\(\int (g+h x)^2 (A+B \log (e (a+b x)^n (c+d x)^{-n}))^2 \, dx\) [303]

Optimal result

Integrand size = 33, antiderivative size = 570 \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx=\frac {B^2 (b c-a d)^2 h^2 n^2 x}{3 b^2 d^2}+\frac {B^2 (b c-a d)^3 h^2 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{3 b^3 d^3}+\frac {B^2 (b c-a d)^3 h^2 n^2 \log (c+d x)}{3 b^3 d^3}+\frac {2 B^2 (b c-a d)^2 h (3 b d g-2 b c h-a d h) n^2 \log (c+d x)}{3 b^3 d^3}-\frac {2 B (b c-a d) h (3 b d g-2 b c h-a d h) n (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{3 b^3 d^2}-\frac {B (b c-a d) h^2 n (c+d x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{3 b d^3}+\frac {2 B (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{3 b^3 d^3}-\frac {(b g-a h)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{3 b^3 h}+\frac {(g+h x)^3 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}{3 h}+\frac {2 B^2 (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{3 b^3 d^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 570, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2573, 2553, 2398, 2404, 2338, 2356, 46, 2351, 31, 2354, 2438} \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx=\frac {2 B n (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (c^2 h^2-3 c d g h+3 d^2 g^2\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{3 b^3 d^3}+\frac {2 B^2 n^2 (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (c^2 h^2-3 c d g h+3 d^2 g^2\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{3 b^3 d^3}-\frac {2 B h n (a+b x) (b c-a d) (-a d h-2 b c h+3 b d g) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{3 b^3 d^2}-\frac {(b g-a h)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{3 b^3 h}-\frac {B h^2 n (c+d x)^2 (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{3 b d^3}+\frac {(g+h x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{3 h}+\frac {2 B^2 h n^2 (b c-a d)^2 \log (c+d x) (-a d h-2 b c h+3 b d g)}{3 b^3 d^3}+\frac {B^2 h^2 n^2 (b c-a d)^3 \log \left (\frac {a+b x}{c+d x}\right )}{3 b^3 d^3}+\frac {B^2 h^2 n^2 (b c-a d)^3 \log (c+d x)}{3 b^3 d^3}+\frac {B^2 h^2 n^2 x (b c-a d)^2}{3 b^2 d^2} \]

[In]

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

[Out]

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

Rule 31

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

Rule 46

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

Rule 2338

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

Rule 2351

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

Rule 2354

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

Rule 2356

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

Rule 2398

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

Rule 2404

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

Rule 2438

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

Rule 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]

Rule 2573

Int[((A_.) + Log[(e_.)*(u_)^(n_.)*(v_)^(mn_)]*(B_.))^(p_.)*(w_.), x_Symbol] :> Subst[Int[w*(A + B*Log[e*(u/v)^
n])^p, x], e*(u/v)^n, e*(u^n/v^n)] /; FreeQ[{e, A, B, n, p}, x] && EqQ[n + mn, 0] && LinearQ[{u, v}, x] &&  !I
ntegerQ[n]

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

Mathematica [A] (verified)

