right ) \int \left (\frac{d (b c-a d)^2}{b^3}+\frac{(b c-a d)^3}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x)}{b^2}+\frac{d (c+d x)^2}{b}\right ) \, dx}{10 b d}+\frac{\left (2 B^2 (b c-a d)^3 g^4 n^2\right ) \int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx}{15 b^2 d}+\frac{\left (B^2 (b c-a d)^4 g^4 n^2\right ) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{5 b^3 d}-\frac{\left (2 B^2 (b c-a d)^5 g^4 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{5 b^5}+\frac{\left (2 B^2 (b c-a d)^5 g^4 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{5 b^4 d}\\ &=-\frac{2 A B (b c-a d)^4 g^4 n x}{5 b^4}+\frac{13 B^2 (b c-a d)^4 g^4 n^2 x}{30 b^4}+\frac{7 B^2 (b c-a d)^3 g^4 n^2 (c+d x)^2}{60 b^3 d}+\frac{B^2 (b c-a d)^2 g^4 n^2 (c+d x)^3}{30 b^2 d}+\frac{13 B^2 (b c-a d)^5 g^4 n^2 \log (a+b x)}{30 b^5 d}-\frac{2 B^2 (b c-a d)^4 g^4 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{5 b^5}-\frac{B (b c-a d)^3 g^4 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac{2 B (b c-a d)^2 g^4 n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{15 b^2 d}-\frac{B (b c-a d) g^4 n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b d}-\frac{2 B (b c-a d)^5 g^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^5 d}+\frac{g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d}+\frac{2 B^2 (b c-a d)^5 g^4 n^2 \log (c+d x)}{5 b^5 d}-\frac{2 B^2 (b c-a d)^5 g^4 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{5 b^5 d}+\frac{\left (2 B^2 (b c-a d)^5 g^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{5 b^5 d}+\frac{\left (2 B^2 (b c-a d)^5 g^4 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{5 b^4 d}\\ &=-\frac{2 A B (b c-a d)^4 g^4 n x}{5 b^4}+\frac{13 B^2 (b c-a d)^4 g^4 n^2 x}{30 b^4}+\frac{7 B^2 (b c-a d)^3 g^4 n^2 (c+d x)^2}{60 b^3 d}+\frac{B^2 (b c-a d)^2 g^4 n^2 (c+d x)^3}{30 b^2 d}+\frac{13 B^2 (b c-a d)^5 g^4 n^2 \log (a+b x)}{30 b^5 d}+\frac{B^2 (b c-a d)^5 g^4 n^2 \log ^2(a+b x)}{5 b^5 d}-\frac{2 B^2 (b c-a d)^4 g^4 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{5 b^5}-\frac{B (b c-a d)^3 g^4 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac{2 B (b c-a d)^2 g^4 n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{15 b^2 d}-\frac{B (b c-a d) g^4 n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b d}-\frac{2 B (b c-a d)^5 g^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^5 d}+\frac{g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d}+\frac{2 B^2 (b c-a d)^5 g^4 n^2 \log (c+d x)}{5 b^5 d}-\frac{2 B^2 (b c-a d)^5 g^4 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{5 b^5 d}+\frac{\left (2 B^2 (b c-a d)^5 g^4 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 )}{5 b^5 d}\\ &=-\frac{2 A B (b c-a d)^4 g^4 n x}{5 b^4}+\frac{13 B^2 (b c-a d)^4 g^4 n^2 x}{30 b^4}+\frac{7 B^2 (b c-a d)^3 g^4 n^2 (c+d