3.158 \(\int \frac{A+B \log (e (\frac{a+b x}{c+d x})^n)}{(a g+b g x)^4 (c i+d i x)^3} \, dx\)

Optimal. Leaf size=587 \[ -\frac{10 b^3 d^2 (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i^3 (a+b x) (b c-a d)^6}-\frac{10 b^2 d^3 \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i^3 (b c-a d)^6}-\frac{b^5 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^4 i^3 (a+b x)^3 (b c-a d)^6}+\frac{5 b^4 d (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^4 i^3 (a+b x)^2 (b c-a d)^6}+\frac{5 b d^4 (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 i^3 (c+d x) (b c-a d)^6}-\frac{d^5 (a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^4 i^3 (c+d x)^2 (b c-a d)^6}-\frac{10 b^3 B d^2 n (c+d x)}{g^4 i^3 (a+b x) (b c-a d)^6}+\frac{5 b^2 B d^3 n \log ^2\left (\frac{a+b x}{c+d x}\right )}{g^4 i^3 (b c-a d)^6}-\frac{b^5 B n (c+d x)^3}{9 g^4 i^3 (a+b x)^3 (b c-a d)^6}+\frac{5 b^4 B d n (c+d x)^2}{4 g^4 i^3 (a+b x)^2 (b c-a d)^6}-\frac{5 b B d^4 n (a+b x)}{g^4 i^3 (c+d x) (b c-a d)^6}+\frac{B d^5 n (a+b x)^2}{4 g^4 i^3 (c+d x)^2 (b c-a d)^6} \]

[Out]

(B*d^5*n*(a + b*x)^2)/(4*(b*c - a*d)^6*g^4*i^3*(c + d*x)^2) - (5*b*B*d^4*n*(a + b*x))/((b*c - a*d)^6*g^4*i^3*(
c + d*x)) - (10*b^3*B*d^2*n*(c + d*x))/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*B*d*n*(c + d*x)^2)/(4*(b*c -
a*d)^6*g^4*i^3*(a + b*x)^2) - (b^5*B*n*(c + d*x)^3)/(9*(b*c - a*d)^6*g^4*i^3*(a + b*x)^3) - (d^5*(a + b*x)^2*
(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^6*g^4*i^3*(c + d*x)^2) + (5*b*d^4*(a + b*x)*(A + B*Log[
e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^6*g^4*i^3*(c + d*x)) - (10*b^3*d^2*(c + d*x)*(A + B*Log[e*((a + b*x)
/(c + d*x))^n]))/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n
]))/(2*(b*c - a*d)^6*g^4*i^3*(a + b*x)^2) - (b^5*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c -
a*d)^6*g^4*i^3*(a + b*x)^3) - (10*b^2*d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(a + b*x)/(c + d*x)])/((
b*c - a*d)^6*g^4*i^3) + (5*b^2*B*d^3*n*Log[(a + b*x)/(c + d*x)]^2)/((b*c - a*d)^6*g^4*i^3)

________________________________________________________________________________________

Rubi [C]  time = 1.69149, antiderivative size = 859, normalized size of antiderivative = 1.46, number of steps used = 38, 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} \[ \frac{5 b^2 B n \log ^2(a+b x) d^3}{(b c-a d)^6 g^4 i^3}+\frac{5 b^2 B n \log ^2(c+d x) d^3}{(b c-a d)^6 g^4 i^3}-\frac{10 b^2 B n \log (a+b x) d^3}{3 (b c-a d)^6 g^4 i^3}-\frac{10 b^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d^3}{(b c-a d)^6 g^4 i^3}-\frac{4 b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d^3}{(b c-a d)^5 g^4 i^3 (c+d x)}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d^3}{2 (b c-a d)^4 g^4 i^3 (c+d x)^2}+\frac{10 b^2 B n \log (c+d x) d^3}{3 (b c-a d)^6 g^4 i^3}-\frac{10 b^2 B n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) d^3}{(b c-a d)^6 g^4 i^3}+\frac{10 b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) d^3}{(b c-a d)^6 g^4 i^3}-\frac{10 b^2 B n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) d^3}{(b c-a d)^6 g^4 i^3}-\frac{10 b^2 B n \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) d^3}{(b c-a d)^6 g^4 i^3}-\frac{10 b^2 B n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) d^3}{(b c-a d)^6 g^4 i^3}+\frac{9 b B n d^3}{2 (b c-a d)^5 g^4 i^3 (c+d x)}+\frac{B n d^3}{4 (b c-a d)^4 g^4 i^3 (c+d x)^2}-\frac{6 b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d^2}{(b c-a d)^5 g^4 i^3 (a+b x)}-\frac{47 b^2 B n d^2}{6 (b c-a d)^5 g^4 i^3 (a+b x)}+\frac{3 b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) d}{2 (b c-a d)^4 g^4 i^3 (a+b x)^2}+\frac{11 b^2 B n d}{12 (b c-a d)^4 g^4 i^3 (a+b x)^2}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^3 g^4 i^3 (a+b x)^3}-\frac{b^2 B n}{9 (b c-a d)^3 g^4 i^3 (a+b x)^3} \]

