\(\int \frac {(a g+b g x)^3 (A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(c i+d i x)^2} \, dx\) [92]

Optimal result

Integrand size = 42, antiderivative size = 722 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\frac {2 A B (b c-a d)^2 g^3 (a+b x)}{d^3 i^2 (c+d x)}-\frac {2 B^2 (b c-a d)^2 g^3 (a+b x)}{d^3 i^2 (c+d x)}+\frac {2 B^2 (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^3 i^2 (c+d x)}-\frac {b B (b c-a d) g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3 i^2}-\frac {6 b B (b c-a d)^2 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4 i^2}-\frac {3 b (b c-a d) g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^2}-\frac {(b c-a d)^2 g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^2 (c+d x)}+\frac {b^3 g^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^4 i^2}-\frac {3 b (b c-a d)^2 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^4 i^2}+\frac {b B^2 (b c-a d)^2 g^3 \log (c+d x)}{d^4 i^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{d^4 i^2}-\frac {6 b B^2 (b c-a d)^2 g^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac {6 b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac {b B^2 (b c-a d)^2 g^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{d^4 i^2}+\frac {6 b B^2 (b c-a d)^2 g^3 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 722, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2562, 2395, 2333, 2332, 2356, 2389, 2379, 2438, 2351, 31, 2355, 2354, 2421, 6724} \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\frac {b^3 g^3 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 d^4 i^2}-\frac {6 b B g^3 (b c-a d)^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^4 i^2}-\frac {3 b g^3 (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^4 i^2}-\frac {6 b B g^3 (b c-a d)^2 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^4 i^2}+\frac {b B g^3 (b c-a d)^2 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^4 i^2}-\frac {3 b g^3 (a+b x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^3 i^2}-\frac {g^3 (a+b x) (b c-a d)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{d^3 i^2 (c+d x)}-\frac {b B g^3 (a+b x) (b c-a d) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{d^3 i^2}+\frac {2 A B g^3 (a+b x) (b c-a d)^2}{d^3 i^2 (c+d x)}-\frac {6 b B^2 g^3 (b c-a d)^2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac {b B^2 g^3 (b c-a d)^2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{d (a+b x)}\right )}{d^4 i^2}+\frac {6 b B^2 g^3 (b c-a d)^2 \operatorname {PolyLog}\left (3,\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}+\frac {b B^2 g^3 (b c-a d)^2 \log (c+d x)}{d^4 i^2}+\frac {2 B^2 g^3 (a+b x) (b c-a d)^2 \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^3 i^2 (c+d x)}-\frac {2 B^2 g^3 (a+b x) (b c-a d)^2}{d^3 i^2 (c+d x)} \]

[In]

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

[Out]

(2*A*B*(b*c - a*d)^2*g^3*(a + b*x))/(d^3*i^2*(c + d*x)) - (2*B^2*(b*c - a*d)^2*g^3*(a + b*x))/(d^3*i^2*(c + d*
x)) + (2*B^2*(b*c - a*d)^2*g^3*(a + b*x)*Log[(e*(a + b*x))/(c + d*x)])/(d^3*i^2*(c + d*x)) - (b*B*(b*c - a*d)*
g^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^3*i^2) - (6*b*B*(b*c - a*d)^2*g^3*Log[(b*c - a*d)/(b*(c
+ d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d^4*i^2) - (3*b*(b*c - a*d)*g^3*(a + b*x)*(A + B*Log[(e*(a +
b*x))/(c + d*x)])^2)/(d^3*i^2) - ((b*c - a*d)^2*g^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^3*i^2
*(c + d*x)) + (b^3*g^3*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*d^4*i^2) - (3*b*(b*c - a*d)^2*g^
3*Log[(b*c - a*d)/(b*(c + d*x))]*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(d^4*i^2) + (b*B^2*(b*c - a*d)^2*g^3*
Log[c + d*x])/(d^4*i^2) + (b*B*(b*c - a*d)^2*g^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[1 - (b*(c + d*x))/(d
*(a + b*x))])/(d^4*i^2) - (6*b*B^2*(b*c - a*d)^2*g^3*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^2) - (6*b
*B*(b*c - a*d)^2*g^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^2) -
(b*B^2*(b*c - a*d)^2*g^3*PolyLog[2, (b*(c + d*x))/(d*(a + b*x))])/(d^4*i^2) + (6*b*B^2*(b*c - a*d)^2*g^3*Poly
Log[3, (d*(a + b*x))/(b*(c + d*x))])/(d^4*i^2)

Rule 31

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

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2333

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.), x_Symbol] :> Simp[x*(a + b*Log[c*x^n])^p, x] - Dist[b*n*p, In
t[(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{a, b, c, n}, x] && GtQ[p, 0] && IntegerQ[2*p]

Rule 2351

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

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 2355

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

Rule 2356

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

Rule 2379

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

Rule 2389

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

Rule 2395

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2421

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

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 2562

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(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] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i
, 0] && IntegersQ[m, q]

