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

Optimal. Leaf size=326 \[ \frac{B d^3 i^3 n \text{PolyLog}\left (2,\frac{b (c+d x)}{d (a+b x)}\right )}{b^4 g^4}-\frac{d^2 i^3 (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^3 g^4 (a+b x)}-\frac{d^3 i^3 \log \left (1-\frac{b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^4}-\frac{d i^3 (c+d x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^2 g^4 (a+b x)^2}-\frac{i^3 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b g^4 (a+b x)^3}-\frac{B d^2 i^3 n (c+d x)}{b^3 g^4 (a+b x)}-\frac{B d i^3 n (c+d x)^2}{4 b^2 g^4 (a+b x)^2}-\frac{B i^3 n (c+d x)^3}{9 b g^4 (a+b x)^3} \]

[Out]

-((B*d^2*i^3*n*(c + d*x))/(b^3*g^4*(a + b*x))) - (B*d*i^3*n*(c + d*x)^2)/(4*b^2*g^4*(a + b*x)^2) - (B*i^3*n*(c
+ d*x)^3)/(9*b*g^4*(a + b*x)^3) - (d^2*i^3*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^3*g^4*(a + b*
x)) - (d*i^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b^2*g^4*(a + b*x)^2) - (i^3*(c + d*x)^3*(A
+ B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b*g^4*(a + b*x)^3) - (d^3*i^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*
Log[1 - (b*(c + d*x))/(d*(a + b*x))])/(b^4*g^4) + (B*d^3*i^3*n*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(b^4*g
^4)

________________________________________________________________________________________

Rubi [A]  time = 0.794242, antiderivative size = 444, normalized size of antiderivative = 1.36, number of steps used = 22, 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} \[ \frac{B d^3 i^3 n \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g^4}+\frac{d^3 i^3 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^4}-\frac{3 d^2 i^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{b^4 g^4 (a+b x)}-\frac{3 d i^3 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^4 g^4 (a+b x)^2}-\frac{i^3 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{3 b^4 g^4 (a+b x)^3}-\frac{11 B d^2 i^3 n (b c-a d)}{6 b^4 g^4 (a+b x)}+\frac{B d^3 i^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^4}-\frac{7 B d i^3 n (b c-a d)^2}{12 b^4 g^4 (a+b x)^2}-\frac{B i^3 n (b c-a d)^3}{9 b^4 g^4 (a+b x)^3}-\frac{B d^3 i^3 n \log ^2(a+b x)}{2 b^4 g^4}-\frac{11 B d^3 i^3 n \log (a+b x)}{6 b^4 g^4}+\frac{11 B d^3 i^3 n \log (c+d x)}{6 b^4 g^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(B*(b*c - a*d)^3*i^3*n)/(9*b^4*g^4*(a + b*x)^3) - (7*B*d*(b*c - a*d)^2*i^3*n)/(12*b^4*g^4*(a + b*x)^2) - (11*
B*d^2*(b*c - a*d)*i^3*n)/(6*b^4*g^4*(a + b*x)) - (11*B*d^3*i^3*n*Log[a + b*x])/(6*b^4*g^4) - (B*d^3*i^3*n*Log[
a + b*x]^2)/(2*b^4*g^4) - ((b*c - a*d)^3*i^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^4*g^4*(a + b*x)^3) -
(3*d*(b*c - a*d)^2*i^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b^4*g^4*(a + b*x)^2) - (3*d^2*(b*c - a*d)*i
^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(b^4*g^4*(a + b*x)) + (d^3*i^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)
/(c + d*x))^n]))/(b^4*g^4) + (11*B*d^3*i^3*n*Log[c + d*x])/(6*b^4*g^4) + (B*d^3*i^3*n*Log[a + b*x]*Log[(b*(c +
d*x))/(b*c - a*d)])/(b^4*g^4) + (B*d^3*i^3*n*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(b^4*g^4)