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)^3),x]

[Out]

-(b^2*B*n)/(9*(b*c - a*d)^3*g^4*i^3*(a + b*x)^3) + (11*b^2*B*d*n)/(12*(b*c - a*d)^4*g^4*i^3*(a + b*x)^2) - (47
*b^2*B*d^2*n)/(6*(b*c - a*d)^5*g^4*i^3*(a + b*x)) + (B*d^3*n)/(4*(b*c - a*d)^4*g^4*i^3*(c + d*x)^2) + (9*b*B*d
^3*n)/(2*(b*c - a*d)^5*g^4*i^3*(c + d*x)) - (10*b^2*B*d^3*n*Log[a + b*x])/(3*(b*c - a*d)^6*g^4*i^3) + (5*b^2*B
*d^3*n*Log[a + b*x]^2)/((b*c - a*d)^6*g^4*i^3) - (b^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c - a*d)^3
*g^4*i^3*(a + b*x)^3) + (3*b^2*d*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^4*g^4*i^3*(a + b*x)^2)
- (6*b^2*d^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c - a*d)^5*g^4*i^3*(a + b*x)) - (d^3*(A + B*Log[e*((
a + b*x)/(c + d*x))^n]))/(2*(b*c - a*d)^4*g^4*i^3*(c + d*x)^2) - (4*b*d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n
]))/((b*c - a*d)^5*g^4*i^3*(c + d*x)) - (10*b^2*d^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/((b*c
- a*d)^6*g^4*i^3) + (10*b^2*B*d^3*n*Log[c + d*x])/(3*(b*c - a*d)^6*g^4*i^3) - (10*b^2*B*d^3*n*Log[-((d*(a + b
*x))/(b*c - a*d))]*Log[c + d*x])/((b*c - a*d)^6*g^4*i^3) + (10*b^2*d^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*
Log[c + d*x])/((b*c - a*d)^6*g^4*i^3) + (5*b^2*B*d^3*n*Log[c + d*x]^2)/((b*c - a*d)^6*g^4*i^3) - (10*b^2*B*d^3
*n*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^6*g^4*i^3) - (10*b^2*B*d^3*n*PolyLog[2, -((d*(a +
b*x))/(b*c - a*d))])/((b*c - a*d)^6*g^4*i^3) - (10*b^2*B*d^3*n*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/((b*c -
a*d)^6*g^4*i^3)

