3.1.73 \(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(f+g x)^3} \, dx\) [73]

Optimal. Leaf size=389 \[ \frac {B (b c-a d) g n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b f-a g)^2 (d f-c g) (f+g x)}+\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 g (b f-a g)^2}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 g (f+g x)^2}+\frac {B^2 (b c-a d)^2 g n^2 \log \left (\frac {f+g x}{c+d x}\right )}{(b f-a g)^2 (d f-c g)^2}+\frac {B (b c-a d) (2 b d f-b c g-a d g) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (1-\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{(b f-a g)^2 (d f-c g)^2}+\frac {B^2 (b c-a d) (2 b d f-b c g-a d g) n^2 \text {Li}_2\left (\frac {(d f-c g) (a+b x)}{(b f-a g) (c+d x)}\right )}{(b f-a g)^2 (d f-c g)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.55, antiderivative size = 389, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2553, 2398, 2404, 2338, 2351, 31, 2354, 2438} \begin {gather*} \frac {B^2 n^2 (b c-a d) (-a d g-b c g+2 b d f) \text {PolyLog}\left (2,\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right )}{(b f-a g)^2 (d f-c g)^2}+\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g (b f-a g)^2}+\frac {B g n (a+b x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(f+g x) (b f-a g)^2 (d f-c g)}+\frac {B n (b c-a d) (-a d g-b c g+2 b d f) \log \left (1-\frac {(a+b x) (d f-c g)}{(c+d x) (b f-a g)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(b f-a g)^2 (d f-c g)^2}-\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g (f+g x)^2}+\frac {B^2 g n^2 (b c-a d)^2 \log \left (\frac {f+g x}{c+d x}\right )}{(b f-a g)^2 (d f-c g)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 31

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

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 2398

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

Rule 2404

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

Rule 2438

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

Rule 2553

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

Rubi steps

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

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Mathematica [A]
time = 1.04, size = 615, normalized size = 1.58 \begin {gather*} -\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {B n (f+g x) \left (2 (b c-a d) g (b f-a g) (d f-c g) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 b^2 (d f-c g)^2 (f+g x) \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 d^2 (b f-a g)^2 (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+2 (b c-a d) g (-2 b d f+b c g+a d g) (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)-2 B (b c-a d) g n (f+g x) (b (d f-c g) \log (a+b x)+(-b d f+a d g) \log (c+d x)+(b c-a d) g \log (f+g x))+b^2 B (d f-c g)^2 n (f+g x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )-B d^2 (b f-a g)^2 n (f+g x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 B (b c-a d) g (-2 b d f+b c g+a d g) n (f+g x) \left (\left (\log \left (\frac {g (a+b x)}{-b f+a g}\right )-\log \left (\frac {g (c+d x)}{-d f+c g}\right )\right ) \log (f+g x)+\text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )-\text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )\right )\right )}{(b f-a g)^2 (d f-c g)^2}}{2 g (f+g x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{\left (g x +f \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2}{{\left (f+g\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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