x)^2}{60 b^3 d}+\frac{B^2 (b c-a d)^2 g^4 n^2 (c+d x)^3}{30 b^2 d}+\frac{13 B^2 (b c-a d)^5 g^4 n^2 \log (a+b x)}{30 b^5 d}+\frac{B^2 (b c-a d)^5 g^4 n^2 \log ^2(a+b x)}{5 b^5 d}-\frac{2 B^2 (b c-a d)^4 g^4 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{5 b^5}-\frac{B (b c-a d)^3 g^4 n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^3 d}-\frac{2 B (b c-a d)^2 g^4 n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{15 b^2 d}-\frac{B (b c-a d) g^4 n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{10 b d}-\frac{2 B (b c-a d)^5 g^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{5 b^5 d}+\frac{g^4 (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{5 d}+\frac{2 B^2 (b c-a d)^5 g^4 n^2 \log (c+d x)}{5 b^5 d}-\frac{2 B^2 (b c-a d)^5 g^4 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{5 b^5 d}-\frac{2 B^2 (b c-a d)^5 g^4 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{5 b^5 d}\\ \end{align*}

Mathematica [A]  time = 0.504332, size = 533, normalized size = 0.98 \[ \frac{g^4 \left ((c+d x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2-\frac{B n (b c-a d) \left (-12 B n (b c-a d)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+12 b^2 (c+d x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+8 b^3 (c+d x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+6 b^4 (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+24 (b c-a d)^4 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+24 A b d x (b c-a d)^3-4 B n (b c-a d)^2 \left (2 b d x (b c-a d)+2 (b c-a d)^2 \log (a+b x)+b^2 (c+d x)^2\right )-B n (b c-a d) \left (3 b^2 (c+d x)^2 (b c-a d)+6 b d x (b c-a d)^2+6 (b c-a d)^3 \log (a+b x)+2 b^3 (c+d x)^3\right )+24 B d (a+b x) (b c-a d)^3 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-24 B n (b c-a d)^4 \log (c+d x)-12 B n (b c-a d)^3 ((b c-a d) \log (a+b x)+b d x)\right )}{12 b^5}\right )}{5 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^4*((c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - (B*(b*c - a*d)*n*(24*A*b*d*(b*c - a*d)^3*x - 12*B
*(b*c - a*d)^3*n*(b*d*x + (b*c - a*d)*Log[a + b*x]) - 4*B*(b*c - a*d)^2*n*(2*b*d*(b*c - a*d)*x + b^2*(c + d*x)
^2 + 2*(b*c - a*d)^2*Log[a + b*x]) - B*(b*c - a*d)*n*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c - a*d)*(c + d*x)^2 +
2*b^3*(c + d*x)^3 + 6*(b*c - a*d)^3*Log[a + b*x]) + 24*B*d*(b*c - a*d)^3*(a + b*x)*Log[e*((a + b*x)/(c + d*x))
^n] + 12*b^2*(b*c - a*d)^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 8*b^3*(b*c - a*d)*(c + d*x)^3*
(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 6*b^4*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 24*(b*c -
a*d)^4*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 24*B*(b*c - a*d)^4*n*Log[c + d*x] - 12*B*(b*c - a
*d)^4*n*(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)])))/(12*b^5)))/(5*d)

