3.308 \(\int \frac{(A+B \log (e (a+b x)^n (c+d x)^{-n}))^2}{(g+h x)^3} \, dx\)

Optimal. Leaf size=393 \[ \frac{B^2 n^2 (b c-a d) (-a d h-b c h+2 b d g) \text{PolyLog}\left (2,\frac{(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{b^2 \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 h (b g-a h)^2}+\frac{B h n (a+b x) (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(g+h x) (b g-a h)^2 (d g-c h)}+\frac{B n (b c-a d) (-a d h-b c h+2 b d g) \log \left (1-\frac{(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(b g-a h)^2 (d g-c h)^2}-\frac{\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{2 h (g+h x)^2}+\frac{B^2 h n^2 (b c-a d)^2 \log \left (\frac{g+h x}{c+d x}\right )}{(b g-a h)^2 (d g-c h)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [B]  time = 1.63057, antiderivative size = 968, normalized size of antiderivative = 2.46, number of steps used = 29, number of rules used = 16, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.485, Rules used = {6742, 2492, 72, 2514, 2488, 2411, 2343, 2333, 2315, 2490, 36, 31, 2494, 2394, 2393, 2391} \[ -\frac{A^2}{2 h (g+h x)^2}+\frac{b^2 B n \log (a+b x) A}{h (b g-a h)^2}-\frac{B d^2 n \log (c+d x) A}{h (d g-c h)^2}-\frac{B \log \left (e (a+b x)^n (c+d x)^{-n}\right ) A}{h (g+h x)^2}+\frac{B (b c-a d) (2 b d g-b c h-a d h) n \log (g+h x) A}{(b g-a h)^2 (d g-c h)^2}-\frac{B (b c-a d) n A}{(b g-a h) (d g-c h) (g+h x)}-\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{2 h (g+h x)^2}-\frac{B^2 (b c-a d)^2 h n^2 \log (c+d x)}{(b g-a h)^2 (d g-c h)^2}-\frac{b^2 B^2 n \log \left (-\frac{b c-a d}{d (a+b x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (b g-a h)^2}+\frac{B^2 d^2 n \log \left (\frac{b c-a d}{b (c+d x)}\right ) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{h (d g-c h)^2}+\frac{B^2 (b c-a d) h n (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h)^2 (d g-c h) (g+h x)}+\frac{B^2 (b c-a d)^2 h n^2 \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}-\frac{B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (-\frac{h (a+b x)}{b g-a h}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac{B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \log \left (-\frac{h (c+d x)}{d g-c h}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac{B^2 (b c-a d) (2 b d g-b c h-a d h) n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log (g+h x)}{(b g-a h)^2 (d g-c h)^2}+\frac{B^2 d^2 n^2 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{h (d g-c h)^2}-\frac{B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \text{PolyLog}\left (2,\frac{b (g+h x)}{b g-a h}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{B^2 (b c-a d) (2 b d g-b c h-a d h) n^2 \text{PolyLog}\left (2,\frac{d (g+h x)}{d g-c h}\right )}{(b g-a h)^2 (d g-c h)^2}+\frac{b^2 B^2 n^2 \text{PolyLog}\left (2,\frac{b c-a d}{d (a+b x)}+1\right )}{h (b g-a h)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6742

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

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 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 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 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 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 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 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 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 2490

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

Rule 36

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]

Rule 31

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

Rule 2494

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

Rule 2394

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

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

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

Rubi steps

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

Mathematica [B]  time = 6.46763, size = 15422, normalized size = 39.24 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Result too large to show

