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

Optimal. Leaf size=786 \[ -\frac{B^2 g i^3 n^2 (b c-a d)^5 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{10 b^4 d^2}-\frac{B g i^3 n (b c-a d)^5 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A+B n\right )}{10 b^4 d^2}+\frac{3 B g i^3 n (c+d x)^2 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^2 d^2}-\frac{B g i^3 n (a+b x) (b c-a d)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{10 b^4 d}+\frac{g i^3 (a+b x)^2 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{20 b^4}-\frac{B g i^3 n (a+b x)^2 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{10 b^4}+\frac{g i^3 (a+b x)^2 (c+d x) (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{10 b^3}+\frac{3 g i^3 (a+b x)^2 (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{20 b^2}+\frac{B g i^3 n (c+d x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{30 b d^2}-\frac{B g i^3 n (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{10 d^2}+\frac{g i^3 (a+b x)^2 (c+d x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 b}+\frac{B^2 g i^3 n^2 (c+d x)^2 (b c-a d)^3}{30 b^2 d^2}-\frac{B^2 g i^3 n^2 (b c-a d)^5 \log \left (\frac{a+b x}{c+d x}\right )}{12 b^4 d^2}-\frac{11 B^2 g i^3 n^2 (b c-a d)^5 \log (c+d x)}{60 b^4 d^2}+\frac{B^2 g i^3 n^2 x (b c-a d)^4}{60 b^3 d}+\frac{B^2 g i^3 n^2 (c+d x)^3 (b c-a d)^2}{30 b d^2} \]

[Out]

(B^2*(b*c - a*d)^4*g*i^3*n^2*x)/(60*b^3*d) + (B^2*(b*c - a*d)^3*g*i^3*n^2*(c + d*x)^2)/(30*b^2*d^2) + (B^2*(b*
c - a*d)^2*g*i^3*n^2*(c + d*x)^3)/(30*b*d^2) - (B*(b*c - a*d)^4*g*i^3*n*(a + b*x)*(A + B*Log[e*((a + b*x)/(c +
d*x))^n]))/(10*b^4*d) - (B*(b*c - a*d)^3*g*i^3*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(10*b^4)
+ (3*B*(b*c - a*d)^3*g*i^3*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(20*b^2*d^2) + (B*(b*c - a*d
)^2*g*i^3*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(30*b*d^2) - (B*(b*c - a*d)*g*i^3*n*(c + d*x)^
4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(10*d^2) + ((b*c - a*d)^3*g*i^3*(a + b*x)^2*(A + B*Log[e*((a + b*x)/
(c + d*x))^n])^2)/(20*b^4) + ((b*c - a*d)^2*g*i^3*(a + b*x)^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])
^2)/(10*b^3) + (3*(b*c - a*d)*g*i^3*(a + b*x)^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(20*b^2)
+ (g*i^3*(a + b*x)^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(5*b) - (B*(b*c - a*d)^5*g*i^3*n*(
A + B*n + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[(b*c - a*d)/(b*(c + d*x))])/(10*b^4*d^2) - (B^2*(b*c - a*d)^5*
g*i^3*n^2*Log[(a + b*x)/(c + d*x)])/(12*b^4*d^2) - (11*B^2*(b*c - a*d)^5*g*i^3*n^2*Log[c + d*x])/(60*b^4*d^2)
- (B^2*(b*c - a*d)^5*g*i^3*n^2*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(10*b^4*d^2)

________________________________________________________________________________________

Rubi [A]  time = 1.92992, antiderivative size = 706, normalized size of antiderivative = 0.9, number of steps used = 52, number of rules used = 13, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.302, Rules used = {2528, 2525, 12, 2486, 31, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 43} \[ \frac{B^2 g i^3 n^2 (b c-a d)^5 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{10 b^4 d^2}+\frac{B g i^3 n (b c-a d)^5 \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{10 b^4 d^2}+\frac{B g i^3 n (c+d x)^2 (b c-a