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Maple [F]  time = 0.441, size = 0, normalized size = 0. \begin{align*} \int \left ( dgx+cg \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((d*g*x+c*g)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,x)

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 3.79233, size = 3888, normalized size = 7.15 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*A*B*d^4*g^4*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/5*A^2*d^4*g^4*x^5 + 2*A*B*c*d^3*g^4*x^4*log(e*(
b*x/(d*x + c) + a/(d*x + c))^n) + A^2*c*d^3*g^4*x^4 + 4*A*B*c^2*d^2*g^4*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c)
)^n) + 2*A^2*c^2*d^2*g^4*x^3 + 4*A*B*c^3*d*g^4*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*A^2*c^3*d*g^4*x^
2 + 1/30*A*B*d^4*g^4*n*(12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4
*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4)) - 1/3*
A*B*c*d^3*g^4*n*(6*a^4*log(b*x + a)/b^4 - 6*c^4*log(d*x + c)/d^4 + (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2
*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3)) + 2*A*B*c^2*d^2*g^4*n*(2*a^3*log(b*x + a)/b^3 - 2*c^
3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2)) - 4*A*B*c^3*d*g^4*n*(a^2*l
og(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) + 2*A*B*c^4*g^4*n*(a*log(b*x + a)/b - c*log(d*x
+ c)/d) + 2*A*B*c^4*g^4*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2*c^4*g^4*x - 1/30*(77*a*b^3*c^4*d*g^4*n^
2 - 94*a^2*b^2*c^3*d^2*g^4*n^2 + 54*a^3*b*c^2*d^3*g^4*n^2 - 12*a^4*c*d^4*g^4*n^2 - (25*g^4*n^2 - 12*g^4*n*log(
e))*b^4*c^5)*B^2*log(d*x + c)/(b^4*d) - 2/5*(b^5*c^5*g^4*n^2 - 5*a*b^4*c^4*d*g^4*n^2 + 10*a^2*b^3*c^3*d^2*g^4*
n^2 - 10*a^3*b^2*c^2*d^3*g^4*n^2 + 5*a^4*b*c*d^4*g^4*n^2 - a^5*d^5*g^4*n^2)*(log(b*x + a)*log((b*d*x + a*d)/(b
*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*B^2/(b^5*d) + 1/60*(12*B^2*b^5*d^5*g^4*x^5*log(e)^2 + 24*B
^2*b^5*c^5*g^4*n^2*log(b*x + a)*log(d*x + c) - 12*B^2*b^5*c^5*g^4*n^2*log(d*x + c)^2 + 6*(a*b^4*d^5*g^4*n*log(
e) - (g^4*n*log(e) - 10*g^4*log(e)^2)*b^5*c*d^4)*B^2*x^4 + 2*((g^4*n^2 - 16*g^4*n*log(e) + 60*g^4*log(e)^2)*b^
5*c^2*d^3 - 2*(g^4*n^2 - 10*g^4*n*log(e))*a*b^4*c*d^4 + (g^4*n^2 - 4*g^4*n*log(e))*a^2*b^3*d^5)*B^2*x^3 + ((13
*g^4*n^2 - 72*g^4*n*log(e) + 120*g^4*log(e)^2)*b^5*c^3*d^2 - 3*(11*g^4*n^2 - 40*g^4*n*log(e))*a*b^4*c^2*d^3 +
3*(9*g^4*n^2 - 20*g^4*n*log(e))*a^2*b^3*c*d^4 - (7*g^4*n^2 - 12*g^4*n*log(e))*a^3*b^2*d^5)*B^2*x^2 - 12*(5*a*b
^4*c^4*d*g^4*n^2 - 10*a^2*b^3*c^3*d^2*g^4*n^2 + 10*a^3*b^2*c^2*d^3*g^4*n^2 - 5*a^4*b*c*d^4*g^4*n^2 + a^5*d^5*g
^4*n^2)*B^2*log(b*x + a)^2 + 2*((23*g^4*n^2 - 48*g^4*n*log(e) + 30*g^4*log(e)^2)*b^5*c^4*d - (79*g^4*n^2 - 120
*g^4*n*log(e))*a*b^4*c^3*d^2 + 6*(17*g^4*n^2 - 20*g^4*n*log(e))*a^2*b^3*c^2*d^3 - (59*g^4*n^2 - 60*g^4*n*log(e
