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

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

[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)

________________________________________________________________________________________

Rubi [B]  time = 1.56109, antiderivative size = 941, normalized size of antiderivative = 2.42, number of steps used = 33, number of rules used = 11, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.344, Rules used = {2525, 12, 2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 72} \[ -\frac{B^2 n^2 \log ^2(a+b x) b^2}{2 g (b f-a g)^2}+\frac{B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) b^2}{g (b f-a g)^2}+\frac{B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) b^2}{g (b f-a g)^2}+\frac{B^2 n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) b^2}{g (b f-a g)^2}+\frac{B^2 (b c-a d) n^2 \log (a+b x) b}{(b f-a g)^2 (d f-c g)}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{2 g (f+g x)^2}-\frac{B^2 d^2 n^2 \log ^2(c+d x)}{2 g (d f-c g)^2}-\frac{B (b c-a d) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b f-a g) (d f-c g) (f+g x)}-\frac{B^2 d (b c-a d) n^2 \log (c+d x)}{(b f-a g) (d f-c g)^2}+\frac{B^2 d^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{g (d f-c g)^2}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{g (d f-c g)^2}+\frac{B^2 (b c-a d)^2 g n^2 \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}-\frac{B^2 (b c-a d) (2 b d f-b c g-a d g) n^2 \log \left (-\frac{g (a+b x)}{b f-a g}\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{B (b c-a d) (2 b d f-b c g-a d g) n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{B^2 (b c-a d) (2 b d f-b c g-a d g) n^2 \log \left (-\frac{g (c+d x)}{d f-c g}\right ) \log (f+g x)}{(b f-a g)^2 (d f-c g)^2}+\frac{B^2 d^2 n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g (d f-c g)^2}-\frac{B^2 (b c-a d) (2 b d f-b c g-a d g) n^2 \text{PolyLog}\left (2,\frac{b (f+g x)}{b f-a g}\right )}{(b f-a g)^2 (d f-c g)^2}+\frac{B^2 (b c-a d) (2 b d f-b c g-a d g) n^2 \text{PolyLog}\left (2,\frac{d (f+g x)}{d f-c g}\right )}{(b f-a g)^2 (d f-c g)^2} \]

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

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2528

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

Rule 2524

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2390

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

Rule 2301

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 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]

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]

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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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 ) \operatorname{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*}

Mathematica [A]  time = 2.23813, size = 615, normalized size = 1.58 \[ -\frac{\frac{B n (f+g x) \left (b^2 B n (f+g x) (d f-c g)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-B d^2 n (f+g x) (b f-a g)^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-2 B g n (f+g x) (b c-a d) (a d g+b c g-2 b d f) \left (\text{PolyLog}\left (2,\frac{b (f+g x)}{b f-a g}\right )-\text{PolyLog}\left (2,\frac{d (f+g x)}{d f-c g}\right )+\log (f+g x) \left (\log \left (\frac{g (a+b x)}{a g-b f}\right )-\log \left (\frac{g (c+d x)}{c g-d f}\right )\right )\right )-2 b^2 (f+g x) \log (a+b x) (d f-c g)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 d^2 (f+g x) (b f-a g)^2 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 g (b c-a d) (b f-a g) (d f-c g) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 g (f+g x) (b c-a d) \log (f+g x) (a d g+b c g-2 b d f) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-2 B g n (f+g x) (b c-a d) (b \log (a+b x) (d f-c g)+\log (c+d x) (a d g-b d f)+g (b c-a d) \log (f+g x))\right )}{(b f-a g)^2 (d f-c g)^2}+\left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g (f+g x)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((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*Lo
g[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))/(2*g*(f + g*x)
^2)

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Maple [F]  time = 0.512, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( gx+f \right ) ^{3}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{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)/(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)

________________________________________________________________________________________

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

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*log(e)^2 + c*g*log(e)^2 + (d*g*x + c*g)*log((b*x + a)^n
)^2 + 2*(d*g*x*log(e) + c*g*log(e))*log((b*x + a)^n) + (d*f*n + (g*n - 2*g*log(e))*d*x - 2*c*g*log(e) - 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(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(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., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{B^{2} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, A B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A^{2}}{g^{3} x^{3} + 3 \, f g^{2} x^{2} + 3 \, f^{2} g x + f^{3}}, x\right ) \end{align*}

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(e*((b*x + a)/(d*x + c))^n)^2 + 2*A*B*log(e*((b*x + a)/(d*x + c))^n) + 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)]  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)/(d*x+c))**n))**2/(g*x+f)**3,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 (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}}{{\left (g x + f\right )}^{3}}\,{d x} \end{align*}

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