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{(134 c+134 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^4} \, dx &=\int \left (\frac{2406104 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)^4}+\frac{7218312 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)^3}+\frac{7218312 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)^2}+\frac{2406104 d^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^4 (a+b x)}\right ) \, dx\\ &=\frac{\left (2406104 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}{b^3 g^4}+\frac{\left (7218312 d^2 (b c-a 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}{b^3 g^4}+\frac{\left (7218312 d (b c-a d)^2\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{b^3 g^4}+\frac{\left (2406104 (b c-a d)^3\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{b^3 g^4}\\ &=-\frac{2406104 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)^3}-\frac{3609156 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)^2}-\frac{7218312 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)}+\frac{2406104 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4}-\frac{\left (2406104 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}{b^4 g^4}+\frac{\left (7218312 B d^2 (b c-a d) n\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^4}+\frac{\left (3609156 B d (b c-a d)^2 n\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^4}+\frac{\left (2406104 B (b c-a d)^3 n\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^4 g^4}\\ &=-\frac{2406104 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)^3}-\frac{3609156 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)^2}-\frac{7218312 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)}+\frac{2406104 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4}-\frac{\left (2406104 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}{b^4 g^4}+\frac{\left (7218312 B d^2 (b c-a d)^2 n\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b^4 g^4}+\frac{\left (3609156 B d (b c-a d)^3 n\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^4 g^4}+\frac{\left (2406104 B (b c-a d)^4 n\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{3 b^4 g^4}\\ &=-\frac{2406104 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)^3}-\frac{3609156 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)^2}-\frac{7218312 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)}+\frac{2406104 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4}-\frac{\left (2406104 B d^3 n\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^3 g^4}+\frac{\left (2406104 B d^4 n\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^4 g^4}+\frac{\left (7218312 B d^2 (b c-a 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}{b^4 g^4}+\frac{\left (3609156 B d (b c-a d)^3 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}{b^4 g^4}+\frac{\left (2406104 B (b c-a d)^4 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}{3 b^4 g^4}\\ &=-\frac{2406104 B (b c-a d)^3 n}{9 b^4 g^4 (a+b x)^3}-\frac{4210682 B d (b c-a d)^2 n}{3 b^4 g^4 (a+b x)^2}-\frac{13233572 B d^2 (b c-a d) n}{3 b^4 g^4 (a+b x)}-\frac{13233572 B d^3 n \log (a+b x)}{3 b^4 g^4}-\frac{2406104 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)^3}-\frac{3609156 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)^2}-\frac{7218312 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)}+\frac{2406104 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4}+\frac{13233572 B d^3 n \log (c+d x)}{3 b^4 g^4}+\frac{2406104 B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^4}-\frac{\left (2406104 B d^3 n\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 g^4}-\frac{\left (2406104 B d^3 n\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 g^4}\\ &=-\frac{2406104 B (b c-a d)^3 n}{9 b^4 g^4 (a+b x)^3}-\frac{4210682 B d (b c-a d)^2 n}{3 b^4 g^4 (a+b x)^2}-\frac{13233572 B d^2 (b c-a d) n}{3 b^4 g^4 (a+b x)}-\frac{13233572 B d^3 n \log (a+b x)}{3 b^4 g^4}-\frac{1203052 B d^3 n \log ^2(a+b x)}{b^4 g^4}-\frac{2406104 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)^3}-\frac{3609156 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)^2}-\frac{7218312 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)}+\frac{2406104 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4}+\frac{13233572 B d^3 n \log (c+d x)}{3 b^4 g^4}+\frac{2406104 B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^4}-\frac{\left (2406104 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 )}{b^4 g^4}\\ &=-\frac{2406104 B (b c-a d)^3 n}{9 b^4 g^4 (a+b x)^3}-\frac{4210682 B d (b c-a d)^2 n}{3 b^4 g^4 (a+b x)^2}-\frac{13233572 B d^2 (b c-a d) n}{3 b^4 g^4 (a+b x)}-\frac{13233572 B d^3 n \log (a+b x)}{3 b^4 g^4}-\frac{1203052 B d^3 n \log ^2(a+b x)}{b^4 g^4}-\frac{2406104 (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^4 g^4 (a+b x)^3}-\frac{3609156 d (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)^2}-\frac{7218312 d^2 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4 (a+b x)}+\frac{2406104 d^3 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 g^4}+\frac{13233572 B d^3 n \log (c+d x)}{3 b^4 g^4}+\frac{2406104 B d^3 n \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 g^4}+\frac{2406104 B d^3 n \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^4 g^4}\\ \end{align*}