))*a^3*b^2*c*d^4 + (13*g^4*n^2 - 12*g^4*n*log(e))*a^4*b*d^5)*B^2*x - 2*(12*(4*g^4*n^2 - 5*g^4*n*log(e))*a*b^4*
c^4*d - 12*(13*g^4*n^2 - 10*g^4*n*log(e))*a^2*b^3*c^3*d^2 + 4*(49*g^4*n^2 - 30*g^4*n*log(e))*a^3*b^2*c^2*d^3 -
(113*g^4*n^2 - 60*g^4*n*log(e))*a^4*b*c*d^4 + (25*g^4*n^2 - 12*g^4*n*log(e))*a^5*d^5)*B^2*log(b*x + a) + 12*(
B^2*b^5*d^5*g^4*x^5 + 5*B^2*b^5*c*d^4*g^4*x^4 + 10*B^2*b^5*c^2*d^3*g^4*x^3 + 10*B^2*b^5*c^3*d^2*g^4*x^2 + 5*B^
2*b^5*c^4*d*g^4*x)*log((b*x + a)^n)^2 + 12*(B^2*b^5*d^5*g^4*x^5 + 5*B^2*b^5*c*d^4*g^4*x^4 + 10*B^2*b^5*c^2*d^3
*g^4*x^3 + 10*B^2*b^5*c^3*d^2*g^4*x^2 + 5*B^2*b^5*c^4*d*g^4*x)*log((d*x + c)^n)^2 + 2*(12*B^2*b^5*d^5*g^4*x^5*
log(e) - 12*B^2*b^5*c^5*g^4*n*log(d*x + c) + 3*(a*b^4*d^5*g^4*n - (g^4*n - 20*g^4*log(e))*b^5*c*d^4)*B^2*x^4 +
4*(5*a*b^4*c*d^4*g^4*n - a^2*b^3*d^5*g^4*n - 2*(2*g^4*n - 15*g^4*log(e))*b^5*c^2*d^3)*B^2*x^3 + 6*(10*a*b^4*c
^2*d^3*g^4*n - 5*a^2*b^3*c*d^4*g^4*n + a^3*b^2*d^5*g^4*n - 2*(3*g^4*n - 10*g^4*log(e))*b^5*c^3*d^2)*B^2*x^2 +
12*(10*a*b^4*c^3*d^2*g^4*n - 10*a^2*b^3*c^2*d^3*g^4*n + 5*a^3*b^2*c*d^4*g^4*n - a^4*b*d^5*g^4*n - (4*g^4*n - 5
*g^4*log(e))*b^5*c^4*d)*B^2*x + 12*(5*a*b^4*c^4*d*g^4*n - 10*a^2*b^3*c^3*d^2*g^4*n + 10*a^3*b^2*c^2*d^3*g^4*n
- 5*a^4*b*c*d^4*g^4*n + a^5*d^5*g^4*n)*B^2*log(b*x + a))*log((b*x + a)^n) - 2*(12*B^2*b^5*d^5*g^4*x^5*log(e) -
12*B^2*b^5*c^5*g^4*n*log(d*x + c) + 3*(a*b^4*d^5*g^4*n - (g^4*n - 20*g^4*log(e))*b^5*c*d^4)*B^2*x^4 + 4*(5*a*
b^4*c*d^4*g^4*n - a^2*b^3*d^5*g^4*n - 2*(2*g^4*n - 15*g^4*log(e))*b^5*c^2*d^3)*B^2*x^3 + 6*(10*a*b^4*c^2*d^3*g
^4*n - 5*a^2*b^3*c*d^4*g^4*n + a^3*b^2*d^5*g^4*n - 2*(3*g^4*n - 10*g^4*log(e))*b^5*c^3*d^2)*B^2*x^2 + 12*(10*a
*b^4*c^3*d^2*g^4*n - 10*a^2*b^3*c^2*d^3*g^4*n + 5*a^3*b^2*c*d^4*g^4*n - a^4*b*d^5*g^4*n - (4*g^4*n - 5*g^4*log
(e))*b^5*c^4*d)*B^2*x + 12*(5*a*b^4*c^4*d*g^4*n - 10*a^2*b^3*c^3*d^2*g^4*n + 10*a^3*b^2*c^2*d^3*g^4*n - 5*a^4*
b*c*d^4*g^4*n + a^5*d^5*g^4*n)*B^2*log(b*x + a) + 12*(B^2*b^5*d^5*g^4*x^5 + 5*B^2*b^5*c*d^4*g^4*x^4 + 10*B^2*b
^5*c^2*d^3*g^4*x^3 + 10*B^2*b^5*c^3*d^2*g^4*x^2 + 5*B^2*b^5*c^4*d*g^4*x)*log((b*x + a)^n))*log((d*x + c)^n))/(
b^5*d)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (A^{2} d^{4} g^{4} x^{4} + 4 \, A^{2} c d^{3} g^{4} x^{3} + 6 \, A^{2} c^{2} d^{2} g^{4} x^{2} + 4 \, A^{2} c^{3} d g^{4} x + A^{2} c^{4} g^{4} +{\left (B^{2} d^{4} g^{4} x^{4} + 4 \, B^{2} c d^{3} g^{4} x^{3} + 6 \, B^{2} c^{2} d^{2} g^{4} x^{2} + 4 \, B^{2} c^{3} d g^{4} x + B^{2} c^{4} g^{4}\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \,{\left (A B d^{4} g^{4} x^{4} + 4 \, A B c d^{3} g^{4} x^{3} + 6 \, A B c^{2} d^{2} g^{4} x^{2} + 4 \, A B c^{3} d g^{4} x + A B c^{4} g^{4}\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d g x + c g\right )}^{4}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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