Time = 1.07 (sec) , antiderivative size = 906, normalized size of antiderivative = 1.59 \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx=\frac {-a B^2 d^3 \left (3 b^2 g^2-3 a b g h+a^2 h^2\right ) n^2 \log ^2(a+b x)+B n \log (a+b x) \left (2 b^3 B c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right ) n \log (c+d x)+2 B \left (3 a b^2 d^3 g^2-3 a^2 b d^3 g h+a^3 d^3 h^2-b^3 c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n \log \left (\frac {b (c+d x)}{b c-a d}\right )+a d \left (2 A d^2 \left (3 b^2 g^2-3 a b g h+a^2 h^2\right )+B \left (-3 a^2 d^2 h^2+a b d h (6 d g+c h)+2 b^2 \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n+2 B d^2 \left (3 b^2 g^2-3 a b g h+a^2 h^2\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )+b \left (-b^2 B^2 c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right ) n^2 \log ^2(c+d x)+B n \log (c+d x) \left (-2 A b^2 c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )+B \left (2 a^2 c d^2 h^2-3 b^2 c^2 h (-2 d g+c h)+a b d \left (-6 d^2 g^2-6 c d g h+c^2 h^2\right )\right ) n-2 b^2 B c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )+d \left (a^2 B d^2 h^2 n (-2 A+B n) x+a b B n \left (A d^2 \left (-6 g^2+6 g h x+h^2 x^2\right )-2 B n \left (3 d^2 g^2+c^2 h^2+c d h (-3 g+h x)\right )\right )+b^2 x \left (B^2 c^2 h^2 n^2+A^2 d^2 \left (3 g^2+3 g h x+h^2 x^2\right )+A B c h n (2 c h-d (6 g+h x))\right )+B \left (-2 a^2 B d^2 h^2 n x+a b B d^2 n \left (-6 g^2+6 g h x+h^2 x^2\right )+b^2 x \left (B c h n (-6 d g+2 c h-d h x)+2 A d^2 \left (3 g^2+3 g h x+h^2 x^2\right )\right )\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )+b^2 B^2 d^2 x \left (3 g^2+3 g h x+h^2 x^2\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )+2 B^2 \left (3 a b^2 d^3 g^2-3 a^2 b d^3 g h+a^3 d^3 h^2-b^3 c \left (3 d^2 g^2-3 c d g h+c^2 h^2\right )\right ) n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )}{3 b^3 d^3} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 90.91 (sec) , antiderivative size = 8443, normalized size of antiderivative = 14.81

method	result	size
risch	\(\text {Expression too large to display}\)	\(8443\)

[In]

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

[Out]

result too large to display

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

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

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1671 vs. \(2 (549) = 1098\).

Time = 0.76 (sec) , antiderivative size = 1671, normalized size of antiderivative = 2.93 \[ \int (g+h x)^2 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

