3.142 \(\int \frac{A+B \log (e (\frac{a+b x}{c+d x})^n)}{(a g+b g x)^4 (c i+d i x)} \, dx\)

Optimal. Leaf size=389 \[ \frac{3 b^2 d (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^4 i (a+b x)^2 (b c-a d)^4}-\frac{b^3 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^4 i (a+b x)^3 (b c-a d)^4}-\frac{d^3 \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i (b c-a d)^4}-\frac{3 b d^2 (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i (a+b x) (b c-a d)^4}+\frac{3 b^2 B d n (c+d x)^2}{4 g^4 i (a+b x)^2 (b c-a d)^4}-\frac{b^3 B n (c+d x)^3}{9 g^4 i (a+b x)^3 (b c-a d)^4}-\frac{3 b B d^2 n (c+d x)}{g^4 i (a+b x) (b c-a d)^4}+\frac{B d^3 n \log ^2\left (\frac{a+b x}{c+d x}\right )}{2 g^4 i (b c-a d)^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 1.07713, antiderivative size = 646, normalized size of antiderivative = 1.66, number of steps used = 30, number of rules used = 11, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.256, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{B d^3 n \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g^4 i (b c-a d)^4}-\frac{B d^3 n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g^4 i (b c-a d)^4}-\frac{d^3 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i (b c-a d)^4}+\frac{d^3 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i (b c-a d)^4}-\frac{d^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i (a+b x) (b c-a d)^3}+\frac{d \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^4 i (a+b x)^2 (b c-a d)^2}-\frac{B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A}{3 g^4 i (a+b x)^3 (b c-a d)}-\frac{11 B d^2 n}{6 g^4 i (a+b x) (b c-a d)^3}+\frac{B d^3 n \log ^2(a+b x)}{2 g^4 i (b c-a d)^4}+\frac{B d^3 n \log ^2(c+d x)}{2 g^4 i (b c-a d)^4}-\frac{11 B d^3 n \log (a+b x)}{6 g^4 i (b c-a d)^4}+\frac{11 B d^3 n \log (c+d x)}{6 g^4 i (b c-a d)^4}-\frac{B d^3 n \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g^4 i (b c-a d)^4}-\frac{B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g^4 i (b c-a d)^4}+\frac{5 B d n}{12 g^4 i (a+b x)^2 (b c-a d)^2}-\frac{B n}{9 g^4 i (a+b x)^3 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^4*(c*i + d*i*x)),x]

[Out]

-(B*n)/(9*(b*c - a*d)*g^4*i*(a + b*x)^3) + (5*B*d*n)/(12*(b*c - a*d)^2*g^4*i*(a + b*x)^2) - (11*B*d^2*n)/(6*(b
*c - a*d)^3*g^4*i*(a + b*x)) - (11*B*d^3*n*Log[a + b*x])/(6*(b*c - a*d)^4*g^4*i) + (B*d^3*n*Log[a + b*x]^2)/(2
*(b*c - a*d)^4*g^4*i) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])/(3*(b*c - a*d)*g^4*i*(a + b*x)^3) + (d*(A + B*L
og[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^2*g^4*i*(a + b*x)^2) - (d^2*(A + B*Log[e*((a + b*x)/(c + d*x))^
n]))/((b*c - a*d)^3*g^4*i*(a + b*x)) - (d^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^
4*g^4*i) + (11*B*d^3*n*Log[c + d*x])/(6*(b*c - a*d)^4*g^4*i) - (B*d^3*n*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[
c + d*x])/((b*c - a*d)^4*g^4*i) + (d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x])/((b*c - a*d)^4*g^4
*i) + (B*d^3*n*Log[c + d*x]^2)/(2*(b*c - a*d)^4*g^4*i) - (B*d^3*n*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])
/((b*c - a*d)^4*g^4*i) - (B*d^3*n*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^4*g^4*i) - (B*d^3*n*P
olyLog[2, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^4*g^4*i)