Mathematica [A]  time = 0.510102, size = 326, normalized size = 1. \[ \frac{i^3 \left (-18 B d^3 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+36 d^3 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+\frac{108 d^2 (a d-b c) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}-\frac{54 d (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{(a+b x)^2}-\frac{12 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{(a+b x)^3}+\frac{66 B d^2 n (a d-b c)}{a+b x}-\frac{21 B d n (b c-a d)^2}{(a+b x)^2}-\frac{4 B n (b c-a d)^3}{(a+b x)^3}-66 B d^3 n \log (a+b x)+66 B d^3 n \log (c+d x)\right )}{36 b^4 g^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(i^3*((-4*B*(b*c - a*d)^3*n)/(a + b*x)^3 - (21*B*d*(b*c - a*d)^2*n)/(a + b*x)^2 + (66*B*d^2*(-(b*c) + a*d)*n)/
(a + b*x) - 66*B*d^3*n*Log[a + b*x] - (12*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x)^3 -
(54*d*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x)^2 + (108*d^2*(-(b*c) + a*d)*(A + B*Log[e
*((a + b*x)/(c + d*x))^n]))/(a + b*x) + 36*d^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 66*B*d^3*
n*Log[c + d*x] - 18*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)])))/(36*b^4*g^4)

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Maple [F]  time = 0.688, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( dix+ci \right ) ^{3}}{ \left ( bgx+ag \right ) ^{4}} \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((d*i*x+c*i)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x)

[Out]

int((d*i*x+c*i)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x, algorithm="maxima")

[Out]