2/3*A*B*h^2*x^3*log((b*x + a)^n*e/(d*x + c)^n) + 1/3*A^2*h^2*x^3 + 2*A*B*g*h*x^2*log((b*x + a)^n*e/(d*x + c)^n
) + A^2*g*h*x^2 + 2*A*B*g^2*x*log((b*x + a)^n*e/(d*x + c)^n) + A^2*g^2*x + 2*(a*e*n*log(b*x + a)/b - c*e*n*log
(d*x + c)/d)*A*B*g^2/e - 2*(a^2*e*n*log(b*x + a)/b^2 - c^2*e*n*log(d*x + c)/d^2 + (b*c*e*n - a*d*e*n)*x/(b*d))
*A*B*g*h/e + 1/3*(2*a^3*e*n*log(b*x + a)/b^3 - 2*c^3*e*n*log(d*x + c)/d^3 - ((b^2*c*d*e*n - a*b*d^2*e*n)*x^2 -
2*(b^2*c^2*e*n - a^2*d^2*e*n)*x)/(b^2*d^2))*A*B*h^2/e + 1/3*(2*a^2*c*d^2*h^2*n^2 - (6*c*d^2*g*h*n^2 - c^2*d*h
^2*n^2)*a*b - (6*c*d^2*g^2*n*log(e) + (3*h^2*n^2 + 2*h^2*n*log(e))*c^3 - 6*(g*h*n^2 + g*h*n*log(e))*c^2*d)*b^2
)*B^2*log(d*x + c)/(b^2*d^3) + 2/3*(3*a*b^2*d^3*g^2*n^2 - 3*a^2*b*d^3*g*h*n^2 + a^3*d^3*h^2*n^2 - (3*c*d^2*g^2
*n^2 - 3*c^2*d*g*h*n^2 + c^3*h^2*n^2)*b^3)*(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^3*d^3) + 1/3*(B^2*b^3*d^3*h^2*x^3*log(e)^2 + 2*(3*c*d^2*g^2*n^2 - 3*c^2*d*g*h*n^2 +
c^3*h^2*n^2)*B^2*b^3*log(b*x + a)*log(d*x + c) - (3*c*d^2*g^2*n^2 - 3*c^2*d*g*h*n^2 + c^3*h^2*n^2)*B^2*b^3*log
(d*x + c)^2 + (a*b^2*d^3*h^2*n*log(e) - (c*d^2*h^2*n*log(e) - 3*d^3*g*h*log(e)^2)*b^3)*B^2*x^2 - (3*a*b^2*d^3*
g^2*n^2 - 3*a^2*b*d^3*g*h*n^2 + a^3*d^3*h^2*n^2)*B^2*log(b*x + a)^2 + ((h^2*n^2 - 2*h^2*n*log(e))*a^2*b*d^3 -
2*(c*d^2*h^2*n^2 - 3*d^3*g*h*n*log(e))*a*b^2 - (6*c*d^2*g*h*n*log(e) - 3*d^3*g^2*log(e)^2 - (h^2*n^2 + 2*h^2*n
*log(e))*c^2*d)*b^3)*B^2*x - ((3*h^2*n^2 - 2*h^2*n*log(e))*a^3*d^3 - (c*d^2*h^2*n^2 + 6*(g*h*n^2 - g*h*n*log(e
))*d^3)*a^2*b + 2*(3*c*d^2*g*h*n^2 - c^2*d*h^2*n^2 - 3*d^3*g^2*n*log(e))*a*b^2)*B^2*log(b*x + a) + (B^2*b^3*d^
3*h^2*x^3 + 3*B^2*b^3*d^3*g*h*x^2 + 3*B^2*b^3*d^3*g^2*x)*log((b*x + a)^n)^2 + (B^2*b^3*d^3*h^2*x^3 + 3*B^2*b^3
*d^3*g*h*x^2 + 3*B^2*b^3*d^3*g^2*x)*log((d*x + c)^n)^2 + (2*B^2*b^3*d^3*h^2*x^3*log(e) - 2*(3*c*d^2*g^2*n - 3*
c^2*d*g*h*n + c^3*h^2*n)*B^2*b^3*log(d*x + c) + (a*b^2*d^3*h^2*n - (c*d^2*h^2*n - 6*d^3*g*h*log(e))*b^3)*B^2*x
^2 + 2*(3*a*b^2*d^3*g*h*n - a^2*b*d^3*h^2*n - (3*c*d^2*g*h*n - c^2*d*h^2*n - 3*d^3*g^2*log(e))*b^3)*B^2*x + 2*
(3*a*b^2*d^3*g^2*n - 3*a^2*b*d^3*g*h*n + a^3*d^3*h^2*n)*B^2*log(b*x + a))*log((b*x + a)^n) - (2*B^2*b^3*d^3*h^
2*x^3*log(e) - 2*(3*c*d^2*g^2*n - 3*c^2*d*g*h*n + c^3*h^2*n)*B^2*b^3*log(d*x + c) + (a*b^2*d^3*h^2*n - (c*d^2*
h^2*n - 6*d^3*g*h*log(e))*b^3)*B^2*x^2 + 2*(3*a*b^2*d^3*g*h*n - a^2*b*d^3*h^2*n - (3*c*d^2*g*h*n - c^2*d*h^2*n
- 3*d^3*g^2*log(e))*b^3)*B^2*x + 2*(3*a*b^2*d^3*g^2*n - 3*a^2*b*d^3*g*h*n + a^3*d^3*h^2*n)*B^2*log(b*x + a) +
2*(B^2*b^3*d^3*h^2*x^3 + 3*B^2*b^3*d^3*g*h*x^2 + 3*B^2*b^3*d^3*g^2*x)*log((b*x + a)^n))*log((d*x + c)^n))/(b^
3*d^3)

Giac [F(-1)]

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

[In]

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

[Out]

Timed out

Mupad [F(-1)]

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

[In]

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

[Out]

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