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 )}{(158 c+158 d x)^3 (a g+b g x)^4} \, dx &=\int \left (\frac{b^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3944312 (b c-a d)^3 g^4 (a+b x)^4}-\frac{3 b^3 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3944312 (b c-a d)^4 g^4 (a+b x)^3}+\frac{3 b^3 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^5 g^4 (a+b x)^2}-\frac{5 b^3 d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^6 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 )}{3944312 (b c-a d)^4 g^4 (c+d x)^3}+\frac{b d^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{986078 (b c-a d)^5 g^4 (c+d x)^2}+\frac{5 b^2 d^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^6 g^4 (c+d x)}\right ) \, dx\\ &=-\frac{\left (5 b^3 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}{1972156 (b c-a d)^6 g^4}+\frac{\left (5 b^2 d^4\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{1972156 (b c-a d)^6 g^4}+\frac{\left (3 b^3 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}{1972156 (b c-a d)^5 g^4}+\frac{\left (b d^4\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{986078 (b c-a d)^5 g^4}-\frac{\left (3 b^3 d\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{3944312 (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)^3} \, dx}{3944312 (b c-a d)^4 g^4}+\frac{b^3 \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{3944312 (b c-a d)^3 g^4}\\ &=-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{11832936 (b c-a d)^3 g^4 (a+b x)^3}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (a+b x)^2}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^5 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (c+d x)^2}-\frac{b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{986078 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^6 g^4}+\frac{5 b^2 d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}+\frac{\left (5 b^2 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}{1972156 (b c-a d)^6 g^4}-\frac{\left (5 b^2 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}{1972156 (b c-a d)^6 g^4}+\frac{\left (3 b^2 B d^2 n\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{1972156 (b c-a d)^5 g^4}+\frac{\left (b B d^3 n\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{986078 (b c-a d)^5 g^4}-\frac{\left (3 b^2 B d n\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{7888624 (b c-a d)^4 g^4}+\frac{\left (B d^3 n\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^3} \, dx}{7888624 (b c-a d)^4 g^4}+\frac{\left (b^2 B n\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{11832936 (b c-a d)^3 g^4}\\ &=-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{11832936 (b c-a d)^3 g^4 (a+b x)^3}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (a+b x)^2}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^5 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (c+d x)^2}-\frac{b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{986078 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^6 g^4}+\frac{5 b^2 d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}+\frac{\left (5 b^2 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}{1972156 (b c-a d)^6 g^4}-\frac{\left (5 b^2 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}{1972156 (b c-a d)^6 g^4}+\frac{\left (3 b^2 B d^2 n\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{1972156 (b c-a d)^4 g^4}+\frac{\left (b B d^3 n\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{986078 (b c-a d)^4 g^4}-\frac{\left (3 b^2 B d n\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{7888624 (b c-a d)^3 g^4}+\frac{\left (B d^3 n\right ) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{7888624 (b c-a d)^3 g^4}+\frac{\left (b^2 B n\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{11832936 (b c-a d)^2 g^4}\\ &=-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{11832936 (b c-a d)^3 g^4 (a+b x)^3}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (a+b x)^2}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^5 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (c+d x)^2}-\frac{b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{986078 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^6 g^4}+\frac{5 b^2 d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}+\frac{\left (5 b^3 B d^3 n\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{1972156 (b c-a d)^6 g^4}-\frac{\left (5 b^3 B d^3 n\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{1972156 (b c-a d)^6 g^4}-\frac{\left (5 b^2 B d^4 n\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{1972156 (b c-a d)^6 g^4}+\frac{\left (5 b^2 B d^4 n\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{1972156 (b c-a d)^6 g^4}+\frac{\left (3 b^2 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}{1972156 (b c-a d)^4 g^4}+\frac{\left (b B d^3 