d)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{20 b^2 d^2}+\frac{A B g i^3 n x (b c-a d)^4}{10 b^3 d}+\frac{B g i^3 n (c+d x)^3 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{30 b d^2}-\frac{g i^3 (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d^2}-\frac{B g i^3 n (c+d x)^4 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{10 d^2}+\frac{b g i^3 (c+d x)^5 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{5 d^2}+\frac{B^2 g i^3 n^2 (c+d x)^2 (b c-a d)^3}{30 b^2 d^2}-\frac{B^2 g i^3 n^2 (b c-a d)^5 \log ^2(a+b x)}{20 b^4 d^2}+\frac{B^2 g i^3 n^2 (b c-a d)^5 \log (a+b x)}{60 b^4 d^2}-\frac{B^2 g i^3 n^2 (b c-a d)^5 \log (c+d x)}{10 b^4 d^2}+\frac{B^2 g i^3 n^2 (b c-a d)^5 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{10 b^4 d^2}+\frac{B^2 g i^3 n (a+b x) (b c-a d)^4 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{10 b^4 d}+\frac{B^2 g i^3 n^2 x (b c-a d)^4}{60 b^3 d}+\frac{B^2 g i^3 n^2 (c+d x)^3 (b c-a d)^2}{30 b d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A*B*(b*c - a*d)^4*g*i^3*n*x)/(10*b^3*d) + (B^2*(b*c - a*d)^4*g*i^3*n^2*x)/(60*b^3*d) + (B^2*(b*c - a*d)^3*g*i
^3*n^2*(c + d*x)^2)/(30*b^2*d^2) + (B^2*(b*c - a*d)^2*g*i^3*n^2*(c + d*x)^3)/(30*b*d^2) + (B^2*(b*c - a*d)^5*g
*i^3*n^2*Log[a + b*x])/(60*b^4*d^2) - (B^2*(b*c - a*d)^5*g*i^3*n^2*Log[a + b*x]^2)/(20*b^4*d^2) + (B^2*(b*c -
a*d)^4*g*i^3*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(10*b^4*d) + (B*(b*c - a*d)^3*g*i^3*n*(c + d*x)^2*(A
+ B*Log[e*((a + b*x)/(c + d*x))^n]))/(20*b^2*d^2) + (B*(b*c - a*d)^2*g*i^3*n*(c + d*x)^3*(A + B*Log[e*((a + b*
x)/(c + d*x))^n]))/(30*b*d^2) - (B*(b*c - a*d)*g*i^3*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(10
*d^2) + (B*(b*c - a*d)^5*g*i^3*n*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(10*b^4*d^2) - ((b*c - a
*d)*g*i^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*d^2) + (b*g*i^3*(c + d*x)^5*(A + B*Log[e*((
a + b*x)/(c + d*x))^n])^2)/(5*d^2) - (B^2*(b*c - a*d)^5*g*i^3*n^2*Log[c + d*x])/(10*b^4*d^2) + (B^2*(b*c - a*d
)^5*g*i^3*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(10*b^4*d^2) + (B^2*(b*c - a*d)^5*g*i^3*n^2*PolyLog
[2, -((d*(a + b*x))/(b*c - a*d))])/(10*b^4*d^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 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 0]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

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]

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (180 c+180 d x)^3 (a g+b g x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\int \left (\frac{(-b c+a d) g (180 c+180 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d}+\frac{b g (180 c+180 d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{180 d}\right ) \, dx\\ &=\frac{(b g) \int (180 c+180 d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{180 d}+\frac{((-b c+a d) g) \int (180 c+180 d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{d}\\ &=-\frac{1458000 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}+\frac{1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{(b B g n) \int \frac{188956800000 (b c-a d) (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{81000 d^2}+\frac{(B (b c-a d) g n) \int \frac{1049760000 (b c-a d) (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{360 d^2}\\ &=-\frac{1458000 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}+\frac{1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{(2332800 