\(\int (f+g x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [67]

Optimal result

Integrand size = 32, antiderivative size = 923 \[ \int (f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {B^2 (b c-a d)^3 g^3 n^2 x}{6 b^3 d^3}+\frac {B^2 (b c-a d)^2 g^2 (4 b d f-3 b c g-a d g) n^2 x}{4 b^3 d^3}+\frac {B^2 (b c-a d)^2 g^3 n^2 (c+d x)^2}{12 b^2 d^4}-\frac {B (b c-a d) g \left (a^2 d^2 g^2-2 a b d g (2 d f-c g)+b^2 \left (6 d^2 f^2-8 c d f g+3 c^2 g^2\right )\right ) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d^3}-\frac {B (b c-a d) g^2 (4 b d f-3 b c g-a d g) n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d^4}-\frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b d^4}-\frac {(b f-a g)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{2 b^4 d^4}+\frac {B^2 (b c-a d)^4 g^3 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{6 b^4 d^4}+\frac {B^2 (b c-a d)^3 g^2 (4 b d f-3 b c g-a d g) n^2 \log \left (\frac {a+b x}{c+d x}\right )}{4 b^4 d^4}+\frac {B^2 (b c-a d)^4 g^3 n^2 \log (c+d x)}{6 b^4 d^4}+\frac {B^2 (b c-a d)^3 g^2 (4 b d f-3 b c g-a d g) n^2 \log (c+d x)}{4 b^4 d^4}+\frac {B^2 (b c-a d)^2 g \left (a^2 d^2 g^2-2 a b d g (2 d f-c g)+b^2 \left (6 d^2 f^2-8 c d f g+3 c^2 g^2\right )\right ) n^2 \log (c+d x)}{2 b^4 d^4}-\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{2 b^4 d^4} \]

[Out]

1/6*B^2*(-a*d+b*c)^3*g^3*n^2*x/b^3/d^3+1/4*B^2*(-a*d+b*c)^2*g^2*(-a*d*g-3*b*c*g+4*b*d*f)*n^2*x/b^3/d^3+1/12*B^
2*(-a*d+b*c)^2*g^3*n^2*(d*x+c)^2/b^2/d^4-1/2*B*(-a*d+b*c)*g*(a^2*d^2*g^2-2*a*b*d*g*(-c*g+2*d*f)+b^2*(3*c^2*g^2
-8*c*d*f*g+6*d^2*f^2))*n*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^4/d^3-1/4*B*(-a*d+b*c)*g^2*(-a*d*g-3*b*c*g+
4*b*d*f)*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^2/d^4-1/6*B*(-a*d+b*c)*g^3*n*(d*x+c)^3*(A+B*ln(e*((b*x+
a)/(d*x+c))^n))/b/d^4-1/4*(-a*g+b*f)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^4/g+1/4*(g*x+f)^4*(A+B*ln(e*((b*x+a
)/(d*x+c))^n))^2/g-1/2*B*(-a*d+b*c)*(-a*d*g-b*c*g+2*b*d*f)*(2*a*b*d^2*f*g-a^2*d^2*g^2-b^2*(c^2*g^2-2*c*d*f*g+2
*d^2*f^2))*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln((-a*d+b*c)/b/(d*x+c))/b^4/d^4+1/6*B^2*(-a*d+b*c)^4*g^3*n^2*ln(
(b*x+a)/(d*x+c))/b^4/d^4+1/4*B^2*(-a*d+b*c)^3*g^2*(-a*d*g-3*b*c*g+4*b*d*f)*n^2*ln((b*x+a)/(d*x+c))/b^4/d^4+1/6
*B^2*(-a*d+b*c)^4*g^3*n^2*ln(d*x+c)/b^4/d^4+1/4*B^2*(-a*d+b*c)^3*g^2*(-a*d*g-3*b*c*g+4*b*d*f)*n^2*ln(d*x+c)/b^
4/d^4+1/2*B^2*(-a*d+b*c)^2*g*(a^2*d^2*g^2-2*a*b*d*g*(-c*g+2*d*f)+b^2*(3*c^2*g^2-8*c*d*f*g+6*d^2*f^2))*n^2*ln(d
*x+c)/b^4/d^4-1/2*B^2*(-a*d+b*c)*(-a*d*g-b*c*g+2*b*d*f)*(2*a*b*d^2*f*g-a^2*d^2*g^2-b^2*(c^2*g^2-2*c*d*f*g+2*d^
2*f^2))*n^2*polylog(2,d*(b*x+a)/b/(d*x+c))/b^4/d^4