-1/6*B*c*d^2*i^3*n*((11*a^2*b^2*c^2 - 7*a^3*b*c*d + 2*a^4*d^2 + 6*(3*b^4*c^2 - 3*a*b^3*c*d + a^2*b^2*d^2)*x^2
+ 3*(9*a*b^3*c^2 - 7*a^2*b^2*c*d + 2*a^3*b*d^2)*x)/((b^8*c^2 - 2*a*b^7*c*d + a^2*b^6*d^2)*g^4*x^3 + 3*(a*b^7*c
^2 - 2*a^2*b^6*c*d + a^3*b^5*d^2)*g^4*x^2 + 3*(a^2*b^6*c^2 - 2*a^3*b^5*c*d + a^4*b^4*d^2)*g^4*x + (a^3*b^5*c^2
- 2*a^4*b^4*c*d + a^5*b^3*d^2)*g^4) + 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(b*x + a)/((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) - 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(d*x + c)/((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)) - 1/18*B*c^3*i^3*n*((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^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 -
2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2
*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g
^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4)) - 1/12*B*c^2*d*i^3*n*(
(5*a*b^2*c^2 - 22*a^2*b*c*d + 5*a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 16*a*b^2*c*d + 5*a^2*
b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4*x^2
+ 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4) - 6*
(3*b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4) + 6*(3*b*c*d^
2 - a*d^3)*log(d*x + c)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4)) + 1/6*A*d^3*i^3*((18*
a*b^2*x^2 + 27*a^2*b*x + 11*a^3)/(b^7*g^4*x^3 + 3*a*b^6*g^4*x^2 + 3*a^2*b^5*g^4*x + a^3*b^4*g^4) + 6*log(b*x +
a)/(b^4*g^4)) + 1/6*B*d^3*i^3*(((18*a*b^2*x^2 + 27*a^2*b*x + 11*a^3 + 6*(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x +
a^3)*log(b*x + a))*log((b*x + a)^n) - (18*a*b^2*x^2 + 27*a^2*b*x + 11*a^3 + 6*(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b
*x + a^3)*log(b*x + a))*log((d*x + c)^n))/(b^7*g^4*x^3 + 3*a*b^6*g^4*x^2 + 3*a^2*b^5*g^4*x + a^3*b^4*g^4) + 6*
integrate(1/6*(6*b^4*d*x^4*log(e) + 6*b^4*c*x^3*log(e) - 11*a^3*b*c*n + 11*a^4*d*n - 18*(a*b^3*c*n - a^2*b^2*d
*n)*x^2 - 27*(a^2*b^2*c*n - a^3*b*d*n)*x - 6*(a^3*b*c*n - a^4*d*n + (b^4*c*n - a*b^3*d*n)*x^3 + 3*(a*b^3*c*n -
a^2*b^2*d*n)*x^2 + 3*(a^2*b^2*c*n - a^3*b*d*n)*x)*log(b*x + a))/(b^8*d*g^4*x^5 + a^4*b^4*c*g^4 + (b^8*c*g^4 +
4*a*b^7*d*g^4)*x^4 + 2*(2*a*b^7*c*g^4 + 3*a^2*b^6*d*g^4)*x^3 + 2*(3*a^2*b^6*c*g^4 + 2*a^3*b^5*d*g^4)*x^2 + (4
*a^3*b^5*c*g^4 + a^4*b^4*d*g^4)*x), x)) - 1/2*(3*b*x + a)*B*c^2*d*i^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(
b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - (3*b^2*x^2 + 3*a*b*x + a^2)*B*c*d^2*i^3*log(e
*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^6*g^4*x^3 + 3*a*b^5*g^4*x^2 + 3*a^2*b^4*g^4*x + a^3*b^3*g^4) - 1/2*(3*b*x
+ a)*A*c^2*d*i^3/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - (3*b^2*x^2 + 3*a*b*x + a^2
)*A*c*d^2*i^3/(b^6*g^4*x^3 + 3*a*b^5*g^4*x^2 + 3*a^2*b^4*g^4*x + a^3*b^3*g^4) - 1/3*B*c^3*i^3*log(e*(b*x/(d*x
+ c) + a/(d*x + c))^n)/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) - 1/3*A*c^3*i^3/(b^4*g^4*
x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{A d^{3} i^{3} x^{3} + 3 \, A c d^{2} i^{3} x^{2} + 3 \, A c^{2} d i^{3} x + A c^{3} i^{3} +{\left (B d^{3} i^{3} x^{3} + 3 \, B c d^{2} i^{3} x^{2} + 3 \, B c^{2} d i^{3} x + B c^{3} i^{3}\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{b^{4} g^{4} x^{4} + 4 \, a b^{3} g^{4} x^{3} + 6 \, a^{2} b^{2} g^{4} x^{2} + 4 \, a^{3} b g^{4} x + a^{4} g^{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x, algorithm="fricas")

[Out]

integral((A*d^3*i^3*x^3 + 3*A*c*d^2*i^3*x^2 + 3*A*c^2*d*i^3*x + A*c^3*i^3 + (B*d^3*i^3*x^3 + 3*B*c*d^2*i^3*x^2
+ 3*B*c^2*d*i^3*x + B*c^3*i^3)*log(e*((b*x + a)/(d*x + c))^n))/(b^4*g^4*x^4 + 4*a*b^3*g^4*x^3 + 6*a^2*b^2*g^4
*x^2 + 4*a^3*b*g^4*x + a^4*g^4), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)**3*(A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**4,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^4,x, algorithm="giac")

[Out]

integrate((d*i*x + c*i)^3*(B*log(e*((b*x + a)/(d*x + c))^n) + A)/(b*g*x + a*g)^4, x)