3.307 \(\int \frac{(A+B \log (e (a+b x)^n (c+d x)^{-n}))^2}{(g+h x)^2} \, dx\)
Optimal. Leaf size=208 \[ \frac{2 B^2 n^2 (b c-a d) \text{PolyLog}\left (2,\frac{(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right )}{(b g-a h) (d g-c h)}+\frac{2 B n (b c-a d) \log \left (1-\frac{(a+b x) (d g-c h)}{(c+d x) (b g-a h)}\right ) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{(b g-a h) (d g-c h)}+\frac{(a+b x) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2}{(g+h x) (b g-a h)} \]
[Out]
((a + b*x)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2)/((b*g - a*h)*(g + h*x)) + (2*B*(b*c - a*d)*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)*(d*g - c
*h)) + (2*B^2*(b*c - a*d)*n^2*PolyLog[2, ((d*g - c*h)*(a + b*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)*(d*g -
c*h))
________________________________________________________________________________________
Rubi [A] time = 0.406977, antiderivative size = 343, normalized size of antiderivative =
1.65, number of steps used = 10, number of rules used = 7, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.212, Rules
used = {6742, 2490, 36, 31, 2503, 2502, 2315} \[ \frac{2 B^2 n^2 (b c-a d) \text{PolyLog}\left (2,1-\frac{(g+h x) (b c-a d)}{(c+d x) (b g-a h)}\right )}{(b g-a h) (d g-c h)}+\frac{2 A B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x) (b g-a h)}-\frac{2 A B n (b c-a d) \log (c+d x)}{(b g-a h) (d g-c h)}+\frac{2 A B n (b c-a d) \log (g+h x)}{(b g-a h) (d g-c h)}+\frac{B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x) (b g-a h)}+\frac{2 B^2 n (b c-a d) \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(g+h x) (b c-a d)}{(c+d x) (b g-a h)}\right )}{(b g-a h) (d g-c h)}-\frac{A^2}{h (g+h x)} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2/(g + h*x)^2,x]
[Out]
-(A^2/(h*(g + h*x))) - (2*A*B*(b*c - a*d)*n*Log[c + d*x])/((b*g - a*h)*(d*g - c*h)) + (2*A*B*(a + b*x)*Log[(e*
(a + b*x)^n)/(c + d*x)^n])/((b*g - a*h)*(g + h*x)) + (B^2*(a + b*x)*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2)/((b*g
- a*h)*(g + h*x)) + (2*A*B*(b*c - a*d)*n*Log[g + h*x])/((b*g - a*h)*(d*g - c*h)) + (2*B^2*(b*c - a*d)*n*Log[(e
*(a + b*x)^n)/(c + d*x)^n]*Log[((b*c - a*d)*(g + h*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)*(d*g - c*h)) + (
2*B^2*(b*c - a*d)*n^2*PolyLog[2, 1 - ((b*c - a*d)*(g + h*x))/((b*g - a*h)*(c + d*x))])/((b*g - a*h)*(d*g - c*h
))
Rule 6742
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
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 2503
Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbol] :> Wi
th[{g = Coeff[Simplify[1/(u*(a + b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -Simp[(Log[e*(f*(
a + b*x)^p*(c + d*x)^q)^r]^s*Log[-(((b*c - a*d)*(g + h*x))/((d*g - c*h)*(a + b*x)))])/(b*g - a*h), 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)*Log[-(((b*c - a*d)*(g + h*x)
)/((d*g - c*h)*(a + b*x)))])/((a + b*x)*(c + d*x)), x], x] /; NeQ[b*g - a*h, 0] && NeQ[d*g - c*h, 0]] /; FreeQ