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 923, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.312, Rules used = {2553, 2398, 2404, 2338, 2356, 46, 2351, 31, 2354, 2438} \[ \int (f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {B^2 g^3 n^2 \log \left (\frac {a+b x}{c+d x}\right ) (b c-a d)^4}{6 b^4 d^4}+\frac {B^2 g^3 n^2 \log (c+d x) (b c-a d)^4}{6 b^4 d^4}+\frac {B^2 g^3 n^2 x (b c-a d)^3}{6 b^3 d^3}+\frac {B^2 g^2 (4 b d f-3 b c g-a d g) n^2 \log \left (\frac {a+b x}{c+d x}\right ) (b c-a d)^3}{4 b^4 d^4}+\frac {B^2 g^2 (4 b d f-3 b c g-a d g) n^2 \log (c+d x) (b c-a d)^3}{4 b^4 d^4}+\frac {B^2 g^3 n^2 (c+d x)^2 (b c-a d)^2}{12 b^2 d^4}+\frac {B^2 g^2 (4 b d f-3 b c g-a d g) n^2 x (b c-a d)^2}{4 b^3 d^3}+\frac {B^2 g \left (\left (6 d^2 f^2-8 c d g f+3 c^2 g^2\right ) b^2-2 a d g (2 d f-c g) b+a^2 d^2 g^2\right ) n^2 \log (c+d x) (b c-a d)^2}{2 b^4 d^4}-\frac {B g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)}{6 b d^4}-\frac {B g^2 (4 b d f-3 b c g-a d g) n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)}{4 b^2 d^4}-\frac {B g \left (\left (6 d^2 f^2-8 c d g f+3 c^2 g^2\right ) b^2-2 a d g (2 d f-c g) b+a^2 d^2 g^2\right ) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)}{2 b^4 d^3}-\frac {B (2 b d f-b c g-a d g) \left (-\left (\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2\right )+2 a d^2 f g b-a^2 d^2 g^2\right ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right ) (b c-a d)}{2 b^4 d^4}-\frac {B^2 (2 b d f-b c g-a d g) \left (-\left (\left (2 d^2 f^2-2 c d g f+c^2 g^2\right ) b^2\right )+2 a d^2 f g b-a^2 d^2 g^2\right ) n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) (b c-a d)}{2 b^4 d^4}-\frac {(b f-a g)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g} \]

[In]

Int[(f + g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2,x]

[Out]