b B (b c-a d) g n) \int \frac{(c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{d^2}+\frac{\left (2916000 B (b c-a d)^2 g n\right ) \int \frac{(c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{d^2}\\ &=-\frac{1458000 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}+\frac{1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{(2332800 b B (b c-a d) g n) \int \left (\frac{d (b c-a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4}+\frac{(b c-a d)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 (a+b x)}+\frac{d (b c-a d)^2 (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^3}+\frac{d (b c-a d) (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac{d (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{d^2}+\frac{\left (2916000 B (b c-a d)^2 g n\right ) \int \left (\frac{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}+\frac{(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 (a+b x)}+\frac{d (b c-a d) (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac{d (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{d^2}\\ &=-\frac{1458000 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}+\frac{1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{(2332800 B (b c-a d) g n) \int (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{d}-\frac{\left (2332800 B (b c-a d)^2 g n\right ) \int (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b d}+\frac{\left (2916000 B (b c-a d)^2 g n\right ) \int (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b d}-\frac{\left (2332800 B (b c-a d)^3 g n\right ) \int (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 d}+\frac{\left (2916000 B (b c-a d)^3 g n\right ) \int (c+d x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^2 d}-\frac{\left (2332800 B (b c-a d)^4 g n\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^3 d}+\frac{\left (2916000 B (b c-a d)^4 g n\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b^3 d}-\frac{\left (2332800 B (b c-a d)^5 g n\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 d^2}+\frac{\left (2916000 B (b c-a d)^5 g n\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 d^2}\\ &=\frac{583200 A B (b c-a d)^4 g n x}{b^3 d}+\frac{291600 B (b c-a d)^3 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 d^2}+\frac{194400 B (b c-a d)^2 g n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b d^2}-\frac{583200 B (b c-a d) g n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac{583200 B (b c-a d)^5 g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 d^2}-\frac{1458000 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}+\frac{1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{\left (2332800 B^2 (b c-a d)^4 g n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b^3 d}+\frac{\left (2916000 B^2 (b c-a d)^4 g n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b^3 d}+\frac{\left (583200 B^2 (b c-a d) g n^2\right ) \int \frac{(b c-a d) (c+d x)^3}{a+b x} \, dx}{d^2}+\frac{\left (777600 B^2 (b c-a d)^2 g n^2\right ) \int \frac{(b c-a d) (c+d x)^2}{a+b x} \, dx}{b d^2}-\frac{\left (972000 B^2 (b c-a d)^2 g n^2\right ) \int \frac{(b c-a d) (c+d x)^2}{a+b x} \, dx}{b d^2}+\frac{\left (1166400 B^2 (b c-a d)^3 g n^2\right ) \int \frac{(b c-a d) (c+d x)}{a+b x} \, dx}{b^2 d^2}-\frac{\left (1458000 B^2 (b c-a d)^3 g n^2\right ) \int \frac{(b c-a d) (c+d x)}{a+b x} \, dx}{b^2 d^2}+\frac{\left (2332800 B^2 (b c-a d)^5 g n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^4 d^2}-\frac{\left (2916000 B^2 (b c-a d)^5 g n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^4 d^2}\\ &=\frac{583200 A B (b c-a d)^4 g n x}{b^3 d}+\frac{583200 B^2 (b c-a d)^4 g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 d}+\frac{291600 B (b c-a d)^3 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 d^2}+\frac{194400 B (b c-a d)^2 g n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b d^2}-\frac{583200 B (b c-a d) g n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac{583200 B (b c-a d)^5 g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 d^2}-\frac{1458000 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}+\frac{1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}+\frac{\left (583200 B^2 (b c-a d)^2 g n^2\right ) \int \frac{(c+d x)^3}{a+b x} \, dx}{d^2}+\frac{\left (777600 B^2 (b c-a d)^3 g n^2\right ) \int \frac{(c+d x)^2}{a+b x} \, dx}{b d^2}-\frac{\left (972000 B^2 (b c-a d)^3 g n^2\right ) \int \frac{(c+d x)^2}{a+b x} \, dx}{b d^2}+\frac{\left (1166400 B^2 (b c-a d)^4 g n^2\right ) \int \frac{c+d x}{a+b x} \, dx}{b^2 d^2}-\frac{\left (1458000 B^2 (b c-a d)^4 g n^2\right ) \int \frac{c+d x}{a+b x} \, dx}{b^2 d^2}+\frac{\left (2332800 B^2 (b c-a d)^5 g n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{b^4 d^2}-\frac{\left (2916000 B^2 (b c-a d)^5 g n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{b^4 d^2}+\frac{\left (2332800 B^2 (b c-a d)^5 g n^2\right ) \int \frac{1}{c+d x} \, dx}{b^4 d}-\frac{\left (2916000 B^2 (b c-a d)^5 g n^2\right ) \int \frac{1}{c+d x} \, dx}{b^4 d}\\ &=\frac{583200 A B (b c-a d)^4 g n x}{b^3 d}+\frac{583200 B^2 (b c-a d)^4 g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 d}+\frac{291600 B (b c-a d)^3 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 d^2}+\frac{194400 B (b c-a d)^2 g n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b d^2}-\frac{583200 B (b c-a d) g n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac{583200 B (b c-a d)^5 g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 d^2}-\frac{1458000 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}+\frac{1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{583200 B^2 (b c-a d)^5 g n^2 \log (c+d x)}{b^4 d^2}+\frac{\left (583200 B^2 (b c-a d)^2 g n^2\right ) \int \left (\frac{d (b c-a d)^2}{b^3}+\frac{(b c-a d)^3}{b^3 (a+b x)}+\frac{d (b c-a d) (c+d x)}{b^2}+\frac{d (c+d x)^2}{b}\right ) \, dx}{d^2}+\frac{\left (777600 B^2 (b c-a d)^3 g n^2\right ) \int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx}{b d^2}-\frac{\left (972000 B^2 (b c-a d)^3 g n^2\right ) \int \left (\frac{d (b c-a d)}{b^2}+\frac{(b c-a d)^2}{b^2 (a+b x)}+\frac{d (c+d x)}{b}\right ) \, dx}{b d^2}+\frac{\left (1166400 B^2 (b c-a d)^4 g n^2\right ) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{b^2 d^2}-\frac{\left (1458000 B^2 (b c-a d)^4 g n^2\right ) \int \left (\frac{d}{b}+\frac{b c-a d}{b (a+b x)}\right ) \, dx}{b^2 d^2}+\frac{\left (2332800 B^2 (b c-a d)^5 g n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^3 d^2}-\frac{\left (2916000 B^2 (b c-a d)^5 g n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{b^3 d^2}-\frac{\left (2332800 B^2 (b c-a d)^5 g n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^4 