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

Optimal. Leaf size=570 \[ \frac {2 B n (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (c^2 h^2-3 c d g h+3 d^2 g^2\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{3 b^3 d^3}+\frac {2 B^2 n^2 (b c-a d) \left (a^2 d^2 h^2-a b d h (3 d g-c h)+b^2 \left (c^2 h^2-3 c d g h+3 d^2 g^2\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{3 b^3 d^3}-\frac {2 B h n (a+b x) (b c-a d) (-a d h-2 b c h+3 b d g) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{3 b^3 d^2}-\frac {(b g-a h)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{3 b^3 h}-\frac {B h^2 n (c+d x)^2 (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{3 b d^3}+\frac {(g+h x)^3 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{3 h}+\frac {2 B^2 h n^2 (b c-a d)^2 \log (c+d x) (-a d h-2 b c h+3 b d g)}{3 b^3 d^3}+\frac {B^2 h^2 n^2 (b c-a d)^3 \log \left (\frac {a+b x}{c+d x}\right )}{3 b^3 d^3}+\frac {B^2 h^2 n^2 (b c-a d)^3 \log (c+d x)}{3 b^3 d^3}+\frac {B^2 h^2 n^2 x (b c-a d)^2}{3 b^2 d^2} \]

[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]  time = 1.29, antiderivative size = 697, normalized size of antiderivative = 1.22, number of steps used = 23, number of rules used = 11, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6742, 2492, 72, 2514, 2486, 31, 2488, 2411, 2343, 2333, 2315} \[ -\frac {2 B^2 n^2 (b g-a h)^3 \text {PolyLog}\left (2,\frac {b c-a d}{d (a+b x)}+1\right )}{3 b^3 h}-\frac {2 B^2 n^2 (d g-c h)^3 \text {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{3 d^3 h}+\frac {a^2 B^2 h^2 n^2 (b c-a d) \log (a+b x)}{3 b^3 d}-\frac {2 A B h n x (b c-a d) (-a d h-b c h+3 b d g)}{3 b^2 d^2}-\frac {2 A B n (b g-a h)^3 \log (a+b x)}{3 b^3 h}+\frac {2 A B (g+h x)^3 \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac {A B h^2 n x^2 (b c-a d)}{3 b d}-\frac {2 B^2 h n (a+b x) (b c-a d) (-a d h-b c h+3 b d g) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b^3 d^2}+\frac {2 B^2 h n^2 (b c-a d)^2 \log (c+d x) (-a d h-b c h+3 b d g)}{3 b^3 d^3}+\frac {B^2 h^2 n^2 x (b c-a d)^2}{3 b^2 d^2}+\frac {2 B^2 n (b g-a h)^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b^3 h}-\frac {B^2 c^2 h^2 n^2 (b c-a d) \log (c+d x)}{3 b d^3}-\frac {2 B^2 n (d g-c h)^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 d^3 h}+\frac {B^2 (g+h x)^3 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{3 h}-\frac {B^2 h^2 n x^2 (b c-a d) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{3 b d}+\frac {A^2 (g+h x)^3}{3 h}+\frac {2 A B n (d g-c h)^3 \log (c+d x)}{3 d^3 h} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*A*B*(b*c - a*d)*h*(3*b*d*g - b*c*h - a*d*h)*n*x)/(3*b^2*d^2) + (B^2*(b*c - a*d)^2*h^2*n^2*x)/(3*b^2*d^2) -
(A*B*(b*c - a*d)*h^2*n*x^2)/(3*b*d) + (A^2*(g + h*x)^3)/(3*h) - (2*A*B*(b*g - a*h)^3*n*Log[a + b*x])/(3*b^3*h
) + (a^2*B^2*(b*c - a*d)*h^2*n^2*Log[a + b*x])/(3*b^3*d) + (2*A*B*(d*g - c*h)^3*n*Log[c + d*x])/(3*d^3*h) - (B
^2*c^2*(b*c - a*d)*h^2*n^2*Log[c + d*x])/(3*b*d^3) + (2*B^2*(b*c - a*d)^2*h*(3*b*d*g - b*c*h - a*d*h)*n^2*Log[
c + d*x])/(3*b^3*d^3) - (B^2*(b*c - a*d)*h^2*n*x^2*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(3*b*d) - (2*B^2*(b*c - a
*d)*h*(3*b*d*g - b*c*h - a*d*h)*n*(a + b*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(3*b^3*d^2) + (2*A*B*(g + h*x)^3
*Log[(e*(a + b*x)^n)/(c + d*x)^n])/(3*h) + (2*B^2*(b*g - a*h)^3*n*Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[(e*(a
+ b*x)^n)/(c + d*x)^n])/(3*b^3*h) - (2*B^2*(d*g - c*h)^3*n*Log[(b*c - a*d)/(b*(c + d*x))]*Log[(e*(a + b*x)^n)/
(c + d*x)^n])/(3*d^3*h) + (B^2*(g + h*x)^3*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)/(3*h) - (2*B^2*(d*g - c*h)^3*n^
2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(3*d^3*h) - (2*B^2*(b*g - a*h)^3*n^2*PolyLog[2, 1 + (b*c - a*d)/(d*
(a + b*x))])/(3*b^3*h)