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

Rule 31

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

Rule 46

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

Rule 2338

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

Rule 2351

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

Rule 2354

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

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2398

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

Rule 2404

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

Rule 2438

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

Rule 2553

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

Rubi steps \begin{align*} \text {integral}& = (b c-a d) \text {Subst}\left (\int \frac {(b f-a g-(d f-c g) x)^3 \left (A+B \log \left (e x^n\right )\right )^2}{(b-d x)^5} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = \frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g}-\frac {(B n) \text {Subst}\left (\int \frac {(b f-a g+(-d f+c g) x)^4 \left (A+B \log \left (e x^n\right )\right )}{x (b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 g} \\ & = \frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g}-\frac {(B n) \text {Subst}\left (\int \left (\frac {(b f-a g)^4 \left (A+B \log \left (e x^n\right )\right )}{b^4 x}+\frac {(b c-a d)^4 g^4 \left (A+B \log \left (e x^n\right )\right )}{b d^3 (b-d x)^4}+\frac {(b c-a d)^3 g^3 (4 b d f-3 b c g-a d g) \left (A+B \log \left (e x^n\right )\right )}{b^2 d^3 (b-d x)^3}+\frac {(b c-a d)^2 g^2 \left (a^2 d^2 g^2-2 a b d g (2 d f-c g)+b^2 \left (6 d^2 f^2-8 c d f g+3 c^2 g^2\right )\right ) \left (A+B \log \left (e x^n\right )\right )}{b^3 d^3 (b-d x)^2}+\frac {(b c-a d) g (2 b d f-b c g-a d g) \left (2 b^2 d^2 f^2-2 b^2 c d f g-2 a b d^2 f g+b^2 c^2 g^2+a^2 d^2 g^2\right ) \left (A+B \log \left (e x^n\right )\right )}{b^4 d^3 (b-d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{2 g} \\ & = \frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g}-\frac {\left (B (b c-a d)^4 g^3 n\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{(b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 b d^3}-\frac {\left (B (b f-a g)^4 n\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 b^4 g}-\frac {\left (B (b c-a d)^3 g^2 (4 b d f-3 b c g-a d g) n\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{(b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 b^2 d^3}+\frac {\left (B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 b^4 d^3}-\frac {\left (B (b c-a d)^2 g \left (a^2 d^2 g^2-2 a b d g (2 d f-c g)+b^2 \left (6 d^2 f^2-8 c d f g+3 c^2 g^2\right )\right ) n\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^n\right )}{(b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 b^3 d^3} \\ & = -\frac {B (b c-a d) g \left (a^2 d^2 g^2-2 a b d g (2 d f-c g)+b^2 \left (6 d^2 f^2-8 c d f g+3 c^2 g^2\right )\right ) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d^3}-\frac {B (b c-a d) g^2 (4 b d f-3 b c g-a d g) n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d^4}-\frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b d^4}-\frac {(b f-a g)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{2 b^4 d^4}+\frac {\left (B^2 (b c-a d)^4 g^3 n^2\right ) \text {Subst}\left (\int \frac {1}{x (b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{6 b d^4}+\frac {\left (B^2 (b c-a d)^3 g^2 (4 b d f-3 b c g-a d g) n^2\right ) \text {Subst}\left (\int \frac {1}{x (b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{4 b^2 d^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 ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 b^4 d^4}+\frac {\left (B^2 (b c-a d)^2 g \left (a^2 d^2 g^2-2 a b d g (2 d f-c g)+b^2 \left (6 d^2 f^2-8 c d f g+3 c^2 g^2\right )\right ) n^2\right ) \text {Subst}\left (\int \frac {1}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 b^4 d^3} \\ & = -\frac {B (b c-a d) g \left (a^2 d^2 g^2-2 a b d g (2 d f-c g)+b^2 \left (6 d^2 f^2-8 c d f g+3 c^2 g^2\right )\right ) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d^3}-\frac {B (b c-a d) g^2 (4 b d f-3 b c g-a d g) n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d^4}-\frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b d^4}-\frac {(b f-a g)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{2 b^4 d^4}+\frac {B^2 (b c-a d)^2 g \left (a^2 d^2 g^2-2 a b d g (2 d f-c g)+b^2 \left (6 d^2 f^2-8 c d f g+3 c^2 g^2\right )\right ) n^2 \log (c+d x)}{2 b^4 d^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 (a+b x)}{b (c+d x)}\right )}{2 b^4 d^4}+\frac {\left (B^2 (b c-a d)^4 g^3 n^2\right ) \text {Subst}\left (\int \left (\frac {1}{b^3 x}+\frac {d}{b (b-d x)^3}+\frac {d}{b^2 (b-d x)^2}+\frac {d}{b^3 (b-d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{6 b d^4}+\frac {\left (B^2 (b c-a d)^3 g^2 (4 b d f-3 b c g-a d g) n^2\right ) \text {Subst}\left (\int \left (\frac {1}{b^2 x}+\frac {d}{b (b-d x)^2}+\frac {d}{b^2 (b-d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{4 b^2 d^4} \\ & = \frac {B^2 (b c-a d)^3 g^3 n^2 x}{6 b^3 d^3}+\frac {B^2 (b c-a d)^2 g^2 (4 b d f-3 b c g-a d g) n^2 x}{4 b^3 d^3}+\frac {B^2 (b c-a d)^2 g^3 n^2 (c+d x)^2}{12 b^2 d^4}-\frac {B (b c-a d) g \left (a^2 d^2 g^2-2 a b d g (2 d f-c g)+b^2 \left (6 d^2 f^2-8 c d f g+3 c^2 g^2\right )\right ) n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d^3}-\frac {B (b c-a d) g^2 (4 b d f-3 b c g-a d g) n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d^4}-\frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b d^4}-\frac {(b f-a g)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^4 g}+\frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 g}-\frac {B (b c-a d) (2 b d f-b c g-a d g) \left (2 a b d^2 f g-a^2 d^2 g^2-b^2 \left (2 d^2 f^2-2 c d f g+c^2 g^2\right )\right ) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{2 b^4 d^4}+\frac {B^2 (b c-a d)^4 g^3 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{6 b^4 d^4}+\frac {B^2 (b c-a d)^3 g^2 (4 b d f-3 b c g-a d g) n^2 \log \left (\frac {a+b x}{c+d x}\right )}{4 b^4 d^4}+\frac {B^2 (b c-a d)^4 g^3 n^2 \log (c+d x)}{6 b^4 d^4}+\frac {B^2 (b c-a d)^3 g^2 (4 b d f-3 b c g-a d g) n^2 \log (c+d x)}{4 b^4 d^4}+\frac {B^2 (b c-a d)^2 g \left (a^2 d^2 g^2-2 a b d g (2 d f-c g)+b^2 \left (6 d^2 f^2-8 c d f g+3 c^2 g^2\right )\right ) n^2 \log (c+d x)}{2 b^4 d^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 (a+b x)}{b (c+d x)}\right )}{2 b^4 d^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.66 (sec) , antiderivative size = 757, normalized size of antiderivative = 0.82 \[ \int (f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {(f+g x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-\frac {B n \left (6 A b d (b c-a d) g^2 \left (a^2 d^2 g^2+a b d g (-4 d f+c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) x+6 B d (b c-a d) g^2 \left (a^2 d^2 g^2+a b d g (-4 d f+c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 b^2 d^2 (b c-a d) g^3 (4 b d f-b c g-a d g) x^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 b^3 d^3 (b c-a d) g^4 x^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+6 d^4 (b f-a g)^4 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 B (b c-a d)^2 g^2 \left (a^2 d^2 g^2+a b d g (-4 d f+c g)+b^2 \left (6 d^2 f^2-4 c d f g+c^2 g^2\right )\right ) n \log (c+d x)-6 b^4 (d f-c g)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+B (b c-a d) g^4 n \left (b d (b c-a d) x (2 b c+2 a d-b d x)+2 a^3 d^3 \log (a+b x)-2 b^3 c^3 \log (c+d x)\right )-3 B (b c-a d) g^3 (-4 b d f+b c g+a d g) n \left (-a^2 d^2 \log (a+b x)+b \left (d (-b c+a d) x+b c^2 \log (c+d x)\right )\right )-3 B d^4 (b f-a g)^4 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+3 b^4 B (d f-c g)^4 n \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{3 b^4 d^4}}{4 g} \]

