\(\int (a g+b g x) (c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\) [170]

Optimal result

Integrand size = 43, antiderivative size = 635 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {B^2 (b c-a d)^3 g i^2 n^2 x}{12 b^2 d}+\frac {B^2 (b c-a d)^2 g i^2 n^2 (c+d x)^2}{12 b d^2}-\frac {B (b c-a d)^3 g i^2 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^3 d}-\frac {B (b c-a d)^2 g i^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^3}+\frac {B (b c-a d)^2 g i^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b d^2}-\frac {B (b c-a d) g i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^2}+\frac {(b c-a d)^2 g i^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{12 b^3}+\frac {(b c-a d) g i^2 (a+b x)^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 b^2}+\frac {g i^2 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}-\frac {B (b c-a d)^4 g i^2 n \left (A+B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{6 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 n^2 \log (c+d x)}{4 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 n^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^3 d^2} \]

[Out]

1/12*B^2*(-a*d+b*c)^3*g*i^2*n^2*x/b^2/d+1/12*B^2*(-a*d+b*c)^2*g*i^2*n^2*(d*x+c)^2/b/d^2-1/6*B*(-a*d+b*c)^3*g*i
^2*n*(b*x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^3/d-1/6*B*(-a*d+b*c)^2*g*i^2*n*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*
x+c))^n))/b^3+1/4*B*(-a*d+b*c)^2*g*i^2*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b/d^2-1/6*B*(-a*d+b*c)*g*i^
2*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^2+1/12*(-a*d+b*c)^2*g*i^2*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c)
)^n))^2/b^3+1/6*(-a*d+b*c)*g*i^2*(b*x+a)^2*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^2+1/4*g*i^2*(b*x+a)^2*(
d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b-1/6*B*(-a*d+b*c)^4*g*i^2*n*(A+B*n+B*ln(e*((b*x+a)/(d*x+c))^n))*ln
((-a*d+b*c)/b/(d*x+c))/b^3/d^2-1/12*B^2*(-a*d+b*c)^4*g*i^2*n^2*ln((b*x+a)/(d*x+c))/b^3/d^2-1/4*B^2*(-a*d+b*c)^
4*g*i^2*n^2*ln(d*x+c)/b^3/d^2-1/6*B^2*(-a*d+b*c)^4*g*i^2*n^2*polylog(2,d*(b*x+a)/b/(d*x+c))/b^3/d^2

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 635, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {2561, 2383, 2381, 2384, 2354, 2438, 2373, 45, 2382, 12, 78} \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=-\frac {B g i^2 n (b c-a d)^4 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A+B n\right )}{6 b^3 d^2}-\frac {B g i^2 n (a+b x) (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 b^3 d}+\frac {g i^2 (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{12 b^3}-\frac {B g i^2 n (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 b^3}+\frac {g i^2 (a+b x)^2 (c+d x) (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{6 b^2}+\frac {B g i^2 n (c+d x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 b d^2}-\frac {B g i^2 n (c+d x)^3 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 d^2}+\frac {g i^2 (a+b x)^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b}-\frac {B^2 g i^2 n^2 (b c-a d)^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^3 d^2}-\frac {B^2 g i^2 n^2 (b c-a d)^4 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^3 d^2}-\frac {B^2 g i^2 n^2 (b c-a d)^4 \log (c+d x)}{4 b^3 d^2}+\frac {B^2 g i^2 n^2 x (b c-a d)^3}{12 b^2 d}+\frac {B^2 g i^2 n^2 (c+d x)^2 (b c-a d)^2}{12 b d^2} \]

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 45

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])

Rule 78

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 2354

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[Log[1 + e*(x/d)]*((a +
b*Log[c*x^n])^p/e), x] - Dist[b*n*(p/e), Int[Log[1 + e*(x/d)]*((a + b*Log[c*x^n])^(p - 1)/x), x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2373

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

Rule 2381

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp
[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + Dist[b*n*(p/(d*(q + 1))), Int[(
f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, q}, x] && EqQ[m
+ q + 2, 0] && IGtQ[p, 0] && LtQ[q, -1]

Rule 2382

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> With[{u = IntHide[
x^m*(d + e*x)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]] /; FreeQ
[{a, b, c, d, e, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[m, 0]