[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0] && LinearQ[Simplify[1/
(u*(a + b*x))], x]
Rule 2502
Int[Log[((e_.)*((c_.) + (d_.)*(x_)))/((a_.) + (b_.)*(x_))]*(u_), x_Symbol] :> With[{g = Coeff[Simplify[1/(u*(a
+ b*x))], x, 0], h = Coeff[Simplify[1/(u*(a + b*x))], x, 1]}, -Dist[(b - d*e)/(h*(b*c - a*d)), Subst[Int[Log[
e*x]/(1 - e*x), x], x, (c + d*x)/(a + b*x)], x] /; EqQ[g*(b - d*e) - h*(a - c*e), 0]] /; FreeQ[{a, b, c, d, e}
, x] && NeQ[b*c - a*d, 0] && LinearQ[Simplify[1/(u*(a + b*x))], x]
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]
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)^2} \, dx &=\int \left (\frac{A^2}{(g+h x)^2}+\frac{2 A B \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2}+\frac{B^2 \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2}\right ) \, dx\\ &=-\frac{A^2}{h (g+h x)}+(2 A B) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2} \, dx+B^2 \int \frac{\log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(g+h x)^2} \, dx\\ &=-\frac{A^2}{h (g+h x)}+\frac{2 A B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}-\frac{(2 A B (b c-a d) n) \int \frac{1}{(c+d x) (g+h x)} \, dx}{b g-a h}-\frac{\left (2 B^2 (b c-a d) n\right ) \int \frac{\log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(c+d x) (g+h x)} \, dx}{b g-a h}\\ &=-\frac{A^2}{h (g+h x)}+\frac{2 A B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{2 B^2 (b c-a d) n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}-\frac{(2 A B d (b c-a d) n) \int \frac{1}{c+d x} \, dx}{(b g-a h) (d g-c h)}+\frac{(2 A B (b c-a d) h n) \int \frac{1}{g+h x} \, dx}{(b g-a h) (d g-c h)}-\frac{\left (2 B^2 (b c-a d)^2 n^2\right ) \int \frac{\log \left (-\frac{(-b c+a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(a+b x) (c+d x)} \, dx}{(b g-a h) (d g-c h)}\\ &=-\frac{A^2}{h (g+h x)}-\frac{2 A B (b c-a d) n \log (c+d x)}{(b g-a h) (d g-c h)}+\frac{2 A B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{2 A B (b c-a d) n \log (g+h x)}{(b g-a h) (d g-c h)}+\frac{2 B^2 (b c-a d) n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac{\left (2 B^2 (b c-a d)^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (-\frac{(-b c+a d) x}{b g-a h}\right )}{1+\frac{(-b c+a d) x}{b g-a h}} \, dx,x,\frac{g+h x}{c+d x}\right )}{(b g-a h)^2 (d g-c h)}\\ &=-\frac{A^2}{h (g+h x)}-\frac{2 A B (b c-a d) n \log (c+d x)}{(b g-a h) (d g-c h)}+\frac{2 A B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{B^2 (a+b x) \log ^2\left (e (a+b x)^n (c+d x)^{-n}\right )}{(b g-a h) (g+h x)}+\frac{2 A B (b c-a d) n \log (g+h x)}{(b g-a h) (d g-c h)}+\frac{2 B^2 (b c-a d) n \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \log \left (\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}+\frac{2 B^2 (b c-a d) n^2 \text{Li}_2\left (1-\frac{(b c-a d) (g+h x)}{(b g-a h) (c+d x)}\right )}{(b g-a h) (d g-c h)}\\ \end{align*}
Mathematica [B] time = 2.28281, size = 3460, normalized size = 16.63 \[ \text{Result too large to show} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^2/(g + h*x)^2,x]