n\right ) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{986078 (b c-a d)^4 g^4}-\frac{\left (3 b^2 B d n\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}{7888624 (b c-a d)^3 g^4}+\frac{\left (B d^3 n\right ) \int \left (\frac{b^3}{(b c-a d)^3 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^3}-\frac{b d}{(b c-a d)^2 (c+d x)^2}-\frac{b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{7888624 (b c-a d)^3 g^4}+\frac{\left (b^2 B n\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}{11832936 (b c-a d)^2 g^4}\\ &=-\frac{b^2 B n}{35498808 (b c-a d)^3 g^4 (a+b x)^3}+\frac{11 b^2 B d n}{47331744 (b c-a d)^4 g^4 (a+b x)^2}-\frac{47 b^2 B d^2 n}{23665872 (b c-a d)^5 g^4 (a+b x)}+\frac{B d^3 n}{15777248 (b c-a d)^4 g^4 (c+d x)^2}+\frac{9 b B d^3 n}{7888624 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 B d^3 n \log (a+b x)}{5916468 (b c-a d)^6 g^4}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{11832936 (b c-a d)^3 g^4 (a+b x)^3}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (a+b x)^2}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^5 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (c+d x)^2}-\frac{b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{986078 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^6 g^4}+\frac{5 b^2 B d^3 n \log (c+d x)}{5916468 (b c-a d)^6 g^4}-\frac{5 b^2 B d^3 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}+\frac{5 b^2 d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}-\frac{5 b^2 B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{1972156 (b c-a d)^6 g^4}+\frac{\left (5 b^2 B d^3 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{1972156 (b c-a d)^6 g^4}+\frac{\left (5 b^2 B d^3 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{1972156 (b c-a d)^6 g^4}+\frac{\left (5 b^3 B d^3 n\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{1972156 (b c-a d)^6 g^4}+\frac{\left (5 b^2 B d^4 n\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{1972156 (b c-a d)^6 g^4}\\ &=-\frac{b^2 B n}{35498808 (b c-a d)^3 g^4 (a+b x)^3}+\frac{11 b^2 B d n}{47331744 (b c-a d)^4 g^4 (a+b x)^2}-\frac{47 b^2 B d^2 n}{23665872 (b c-a d)^5 g^4 (a+b x)}+\frac{B d^3 n}{15777248 (b c-a d)^4 g^4 (c+d x)^2}+\frac{9 b B d^3 n}{7888624 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 B d^3 n \log (a+b x)}{5916468 (b c-a d)^6 g^4}+\frac{5 b^2 B d^3 n \log ^2(a+b x)}{3944312 (b c-a d)^6 g^4}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{11832936 (b c-a d)^3 g^4 (a+b x)^3}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (a+b x)^2}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^5 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (c+d x)^2}-\frac{b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{986078 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^6 g^4}+\frac{5 b^2 B d^3 n \log (c+d x)}{5916468 (b c-a d)^6 g^4}-\frac{5 b^2 B d^3 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}+\frac{5 b^2 d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}+\frac{5 b^2 B d^3 n \log ^2(c+d x)}{3944312 (b c-a d)^6 g^4}-\frac{5 b^2 B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{1972156 (b c-a d)^6 g^4}+\frac{\left (5 b^2 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 )}{1972156 (b c-a d)^6 g^4}+\frac{\left (5 b^2 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 )}{1972156 (b c-a d)^6 g^4}\\ &=-\frac{b^2 B n}{35498808 (b c-a d)^3 g^4 (a+b x)^3}+\frac{11 b^2 B d n}{47331744 (b c-a d)^4 g^4 (a+b x)^2}-\frac{47 b^2 B d^2 n}{23665872 (b c-a d)^5 g^4 (a+b x)}+\frac{B d^3 n}{15777248 (b c-a d)^4 g^4 (c+d x)^2}+\frac{9 b B d^3 n}{7888624 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 B d^3 n \log (a+b x)}{5916468 (b c-a d)^6 g^4}+\frac{5 b^2 B d^3 n \log ^2(a+b x)}{3944312 (b c-a d)^6 g^4}-\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{11832936 (b c-a d)^3 g^4 (a+b x)^3}+\frac{3 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (a+b x)^2}-\frac{3 b^2 d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^5 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{7888624 (b c-a d)^4 g^4 (c+d x)^2}-\frac{b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{986078 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{1972156 (b c-a d)^6 g^4}+\frac{5 b^2 B d^3 n \log (c+d x)}{5916468 (b c-a d)^6 g^4}-\frac{5 b^2 B d^3 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}+\frac{5 b^2 d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{1972156 (b c-a d)^6 g^4}+\frac{5 b^2 B d^3 n \log ^2(c+d x)}{3944312 (b c-a d)^6 g^4}-\frac{5 b^2 B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{1972156 (b c-a d)^6 g^4}-\frac{5 b^2 B d^3 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{1972156 (b c-a d)^6 g^4}-\frac{5 b^2 B d^3 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{1972156 (b c-a d)^6 g^4}\\ \end{align*}