d}+\frac{\left (2916000 B^2 (b c-a d)^5 g n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b^4 d}\\ &=\frac{583200 A B (b c-a d)^4 g n x}{b^3 d}+\frac{97200 B^2 (b c-a d)^4 g n^2 x}{b^3 d}+\frac{194400 B^2 (b c-a d)^3 g n^2 (c+d x)^2}{b^2 d^2}+\frac{194400 B^2 (b c-a d)^2 g n^2 (c+d x)^3}{b d^2}+\frac{97200 B^2 (b c-a d)^5 g n^2 \log (a+b x)}{b^4 d^2}+\frac{583200 B^2 (b c-a d)^4 g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 d}+\frac{291600 B (b c-a d)^3 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 d^2}+\frac{194400 B (b c-a d)^2 g n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b d^2}-\frac{583200 B (b c-a d) g n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac{583200 B (b c-a d)^5 g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 d^2}-\frac{1458000 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}+\frac{1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{583200 B^2 (b c-a d)^5 g n^2 \log (c+d x)}{b^4 d^2}+\frac{583200 B^2 (b c-a d)^5 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 d^2}+\frac{\left (2332800 B^2 (b c-a d)^5 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 d^2}-\frac{\left (2916000 B^2 (b c-a d)^5 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b^4 d^2}+\frac{\left (2332800 B^2 (b c-a d)^5 g n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 d^2}-\frac{\left (2916000 B^2 (b c-a d)^5 g n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^3 d^2}\\ &=\frac{583200 A B (b c-a d)^4 g n x}{b^3 d}+\frac{97200 B^2 (b c-a d)^4 g n^2 x}{b^3 d}+\frac{194400 B^2 (b c-a d)^3 g n^2 (c+d x)^2}{b^2 d^2}+\frac{194400 B^2 (b c-a d)^2 g n^2 (c+d x)^3}{b d^2}+\frac{97200 B^2 (b c-a d)^5 g n^2 \log (a+b x)}{b^4 d^2}-\frac{291600 B^2 (b c-a d)^5 g n^2 \log ^2(a+b x)}{b^4 d^2}+\frac{583200 B^2 (b c-a d)^4 g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 d}+\frac{291600 B (b c-a d)^3 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 d^2}+\frac{194400 B (b c-a d)^2 g n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b d^2}-\frac{583200 B (b c-a d) g n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac{583200 B (b c-a d)^5 g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 d^2}-\frac{1458000 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}+\frac{1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{583200 B^2 (b c-a d)^5 g n^2 \log (c+d x)}{b^4 d^2}+\frac{583200 B^2 (b c-a d)^5 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 d^2}+\frac{\left (2332800 B^2 (b c-a d)^5 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 d^2}-\frac{\left (2916000 B^2 (b c-a d)^5 g n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^4 d^2}\\ &=\frac{583200 A B (b c-a d)^4 g n x}{b^3 d}+\frac{97200 B^2 (b c-a d)^4 g n^2 x}{b^3 d}+\frac{194400 B^2 (b c-a d)^3 g n^2 (c+d x)^2}{b^2 d^2}+\frac{194400 B^2 (b c-a d)^2 g n^2 (c+d x)^3}{b d^2}+\frac{97200 B^2 (b c-a d)^5 g n^2 \log (a+b x)}{b^4 d^2}-\frac{291600 B^2 (b c-a d)^5 g n^2 \log ^2(a+b x)}{b^4 d^2}+\frac{583200 B^2 (b c-a d)^4 g n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b^4 d}+\frac{291600 B (b c-a d)^3 g n (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^2 d^2}+\frac{194400 B (b c-a d)^2 g n (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b d^2}-\frac{583200 B (b c-a d) g n (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac{583200 B (b c-a d)^5 g n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{b^4 d^2}-\frac{1458000 (b c-a d) g (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}+\frac{1166400 b g (c+d x)^5 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac{583200 B^2 (b c-a d)^5 g n^2 \log (c+d x)}{b^4 d^2}+\frac{583200 B^2 (b c-a d)^5 g n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b^4 d^2}+\frac{583200 B^2 (b c-a d)^5 g n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b^4 d^2}\\ \end{align*}