[Out]
(-(A^2*b*d*g^2) + A^2*b*c*g*h + a*A^2*d*g*h - a*A^2*c*h^2 + 2*A*b*B*d*g^2*n*Log[a + b*x] - 2*A*b*B*c*g*h*n*Log
[a + b*x] + 2*A*b*B*d*g*h*n*x*Log[a + b*x] - 2*A*b*B*c*h^2*n*x*Log[a + b*x] - b*B^2*d*g^2*n^2*Log[a + b*x]^2 +
b*B^2*c*g*h*n^2*Log[a + b*x]^2 - b*B^2*d*g*h*n^2*x*Log[a + b*x]^2 + b*B^2*c*h^2*n^2*x*Log[a + b*x]^2 - 2*A*b*
B*d*g^2*n*Log[c + d*x] + 2*a*A*B*d*g*h*n*Log[c + d*x] - 2*A*b*B*d*g*h*n*x*Log[c + d*x] + 2*a*A*B*d*h^2*n*x*Log
[c + d*x] + 2*b*B^2*d*g^2*n^2*Log[a + b*x]*Log[c + d*x] - 2*a*B^2*d*g*h*n^2*Log[a + b*x]*Log[c + d*x] + 2*b*B^
2*d*g*h*n^2*x*Log[a + b*x]*Log[c + d*x] - 2*a*B^2*d*h^2*n^2*x*Log[a + b*x]*Log[c + d*x] - b*B^2*d*g^2*n^2*Log[
c + d*x]^2 + a*B^2*d*g*h*n^2*Log[c + d*x]^2 - b*B^2*d*g*h*n^2*x*Log[c + d*x]^2 + a*B^2*d*h^2*n^2*x*Log[c + d*x
]^2 - 2*b*B^2*c*g*h*n^2*Log[a + b*x]*Log[(h*(c + d*x))/(-(d*g) + c*h)] + 2*a*B^2*d*g*h*n^2*Log[a + b*x]*Log[(h
*(c + d*x))/(-(d*g) + c*h)] - 2*b*B^2*c*h^2*n^2*x*Log[a + b*x]*Log[(h*(c + d*x))/(-(d*g) + c*h)] + 2*a*B^2*d*h
^2*n^2*x*Log[a + b*x]*Log[(h*(c + d*x))/(-(d*g) + c*h)] + b*B^2*c*g*h*n^2*Log[(h*(c + d*x))/(-(d*g) + c*h)]^2
- a*B^2*d*g*h*n^2*Log[(h*(c + d*x))/(-(d*g) + c*h)]^2 + b*B^2*c*h^2*n^2*x*Log[(h*(c + d*x))/(-(d*g) + c*h)]^2
- a*B^2*d*h^2*n^2*x*Log[(h*(c + d*x))/(-(d*g) + c*h)]^2 - 2*b*B^2*c*g*h*n^2*Log[(-(b*c) + a*d)/(d*(a + b*x))]*
Log[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))] + 2*a*B^2*d*g*h*n^2*Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log
[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))] - 2*b*B^2*c*h^2*n^2*x*Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[
((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))] + 2*a*B^2*d*h^2*n^2*x*Log[(-(b*c) + a*d)/(d*(a + b*x))]*Log[(
(b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))] - 2*b*B^2*c*g*h*n^2*Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[((b*
g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))] + 2*a*B^2*d*g*h*n^2*Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[((b*g -
a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))] - 2*b*B^2*c*h^2*n^2*x*Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[((b*g -
a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))] + 2*a*B^2*d*h^2*n^2*x*Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[((b*g - a
*h)*(c + d*x))/((d*g - c*h)*(a + b*x))] + b*B^2*c*g*h*n^2*Log[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]
^2 - a*B^2*d*g*h*n^2*Log[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x))]^2 + b*B^2*c*h^2*n^2*x*Log[((b*g - a*
h)*(c + d*x))/((d*g - c*h)*(a + b*x))]^2 - a*B^2*d*h^2*n^2*x*Log[((b*g - a*h)*(c + d*x))/((d*g - c*h)*(a + b*x
))]^2 - 2*A*b*B*d*g^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 2*A*b*B*c*g*h*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 2*a*
A*B*d*g*h*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 2*a*A*B*c*h^2*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 2*b*B^2*d*g^2*n*
