3.18 \(\int \frac{(A+B \log (e (\frac{a+b x}{c+d x})^n))^2}{(a g+b g x)^5} \, dx\)

Optimal. Leaf size=615 \[ -\frac{b^3 (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 g^5 (a+b x)^4 (b c-a d)^4}-\frac{b^3 B n (c+d x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{8 g^5 (a+b x)^4 (b c-a d)^4}+\frac{b^2 d (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^5 (a+b x)^3 (b c-a d)^4}+\frac{2 b^2 B d n (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^5 (a+b x)^3 (b c-a d)^4}-\frac{3 b d^2 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g^5 (a+b x)^2 (b c-a d)^4}-\frac{3 b B d^2 n (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^5 (a+b x)^2 (b c-a d)^4}+\frac{d^3 (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^5 (a+b x) (b c-a d)^4}+\frac{2 B d^3 n (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^5 (a+b x) (b c-a d)^4}-\frac{b^3 B^2 n^2 (c+d x)^4}{32 g^5 (a+b x)^4 (b c-a d)^4}+\frac{2 b^2 B^2 d n^2 (c+d x)^3}{9 g^5 (a+b x)^3 (b c-a d)^4}-\frac{3 b B^2 d^2 n^2 (c+d x)^2}{4 g^5 (a+b x)^2 (b c-a d)^4}+\frac{2 B^2 d^3 n^2 (c+d x)}{g^5 (a+b x) (b c-a d)^4} \]

[Out]

(2*B^2*d^3*n^2*(c + d*x))/((b*c - a*d)^4*g^5*(a + b*x)) - (3*b*B^2*d^2*n^2*(c + d*x)^2)/(4*(b*c - a*d)^4*g^5*(
a + b*x)^2) + (2*b^2*B^2*d*n^2*(c + d*x)^3)/(9*(b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*B^2*n^2*(c + d*x)^4)/(32*
(b*c - a*d)^4*g^5*(a + b*x)^4) + (2*B*d^3*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^4*g
^5*(a + b*x)) - (3*b*B*d^2*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^4*g^5*(a + b*x
)^2) + (2*b^2*B*d*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c - a*d)^4*g^5*(a + b*x)^3) - (b
^3*B*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(8*(b*c - a*d)^4*g^5*(a + b*x)^4) + (d^3*(c + d*x)*
(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^4*g^5*(a + b*x)) - (3*b*d^2*(c + d*x)^2*(A + B*Log[e*((
a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^4*g^5*(a + b*x)^2) + (b^2*d*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c +
d*x))^n])^2)/((b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*
(b*c - a*d)^4*g^5*(a + b*x)^4)

________________________________________________________________________________________

Rubi [C]  time = 1.31405, antiderivative size = 826, normalized size of antiderivative = 1.34, number of steps used = 36, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314, Rules used = {2525, 12, 2528, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{B^2 n^2 \log ^2(a+b x) d^4}{4 b (b c-a d)^4 g^5}-\frac{B^2 n^2 \log ^2(c+d x) d^4}{4 b (b c-a d)^4 g^5}+\frac{25 B^2 n^2 \log (a+b x) d^4}{24 b (b c-a d)^4 g^5}+\frac{B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d^4}{2 b (b c-a d)^4 g^5}-\frac{25 B^2 n^2 \log (c+d x) d^4}{24 b (b c-a d)^4 g^5}+\frac{B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) d^4}{2 b (b c-a d)^4 g^5}-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) d^4}{2 b (b c-a d)^4 g^5}+\frac{B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) d^4}{2 b (b c-a d)^4 g^5}+\frac{B^2 n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) d^4}{2 b (b c-a d)^4 g^5}+\frac{B^2 n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) d^4}{2 b (b c-a d)^4 g^5}+\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d^3}{2 b (b c-a d)^3 g^5 (a+b x)}+\frac{25 B^2 n^2 d^3}{24 b (b c-a d)^3 g^5 (a+b x)}-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d^2}{4 b (b c-a d)^2 g^5 (a+b x)^2}-\frac{13 B^2 n^2 d^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d}{6 b (b c-a d) g^5 (a+b x)^3}+\frac{7 B^2 n^2 d}{72 b (b c-a d) g^5 (a+b x)^3}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8 b g^5 (a+b x)^4}-\frac{B^2 n^2}{32 b g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^5,x]