Mathematica [A]  time = 0.726238, size = 945, normalized size = 1.2 \[ \frac{g i^3 \left (4 b \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 (c+d x)^5-5 (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 (c+d x)^4+\frac{5 B (b c-a d)^2 n \left (6 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^3-6 B n \log (c+d x) (b c-a d)^3-3 B 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 ) (b c-a d)^3+6 A b d x (b c-a d)^2-3 B n (b d x+(b c-a d) \log (a+b x)) (b c-a d)^2+6 B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) (b c-a d)^2-B n \left (2 \log (a+b x) (b c-a d)^2+2 b d x (b c-a d)+b^2 (c+d x)^2\right ) (b c-a d)+3 b^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)+2 b^3 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )\right )}{3 b^4}-\frac{B (b c-a d) n \left (24 \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^4-24 B n \log (c+d x) (b c-a d)^4-12 B 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 ) (b c-a d)^4+24 A b d x (b c-a d)^3-12 B n (b d x+(b c-a d) \log (a+b x)) (b c-a d)^3+24 B d (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) (b c-a d)^3-4 B n \left (2 \log (a+b x) (b c-a d)^2+2 b d x (b c-a d)+b^2 (c+d x)^2\right ) (b c-a d)^2+12 b^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2-B n \left (6 \log (a+b x) (b c-a d)^3+6 b d x (b c-a d)^2+3 b^2 (c+d x)^2 (b c-a d)+2 b^3 (c+d x)^3\right ) (b c-a d)+8 b^3 (c+d x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)+6 b^4 (c+d x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )\right )}{3 b^4}\right )}{20 d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g*i^3*(-5*(b*c - a*d)*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 4*b*(c + d*x)^5*(A + B*Log[e*((a
+ b*x)/(c + d*x))^n])^2 + (5*B*(b*c - a*d)^2*n*(6*A*b*d*(b*c - a*d)^2*x - 3*B*(b*c - a*d)^2*n*(b*d*x + (b*c -
a*d)*Log[a + b*x]) - B*(b*c - a*d)*n*(2*b*d*(b*c - a*d)*x + b^2*(c + d*x)^2 + 2*(b*c - a*d)^2*Log[a + b*x]) +
6*B*d*(b*c - a*d)^2*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + 3*b^2*(b*c - a*d)*(c + d*x)^2*(A + B*Log[e*((a
+ b*x)/(c + d*x))^n]) + 2*b^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 6*(b*c - a*d)^3*Log[a + b*
x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 6*B*(b*c - a*d)^3*n*Log[c + d*x] - 3*B*(b*c - a*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)])))/(3*b^4) -
(B*(b*c - a*d)*n*(24*A*b*d*(b*c - a*d)^3*x - 12*B*(b*c - a*d)^3*n*(b*d*x + (b*c - a*d)*Log[a + b*x]) - 4*B*(b
*c - a*d)^2*n*(2*b*d*(b*c - a*d)*x + b^2*(c + d*x)^2 + 2*(b*c - a*d)^2*Log[a + b*x]) - B*(b*c - a*d)*n*(6*b*d*
(b*c - a*d)^2*x + 3*b^2*(b*c - a*d)*(c + d*x)^2 + 2*b^3*(c + d*x)^3 + 6*(b*c - a*d)^3*Log[a + b*x]) + 24*B*d*(
b*c - a*d)^3*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n] + 12*b^2*(b*c - a*d)^2*(c + d*x)^2*(A + B*Log[e*((a + b*
x)/(c + d*x))^n]) + 8*b^3*(b*c - a*d)*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 6*b^4*(c + d*x)^4*(