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 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 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{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(142 c+142 d x) (a g+b g x)^4} \, dx &=\int \left (\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d) g^4 (a+b x)^4}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^2 g^4 (a+b x)^3}+\frac{b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)^2}-\frac{b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4 (a+b x)}+\frac{d^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4 (c+d x)}\right ) \, dx\\ &=-\frac{\left (b d^3\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{142 (b c-a d)^4 g^4}+\frac{d^4 \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{142 (b c-a d)^4 g^4}+\frac{\left (b d^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{142 (b c-a d)^3 g^4}-\frac{(b d) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{142 (b c-a d)^2 g^4}+\frac{b \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{142 (b c-a d) g^4}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac{\left (B d^3 n\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}{142 (b c-a d)^4 g^4}-\frac{\left (B d^3 n\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}{142 (b c-a d)^4 g^4}+\frac{\left (B d^2 n\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{142 (b c-a d)^3 g^4}-\frac{(B d n) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{284 (b c-a d)^2 g^4}+\frac{(B n) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{426 (b c-a d) g^4}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac{(B n) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{426 g^4}+\frac{\left (B d^3 n\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{142 (b c-a d)^4 g^4}-\frac{\left (B d^3 n\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{142 (b c-a d)^4 g^4}+\frac{\left (B d^2 n\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{142 (b c-a d)^2 g^4}-\frac{(B d n) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{284 (b c-a d) g^4}\\ &=-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac{(B n) \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}{426 g^4}+\frac{\left (b B d^3 n\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{142 (b c-a d)^4 g^4}-\frac{\left (b B d^3 n\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{142 (b c-a d)^4 g^4}-\frac{\left (B d^4 n\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{142 (b c-a d)^4 g^4}+\frac{\left (B d^4 n\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{142 (b c-a d)^4 g^4}+\frac{\left (B d^2 n\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}{142 (b c-a d)^2 g^4}-\frac{(B d n) \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}{284 (b c-a d) g^4}\\ &=-\frac{B n}{1278 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d n}{1704 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 n}{852 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 n \log (a+b x)}{852 (b c-a d)^4 g^4}-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac{11 B d^3 n \log (c+d x)}{852 (b c-a d)^4 g^4}-\frac{B d^3 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}-\frac{B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}+\frac{\left (B d^3 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{142 (b c-a d)^4 g^4}+\frac{\left (B d^3 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{142 (b c-a d)^4 g^4}+\frac{\left (b B d^3 n\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{142 (b c-a d)^4 g^4}+\frac{\left (B d^4 n\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{142 (b c-a d)^4 g^4}\\ &=-\frac{B n}{1278 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d n}{1704 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 n}{852 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 n \log (a+b x)}{852 (b c-a d)^4 g^4}+\frac{B d^3 n \log ^2(a+b x)}{284 (b c-a d)^4 g^4}-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac{11 B d^3 n \log (c+d x)}{852 (b c-a d)^4 g^4}-\frac{B d^3 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac{B d^3 n \log ^2(c+d x)}{284 (b c-a d)^4 g^4}-\frac{B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}+\frac{\left (B d^3 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{142 (b c-a d)^4 g^4}+\frac{\left (B d^3 n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{142 (b c-a d)^4 g^4}\\ &=-\frac{B n}{1278 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d n}{1704 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2 n}{852 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 n \log (a+b x)}{852 (b c-a d)^4 g^4}+\frac{B d^3 n \log ^2(a+b x)}{284 (b c-a d)^4 g^4}-\frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{426 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{284 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{142 (b c-a d)^4 g^4}+\frac{11 B d^3 n \log (c+d x)}{852 (b c-a d)^4 g^4}-\frac{B d^3 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{142 (b c-a d)^4 g^4}+\frac{B d^3 n \log ^2(c+d x)}{284 (b c-a d)^4 g^4}-\frac{B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}-\frac{B d^3 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}-\frac{B d^3 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{142 (b c-a d)^4 g^4}\\ \end{align*}

Mathematica [C]  time = 0.754456, size = 518, normalized size = 1.33 \[ \frac{-36 B d^3 n \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )-36 B d^3 n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\frac{36 A d^2 (a d-b c)}{a+b x}+\frac{18 A d (b c-a d)^2}{(a+b x)^2}-\frac{12 A (b c-a d)^3}{(a+b x)^3}-36 A d^3 \log (a+b x)-36 B d^3 \log (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+36 B d^3 \log (c+d x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\frac{36 B d^2 (a d-b c) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x}+\frac{66 B d^2 n (a d-b c)}{a+b x}-36 B d^3 n \log (c+d x) \log \left (\frac{d (a+b x)}{a d-b c}\right )-36 B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )+\frac{18 B d (b c-a d)^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2}-\frac{12 B (b c-a d)^3 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3}+\frac{15 B d n (b c-a d)^2}{(a+b x)^2}-\frac{4 B n (b c-a d)^3}{(a+b x)^3}+18 B d^3 n \log ^2(a+b x)-66 B d^3 n \log (a+b x)+36 A d^3 \log (c+d x)+18 B d^3 n \log ^2(c+d x)+66 B d^3 n \log (c+d x)}{36 g^4 i (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])/((a*g + b*g*x)^4*(c*i + d*i*x)),x]