[Out]

-(B^2*n^2)/(32*b*g^5*(a + b*x)^4) + (7*B^2*d*n^2)/(72*b*(b*c - a*d)*g^5*(a + b*x)^3) - (13*B^2*d^2*n^2)/(48*b*
(b*c - a*d)^2*g^5*(a + b*x)^2) + (25*B^2*d^3*n^2)/(24*b*(b*c - a*d)^3*g^5*(a + b*x)) + (25*B^2*d^4*n^2*Log[a +
b*x])/(24*b*(b*c - a*d)^4*g^5) - (B^2*d^4*n^2*Log[a + b*x]^2)/(4*b*(b*c - a*d)^4*g^5) - (B*n*(A + B*Log[e*((a
+ b*x)/(c + d*x))^n]))/(8*b*g^5*(a + b*x)^4) + (B*d*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*b*(b*c - a*d
)*g^5*(a + b*x)^3) - (B*d^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*b*(b*c - a*d)^2*g^5*(a + b*x)^2) + (B
*d^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b*(b*c - a*d)^3*g^5*(a + b*x)) + (B*d^4*n*Log[a + b*x]*(A +
B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b*(b*c - a*d)^4*g^5) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(4*b*g^5
*(a + b*x)^4) - (25*B^2*d^4*n^2*Log[c + d*x])/(24*b*(b*c - a*d)^4*g^5) + (B^2*d^4*n^2*Log[-((d*(a + b*x))/(b*c
- a*d))]*Log[c + d*x])/(2*b*(b*c - a*d)^4*g^5) - (B*d^4*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x]
)/(2*b*(b*c - a*d)^4*g^5) - (B^2*d^4*n^2*Log[c + d*x]^2)/(4*b*(b*c - a*d)^4*g^5) + (B^2*d^4*n^2*Log[a + b*x]*L
og[(b*(c + d*x))/(b*c - a*d)])/(2*b*(b*c - a*d)^4*g^5) + (B^2*d^4*n^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))]
)/(2*b*(b*c - a*d)^4*g^5) + (B^2*d^4*n^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(2*b*(b*c - a*d)^4*g^5)

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 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 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]