[In]

Integrate[(f + g*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2,x]

[Out]

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

Maple [F]

\[\int \left (g x +f \right )^{3} {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}d x\]

[In]

int((g*x+f)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,x)

[Out]

int((g*x+f)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2,x)

Fricas [F]

\[ \int (f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (g x + f\right )}^{3} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} \,d x } \]

[In]

integrate((g*x+f)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="fricas")

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int (f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Timed out} \]

[In]

integrate((g*x+f)**3*(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2651 vs. \(2 (892) = 1784\).

Time = 0.71 (sec) , antiderivative size = 2651, normalized size of antiderivative = 2.87 \[ \int (f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

integrate((g*x+f)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")

[Out]

1/2*A*B*g^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A^2*g^3*x^4 + 2*A*B*f*g^2*x^3*log(e*(b*x/(d*x + c
) + a/(d*x + c))^n) + A^2*f*g^2*x^3 + 3*A*B*f^2*g*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/2*A^2*f^2*g*x
^2 - 1/12*A*B*g^3*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)) + A*B*f*g^2*n*(2*a^3*log(b*x + a)/b^3 - 2*c^3*lo
g(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)) - 3*A*B*f^2*g*n*(a^2*log(b*x +
a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) + 2*A*B*f^3*n*(a*log(b*x + a)/b - c*log(d*x + c)/d) + 2*
A*B*f^3*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2*f^3*x - 1/12*(6*a^3*c*d^3*g^3*n^2 - 3*(8*c*d^3*f*g^2*n^
2 - c^2*d^2*g^3*n^2)*a^2*b + 2*(18*c*d^3*f^2*g*n^2 - 6*c^2*d^2*f*g^2*n^2 + c^3*d*g^3*n^2)*a*b^2 + (24*c*d^3*f^
3*n*log(e) - (11*g^3*n^2 + 6*g^3*n*log(e))*c^4 + 12*(3*f*g^2*n^2 + 2*f*g^2*n*log(e))*c^3*d - 36*(f^2*g*n^2 + f
^2*g*n*log(e))*c^2*d^2)*b^3)*B^2*log(d*x + c)/(b^3*d^4) + 1/2*(4*a*b^3*d^4*f^3*n^2 - 6*a^2*b^2*d^4*f^2*g*n^2 +
4*a^3*b*d^4*f*g^2*n^2 - a^4*d^4*g^3*n^2 - (4*c*d^3*f^3*n^2 - 6*c^2*d^2*f^2*g*n^2 + 4*c^3*d*f*g^2*n^2 - c^4*g^
3*n^2)*b^4)*(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^4*d^4
) + 1/12*(3*B^2*b^4*d^4*g^3*x^4*log(e)^2 + 6*(4*c*d^3*f^3*n^2 - 6*c^2*d^2*f^2*g*n^2 + 4*c^3*d*f*g^2*n^2 - c^4*
g^3*n^2)*B^2*b^4*log(b*x + a)*log(d*x + c) - 3*(4*c*d^3*f^3*n^2 - 6*c^2*d^2*f^2*g*n^2 + 4*c^3*d*f*g^2*n^2 - c^
4*g^3*n^2)*B^2*b^4*log(d*x + c)^2 + 2*(a*b^3*d^4*g^3*n*log(e) - (c*d^3*g^3*n*log(e) - 6*d^4*f*g^2*log(e)^2)*b^
4)*B^2*x^3 + ((g^3*n^2 - 3*g^3*n*log(e))*a^2*b^2*d^4 - 2*(c*d^3*g^3*n^2 - 6*d^4*f*g^2*n*log(e))*a*b^3 - (12*c*