Rule 2383

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_), x_Symbol] :> Simp
[(-(f*x)^(m + 1))*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])^p/(d*f*(q + 1))), x] + (Dist[(m + q + 2)/(d*(q + 1)),
Int[(f*x)^m*(d + e*x)^(q + 1)*(a + b*Log[c*x^n])^p, x], x] + Dist[b*n*(p/(d*(q + 1))), Int[(f*x)^m*(d + e*x)^(
q + 1)*(a + b*Log[c*x^n])^(p - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, n}, x] && ILtQ[m + q + 2, 0] && IGtQ[p,
0] && LtQ[q, -1] && GtQ[m, 0]

Rule 2384

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(f*x
)^m*(d + e*x)^(q + 1)*((a + b*Log[c*x^n])/(e*(q + 1))), x] - Dist[f/(e*(q + 1)), Int[(f*x)^(m - 1)*(d + e*x)^(
q + 1)*(a*m + b*n + b*m*Log[c*x^n]), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && ILtQ[q, -1] && GtQ[m, 0]

Rule 2438

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 2561

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[x^m*((A +
B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i,
A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]

Rubi steps \begin{align*} \text {integral}& = \left ((b c-a d)^4 g i^2\right ) \text {Subst}\left (\int \frac {x \left (A+B \log \left (e x^n\right )\right )^2}{(b-d x)^5} \, dx,x,\frac {a+b x}{c+d x}\right ) \\ & = \frac {g i^2 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}+\frac {\left ((b c-a d)^4 g i^2\right ) \text {Subst}\left (\int \frac {x \left (A+B \log \left (e x^n\right )\right )^2}{(b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 b}-\frac {\left (B (b c-a d)^4 g i^2 n\right ) \text {Subst}\left (\int \frac {x \left (A+B \log \left (e x^n\right )\right )}{(b-d x)^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 b} \\ & = \frac {B (b c-a d)^2 g i^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b d^2}-\frac {B (b c-a d) g i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^2}+\frac {(b c-a d) g i^2 (a+b x)^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 b^2}+\frac {g i^2 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}+\frac {\left ((b c-a d)^4 g i^2\right ) \text {Subst}\left (\int \frac {x \left (A+B \log \left (e x^n\right )\right )^2}{(b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{6 b^2}-\frac {\left (B (b c-a d)^4 g i^2 n\right ) \text {Subst}\left (\int \frac {x \left (A+B \log \left (e x^n\right )\right )}{(b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{3 b^2}+\frac {\left (B^2 (b c-a d)^4 g i^2 n^2\right ) \text {Subst}\left (\int \frac {-b+3 d x}{6 d^2 x (b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{2 b} \\ & = -\frac {B (b c-a d)^2 g i^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^3}+\frac {B (b c-a d)^2 g i^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b d^2}-\frac {B (b c-a d) g i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^2}+\frac {(b c-a d)^2 g i^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{12 b^3}+\frac {(b c-a d) g i^2 (a+b x)^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 b^2}+\frac {g i^2 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}-\frac {\left (B (b c-a d)^4 g i^2 n\right ) \text {Subst}\left (\int \frac {x \left (A+B \log \left (e x^n\right )\right )}{(b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{6 b^3}+\frac {\left (B^2 (b c-a d)^4 g i^2 n^2\right ) \text {Subst}\left (\int \frac {x}{(b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{6 b^3}+\frac {\left (B^2 (b c-a d)^4 g i^2 n^2\right ) \text {Subst}\left (\int \frac {-b+3 d x}{x (b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{12 b d^2} \\ & = -\frac {B (b c-a d)^3 g i^2 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^3 d}-\frac {B (b c-a d)^2 g i^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^3}+\frac {B (b c-a d)^2 g i^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b d^2}-\frac {B (b c-a d) g i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^2}+\frac {(b c-a d)^2 g i^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{12 b^3}+\frac {(b c-a d) g i^2 (a+b x)^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 b^2}+\frac {g i^2 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}+\frac {\left (B (b c-a d)^4 g i^2 n\right ) \text {Subst}\left (\int \frac {A+B n+B \log \left (e x^n\right )}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{6 b^3 d}+\frac {\left (B^2 (b c-a d)^4 g i^2 n^2\right ) \text {Subst}\left (\int \left (\frac {b}{d (-b+d x)^2}+\frac {1}{d (-b+d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{6 b^3}+\frac {\left (B^2 (b c-a d)^4 g i^2 n^2\right ) \text {Subst}\left (\int \left (-\frac {1}{b^2 x}+\frac {2 d}{(b-d x)^3}-\frac {d}{b (b-d x)^2}-\frac {d}{b^2 (b-d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{12 b d^2} \\ & = \frac {B^2 (b c-a d)^3 g i^2 n^2 x}{12 b^2 d}+\frac {B^2 (b c-a d)^2 g i^2 n^2 (c+d x)^2}{12 b d^2}-\frac {B (b c-a d)^3 g i^2 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^3 d}-\frac {B (b c-a d)^2 g i^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^3}+\frac {B (b c-a d)^2 g i^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b d^2}-\frac {B (b c-a d) g i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^2}+\frac {(b c-a d)^2 g i^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{12 b^3}+\frac {(b c-a d) g i^2 (a+b x)^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 b^2}+\frac {g i^2 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}-\frac {B (b c-a d)^4 g i^2 n \left (A+B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{6 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 n^2 \log (c+d x)}{4 b^3 d^2}+\frac {\left (B^2 (b c-a d)^4 g i^2 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{6 b^3 d^2} \\ & = \frac {B^2 (b c-a d)^3 g i^2 n^2 x}{12 b^2 d}+\frac {B^2 (b c-a d)^2 g i^2 n^2 (c+d x)^2}{12 b d^2}-\frac {B (b c-a d)^3 g i^2 n (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^3 d}-\frac {B (b c-a d)^2 g i^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b^3}+\frac {B (b c-a d)^2 g i^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b d^2}-\frac {B (b c-a d) g i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 d^2}+\frac {(b c-a d)^2 g i^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{12 b^3}+\frac {(b c-a d) g i^2 (a+b x)^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 b^2}+\frac {g i^2 (a+b x)^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b}-\frac {B (b c-a d)^4 g i^2 n \left (A+B n+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right )}{6 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 n^2 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 n^2 \log (c+d x)}{4 b^3 d^2}-\frac {B^2 (b c-a d)^4 g i^2 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{6 b^3 d^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 713, normalized size of antiderivative = 1.12 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\frac {g i^2 \left (-4 (b c-a d) (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+3 b (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+\frac {4 B (b c-a d)^2 n \left (2 A b d (b c-a d) x-B (b c-a d) n (b d x+(b c-a d) \log (a+b x))+2 B d (b c-a d) (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+b^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 (b c-a d)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-2 B (b c-a d)^2 n \log (c+d x)-B (b c-a d)^2 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b^3}-\frac {B (b c-a d) n \left (6 A b d (b c-a d)^2 x-3 B (b c-a d)^2 n (b d x+(b c-a d) \log (a+b x))-B (b c-a d) n \left (2 b d (b c-a d) x+b^2 (c+d x)^2+2 (b c-a d)^2 \log (a+b x)\right )+6 B d (b c-a d)^2 (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+3 b^2 (b c-a d) (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+2 b^3 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+6 (b c-a d)^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-6 B (b c-a d)^3 n \log (c+d x)-3 B (b c-a d)^3 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )\right )}{b^3}\right )}{12 d^2} \]

[In]

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

[Out]

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

Maple [F]

\[\int \left (b g x +a g \right ) \left (d i x +c i \right )^{2} {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )}^{2}d x\]

[In]

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

Fricas [F]

\[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )} {\left (d i x + c i\right )}^{2} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2662 vs. \(2 (608) = 1216\).

Time = 0.74 (sec) , antiderivative size = 2662, normalized size of antiderivative = 4.19 \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

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

Giac [F]

\[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )} {\left (d i x + c i\right )}^{2} {\left (B \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2} \,d x } \]

[In]

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

Mupad [F(-1)]

Timed out. \[ \int (a g+b g x) (c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx=\int \left (a\,g+b\,g\,x\right )\,{\left (c\,i+d\,i\,x\right )}^2\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]

[In]

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

[Out]

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