Mathematica [C]  time = 2.08042, size = 671, normalized size = 1.14 \[ -\frac{-180 b^2 B d^3 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+180 b^2 B d^3 n \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+360 b^2 d^3 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-360 b^2 d^3 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+\frac{216 b^2 d^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac{54 b^2 d (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{(a+b x)^2}+\frac{12 b^2 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{(a+b x)^3}+\frac{144 b d^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}+\frac{18 d^3 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{(c+d x)^2}+\frac{216 b^3 B c d^2 n}{a+b x}+\frac{66 b^2 B d^2 n (b c-a d)}{a+b x}-\frac{33 b^2 B d n (b c-a d)^2}{(a+b x)^2}+\frac{4 b^2 B n (b c-a d)^3}{(a+b x)^3}-\frac{216 a b^2 B d^3 n}{a+b x}+120 b^2 B d^3 n \log (a+b x)+\frac{144 a b B d^4 n}{c+d x}-\frac{18 b B d^3 n (b c-a d)}{c+d x}-\frac{9 B d^3 n (b c-a d)^2}{(c+d x)^2}-\frac{144 b^2 B c d^3 n}{c+d x}-120 b^2 B d^3 n \log (c+d x)}{36 g^4 i^3 (b c-a d)^6} \]

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)^3),x]

[Out]

-((4*b^2*B*(b*c - a*d)^3*n)/(a + b*x)^3 - (33*b^2*B*d*(b*c - a*d)^2*n)/(a + b*x)^2 + (216*b^3*B*c*d^2*n)/(a +
b*x) - (216*a*b^2*B*d^3*n)/(a + b*x) + (66*b^2*B*d^2*(b*c - a*d)*n)/(a + b*x) - (9*B*d^3*(b*c - a*d)^2*n)/(c +
d*x)^2 - (144*b^2*B*c*d^3*n)/(c + d*x) + (144*a*b*B*d^4*n)/(c + d*x) - (18*b*B*d^3*(b*c - a*d)*n)/(c + d*x) +
120*b^2*B*d^3*n*Log[a + b*x] + (12*b^2*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x)^3 - (5
4*b^2*d*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x)^2 + (216*b^2*d^2*(b*c - a*d)*(A + B*Lo
g[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) + (18*d^3*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c +
d*x)^2 + (144*b*d^3*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x) + 360*b^2*d^3*Log[a + b*x]*(
A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 120*b^2*B*d^3*n*Log[c + d*x] - 360*b^2*d^3*(A + B*Log[e*((a + b*x)/(c
+ d*x))^n])*Log[c + d*x] - 180*b^2*B*d^3*n*(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)]) + 180*b^2*B*d^3*n*((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)]))/(36*(b*c - a*d)^6*g^4*i^3)

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Maple [F]  time = 0.726, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{4} \left ( dix+ci \right ) ^{3}} \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)^3,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)^3,x)

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Maxima [B]  time = 3.15538, size = 5156, normalized size = 8.78 \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)^3,x, algorithm="maxima")

[Out]

-1/6*B*((60*b^4*d^4*x^4 + 2*b^4*c^4 - 13*a*b^3*c^3*d + 47*a^2*b^2*c^2*d^2 + 27*a^3*b*c*d^3 - 3*a^4*d^4 + 30*(3
*b^4*c*d^3 + 5*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 + 23*a*b^3*c*d^3 + 11*a^2*b^2*d^4)*x^2 - 5*(b^4*c^3*d - 11*a
*b^3*c^2*d^2 - 35*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x)/((b^8*c^5*d^2 - 5*a*b^7*c^4*d^3 + 10*a^2*b^6*c^3*d^4 - 10*a^
3*b^5*c^2*d^5 + 5*a^4*b^4*c*d^6 - a^5*b^3*d^7)*g^4*i^3*x^5 + (2*b^8*c^6*d - 7*a*b^7*c^5*d^2 + 5*a^2*b^6*c^4*d^
3 + 10*a^3*b^5*c^3*d^4 - 20*a^4*b^4*c^2*d^5 + 13*a^5*b^3*c*d^6 - 3*a^6*b^2*d^7)*g^4*i^3*x^4 + (b^8*c^7 + a*b^7
*c^6*d - 17*a^2*b^6*c^5*d^2 + 35*a^3*b^5*c^4*d^3 - 25*a^4*b^4*c^3*d^4 - a^5*b^3*c^2*d^5 + 9*a^6*b^2*c*d^6 - 3*
a^7*b*d^7)*g^4*i^3*x^3 + (3*a*b^7*c^7 - 9*a^2*b^6*c^6*d + a^3*b^5*c^5*d^2 + 25*a^4*b^4*c^4*d^3 - 35*a^5*b^3*c^
3*d^4 + 17*a^6*b^2*c^2*d^5 - a^7*b*c*d^6 - a^8*d^7)*g^4*i^3*x^2 + (3*a^2*b^6*c^7 - 13*a^3*b^5*c^6*d + 20*a^4*b
^4*c^5*d^2 - 10*a^5*b^3*c^4*d^3 - 5*a^6*b^2*c^3*d^4 + 7*a^7*b*c^2*d^5 - 2*a^8*c*d^6)*g^4*i^3*x + (a^3*b^5*c^7
- 5*a^4*b^4*c^6*d + 10*a^5*b^3*c^5*d^2 - 10*a^6*b^2*c^4*d^3 + 5*a^7*b*c^3*d^4 - a^8*c^2*d^5)*g^4*i^3) + 60*b^2
*d^3*log(b*x + a)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6
*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3) - 60*b^2*d^3*log(d*x + c)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20
*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3))*log(e*(b*x/(d*x + c) + a/(d*x + c))
^n) - 1/36*(4*b^5*c^5 - 45*a*b^4*c^4*d + 360*a^2*b^3*c^3*d^2 - 490*a^3*b^2*c^2*d^3 + 180*a^4*b*c*d^4 - 9*a^5*d
^5 + 120*(b^5*c*d^4 - a*b^4*d^5)*x^4 + 120*(3*b^5*c^2*d^3 - 2*a*b^4*c*d^4 - a^2*b^3*d^5)*x^3 + 20*(11*b^5*c^3*
d^2 + 21*a*b^4*c^2*d^3 - 39*a^2*b^3*c*d^4 + 7*a^3*b^2*d^5)*x^2 - 180*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c
*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d
^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(b*x + a)^2 - 180*(b^5*d^5*x^5 + a^3*b^2*c
^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^
3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(d*x + c)^2 - 5*(5*b^5*c^
4*d - 108*a*b^4*c^3*d^2 + 78*a^2*b^3*c^2*d^3 + 52*a^3*b^2*c*d^4 - 27*a^4*b*d^5)*x + 120*(b^5*d^5*x^5 + a^3*b^2
*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*
d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(b*x + a) - 120*(b^5*d^
5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3
+ (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x - 3*(b^5*d^5
*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 +
(3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(b*x + a)
)*log(d*x + c))*B*n/(a^3*b^6*c^8*g^4*i^3 - 6*a^4*b^5*c^7*d*g^4*i^3 + 15*a^5*b^4*c^6*d^2*g^4*i^3 - 20*a^6*b^3*c
^5*d^3*g^4*i^3 + 15*a^7*b^2*c^4*d^4*g^4*i^3 - 6*a^8*b*c^3*d^5*g^4*i^3 + a^9*c^2*d^6*g^4*i^3 + (b^9*c^6*d^2*g^4
*i^3 - 6*a*b^8*c^5*d^3*g^4*i^3 + 15*a^2*b^7*c^4*d^4*g^4*i^3 - 20*a^3*b^6*c^3*d^5*g^4*i^3 + 15*a^4*b^5*c^2*d^6*
g^4*i^3 - 6*a^5*b^4*c*d^7*g^4*i^3 + a^6*b^3*d^8*g^4*i^3)*x^5 + (2*b^9*c^7*d*g^4*i^3 - 9*a*b^8*c^6*d^2*g^4*i^3
+ 12*a^2*b^7*c^5*d^3*g^4*i^3 + 5*a^3*b^6*c^4*d^4*g^4*i^3 - 30*a^4*b^5*c^3*d^5*g^4*i^3 + 33*a^5*b^4*c^2*d^6*g^4
*i^3 - 16*a^6*b^3*c*d^7*g^4*i^3 + 3*a^7*b^2*d^8*g^4*i^3)*x^4 + (b^9*c^8*g^4*i^3 - 18*a^2*b^7*c^6*d^2*g^4*i^3 +
52*a^3*b^6*c^5*d^3*g^4*i^3 - 60*a^4*b^5*c^4*d^4*g^4*i^3 + 24*a^5*b^4*c^3*d^5*g^4*i^3 + 10*a^6*b^3*c^2*d^6*g^4
*i^3 - 12*a^7*b^2*c*d^7*g^4*i^3 + 3*a^8*b*d^8*g^4*i^3)*x^3 + (3*a*b^8*c^8*g^4*i^3 - 12*a^2*b^7*c^7*d*g^4*i^3 +
10*a^3*b^6*c^6*d^2*g^4*i^3 + 24*a^4*b^5*c^5*d^3*g^4*i^3 - 60*a^5*b^4*c^4*d^4*g^4*i^3 + 52*a^6*b^3*c^3*d^5*g^4
*i^3 - 18*a^7*b^2*c^2*d^6*g^4*i^3 + a^9*d^8*g^4*i^3)*x^2 + (3*a^2*b^7*c^8*g^4*i^3 - 16*a^3*b^6*c^7*d*g^4*i^3 +
33*a^4*b^5*c^6*d^2*g^4*i^3 - 30*a^5*b^4*c^5*d^3*g^4*i^3 + 5*a^6*b^3*c^4*d^4*g^4*i^3 + 12*a^7*b^2*c^3*d^5*g^4*
i^3 - 9*a^8*b*c^2*d^6*g^4*i^3 + 2*a^9*c*d^7*g^4*i^3)*x) - 1/6*A*((60*b^4*d^4*x^4 + 2*b^4*c^4 - 13*a*b^3*c^3*d
+ 47*a^2*b^2*c^2*d^2 + 27*a^3*b*c*d^3 - 3*a^4*d^4 + 30*(3*b^4*c*d^3 + 5*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 + 2
3*a*b^3*c*d^3 + 11*a^2*b^2*d^4)*x^2 - 5*(b^4*c^3*d - 11*a*b^3*c^2*d^2 - 35*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x)/((b
^8*c^5*d^2 - 5*a*b^7*c^4*d^3 + 10*a^2*b^6*c^3*d^4 - 10*a^3*b^5*c^2*d^5 + 5*a^4*b^4*c*d^6 - a^5*b^3*d^7)*g^4*i^
3*x^5 + (2*b^8*c^6*d - 7*a*b^7*c^5*d^2 + 5*a^2*b^6*c^4*d^3 + 10*a^3*b^5*c^3*d^4 - 20*a^4*b^4*c^2*d^5 + 13*a^5*
b^3*c*d^6 - 3*a^6*b^2*d^7)*g^4*i^3*x^4 + (b^8*c^7 + a*b^7*c^6*d - 17*a^2*b^6*c^5*d^2 + 35*a^3*b^5*c^4*d^3 - 25
*a^4*b^4*c^3*d^4 - a^5*b^3*c^2*d^5 + 9*a^6*b^2*c*d^6 - 3*a^7*b*d^7)*g^4*i^3*x^3 + (3*a*b^7*c^7 - 9*a^2*b^6*c^6
*d + a^3*b^5*c^5*d^2 + 25*a^4*b^4*c^4*d^3 - 35*a^5*b^3*c^3*d^4 + 17*a^6*b^2*c^2*d^5 - a^7*b*c*d^6 - a^8*d^7)*g
^4*i^3*x^2 + (3*a^2*b^6*c^7 - 13*a^3*b^5*c^6*d + 20*a^4*b^4*c^5*d^2 - 10*a^5*b^3*c^4*d^3 - 5*a^6*b^2*c^3*d^4 +
7*a^7*b*c^2*d^5 - 2*a^8*c*d^6)*g^4*i^3*x + (a^3*b^5*c^7 - 5*a^4*b^4*c^6*d + 10*a^5*b^3*c^5*d^2 - 10*a^6*b^2*c
^4*d^3 + 5*a^7*b*c^3*d^4 - a^8*c^2*d^5)*g^4*i^3) + 60*b^2*d^3*log(b*x + a)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*
b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3) - 60*b^2*d^3*log(d*x
+ c)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5
+ a^6*d^6)*g^4*i^3))