d^3*f*g^2*n*log(e) - 18*d^4*f^2*g*log(e)^2 - (g^3*n^2 + 3*g^3*n*log(e))*c^2*d^2)*b^4)*B^2*x^2 - 3*(4*a*b^3*d^4
*f^3*n^2 - 6*a^2*b^2*d^4*f^2*g*n^2 + 4*a^3*b*d^4*f*g^2*n^2 - a^4*d^4*g^3*n^2)*B^2*log(b*x + a)^2 - ((5*g^3*n^2
- 6*g^3*n*log(e))*a^3*b*d^4 - (5*c*d^3*g^3*n^2 + 12*(f*g^2*n^2 - 2*f*g^2*n*log(e))*d^4)*a^2*b^2 + (24*c*d^3*f
*g^2*n^2 - 5*c^2*d^2*g^3*n^2 - 36*d^4*f^2*g*n*log(e))*a*b^3 + (36*c*d^3*f^2*g*n*log(e) - 12*d^4*f^3*log(e)^2 +
(5*g^3*n^2 + 6*g^3*n*log(e))*c^3*d - 12*(f*g^2*n^2 + 2*f*g^2*n*log(e))*c^2*d^2)*b^4)*B^2*x + ((11*g^3*n^2 - 6
*g^3*n*log(e))*a^4*d^4 - 2*(c*d^3*g^3*n^2 + 6*(3*f*g^2*n^2 - 2*f*g^2*n*log(e))*d^4)*a^3*b + 3*(4*c*d^3*f*g^2*n
^2 - c^2*d^2*g^3*n^2 + 12*(f^2*g*n^2 - f^2*g*n*log(e))*d^4)*a^2*b^2 - 6*(6*c*d^3*f^2*g*n^2 - 4*c^2*d^2*f*g^2*n
^2 + c^3*d*g^3*n^2 - 4*d^4*f^3*n*log(e))*a*b^3)*B^2*log(b*x + a) + 3*(B^2*b^4*d^4*g^3*x^4 + 4*B^2*b^4*d^4*f*g^
2*x^3 + 6*B^2*b^4*d^4*f^2*g*x^2 + 4*B^2*b^4*d^4*f^3*x)*log((b*x + a)^n)^2 + 3*(B^2*b^4*d^4*g^3*x^4 + 4*B^2*b^4
*d^4*f*g^2*x^3 + 6*B^2*b^4*d^4*f^2*g*x^2 + 4*B^2*b^4*d^4*f^3*x)*log((d*x + c)^n)^2 + (6*B^2*b^4*d^4*g^3*x^4*lo
g(e) - 6*(4*c*d^3*f^3*n - 6*c^2*d^2*f^2*g*n + 4*c^3*d*f*g^2*n - c^4*g^3*n)*B^2*b^4*log(d*x + c) + 2*(a*b^3*d^4
*g^3*n - (c*d^3*g^3*n - 12*d^4*f*g^2*log(e))*b^4)*B^2*x^3 + 3*(4*a*b^3*d^4*f*g^2*n - a^2*b^2*d^4*g^3*n - (4*c*
d^3*f*g^2*n - c^2*d^2*g^3*n - 12*d^4*f^2*g*log(e))*b^4)*B^2*x^2 + 6*(6*a*b^3*d^4*f^2*g*n - 4*a^2*b^2*d^4*f*g^2
*n + a^3*b*d^4*g^3*n - (6*c*d^3*f^2*g*n - 4*c^2*d^2*f*g^2*n + c^3*d*g^3*n - 4*d^4*f^3*log(e))*b^4)*B^2*x + 6*(
4*a*b^3*d^4*f^3*n - 6*a^2*b^2*d^4*f^2*g*n + 4*a^3*b*d^4*f*g^2*n - a^4*d^4*g^3*n)*B^2*log(b*x + a))*log((b*x +
a)^n) - (6*B^2*b^4*d^4*g^3*x^4*log(e) - 6*(4*c*d^3*f^3*n - 6*c^2*d^2*f^2*g*n + 4*c^3*d*f*g^2*n - c^4*g^3*n)*B^
2*b^4*log(d*x + c) + 2*(a*b^3*d^4*g^3*n - (c*d^3*g^3*n - 12*d^4*f*g^2*log(e))*b^4)*B^2*x^3 + 3*(4*a*b^3*d^4*f*
g^2*n - a^2*b^2*d^4*g^3*n - (4*c*d^3*f*g^2*n - c^2*d^2*g^3*n - 12*d^4*f^2*g*log(e))*b^4)*B^2*x^2 + 6*(6*a*b^3*
d^4*f^2*g*n - 4*a^2*b^2*d^4*f*g^2*n + a^3*b*d^4*g^3*n - (6*c*d^3*f^2*g*n - 4*c^2*d^2*f*g^2*n + c^3*d*g^3*n - 4
*d^4*f^3*log(e))*b^4)*B^2*x + 6*(4*a*b^3*d^4*f^3*n - 6*a^2*b^2*d^4*f^2*g*n + 4*a^3*b*d^4*f*g^2*n - a^4*d^4*g^3
*n)*B^2*log(b*x + a) + 6*(B^2*b^4*d^4*g^3*x^4 + 4*B^2*b^4*d^4*f*g^2*x^3 + 6*B^2*b^4*d^4*f^2*g*x^2 + 4*B^2*b^4*
d^4*f^3*x)*log((b*x + a)^n))*log((d*x + c)^n))/(b^4*d^4)

Giac [F(-1)]

Timed out. \[ \int (f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Timed out} \]

[In]

integrate((g*x+f)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int (f+g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int {\left (f+g\,x\right )}^3\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]

[In]

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

[Out]

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