Log[a + b*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 2*b*B^2*c*g*h*n*Log[a + b*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n]
+ 2*b*B^2*d*g*h*n*x*Log[a + b*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 2*b*B^2*c*h^2*n*x*Log[a + b*x]*Log[(e*(a +
b*x)^n)/(c + d*x)^n] - 2*b*B^2*d*g^2*n*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 2*a*B^2*d*g*h*n*Log[c
+ d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - 2*b*B^2*d*g*h*n*x*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] + 2*
a*B^2*d*h^2*n*x*Log[c + d*x]*Log[(e*(a + b*x)^n)/(c + d*x)^n] - b*B^2*d*g^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2
+ b*B^2*c*g*h*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 + a*B^2*d*g*h*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 - a*B^2*c*h
^2*Log[(e*(a + b*x)^n)/(c + d*x)^n]^2 - 2*A*b*B*d*g^2*n*Log[(b*(g + h*x))/(b*g - a*h)] + 2*A*b*B*c*g*h*n*Log[(
b*(g + h*x))/(b*g - a*h)] - 2*A*b*B*d*g*h*n*x*Log[(b*(g + h*x))/(b*g - a*h)] + 2*A*b*B*c*h^2*n*x*Log[(b*(g + h
*x))/(b*g - a*h)] + 2*b*B^2*d*g^2*n^2*Log[a + b*x]*Log[(b*(g + h*x))/(b*g - a*h)] - 2*a*B^2*d*g*h*n^2*Log[a +
b*x]*Log[(b*(g + h*x))/(b*g - a*h)] + 2*b*B^2*d*g*h*n^2*x*Log[a + b*x]*Log[(b*(g + h*x))/(b*g - a*h)] - 2*a*B^
2*d*h^2*n^2*x*Log[a + b*x]*Log[(b*(g + h*x))/(b*g - a*h)] - 2*b*B^2*d*g^2*n^2*Log[(h*(c + d*x))/(-(d*g) + c*h)
]*Log[(b*(g + h*x))/(b*g - a*h)] + 2*b*B^2*c*g*h*n^2*Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[(b*(g + h*x))/(b*g
- a*h)] - 2*b*B^2*d*g*h*n^2*x*Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[(b*(g + h*x))/(b*g - a*h)] + 2*b*B^2*c*h^2
*n^2*x*Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[(b*(g + h*x))/(b*g - a*h)] - 2*b*B^2*d*g^2*n*Log[(e*(a + b*x)^n)/
(c + d*x)^n]*Log[(b*(g + h*x))/(b*g - a*h)] + 2*b*B^2*c*g*h*n*Log[(e*(a + b*x)^n)/(c + d*x)^n]*Log[(b*(g + h*x
))/(b*g - a*h)] - 2*b*B^2*d*g*h*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n]*Log[(b*(g + h*x))/(b*g - a*h)] + 2*b*B^2*
c*h^2*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n]*Log[(b*(g + h*x))/(b*g - a*h)] + 2*A*b*B*d*g^2*n*Log[(d*(g + h*x))/
(d*g - c*h)] - 2*a*A*B*d*g*h*n*Log[(d*(g + h*x))/(d*g - c*h)] + 2*A*b*B*d*g*h*n*x*Log[(d*(g + h*x))/(d*g - c*h
)] - 2*a*A*B*d*h^2*n*x*Log[(d*(g + h*x))/(d*g - c*h)] - 2*b*B^2*d*g^2*n^2*Log[a + b*x]*Log[(d*(g + h*x))/(d*g
- c*h)] + 2*a*B^2*d*g*h*n^2*Log[a + b*x]*Log[(d*(g + h*x))/(d*g - c*h)] - 2*b*B^2*d*g*h*n^2*x*Log[a + b*x]*Log
[(d*(g + h*x))/(d*g - c*h)] + 2*a*B^2*d*h^2*n^2*x*Log[a + b*x]*Log[(d*(g + h*x))/(d*g - c*h)] + 2*b*B^2*d*g^2*
n^2*Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[(d*(g + h*x))/(d*g - c*h)] - 2*b*B^2*c*g*h*n^2*Log[(h*(c + d*x))/(-(
d*g) + c*h)]*Log[(d*(g + h*x))/(d*g - c*h)] + 2*b*B^2*d*g*h*n^2*x*Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[(d*(g
+ h*x))/(d*g - c*h)] - 2*b*B^2*c*h^2*n^2*x*Log[(h*(c + d*x))/(-(d*g) + c*h)]*Log[(d*(g + h*x))/(d*g - c*h)] +
2*b*B^2*d*g^2*n*Log[(e*(a + b*x)^n)/(c + d*x)^n]*Log[(d*(g + h*x))/(d*g - c*h)] - 2*a*B^2*d*g*h*n*Log[(e*(a +