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Maple [F]  time = 1.788, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( hx+g \right ) ^{3}} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{1}{2} \, B^{2}{\left (\frac{\log \left ({\left (d x + c\right )}^{n}\right )^{2}}{h^{3} x^{2} + 2 \, g h^{2} x + g^{2} h} + 2 \, \int -\frac{d h x \log \left (e\right )^{2} + c h \log \left (e\right )^{2} +{\left (d h x + c h\right )} \log \left ({\left (b x + a\right )}^{n}\right )^{2} + 2 \,{\left (d h x \log \left (e\right ) + c h \log \left (e\right )\right )} \log \left ({\left (b x + a\right )}^{n}\right ) +{\left (d g n +{\left (h n - 2 \, h \log \left (e\right )\right )} d x - 2 \, c h \log \left (e\right ) - 2 \,{\left (d h x + c h\right )} \log \left ({\left (b x + a\right )}^{n}\right )\right )} \log \left ({\left (d x + c\right )}^{n}\right )}{d h^{4} x^{4} + c g^{3} h +{\left (3 \, d g h^{3} + c h^{4}\right )} x^{3} + 3 \,{\left (d g^{2} h^{2} + c g h^{3}\right )} x^{2} +{\left (d g^{3} h + 3 \, c g^{2} h^{2}\right )} x}\,{d x}\right )} + \frac{{\left (\frac{b^{2} e n \log \left (b x + a\right )}{b^{2} g^{2} h - 2 \, a b g h^{2} + a^{2} h^{3}} - \frac{d^{2} e n \log \left (d x + c\right )}{d^{2} g^{2} h - 2 \, c d g h^{2} + c^{2} h^{3}} - \frac{{\left (2 \, a b d^{2} e g n - a^{2} d^{2} e h n -{\left (2 \, c d e g n - c^{2} e h n\right )} b^{2}\right )} \log \left (h x + g\right )}{{\left (d^{2} g^{2} h^{2} - 2 \, c d g h^{3} + c^{2} h^{4}\right )} a^{2} - 2 \,{\left (d^{2} g^{3} h - 2 \, c d g^{2} h^{2} + c^{2} g h^{3}\right )} a b +{\left (d^{2} g^{4} - 2 \, c d g^{3} h + c^{2} g^{2} h^{2}\right )} b^{2}} + \frac{b c e n - a d e n}{{\left (d g^{2} h - c g h^{2}\right )} a -{\left (d g^{3} - c g^{2} h\right )} b +{\left ({\left (d g h^{2} - c h^{3}\right )} a -{\left (d g^{2} h - c g h^{2}\right )} b\right )} x}\right )} A B}{e} - \frac{A B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{h^{3} x^{2} + 2 \, g h^{2} x + g^{2} h} - \frac{A^{2}}{2 \,{\left (h^{3} x^{2} + 2 \, g h^{2} x + g^{2} h\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*B^2*(log((d*x + c)^n)^2/(h^3*x^2 + 2*g*h^2*x + g^2*h) + 2*integrate(-(d*h*x*log(e)^2 + c*h*log(e)^2 + (d*
h*x + c*h)*log((b*x + a)^n)^2 + 2*(d*h*x*log(e) + c*h*log(e))*log((b*x + a)^n) + (d*g*n + (h*n - 2*h*log(e))*d
*x - 2*c*h*log(e) - 2*(d*h*x + c*h)*log((b*x + a)^n))*log((d*x + c)^n))/(d*h^4*x^4 + c*g^3*h + (3*d*g*h^3 + c*
h^4)*x^3 + 3*(d*g^2*h^2 + c*g*h^3)*x^2 + (d*g^3*h + 3*c*g^2*h^2)*x), x)) + (b^2*e*n*log(b*x + a)/(b^2*g^2*h -
2*a*b*g*h^2 + a^2*h^3) - d^2*e*n*log(d*x + c)/(d^2*g^2*h - 2*c*d*g*h^2 + c^2*h^3) - (2*a*b*d^2*e*g*n - a^2*d^2
*e*h*n - (2*c*d*e*g*n - c^2*e*h*n)*b^2)*log(h*x + g)/((d^2*g^2*h^2 - 2*c*d*g*h^3 + c^2*h^4)*a^2 - 2*(d^2*g^3*h
- 2*c*d*g^2*h^2 + c^2*g*h^3)*a*b + (d^2*g^4 - 2*c*d*g^3*h + c^2*g^2*h^2)*b^2) + (b*c*e*n - a*d*e*n)/((d*g^2*h
- c*g*h^2)*a - (d*g^3 - c*g^2*h)*b + ((d*g*h^2 - c*h^3)*a - (d*g^2*h - c*g*h^2)*b)*x))*A*B/e - A*B*log((b*x +
a)^n*e/(d*x + c)^n)/(h^3*x^2 + 2*g*h^2*x + g^2*h) - 1/2*A^2/(h^3*x^2 + 2*g*h^2*x + g^2*h)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError