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

Optimal. Leaf size=380 \[ -\frac{b^3 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^2 (a+b x)^2 (b c-a d)^4}+\frac{3 b^2 d (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^2 (a+b x) (b c-a d)^4}+\frac{3 b d^2 \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^2 (b c-a d)^4}-\frac{d^3 (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^2 (c+d x) (b c-a d)^4}-\frac{b^3 B n (c+d x)^2}{4 g^3 i^2 (a+b x)^2 (b c-a d)^4}+\frac{3 b^2 B d n (c+d x)}{g^3 i^2 (a+b x) (b c-a d)^4}+\frac{B d^3 n (a+b x)}{g^3 i^2 (c+d x) (b c-a d)^4}-\frac{3 b B d^2 n \log ^2\left (\frac{a+b x}{c+d x}\right )}{2 g^3 i^2 (b c-a d)^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 1.08723, antiderivative size = 656, normalized size of antiderivative = 1.73, 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} \[ \frac{3 b B d^2 n \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g^3 i^2 (b c-a d)^4}+\frac{3 b B d^2 n \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g^3 i^2 (b c-a d)^4}+\frac{3 b d^2 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^2 (b c-a d)^4}+\frac{d^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^2 (c+d x) (b c-a d)^3}-\frac{3 b d^2 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^2 (b c-a d)^4}+\frac{2 b d \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^2 (a+b x) (b c-a d)^3}-\frac{b \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^2 (a+b x)^2 (b c-a d)^2}-\frac{B d^2 n}{g^3 i^2 (c+d x) (b c-a d)^3}-\frac{3 b B d^2 n \log ^2(a+b x)}{2 g^3 i^2 (b c-a d)^4}-\frac{3 b B d^2 n \log ^2(c+d x)}{2 g^3 i^2 (b c-a d)^4}+\frac{3 b B d^2 n \log (a+b x)}{2 g^3 i^2 (b c-a d)^4}-\frac{3 b B d^2 n \log (c+d x)}{2 g^3 i^2 (b c-a d)^4}+\frac{3 b B d^2 n \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g^3 i^2 (b c-a d)^4}+\frac{3 b B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g^3 i^2 (b c-a d)^4}+\frac{5 b B d n}{2 g^3 i^2 (a+b x) (b c-a d)^3}-\frac{b B n}{4 g^3 i^2 (a+b x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 )}{(149 c+149 d x)^2 (a g+b g x)^3} \, dx &=\int \left (\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^2 g^3 (a+b x)^3}-\frac{2 b^2 d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (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 )}{22201 (b c-a d)^4 g^3 (a+b x)}-\frac{d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (c+d x)^2}-\frac{3 b d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3 (c+d x)}\right ) \, dx\\ &=\frac{\left (3 b^2 d^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{22201 (b c-a d)^4 g^3}-\frac{\left (3 b d^3\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{22201 (b c-a d)^4 g^3}-\frac{\left (2 b^2 d\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{22201 (b c-a d)^3 g^3}-\frac{d^3 \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{22201 (b c-a d)^3 g^3}+\frac{b^2 \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{22201 (b c-a d)^2 g^3}\\ &=-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac{2 b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac{3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac{\left (3 b B d^2 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}{22201 (b c-a d)^4 g^3}+\frac{\left (3 b B d^2 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}{22201 (b c-a d)^4 g^3}-\frac{(2 b B d n) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{22201 (b c-a d)^3 g^3}-\frac{\left (B d^2 n\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{22201 (b c-a d)^3 g^3}+\frac{(b B n) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{44402 (b c-a d)^2 g^3}\\ &=-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac{2 b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac{3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac{\left (3 b B d^2 n\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{22201 (b c-a d)^4 g^3}+\frac{\left (3 b B d^2 n\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{22201 (b c-a d)^4 g^3}-\frac{(2 b B d n) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{22201 (b c-a d)^2 g^3}-\frac{\left (B d^2 n\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{22201 (b c-a d)^2 g^3}+\frac{(b B n) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{44402 (b c-a d) g^3}\\ &=-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac{2 b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac{3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac{\left (3 b^2 B d^2 n\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{22201 (b c-a d)^4 g^3}+\frac{\left (3 b^2 B d^2 n\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{22201 (b c-a d)^4 g^3}+\frac{\left (3 b B d^3 n\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{22201 (b c-a d)^4 g^3}-\frac{\left (3 b B d^3 n\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{22201 (b c-a d)^4 g^3}-\frac{(2 b B d n) \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}{22201 (b c-a d)^2 g^3}-\frac{\left (B d^2 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}{22201 (b c-a d)^2 g^3}+\frac{(b B 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}{44402 (b c-a d) g^3}\\ &=-\frac{b B n}{88804 (b c-a d)^2 g^3 (a+b x)^2}+\frac{5 b B d n}{44402 (b c-a d)^3 g^3 (a+b x)}-\frac{B d^2 n}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac{3 b B d^2 n \log (a+b x)}{44402 (b c-a d)^4 g^3}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac{2 b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac{3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac{3 b B d^2 n \log (c+d x)}{44402 (b c-a d)^4 g^3}+\frac{3 b B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}+\frac{3 b B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{22201 (b c-a d)^4 g^3}-\frac{\left (3 b B d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{22201 (b c-a d)^4 g^3}-\frac{\left (3 b B d^2 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{22201 (b c-a d)^4 g^3}-\frac{\left (3 b^2 B d^2 n\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{22201 (b c-a d)^4 g^3}-\frac{\left (3 b B d^3 n\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{22201 (b c-a d)^4 g^3}\\ &=-\frac{b B n}{88804 (b c-a d)^2 g^3 (a+b x)^2}+\frac{5 b B d n}{44402 (b c-a d)^3 g^3 (a+b x)}-\frac{B d^2 n}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac{3 b B d^2 n \log (a+b x)}{44402 (b c-a d)^4 g^3}-\frac{3 b B d^2 n \log ^2(a+b x)}{44402 (b c-a d)^4 g^3}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac{2 b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac{3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac{3 b B d^2 n \log (c+d x)}{44402 (b c-a d)^4 g^3}+\frac{3 b B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac{3 b B d^2 n \log ^2(c+d x)}{44402 (b c-a d)^4 g^3}+\frac{3 b B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{22201 (b c-a d)^4 g^3}-\frac{\left (3 b B d^2 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 )}{22201 (b c-a d)^4 g^3}-\frac{\left (3 b B d^2 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 )}{22201 (b c-a d)^4 g^3}\\ &=-\frac{b B n}{88804 (b c-a d)^2 g^3 (a+b x)^2}+\frac{5 b B d n}{44402 (b c-a d)^3 g^3 (a+b x)}-\frac{B d^2 n}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac{3 b B d^2 n \log (a+b x)}{44402 (b c-a d)^4 g^3}-\frac{3 b B d^2 n \log ^2(a+b x)}{44402 (b c-a d)^4 g^3}-\frac{b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{44402 (b c-a d)^2 g^3 (a+b x)^2}+\frac{2 b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^3 g^3 (c+d x)}+\frac{3 b d^2 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{22201 (b c-a d)^4 g^3}-\frac{3 b B d^2 n \log (c+d x)}{44402 (b c-a d)^4 g^3}+\frac{3 b B d^2 n \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac{3 b d^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{22201 (b c-a d)^4 g^3}-\frac{3 b B d^2 n \log ^2(c+d x)}{44402 (b c-a d)^4 g^3}+\frac{3 b B d^2 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{22201 (b c-a d)^4 g^3}+\frac{3 b B d^2 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{22201 (b c-a d)^4 g^3}+\frac{3 b B d^2 n \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{22201 (b c-a d)^4 g^3}\\ \end{align*}

Mathematica [C]  time = 0.844032, size = 478, normalized size = 1.26 \[ \frac{-6 b B d^2 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+6 b B d^2 n \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+12 b d^2 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+\frac{4 d^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}-12 b d^2 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+\frac{8 b d (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac{2 b (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{(a+b x)^2}+\frac{8 b^2 B c d n}{a+b x}+\frac{2 b B d n (b c-a d)}{a+b x}-\frac{b B n (b c-a d)^2}{(a+b x)^2}-\frac{8 a b B d^2 n}{a+b x}+6 b B d^2 n \log (a+b x)+\frac{4 a B d^3 n}{c+d x}-\frac{4 b B c d^2 n}{c+d x}-6 b B d^2 n \log (c+d x)}{4 g^3 i^2 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((b*B*(b*c - a*d)^2*n)/(a + b*x)^2) + (8*b^2*B*c*d*n)/(a + b*x) - (8*a*b*B*d^2*n)/(a + b*x) + (2*b*B*d*(b*c
- a*d)*n)/(a + b*x) - (4*b*B*c*d^2*n)/(c + d*x) + (4*a*B*d^3*n)/(c + d*x) + 6*b*B*d^2*n*Log[a + b*x] - (2*b*(b
*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x)^2 + (8*b*d*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(
c + d*x))^n]))/(a + b*x) + (4*d^2*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x) + 12*b*d^2*Log
[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 6*b*B*d^2*n*Log[c + d*x] - 12*b*d^2*(A + B*Log[e*((a + b*x)
/(c + d*x))^n])*Log[c + d*x] - 6*b*B*d^2*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)]) + 6*b*B*d^2*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)]))/(4*(b*c - a*d)^4*g^3*i^2)

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Maple [F]  time = 0.746, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( bgx+ag \right ) ^{3} \left ( dix+ci \right ) ^{2}} \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)^3/(d*i*x+c*i)^2,x)

[Out]

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

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Maxima [B]  time = 1.75279, size = 2327, normalized size = 6.12 \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)^3/(d*i*x+c*i)^2,x, algorithm="maxima")

[Out]

1/2*B*((6*b^2*d^2*x^2 - b^2*c^2 + 5*a*b*c*d + 2*a^2*d^2 + 3*(b^2*c*d + 3*a*b*d^2)*x)/((b^5*c^3*d - 3*a*b^4*c^2
*d^2 + 3*a^2*b^3*c*d^3 - a^3*b^2*d^4)*g^3*i^2*x^3 + (b^5*c^4 - a*b^4*c^3*d - 3*a^2*b^3*c^2*d^2 + 5*a^3*b^2*c*d
^3 - 2*a^4*b*d^4)*g^3*i^2*x^2 + (2*a*b^4*c^4 - 5*a^2*b^3*c^3*d + 3*a^3*b^2*c^2*d^2 + a^4*b*c*d^3 - a^5*d^4)*g^
3*i^2*x + (a^2*b^3*c^4 - 3*a^3*b^2*c^3*d + 3*a^4*b*c^2*d^2 - a^5*c*d^3)*g^3*i^2) + 6*b*d^2*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^3*i^2) - 6*b*d^2*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^3*i^2))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)
- 1/4*(b^3*c^3 - 12*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3 - 6*(b^3*c*d^2 - a*b^2*d^3)*x^2 + 6*(b^3*d^3*x^3
+ a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*log(b*x + a)^2 + 6*(b^3*d^3*x^
3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)*x)*log(d*x + c)^2 - 3*(3*b^3*c^2
*d - 2*a*b^2*c*d^2 - a^2*b*d^3)*x - 6*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*
d^2 + a^2*b*d^3)*x)*log(b*x + a) + 6*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d
^2 + a^2*b*d^3)*x - 2*(b^3*d^3*x^3 + a^2*b*c*d^2 + (b^3*c*d^2 + 2*a*b^2*d^3)*x^2 + (2*a*b^2*c*d^2 + a^2*b*d^3)
*x)*log(b*x + a))*log(d*x + c))*B*n/(a^2*b^4*c^5*g^3*i^2 - 4*a^3*b^3*c^4*d*g^3*i^2 + 6*a^4*b^2*c^3*d^2*g^3*i^2
- 4*a^5*b*c^2*d^3*g^3*i^2 + a^6*c*d^4*g^3*i^2 + (b^6*c^4*d*g^3*i^2 - 4*a*b^5*c^3*d^2*g^3*i^2 + 6*a^2*b^4*c^2*
d^3*g^3*i^2 - 4*a^3*b^3*c*d^4*g^3*i^2 + a^4*b^2*d^5*g^3*i^2)*x^3 + (b^6*c^5*g^3*i^2 - 2*a*b^5*c^4*d*g^3*i^2 -
2*a^2*b^4*c^3*d^2*g^3*i^2 + 8*a^3*b^3*c^2*d^3*g^3*i^2 - 7*a^4*b^2*c*d^4*g^3*i^2 + 2*a^5*b*d^5*g^3*i^2)*x^2 + (
2*a*b^5*c^5*g^3*i^2 - 7*a^2*b^4*c^4*d*g^3*i^2 + 8*a^3*b^3*c^3*d^2*g^3*i^2 - 2*a^4*b^2*c^2*d^3*g^3*i^2 - 2*a^5*
b*c*d^4*g^3*i^2 + a^6*d^5*g^3*i^2)*x) + 1/2*A*((6*b^2*d^2*x^2 - b^2*c^2 + 5*a*b*c*d + 2*a^2*d^2 + 3*(b^2*c*d +
3*a*b*d^2)*x)/((b^5*c^3*d - 3*a*b^4*c^2*d^2 + 3*a^2*b^3*c*d^3 - a^3*b^2*d^4)*g^3*i^2*x^3 + (b^5*c^4 - a*b^4*c
^3*d - 3*a^2*b^3*c^2*d^2 + 5*a^3*b^2*c*d^3 - 2*a^4*b*d^4)*g^3*i^2*x^2 + (2*a*b^4*c^4 - 5*a^2*b^3*c^3*d + 3*a^3
*b^2*c^2*d^2 + a^4*b*c*d^3 - a^5*d^4)*g^3*i^2*x + (a^2*b^3*c^4 - 3*a^3*b^2*c^3*d + 3*a^4*b*c^2*d^2 - a^5*c*d^3
)*g^3*i^2) + 6*b*d^2*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^3
*i^2) - 6*b*d^2*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^3*i^2)
)

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Fricas [B]  time = 0.588953, size = 1956, normalized size = 5.15 \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)^3/(d*i*x+c*i)^2,x, algorithm="fricas")

[Out]

-1/4*(2*A*b^3*c^3 - 12*A*a*b^2*c^2*d + 6*A*a^2*b*c*d^2 + 4*A*a^3*d^3 - 6*(2*A*b^3*c*d^2 - 2*A*a*b^2*d^3 + (B*b
^3*c*d^2 - B*a*b^2*d^3)*n)*x^2 - 6*(B*b^3*d^3*n*x^3 + B*a^2*b*c*d^2*n + (B*b^3*c*d^2 + 2*B*a*b^2*d^3)*n*x^2 +
(2*B*a*b^2*c*d^2 + B*a^2*b*d^3)*n*x)*log((b*x + a)/(d*x + c))^2 + (B*b^3*c^3 - 12*B*a*b^2*c^2*d + 15*B*a^2*b*c
*d^2 - 4*B*a^3*d^3)*n - 3*(2*A*b^3*c^2*d + 4*A*a*b^2*c*d^2 - 6*A*a^2*b*d^3 + (3*B*b^3*c^2*d - 2*B*a*b^2*c*d^2
- B*a^2*b*d^3)*n)*x + 2*(B*b^3*c^3 - 6*B*a*b^2*c^2*d + 3*B*a^2*b*c*d^2 + 2*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 + 2*B*a*b^2*c*d^2 - 3*B*a^2*b*d^3)*x - 6*(B*b^3*d^3*x^3 + B*a^2*b*c*d^2 + (B*b^3*c
*d^2 + 2*B*a*b^2*d^3)*x^2 + (2*B*a*b^2*c*d^2 + B*a^2*b*d^3)*x)*log((b*x + a)/(d*x + c)))*log(e) - 2*(6*A*a^2*b
*c*d^2 + 3*(B*b^3*d^3*n + 2*A*b^3*d^3)*x^3 + 3*(3*B*b^3*c*d^2*n + 2*A*b^3*c*d^2 + 4*A*a*b^2*d^3)*x^2 - (B*b^3*
c^3 - 6*B*a*b^2*c^2*d + 2*B*a^3*d^3)*n + 3*(4*A*a*b^2*c*d^2 + 2*A*a^2*b*d^3 + (B*b^3*c^2*d + 4*B*a*b^2*c*d^2 -
2*B*a^2*b*d^3)*n)*x)*log((b*x + a)/(d*x + c)))/((b^6*c^4*d - 4*a*b^5*c^3*d^2 + 6*a^2*b^4*c^2*d^3 - 4*a^3*b^3*
c*d^4 + a^4*b^2*d^5)*g^3*i^2*x^3 + (b^6*c^5 - 2*a*b^5*c^4*d - 2*a^2*b^4*c^3*d^2 + 8*a^3*b^3*c^2*d^3 - 7*a^4*b^
2*c*d^4 + 2*a^5*b*d^5)*g^3*i^2*x^2 + (2*a*b^5*c^5 - 7*a^2*b^4*c^4*d + 8*a^3*b^3*c^3*d^2 - 2*a^4*b^2*c^2*d^3 -
2*a^5*b*c*d^4 + a^6*d^5)*g^3*i^2*x + (a^2*b^4*c^5 - 4*a^3*b^3*c^4*d + 6*a^4*b^2*c^3*d^2 - 4*a^5*b*c^2*d^3 + a^
6*c*d^4)*g^3*i^2)

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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)**3/(d*i*x+c*i)**2,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 )}^{3}{\left (d i x + c i\right )}^{2}}\,{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)^3/(d*i*x+c*i)^2,x, algorithm="giac")

[Out]

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