Rule 31

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

Rule 72

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

Rule 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)/(x_))^(q_.)*(x_)^(m_.), x_Symbol] :> Int[(e + d*
x)^q*(a + b*Log[c*x^n])^p, x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && EqQ[m, q] && IntegerQ[q]

Rule 2343

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

Rule 2411

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]

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 2488

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

Rule 2492

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

Rule 2514

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(RFx_), x_Symbol] :>
With[{u = ExpandIntegrand[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a,
b, c, d, e, f, p, q, r, s}, x] && RationalFunctionQ[RFx, x] && IGtQ[s, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

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

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Mathematica [A]  time = 1.84, size = 906, normalized size = 1.59 \[ \frac {-a B^2 \left (3 b^2 g^2-3 a b h g+a^2 h^2\right ) n^2 \log ^2(a+b x) d^3+B n \log (a+b x) \left (2 B c \left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) n \log (c+d x) b^3+2 B \left (-c \left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^3+3 a d^3 g^2 b^2-3 a^2 d^3 g h b+a^3 d^3 h^2\right ) n \log \left (\frac {b (c+d x)}{b c-a d}\right )+a d \left (2 A \left (3 b^2 g^2-3 a b h g+a^2 h^2\right ) d^2+2 B \left (3 b^2 g^2-3 a b h g+a^2 h^2\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) d^2+B \left (2 \left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^2+a d h (6 d g+c h) b-3 a^2 d^2 h^2\right ) n\right )\right )+b \left (-b^2 B^2 c \left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) n^2 \log ^2(c+d x)+B n \left (-2 A c \left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^2-2 B c \left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) b^2+B \left (-3 b^2 h (c h-2 d g) c^2+2 a^2 d^2 h^2 c+a b d \left (-6 d^2 g^2-6 c d h g+c^2 h^2\right )\right ) n\right ) \log (c+d x)+d \left (B^2 d^2 x \left (3 g^2+3 h x g+h^2 x^2\right ) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right ) b^2+x \left (A^2 \left (3 g^2+3 h x g+h^2 x^2\right ) d^2+B^2 c^2 h^2 n^2+A B c h n (2 c h-d (6 g+h x))\right ) b^2+a B n \left (A d^2 \left (-6 g^2+6 h x g+h^2 x^2\right )-2 B n \left (3 d^2 g^2+c^2 h^2+c d h (h x-3 g)\right )\right ) b+a^2 B d^2 h^2 n (B n-2 A) x+B \left (x \left (2 A \left (3 g^2+3 h x g+h^2 x^2\right ) d^2+B c h n (-6 d g+2 c h-d h x)\right ) b^2+a B d^2 n \left (-6 g^2+6 h x g+h^2 x^2\right ) b-2 a^2 B d^2 h^2 n x\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )\right )+2 B^2 \left (-c \left (3 d^2 g^2-3 c d h g+c^2 h^2\right ) b^3+3 a d^3 g^2 b^2-3 a^2 d^3 g h b+a^3 d^3 h^2\right ) n^2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )}{3 b^3 d^3} \]

Antiderivative was successfully verified.

[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)

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fricas [F]  time = 1.14, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} h^{2} x^{2} + 2 \, A^{2} g h x + A^{2} g^{2} + {\left (B^{2} h^{2} x^{2} + 2 \, B^{2} g h x + B^{2} g^{2}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 2 \, {\left (A B h^{2} x^{2} + 2 \, A B g h x + A B g^{2}\right )} \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [C]  time = 4.82, size = 22955, normalized size = 40.27 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

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maxima [B]  time = 6.79, size = 1671, normalized size = 2.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (g+h\,x\right )}^2\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: HeuristicGCDFailed

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