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Fricas [B]  time = 0.76607, size = 4535, normalized size = 7.73 \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)^3,x, algorithm="fricas")

[Out]

-1/36*(12*A*b^5*c^5 - 90*A*a*b^4*c^4*d + 360*A*a^2*b^3*c^3*d^2 - 120*A*a^3*b^2*c^2*d^3 - 180*A*a^4*b*c*d^4 + 1
8*A*a^5*d^5 + 120*(3*A*b^5*c*d^4 - 3*A*a*b^4*d^5 + (B*b^5*c*d^4 - B*a*b^4*d^5)*n)*x^4 + 60*(9*A*b^5*c^2*d^3 +
6*A*a*b^4*c*d^4 - 15*A*a^2*b^3*d^5 + 2*(3*B*b^5*c^2*d^3 - 2*B*a*b^4*c*d^4 - B*a^2*b^3*d^5)*n)*x^3 + 20*(6*A*b^
5*c^3*d^2 + 63*A*a*b^4*c^2*d^3 - 36*A*a^2*b^3*c*d^4 - 33*A*a^3*b^2*d^5 + (11*B*b^5*c^3*d^2 + 21*B*a*b^4*c^2*d^
3 - 39*B*a^2*b^3*c*d^4 + 7*B*a^3*b^2*d^5)*n)*x^2 + 180*(B*b^5*d^5*n*x^5 + B*a^3*b^2*c^2*d^3*n + (2*B*b^5*c*d^4
+ 3*B*a*b^4*d^5)*n*x^4 + (B*b^5*c^2*d^3 + 6*B*a*b^4*c*d^4 + 3*B*a^2*b^3*d^5)*n*x^3 + (3*B*a*b^4*c^2*d^3 + 6*B
*a^2*b^3*c*d^4 + B*a^3*b^2*d^5)*n*x^2 + (3*B*a^2*b^3*c^2*d^3 + 2*B*a^3*b^2*c*d^4)*n*x)*log((b*x + a)/(d*x + c)
)^2 + (4*B*b^5*c^5 - 45*B*a*b^4*c^4*d + 360*B*a^2*b^3*c^3*d^2 - 490*B*a^3*b^2*c^2*d^3 + 180*B*a^4*b*c*d^4 - 9*
B*a^5*d^5)*n - 5*(6*A*b^5*c^4*d - 72*A*a*b^4*c^3*d^2 - 144*A*a^2*b^3*c^2*d^3 + 192*A*a^3*b^2*c*d^4 + 18*A*a^4*
b*d^5 + (5*B*b^5*c^4*d - 108*B*a*b^4*c^3*d^2 + 78*B*a^2*b^3*c^2*d^3 + 52*B*a^3*b^2*c*d^4 - 27*B*a^4*b*d^5)*n)*
x + 6*(2*B*b^5*c^5 - 15*B*a*b^4*c^4*d + 60*B*a^2*b^3*c^3*d^2 - 20*B*a^3*b^2*c^2*d^3 - 30*B*a^4*b*c*d^4 + 3*B*a
^5*d^5 + 60*(B*b^5*c*d^4 - B*a*b^4*d^5)*x^4 + 30*(3*B*b^5*c^2*d^3 + 2*B*a*b^4*c*d^4 - 5*B*a^2*b^3*d^5)*x^3 + 1
0*(2*B*b^5*c^3*d^2 + 21*B*a*b^4*c^2*d^3 - 12*B*a^2*b^3*c*d^4 - 11*B*a^3*b^2*d^5)*x^2 - 5*(B*b^5*c^4*d - 12*B*a
*b^4*c^3*d^2 - 24*B*a^2*b^3*c^2*d^3 + 32*B*a^3*b^2*c*d^4 + 3*B*a^4*b*d^5)*x + 60*(B*b^5*d^5*x^5 + B*a^3*b^2*c^
2*d^3 + (2*B*b^5*c*d^4 + 3*B*a*b^4*d^5)*x^4 + (B*b^5*c^2*d^3 + 6*B*a*b^4*c*d^4 + 3*B*a^2*b^3*d^5)*x^3 + (3*B*a
*b^4*c^2*d^3 + 6*B*a^2*b^3*c*d^4 + B*a^3*b^2*d^5)*x^2 + (3*B*a^2*b^3*c^2*d^3 + 2*B*a^3*b^2*c*d^4)*x)*log((b*x
+ a)/(d*x + c)))*log(e) + 6*(60*A*a^3*b^2*c^2*d^3 + 20*(B*b^5*d^5*n + 3*A*b^5*d^5)*x^5 + 20*(5*B*b^5*c*d^4*n +
6*A*b^5*c*d^4 + 9*A*a*b^4*d^5)*x^4 + 10*(6*A*b^5*c^2*d^3 + 36*A*a*b^4*c*d^4 + 18*A*a^2*b^3*d^5 + (11*B*b^5*c^
2*d^3 + 18*B*a*b^4*c*d^4 - 9*B*a^2*b^3*d^5)*n)*x^3 + 10*(18*A*a*b^4*c^2*d^3 + 36*A*a^2*b^3*c*d^4 + 6*A*a^3*b^2
*d^5 + (2*B*b^5*c^3*d^2 + 27*B*a*b^4*c^2*d^3 - 9*B*a^3*b^2*d^5)*n)*x^2 + (2*B*b^5*c^5 - 15*B*a*b^4*c^4*d + 60*
B*a^2*b^3*c^3*d^2 - 30*B*a^4*b*c*d^4 + 3*B*a^5*d^5)*n + 5*(36*A*a^2*b^3*c^2*d^3 + 24*A*a^3*b^2*c*d^4 - (B*b^5*
c^4*d - 12*B*a*b^4*c^3*d^2 - 36*B*a^2*b^3*c^2*d^3 + 24*B*a^3*b^2*c*d^4 + 3*B*a^4*b*d^5)*n)*x)*log((b*x + a)/(d
*x + c)))/((b^9*c^6*d^2 - 6*a*b^8*c^5*d^3 + 15*a^2*b^7*c^4*d^4 - 20*a^3*b^6*c^3*d^5 + 15*a^4*b^5*c^2*d^6 - 6*a
^5*b^4*c*d^7 + a^6*b^3*d^8)*g^4*i^3*x^5 + (2*b^9*c^7*d - 9*a*b^8*c^6*d^2 + 12*a^2*b^7*c^5*d^3 + 5*a^3*b^6*c^4*
d^4 - 30*a^4*b^5*c^3*d^5 + 33*a^5*b^4*c^2*d^6 - 16*a^6*b^3*c*d^7 + 3*a^7*b^2*d^8)*g^4*i^3*x^4 + (b^9*c^8 - 18*
a^2*b^7*c^6*d^2 + 52*a^3*b^6*c^5*d^3 - 60*a^4*b^5*c^4*d^4 + 24*a^5*b^4*c^3*d^5 + 10*a^6*b^3*c^2*d^6 - 12*a^7*b
^2*c*d^7 + 3*a^8*b*d^8)*g^4*i^3*x^3 + (3*a*b^8*c^8 - 12*a^2*b^7*c^7*d + 10*a^3*b^6*c^6*d^2 + 24*a^4*b^5*c^5*d^
3 - 60*a^5*b^4*c^4*d^4 + 52*a^6*b^3*c^3*d^5 - 18*a^7*b^2*c^2*d^6 + a^9*d^8)*g^4*i^3*x^2 + (3*a^2*b^7*c^8 - 16*
a^3*b^6*c^7*d + 33*a^4*b^5*c^6*d^2 - 30*a^5*b^4*c^5*d^3 + 5*a^6*b^3*c^4*d^4 + 12*a^7*b^2*c^3*d^5 - 9*a^8*b*c^2
*d^6 + 2*a^9*c*d^7)*g^4*i^3*x + (a^3*b^6*c^8 - 6*a^4*b^5*c^7*d + 15*a^5*b^4*c^6*d^2 - 20*a^6*b^3*c^5*d^3 + 15*
a^7*b^2*c^4*d^4 - 6*a^8*b*c^3*d^5 + a^9*c^2*d^6)*g^4*i^3)

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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)**3,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 )}^{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))/(b*g*x+a*g)^4/(d*i*x+c*i)^3,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)^3), x)