b*x)^n)/(c + d*x)^n]*Log[(d*(g + h*x))/(d*g - c*h)] + 2*b*B^2*d*g*h*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n]*Log[(
d*(g + h*x))/(d*g - c*h)] - 2*a*B^2*d*h^2*n*x*Log[(e*(a + b*x)^n)/(c + d*x)^n]*Log[(d*(g + h*x))/(d*g - c*h)]
+ 2*B^2*(b*c - a*d)*h*n^2*(g + h*x)*PolyLog[2, (h*(a + b*x))/(-(b*g) + a*h)] - 2*B^2*(b*c - a*d)*h*n^2*(g + h*
x)*PolyLog[2, (h*(c + d*x))/(-(d*g) + c*h)] - 2*b*B^2*c*g*h*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))] + 2*a*
B^2*d*g*h*n^2*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))] - 2*b*B^2*c*h^2*n^2*x*PolyLog[2, (b*(c + d*x))/(d*(a + b
*x))] + 2*a*B^2*d*h^2*n^2*x*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(h*(-(b*g) + a*h)*(-(d*g) + c*h)*(g + h*x
))
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Maple [F] time = 2.712, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( hx+g \right ) ^{2}} \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)^2,x)
[Out]
int((A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^2/(h*x+g)^2,x)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -B^{2}{\left (\frac{\log \left ({\left (d x + c\right )}^{n}\right )^{2}}{h^{2} x + g h} + \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 ) + 2 \,{\left (d g n +{\left (h n - h \log \left (e\right )\right )} d x - c h \log \left (e\right ) -{\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^{3} x^{3} + c g^{2} h +{\left (2 \, d g h^{2} + c h^{3}\right )} x^{2} +{\left (d g^{2} h + 2 \, c g h^{2}\right )} x}\,{d x}\right )} + \frac{2 \,{\left (\frac{b e n \log \left (b x + a\right )}{b g h - a h^{2}} - \frac{d e n \log \left (d x + c\right )}{d g h - c h^{2}} - \frac{{\left (b c e n - a d e n\right )} \log \left (h x + g\right )}{{\left (d g h - c h^{2}\right )} a -{\left (d g^{2} - c g h\right )} b}\right )} A B}{e} - \frac{2 \, A B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{h^{2} x + g h} - \frac{A^{2}}{h^{2} x + g h} \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)^2,x, algorithm="maxima")
[Out]
-B^2*(log((d*x + c)^n)^2/(h^2*x + g*h) + 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) + 2*(d*g*n + (h*n - h*log(e))*d*x - c*h*log(e) - (d*h
*x + c*h)*log((b*x + a)^n))*log((d*x + c)^n))/(d*h^3*x^3 + c*g^2*h + (2*d*g*h^2 + c*h^3)*x^2 + (d*g^2*h + 2*c*
g*h^2)*x), x)) + 2*(b*e*n*log(b*x + a)/(b*g*h - a*h^2) - d*e*n*log(d*x + c)/(d*g*h - c*h^2) - (b*c*e*n - a*d*e
*n)*log(h*x + g)/((d*g*h - c*h^2)*a - (d*g^2 - c*g*h)*b))*A*B/e - 2*A*B*log((b*x + a)^n*e/(d*x + c)^n)/(h^2*x
+ g*h) - A^2/(h^2*x + g*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^{2} x^{2} + 2 \, g h x + g^{2}}, 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)^2,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^2*x^2 + 2*g*h*
x + g^2), 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)**2,x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{2}}{{\left (h x + g\right )}^{2}}\,{d x} \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)^2,x, algorithm="giac")
[Out]
integrate((B*log((b*x + a)^n*e/(d*x + c)^n) + A)^2/(h*x + g)^2, x)