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}{(a g+b g x)^5} \, dx &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}+\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 )}{g^4 (a+b x)^5 (c+d x)} \, dx}{2 b g}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}+\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)^5 (c+d x)} \, dx}{2 b g^5}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac{(B (b c-a d) n) \int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^5}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 (a+b x)}-\frac{d^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{2 b g^5}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac{(B n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^5} \, dx}{2 g^5}+\frac{\left (B d^4 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{2 (b c-a d)^4 g^5}-\frac{\left (B d^5 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}-\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 )}{(a+b x)^2} \, dx}{2 (b c-a d)^3 g^5}+\frac{\left (B d^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{2 (b c-a d)^2 g^5}-\frac{(B d n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{2 (b c-a d) g^5}\\ &=-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8 b g^5 (a+b x)^4}+\frac{B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}+\frac{\left (B^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^5 (c+d x)} \, dx}{8 b g^5}-\frac{\left (B^2 d^4 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}{2 b (b c-a d)^4 g^5}+\frac{\left (B^2 d^4 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}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d^3 n^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{2 b (b c-a d)^3 g^5}+\frac{\left (B^2 d^2 n^2\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{4 b (b c-a d)^2 g^5}-\frac{\left (B^2 d n^2\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{6 b (b c-a d) g^5}\\ &=-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8 b g^5 (a+b x)^4}+\frac{B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d n^2\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{6 b g^5}-\frac{\left (B^2 d^4 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}{2 b (b c-a d)^4 g^5}+\frac{\left (B^2 d^4 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}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d^3 n^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{2 b (b c-a d)^2 g^5}+\frac{\left (B^2 d^2 n^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{4 b (b c-a d) g^5}+\frac{\left (B^2 (b c-a d) n^2\right ) \int \frac{1}{(a+b x)^5 (c+d x)} \, dx}{8 b g^5}\\ &=-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8 b g^5 (a+b x)^4}+\frac{B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{6 b g^5}-\frac{\left (B^2 d^4 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{2 (b c-a d)^4 g^5}+\frac{\left (B^2 d^4 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{2 (b c-a d)^4 g^5}+\frac{\left (B^2 d^5 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d^5 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d^3 n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{2 b (b c-a d)^2 g^5}+\frac{\left (B^2 d^2 n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{4 b (b c-a d) g^5}+\frac{\left (B^2 (b c-a d) n^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^5}-\frac{b d}{(b c-a d)^2 (a+b x)^4}+\frac{b d^2}{(b c-a d)^3 (a+b x)^3}-\frac{b d^3}{(b c-a d)^4 (a+b x)^2}+\frac{b d^4}{(b c-a d)^5 (a+b x)}-\frac{d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{8 b g^5}\\ &=-\frac{B^2 n^2}{32 b g^5 (a+b x)^4}+\frac{7 B^2 d n^2}{72 b (b c-a d) g^5 (a+b x)^3}-\frac{13 B^2 d^2 n^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{25 B^2 d^3 n^2}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac{25 B^2 d^4 n^2 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8 b g^5 (a+b x)^4}+\frac{B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{25 B^2 d^4 n^2 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac{B^2 d^4 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac{B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}+\frac{B^2 d^4 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d^4 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{2 (b c-a d)^4 g^5}-\frac{\left (B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d^4 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d^5 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}\\ &=-\frac{B^2 n^2}{32 b g^5 (a+b x)^4}+\frac{7 B^2 d n^2}{72 b (b c-a d) g^5 (a+b x)^3}-\frac{13 B^2 d^2 n^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{25 B^2 d^3 n^2}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac{25 B^2 d^4 n^2 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac{B^2 d^4 n^2 \log ^2(a+b x)}{4 b (b c-a d)^4 g^5}-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8 b g^5 (a+b x)^4}+\frac{B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{25 B^2 d^4 n^2 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac{B^2 d^4 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac{B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac{B^2 d^4 n^2 \log ^2(c+d x)}{4 b (b c-a d)^4 g^5}+\frac{B^2 d^4 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d^4 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 )}{2 b (b c-a d)^4 g^5}-\frac{\left (B^2 d^4 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 )}{2 b (b c-a d)^4 g^5}\\ &=-\frac{B^2 n^2}{32 b g^5 (a+b x)^4}+\frac{7 B^2 d n^2}{72 b (b c-a d) g^5 (a+b x)^3}-\frac{13 B^2 d^2 n^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{25 B^2 d^3 n^2}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac{25 B^2 d^4 n^2 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac{B^2 d^4 n^2 \log ^2(a+b x)}{4 b (b c-a d)^4 g^5}-\frac{B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{8 b g^5 (a+b x)^4}+\frac{B d n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}-\frac{B d^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}+\frac{B d^3 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}+\frac{B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^4 g^5}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac{25 B^2 d^4 n^2 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac{B^2 d^4 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac{B d^4 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac{B^2 d^4 n^2 \log ^2(c+d x)}{4 b (b c-a d)^4 g^5}+\frac{B^2 d^4 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}+\frac{B^2 d^4 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}+\frac{B^2 d^4 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}\\ \end{align*}