[Out]

((-12*A*(b*c - a*d)^3)/(a + b*x)^3 - (4*B*(b*c - a*d)^3*n)/(a + b*x)^3 + (18*A*d*(b*c - a*d)^2)/(a + b*x)^2 +
(15*B*d*(b*c - a*d)^2*n)/(a + b*x)^2 + (36*A*d^2*(-(b*c) + a*d))/(a + b*x) + (66*B*d^2*(-(b*c) + a*d)*n)/(a +
b*x) - 36*A*d^3*Log[a + b*x] - 66*B*d^3*n*Log[a + b*x] + 18*B*d^3*n*Log[a + b*x]^2 - (12*B*(b*c - a*d)^3*Log[e
*((a + b*x)/(c + d*x))^n])/(a + b*x)^3 + (18*B*d*(b*c - a*d)^2*Log[e*((a + b*x)/(c + d*x))^n])/(a + b*x)^2 + (
36*B*d^2*(-(b*c) + a*d)*Log[e*((a + b*x)/(c + d*x))^n])/(a + b*x) - 36*B*d^3*Log[a + b*x]*Log[e*((a + b*x)/(c
+ d*x))^n] + 36*A*d^3*Log[c + d*x] + 66*B*d^3*n*Log[c + d*x] - 36*B*d^3*n*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Lo
g[c + d*x] + 36*B*d^3*Log[e*((a + b*x)/(c + d*x))^n]*Log[c + d*x] + 18*B*d^3*n*Log[c + d*x]^2 - 36*B*d^3*n*Log
[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)] - 36*B*d^3*n*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] - 36*B*d^3*n*Po
lyLog[2, (b*(c + d*x))/(b*c - a*d)])/(36*(b*c - a*d)^4*g^4*i)

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Maple [F]  time = 0.773, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{4} \left ( dix+ci \right ) } \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) }\, 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))/(b*g*x+a*g)^4/(d*i*x+c*i),x)

[Out]

int((A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4/(d*i*x+c*i),x)

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Maxima [B]  time = 1.79299, size = 1987, normalized size = 5.11 \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))/(b*g*x+a*g)^4/(d*i*x+c*i),x, algorithm="maxima")

[Out]

-1/6*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^3 - 3*a*b^5*c
^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4*i*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d
^3)*g^4*i*x^2 + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*g^4*i*x + (a^3*b^3*c^3 - 3*a^4
*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*g^4*i) + 6*d^3*log(b*x + a)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^
2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^4*i) - 6*d^3*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3
*b*c*d^3 + a^4*d^4)*g^4*i))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) - 1/36*(4*b^3*c^3 - 27*a*b^2*c^2*d + 108*a^
2*b*c*d^2 - 85*a^3*d^3 + 66*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x +
a^3*d^3)*log(b*x + a)^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(d*x + c)^2 - 3*(5*b
^3*c^2*d - 54*a*b^2*c*d^2 + 49*a^2*b*d^3)*x + 66*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log
(b*x + a) - 6*(11*b^3*d^3*x^3 + 33*a*b^2*d^3*x^2 + 33*a^2*b*d^3*x + 11*a^3*d^3 - 6*(b^3*d^3*x^3 + 3*a*b^2*d^3*
x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a))*log(d*x + c))*B*n/(a^3*b^4*c^4*g^4*i - 4*a^4*b^3*c^3*d*g^4*i + 6*
a^5*b^2*c^2*d^2*g^4*i - 4*a^6*b*c*d^3*g^4*i + a^7*d^4*g^4*i + (b^7*c^4*g^4*i - 4*a*b^6*c^3*d*g^4*i + 6*a^2*b^5
*c^2*d^2*g^4*i - 4*a^3*b^4*c*d^3*g^4*i + a^4*b^3*d^4*g^4*i)*x^3 + 3*(a*b^6*c^4*g^4*i - 4*a^2*b^5*c^3*d*g^4*i +
6*a^3*b^4*c^2*d^2*g^4*i - 4*a^4*b^3*c*d^3*g^4*i + a^5*b^2*d^4*g^4*i)*x^2 + 3*(a^2*b^5*c^4*g^4*i - 4*a^3*b^4*c
^3*d*g^4*i + 6*a^4*b^3*c^2*d^2*g^4*i - 4*a^5*b^2*c*d^3*g^4*i + a^6*b*d^4*g^4*i)*x) - 1/6*A*((6*b^2*d^2*x^2 + 2
*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a
^3*b^3*d^3)*g^4*i*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3)*g^4*i*x^2 + 3*(a^2*b^4
*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*g^4*i*x + (a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2
- a^6*d^3)*g^4*i) + 6*d^3*log(b*x + a)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^
4)*g^4*i) - 6*d^3*log(d*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^4*i)
)