A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 24*(b*c - a*d)^4*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) -
24*B*(b*c - a*d)^4*n*Log[c + d*x] - 12*B*(b*c - a*d)^4*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)])))/(3*b^4)))/(20*d^2)

________________________________________________________________________________________

Maple [F]  time = 0.534, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) \left ( dix+ci \right ) ^{3} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 4.63509, size = 5027, normalized size = 6.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*A*B*b*d^3*g*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/5*A^2*b*d^3*g*i^3*x^5 + 3/2*A*B*b*c*d^2*g*i
^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/2*A*B*a*d^3*g*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)
+ 3/4*A^2*b*c*d^2*g*i^3*x^4 + 1/4*A^2*a*d^3*g*i^3*x^4 + 2*A*B*b*c^2*d*g*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x
+ c))^n) + 2*A*B*a*c*d^2*g*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2*b*c^2*d*g*i^3*x^3 + A^2*a*c*d
^2*g*i^3*x^3 + A*B*b*c^3*g*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3*A*B*a*c^2*d*g*i^3*x^2*log(e*(b*x
/(d*x + c) + a/(d*x + c))^n) + 1/2*A^2*b*c^3*g*i^3*x^2 + 3/2*A^2*a*c^2*d*g*i^3*x^2 + 1/30*A*B*b*d^3*g*i^3*n*(1
2*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d
^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4)) - 1/4*A*B*b*c*d^2*g*i^3*n*(6*a^
4*log(b*x + a)/b^4 - 6*c^4*log(d*x + c)/d^4 + (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 +
6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3)) - 1/12*A*B*a*d^3*g*i^3*n*(6*a^4*log(b*x + a)/b^4 - 6*c^4*log(d*x + c)/d^4
+ (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3)) + A*B*
b*c^2*d*g*i^3*n*(2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2
*d^2)*x)/(b^2*d^2)) + A*B*a*c*d^2*g*i^3*n*(2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d
^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2)) - A*B*b*c^3*g*i^3*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2
+ (b*c - a*d)*x/(b*d)) - 3*A*B*a*c^2*d*g*i^3*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(
b*d)) + 2*A*B*a*c^3*g*i^3*n*(a*log(b*x + a)/b - c*log(d*x + c)/d) + 2*A*B*a*c^3*g*i^3*x*log(e*(b*x/(d*x + c) +
a/(d*x + c))^n) + A^2*a*c^3*g*i^3*x - 1/60*(47*a^2*b^2*c^3*d^2*g*i^3*n^2 - 27*a^3*b*c^2*d^3*g*i^3*n^2 + 6*a^4
*c*d^4*g*i^3*n^2 + (5*g*i^3*n^2 - 6*g*i^3*n*log(e))*b^4*c^5 - (31*g*i^3*n^2 - 30*g*i^3*n*log(e))*a*b^3*c^4*d)*
B^2*log(d*x + c)/(b^3*d^2) + 1/10*(b^5*c^5*g*i^3*n^2 - 5*a*b^4*c^4*d*g*i^3*n^2 + 10*a^2*b^3*c^3*d^2*g*i^3*n^2
- 10*a^3*b^2*c^2*d^3*g*i^3*n^2 + 5*a^4*b*c*d^4*g*i^3*n^2 - a^5*d^5*g*i^3*n^2)*(log(b*x + a)*log((b*d*x + a*d)/
(b*c - a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*B^2/(b^4*d^2) + 1/60*(12*B^2*b^5*d^5*g*i^3*x^5*log(e)^2
- 3*((2*g*i^3*n*log(e) - 15*g*i^3*log(e)^2)*b^5*c*d^4 - (2*g*i^3*n*log(e) + 5*g*i^3*log(e)^2)*a*b^4*d^5)*B^2*x
^4 + 2*((g*i^3*n^2 - 11*g*i^3*n*log(e) + 30*g*i^3*log(e)^2)*b^5*c^2*d^3 - 2*(g*i^3*n^2 - 5*g*i^3*n*log(e) - 15
*g*i^3*log(e)^2)*a*b^4*c*d^4 + (g*i^3*n^2 + g*i^3*n*log(e))*a^2*b^3*d^5)*B^2*x^3 + ((8*g*i^3*n^2 - 27*g*i^3*n*
log(e) + 30*g*i^3*log(e)^2)*b^5*c^3*d^2 - 3*(6*g*i^3*n^2 - 5*g*i^3*n*log(e) - 30*g*i^3*log(e)^2)*a*b^4*c^2*d^3