Mathematica [C]  time = 1.18765, size = 776, normalized size = 1.26 \[ -\frac{\frac{B n \left (72 B d^4 n (a+b x)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-72 B d^4 n (a+b x)^4 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+72 d^2 (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+144 d^3 (a+b x)^3 (a d-b c) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-144 d^4 (a+b x)^4 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+144 d^4 (a+b x)^4 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+36 (b c-a d)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+48 d (a+b x) (a d-b c)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-144 B d^3 n (a+b x)^3 (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)+36 B d^2 n (a+b x)^2 \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )-8 B d n (a+b x) \left (6 d^2 (a+b x)^2 (b c-a d)-6 d^3 (a+b x)^3 \log (c+d x)-3 d (a+b x) (b c-a d)^2+2 (b c-a d)^3+6 d^3 (a+b x)^3 \log (a+b x)\right )+3 B n \left (6 d^2 (a+b x)^2 (b c-a d)^2+12 d^3 (a+b x)^3 (a d-b c)+12 d^4 (a+b x)^4 \log (c+d x)+4 d (a+b x) (a d-b c)^3+3 (b c-a d)^4-12 d^4 (a+b x)^4 \log (a+b x)\right )\right )}{(b c-a d)^4}+72 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{288 b g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^5,x]

[Out]

-(72*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + (B*n*(36*(b*c - a*d)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])
+ 48*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 72*d^2*(b*c - a*d)^2*(a + b*x)^2*(A
+ B*Log[e*((a + b*x)/(c + d*x))^n]) + 144*d^3*(-(b*c) + a*d)*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n
]) - 144*d^4*(a + b*x)^4*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 144*d^4*(a + b*x)^4*(A + B*Log[
e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - 144*B*d^3*n*(a + b*x)^3*(b*c - a*d + d*(a + b*x)*Log[a + b*x] - d*(
a + b*x)*Log[c + d*x]) + 36*B*d^2*n*(a + b*x)^2*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x) - 2*d^2*(a + b*x
)^2*Log[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d*x]) - 8*B*d*n*(a + b*x)*(2*(b*c - a*d)^3 - 3*d*(b*c - a*d)^2*(a
+ b*x) + 6*d^2*(b*c - a*d)*(a + b*x)^2 + 6*d^3*(a + b*x)^3*Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[c + d*x]) + 3
*B*n*(3*(b*c - a*d)^4 + 4*d*(-(b*c) + a*d)^3*(a + b*x) + 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 12*d^3*(-(b*c) + a*
d)*(a + b*x)^3 - 12*d^4*(a + b*x)^4*Log[a + b*x] + 12*d^4*(a + b*x)^4*Log[c + d*x]) + 72*B*d^4*n*(a + b*x)^4*(
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)]) -
72*B*d^4*n*(a + b*x)^4*((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)])))/(b*c - a*d)^4)/(288*b*g^5*(a + b*x)^4)

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Maple [F]  time = 0.451, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{5}} \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/(b*g*x+a*g)^5,x)

[Out]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^5,x)

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Maxima [B]  time = 2.03168, size = 2884, normalized size = 4.69 \begin{align*} \text{result too large to display} \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/(b*g*x+a*g)^5,x, algorithm="maxima")

[Out]