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Fricas [B]  time = 0.599218, size = 1809, normalized size = 4.65 \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))/(b*g*x+a*g)^4/(d*i*x+c*i),x, algorithm="fricas")

[Out]

-1/36*(12*A*b^3*c^3 - 54*A*a*b^2*c^2*d + 108*A*a^2*b*c*d^2 - 66*A*a^3*d^3 + 6*(6*A*b^3*c*d^2 - 6*A*a*b^2*d^3 +
11*(B*b^3*c*d^2 - B*a*b^2*d^3)*n)*x^2 + 18*(B*b^3*d^3*n*x^3 + 3*B*a*b^2*d^3*n*x^2 + 3*B*a^2*b*d^3*n*x + B*a^3
*d^3*n)*log((b*x + a)/(d*x + c))^2 + (4*B*b^3*c^3 - 27*B*a*b^2*c^2*d + 108*B*a^2*b*c*d^2 - 85*B*a^3*d^3)*n - 3
*(6*A*b^3*c^2*d - 36*A*a*b^2*c*d^2 + 30*A*a^2*b*d^3 + (5*B*b^3*c^2*d - 54*B*a*b^2*c*d^2 + 49*B*a^2*b*d^3)*n)*x
+ 6*(2*B*b^3*c^3 - 9*B*a*b^2*c^2*d + 18*B*a^2*b*c*d^2 - 11*B*a^3*d^3 + 6*(B*b^3*c*d^2 - B*a*b^2*d^3)*x^2 - 3*
(B*b^3*c^2*d - 6*B*a*b^2*c*d^2 + 5*B*a^2*b*d^3)*x + 6*(B*b^3*d^3*x^3 + 3*B*a*b^2*d^3*x^2 + 3*B*a^2*b*d^3*x + B
*a^3*d^3)*log((b*x + a)/(d*x + c)))*log(e) + 6*(6*A*a^3*d^3 + (11*B*b^3*d^3*n + 6*A*b^3*d^3)*x^3 + 3*(6*A*a*b^
2*d^3 + (2*B*b^3*c*d^2 + 9*B*a*b^2*d^3)*n)*x^2 + (2*B*b^3*c^3 - 9*B*a*b^2*c^2*d + 18*B*a^2*b*c*d^2)*n + 3*(6*A
*a^2*b*d^3 - (B*b^3*c^2*d - 6*B*a*b^2*c*d^2 - 6*B*a^2*b*d^3)*n)*x)*log((b*x + a)/(d*x + c)))/((b^7*c^4 - 4*a*b
^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^4*i*x^3 + 3*(a*b^6*c^4 - 4*a^2*b^5*c^3*d + 6*a
^3*b^4*c^2*d^2 - 4*a^4*b^3*c*d^3 + a^5*b^2*d^4)*g^4*i*x^2 + 3*(a^2*b^5*c^4 - 4*a^3*b^4*c^3*d + 6*a^4*b^3*c^2*d
^2 - 4*a^5*b^2*c*d^3 + a^6*b*d^4)*g^4*i*x + (a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3
+ a^7*d^4)*g^4*i)

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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))/(b*g*x+a*g)**4/(d*i*x+c*i),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A}{{\left (b g x + a g\right )}^{4}{\left (d i x + c i\right )}}\,{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))/(b*g*x+a*g)^4/(d*i*x+c*i),x, algorithm="giac")

[Out]

integrate((B*log(e*((b*x + a)/(d*x + c))^n) + A)/((b*g*x + a*g)^4*(d*i*x + c*i)), x)