+ 3*(4*g*i^3*n^2 + 5*g*i^3*n*log(e))*a^2*b^3*c*d^4 - (2*g*i^3*n^2 + 3*g*i^3*n*log(e))*a^3*b^2*d^5)*B^2*x^2 -
3*(10*a^2*b^3*c^3*d^2*g*i^3*n^2 - 10*a^3*b^2*c^2*d^3*g*i^3*n^2 + 5*a^4*b*c*d^4*g*i^3*n^2 - a^5*d^5*g*i^3*n^2)*
B^2*log(b*x + a)^2 - 6*(b^5*c^5*g*i^3*n^2 - 5*a*b^4*c^4*d*g*i^3*n^2)*B^2*log(b*x + a)*log(d*x + c) + 3*(b^5*c^
5*g*i^3*n^2 - 5*a*b^4*c^4*d*g*i^3*n^2)*B^2*log(d*x + c)^2 + ((11*g*i^3*n^2 - 6*g*i^3*n*log(e))*b^5*c^4*d - 2*(
14*g*i^3*n^2 + 15*g*i^3*n*log(e) - 30*g*i^3*log(e)^2)*a*b^4*c^3*d^2 + 12*(2*g*i^3*n^2 + 5*g*i^3*n*log(e))*a^2*
b^3*c^2*d^3 - 2*(4*g*i^3*n^2 + 15*g*i^3*n*log(e))*a^3*b^2*c*d^4 + (g*i^3*n^2 + 6*g*i^3*n*log(e))*a^4*b*d^5)*B^
2*x - (6*a*b^4*c^4*d*g*i^3*n^2 + 3*(g*i^3*n^2 - 20*g*i^3*n*log(e))*a^2*b^3*c^3*d^2 - (23*g*i^3*n^2 - 60*g*i^3*
n*log(e))*a^3*b^2*c^2*d^3 + (19*g*i^3*n^2 - 30*g*i^3*n*log(e))*a^4*b*c*d^4 - (5*g*i^3*n^2 - 6*g*i^3*n*log(e))*
a^5*d^5)*B^2*log(b*x + a) + 3*(4*B^2*b^5*d^5*g*i^3*x^5 + 20*B^2*a*b^4*c^3*d^2*g*i^3*x + 5*(3*b^5*c*d^4*g*i^3 +
a*b^4*d^5*g*i^3)*B^2*x^4 + 20*(b^5*c^2*d^3*g*i^3 + a*b^4*c*d^4*g*i^3)*B^2*x^3 + 10*(b^5*c^3*d^2*g*i^3 + 3*a*b
^4*c^2*d^3*g*i^3)*B^2*x^2)*log((b*x + a)^n)^2 + 3*(4*B^2*b^5*d^5*g*i^3*x^5 + 20*B^2*a*b^4*c^3*d^2*g*i^3*x + 5*
(3*b^5*c*d^4*g*i^3 + a*b^4*d^5*g*i^3)*B^2*x^4 + 20*(b^5*c^2*d^3*g*i^3 + a*b^4*c*d^4*g*i^3)*B^2*x^3 + 10*(b^5*c
^3*d^2*g*i^3 + 3*a*b^4*c^2*d^3*g*i^3)*B^2*x^2)*log((d*x + c)^n)^2 + (24*B^2*b^5*d^5*g*i^3*x^5*log(e) - 6*((g*i
^3*n - 15*g*i^3*log(e))*b^5*c*d^4 - (g*i^3*n + 5*g*i^3*log(e))*a*b^4*d^5)*B^2*x^4 + 2*(a^2*b^3*d^5*g*i^3*n - (
11*g*i^3*n - 60*g*i^3*log(e))*b^5*c^2*d^3 + 10*(g*i^3*n + 6*g*i^3*log(e))*a*b^4*c*d^4)*B^2*x^3 + 3*(5*a^2*b^3*
c*d^4*g*i^3*n - a^3*b^2*d^5*g*i^3*n - (9*g*i^3*n - 20*g*i^3*log(e))*b^5*c^3*d^2 + 5*(g*i^3*n + 12*g*i^3*log(e)
)*a*b^4*c^2*d^3)*B^2*x^2 - 6*(b^5*c^4*d*g*i^3*n - 10*a^2*b^3*c^2*d^3*g*i^3*n + 5*a^3*b^2*c*d^4*g*i^3*n - a^4*b
*d^5*g*i^3*n + 5*(g*i^3*n - 4*g*i^3*log(e))*a*b^4*c^3*d^2)*B^2*x + 6*(10*a^2*b^3*c^3*d^2*g*i^3*n - 10*a^3*b^2*
c^2*d^3*g*i^3*n + 5*a^4*b*c*d^4*g*i^3*n - a^5*d^5*g*i^3*n)*B^2*log(b*x + a) + 6*(b^5*c^5*g*i^3*n - 5*a*b^4*c^4
*d*g*i^3*n)*B^2*log(d*x + c))*log((b*x + a)^n) - (24*B^2*b^5*d^5*g*i^3*x^5*log(e) - 6*((g*i^3*n - 15*g*i^3*log
(e))*b^5*c*d^4 - (g*i^3*n + 5*g*i^3*log(e))*a*b^4*d^5)*B^2*x^4 + 2*(a^2*b^3*d^5*g*i^3*n - (11*g*i^3*n - 60*g*i
^3*log(e))*b^5*c^2*d^3 + 10*(g*i^3*n + 6*g*i^3*log(e))*a*b^4*c*d^4)*B^2*x^3 + 3*(5*a^2*b^3*c*d^4*g*i^3*n - a^3
*b^2*d^5*g*i^3*n - (9*g*i^3*n - 20*g*i^3*log(e))*b^5*c^3*d^2 + 5*(g*i^3*n + 12*g*i^3*log(e))*a*b^4*c^2*d^3)*B^
2*x^2 - 6*(b^5*c^4*d*g*i^3*n - 10*a^2*b^3*c^2*d^3*g*i^3*n + 5*a^3*b^2*c*d^4*g*i^3*n - a^4*b*d^5*g*i^3*n + 5*(g
*i^3*n - 4*g*i^3*log(e))*a*b^4*c^3*d^2)*B^2*x + 6*(10*a^2*b^3*c^3*d^2*g*i^3*n - 10*a^3*b^2*c^2*d^3*g*i^3*n + 5
*a^4*b*c*d^4*g*i^3*n - a^5*d^5*g*i^3*n)*B^2*log(b*x + a) + 6*(b^5*c^5*g*i^3*n - 5*a*b^4*c^4*d*g*i^3*n)*B^2*log
(d*x + c) + 6*(4*B^2*b^5*d^5*g*i^3*x^5 + 20*B^2*a*b^4*c^3*d^2*g*i^3*x + 5*(3*b^5*c*d^4*g*i^3 + a*b^4*d^5*g*i^3
)*B^2*x^4 + 20*(b^5*c^2*d^3*g*i^3 + a*b^4*c*d^4*g*i^3)*B^2*x^3 + 10*(b^5*c^3*d^2*g*i^3 + 3*a*b^4*c^2*d^3*g*i^3
)*B^2*x^2)*log((b*x + a)^n))*log((d*x + c)^n))/(b^4*d^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b g x + a g\right )}{\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 )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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