1/24*A*B*n*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^
2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3
*b^5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 -
3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2
- a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) + 12*d^4*log(b*x +
a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b
^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5)) + 1/288*(12*n*((12*b^3*d^3*x^3
- 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d -
5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*g^5*x^4 + 4*(a*b^
7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^
4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*g^5*x + (a^
4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*
d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*
a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) - (9*b^4*c^4 - 64*
a*b^3*c^3*d + 216*a^2*b^2*c^2*d^2 - 576*a^3*b*c*d^3 + 415*a^4*d^4 - 300*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 6*(13*b^
4*c^2*d^2 - 176*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4
*a^3*b*d^4*x + a^4*d^4)*log(b*x + a)^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x
+ a^4*d^4)*log(d*x + c)^2 - 4*(7*b^4*c^3*d - 60*a*b^3*c^2*d^2 + 324*a^2*b^2*c*d^3 - 271*a^3*b*d^4)*x - 300*(b
^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a) + 12*(25*b^4*d^4*x^4
+ 100*a*b^3*d^4*x^3 + 150*a^2*b^2*d^4*x^2 + 100*a^3*b*d^4*x + 25*a^4*d^4 - 12*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 +
6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a))*log(d*x + c))*n^2/(a^4*b^5*c^4*g^5 - 4*a^5*b^4*c^3
*d*g^5 + 6*a^6*b^3*c^2*d^2*g^5 - 4*a^7*b^2*c*d^3*g^5 + a^8*b*d^4*g^5 + (b^9*c^4*g^5 - 4*a*b^8*c^3*d*g^5 + 6*a^
2*b^7*c^2*d^2*g^5 - 4*a^3*b^6*c*d^3*g^5 + a^4*b^5*d^4*g^5)*x^4 + 4*(a*b^8*c^4*g^5 - 4*a^2*b^7*c^3*d*g^5 + 6*a^
3*b^6*c^2*d^2*g^5 - 4*a^4*b^5*c*d^3*g^5 + a^5*b^4*d^4*g^5)*x^3 + 6*(a^2*b^7*c^4*g^5 - 4*a^3*b^6*c^3*d*g^5 + 6*
a^4*b^5*c^2*d^2*g^5 - 4*a^5*b^4*c*d^3*g^5 + a^6*b^3*d^4*g^5)*x^2 + 4*(a^3*b^6*c^4*g^5 - 4*a^4*b^5*c^3*d*g^5 +
6*a^5*b^4*c^2*d^2*g^5 - 4*a^6*b^3*c*d^3*g^5 + a^7*b^2*d^4*g^5)*x))*B^2 - 1/4*B^2*log(e*(b*x/(d*x + c) + a/(d*x
+ c))^n)^2/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) - 1/2*A*B*log(e*
(b*x/(d*x + c) + a/(d*x + c))^n)/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*
g^5) - 1/4*A^2/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5)

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Fricas [B]  time = 1.15835, size = 3615, normalized size = 5.88 \begin{align*} \text{result too large to display} \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/(b*g*x+a*g)^5,x, algorithm="fricas")

[Out]

-1/288*(72*A^2*b^4*c^4 - 288*A^2*a*b^3*c^3*d + 432*A^2*a^2*b^2*c^2*d^2 - 288*A^2*a^3*b*c*d^3 + 72*A^2*a^4*d^4
- 12*(25*(B^2*b^4*c*d^3 - B^2*a*b^3*d^4)*n^2 + 12*(A*B*b^4*c*d^3 - A*B*a*b^3*d^4)*n)*x^3 + (9*B^2*b^4*c^4 - 64
*B^2*a*b^3*c^3*d + 216*B^2*a^2*b^2*c^2*d^2 - 576*B^2*a^3*b*c*d^3 + 415*B^2*a^4*d^4)*n^2 + 6*((13*B^2*b^4*c^2*d
^2 - 176*B^2*a*b^3*c*d^3 + 163*B^2*a^2*b^2*d^4)*n^2 + 12*(A*B*b^4*c^2*d^2 - 8*A*B*a*b^3*c*d^3 + 7*A*B*a^2*b^2*
d^4)*n)*x^2 + 72*(B^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2*d^2 - 4*B^2*a^3*b*c*d^3 + B^2*a^4*d^4)*l
og(e)^2 - 72*(B^2*b^4*d^4*n^2*x^4 + 4*B^2*a*b^3*d^4*n^2*x^3 + 6*B^2*a^2*b^2*d^4*n^2*x^2 + 4*B^2*a^3*b*d^4*n^2*
x - (B^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2*d^2 - 4*B^2*a^3*b*c*d^3)*n^2)*log((b*x + a)/(d*x + c)
)^2 + 12*(3*A*B*b^4*c^4 - 16*A*B*a*b^3*c^3*d + 36*A*B*a^2*b^2*c^2*d^2 - 48*A*B*a^3*b*c*d^3 + 25*A*B*a^4*d^4)*n
- 4*((7*B^2*b^4*c^3*d - 60*B^2*a*b^3*c^2*d^2 + 324*B^2*a^2*b^2*c*d^3 - 271*B^2*a^3*b*d^4)*n^2 + 12*(A*B*b^4*c
^3*d - 6*A*B*a*b^3*c^2*d^2 + 18*A*B*a^2*b^2*c*d^3 - 13*A*B*a^3*b*d^4)*n)*x + 12*(12*A*B*b^4*c^4 - 48*A*B*a*b^3
*c^3*d + 72*A*B*a^2*b^2*c^2*d^2 - 48*A*B*a^3*b*c*d^3 + 12*A*B*a^4*d^4 - 12*(B^2*b^4*c*d^3 - B^2*a*b^3*d^4)*n*x
^3 + 6*(B^2*b^4*c^2*d^2 - 8*B^2*a*b^3*c*d^3 + 7*B^2*a^2*b^2*d^4)*n*x^2 - 4*(B^2*b^4*c^3*d - 6*B^2*a*b^3*c^2*d^
2 + 18*B^2*a^2*b^2*c*d^3 - 13*B^2*a^3*b*d^4)*n*x + (3*B^2*b^4*c^4 - 16*B^2*a*b^3*c^3*d + 36*B^2*a^2*b^2*c^2*d^
2 - 48*B^2*a^3*b*c*d^3 + 25*B^2*a^4*d^4)*n - 12*(B^2*b^4*d^4*n*x^4 + 4*B^2*a*b^3*d^4*n*x^3 + 6*B^2*a^2*b^2*d^4
*n*x^2 + 4*B^2*a^3*b*d^4*n*x - (B^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2*d^2 - 4*B^2*a^3*b*c*d^3)*n
)*log((b*x + a)/(d*x + c)))*log(e) - 12*((25*B^2*b^4*d^4*n^2 + 12*A*B*b^4*d^4*n)*x^4 + 4*(12*A*B*a*b^3*d^4*n +
(3*B^2*b^4*c*d^3 + 22*B^2*a*b^3*d^4)*n^2)*x^3 - (3*B^2*b^4*c^4 - 16*B^2*a*b^3*c^3*d + 36*B^2*a^2*b^2*c^2*d^2
- 48*B^2*a^3*b*c*d^3)*n^2 + 6*(12*A*B*a^2*b^2*d^4*n - (B^2*b^4*c^2*d^2 - 8*B^2*a*b^3*c*d^3 - 18*B^2*a^2*b^2*d^
4)*n^2)*x^2 - 12*(A*B*b^4*c^4 - 4*A*B*a*b^3*c^3*d + 6*A*B*a^2*b^2*c^2*d^2 - 4*A*B*a^3*b*c*d^3)*n + 4*(12*A*B*a
^3*b*d^4*n + (B^2*b^4*c^3*d - 6*B^2*a*b^3*c^2*d^2 + 18*B^2*a^2*b^2*c*d^3 + 12*B^2*a^3*b*d^4)*n^2)*x)*log((b*x
+ a)/(d*x + c)))/((b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*g^5*x^4 + 4*(a
*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*g^5*x^3 + 6*(a^2*b^7*c^4 - 4*a
^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a^5*b^4*c*d^3 + a^6*b^3*d^4)*g^5*x^2 + 4*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d +
6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7*b^2*d^4)*g^5*x + (a^4*b^5*c^4 - 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2
- 4*a^7*b^2*c*d^3 + a^8*b*d^4)*g^5)

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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/(b*g*x+a*g)**5,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 (b g x + a g\right )}^{5}}\,{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/(b*g*x+a*g)^5,x, algorithm="giac")

[Out]

integrate((B*log(e*((b*x + a)/(d*x + c))^n) + A)^2/(b*g*x + a*g)^5, x)