Rule 6724

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left ((b c-a d)^2 g^3\right ) \text {Subst}\left (\int \frac {x^3 (A+B \log (e x))^2}{(b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{i^2} \\ & = \frac {\left ((b c-a d)^2 g^3\right ) \text {Subst}\left (\int \left (-\frac {(A+B \log (e x))^2}{d^3}+\frac {b^3 (A+B \log (e x))^2}{d^3 (b-d x)^3}-\frac {3 b^2 (A+B \log (e x))^2}{d^3 (b-d x)^2}+\frac {3 b (A+B \log (e x))^2}{d^3 (b-d x)}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{i^2} \\ & = -\frac {\left ((b c-a d)^2 g^3\right ) \text {Subst}\left (\int (A+B \log (e x))^2 \, dx,x,\frac {a+b x}{c+d x}\right )}{d^3 i^2}+\frac {\left (3 b (b c-a d)^2 g^3\right ) \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{d^3 i^2}-\frac {\left (3 b^2 (b c-a d)^2 g^3\right ) \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{(b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{d^3 i^2}+\frac {\left (b^3 (b c-a d)^2 g^3\right ) \text {Subst}\left (\int \frac {(A+B \log (e x))^2}{(b-d x)^3} \, dx,x,\frac {a+b x}{c+d x}\right )}{d^3 i^2} \\ & = -\frac {3 b (b c-a d) g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^2}-\frac {(b c-a d)^2 g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^2 (c+d x)}+\frac {b^3 g^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^4 i^2}-\frac {3 b (b c-a d)^2 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^4 i^2}+\frac {\left (6 b B (b c-a d)^2 g^3\right ) \text {Subst}\left (\int \frac {(A+B \log (e x)) \log \left (1-\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{d^4 i^2}-\frac {\left (b^3 B (b c-a d)^2 g^3\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{x (b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{d^4 i^2}+\frac {\left (2 B (b c-a d)^2 g^3\right ) \text {Subst}\left (\int (A+B \log (e x)) \, dx,x,\frac {a+b x}{c+d x}\right )}{d^3 i^2}+\frac {\left (6 b B (b c-a d)^2 g^3\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{d^3 i^2} \\ & = \frac {2 A B (b c-a d)^2 g^3 (a+b x)}{d^3 i^2 (c+d x)}-\frac {6 b B (b c-a d)^2 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4 i^2}-\frac {3 b (b c-a d) g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^2}-\frac {(b c-a d)^2 g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^2 (c+d x)}+\frac {b^3 g^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^4 i^2}-\frac {3 b (b c-a d)^2 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^4 i^2}-\frac {6 b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac {\left (b^2 B (b c-a d)^2 g^3\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{x (b-d x)} \, dx,x,\frac {a+b x}{c+d x}\right )}{d^4 i^2}+\frac {\left (6 b B^2 (b c-a d)^2 g^3\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 )}{d^4 i^2}+\frac {\left (6 b B^2 (b c-a d)^2 g^3\right ) \text {Subst}\left (\int \frac {\text {Li}_2\left (\frac {d x}{b}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{d^4 i^2}-\frac {\left (b^2 B (b c-a d)^2 g^3\right ) \text {Subst}\left (\int \frac {A+B \log (e x)}{(b-d x)^2} \, dx,x,\frac {a+b x}{c+d x}\right )}{d^3 i^2}+\frac {\left (2 B^2 (b c-a d)^2 g^3\right ) \text {Subst}\left (\int \log (e x) \, dx,x,\frac {a+b x}{c+d x}\right )}{d^3 i^2} \\ & = \frac {2 A B (b c-a d)^2 g^3 (a+b x)}{d^3 i^2 (c+d x)}-\frac {2 B^2 (b c-a d)^2 g^3 (a+b x)}{d^3 i^2 (c+d x)}+\frac {2 B^2 (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^3 i^2 (c+d x)}-\frac {b B (b c-a d) g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3 i^2}-\frac {6 b B (b c-a d)^2 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4 i^2}-\frac {3 b (b c-a d) g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^2}-\frac {(b c-a d)^2 g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^2 (c+d x)}+\frac {b^3 g^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^4 i^2}-\frac {3 b (b c-a d)^2 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^4 i^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{d^4 i^2}-\frac {6 b B^2 (b c-a d)^2 g^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac {6 b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}+\frac {6 b B^2 (b c-a d)^2 g^3 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac {\left (b B^2 (b c-a d)^2 g^3\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {b}{d x}\right )}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{d^4 i^2}+\frac {\left (b B^2 (b c-a d)^2 g^3\right ) \text {Subst}\left (\int \frac {1}{b-d x} \, dx,x,\frac {a+b x}{c+d x}\right )}{d^3 i^2} \\ & = \frac {2 A B (b c-a d)^2 g^3 (a+b x)}{d^3 i^2 (c+d x)}-\frac {2 B^2 (b c-a d)^2 g^3 (a+b x)}{d^3 i^2 (c+d x)}+\frac {2 B^2 (b c-a d)^2 g^3 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{d^3 i^2 (c+d x)}-\frac {b B (b c-a d) g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^3 i^2}-\frac {6 b B (b c-a d)^2 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{d^4 i^2}-\frac {3 b (b c-a d) g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^2}-\frac {(b c-a d)^2 g^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^3 i^2 (c+d x)}+\frac {b^3 g^3 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 d^4 i^2}-\frac {3 b (b c-a d)^2 g^3 \log \left (\frac {b c-a d}{b (c+d x)}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{d^4 i^2}+\frac {b B^2 (b c-a d)^2 g^3 \log (c+d x)}{d^4 i^2}+\frac {b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right )}{d^4 i^2}-\frac {6 b B^2 (b c-a d)^2 g^3 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac {6 b B (b c-a d)^2 g^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2}-\frac {b B^2 (b c-a d)^2 g^3 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{d^4 i^2}+\frac {6 b B^2 (b c-a d)^2 g^3 \text {Li}_3\left (\frac {d (a+b x)}{b (c+d x)}\right )}{d^4 i^2} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(4443\) vs. \(2(722)=1444\).

Time = 6.29 (sec) , antiderivative size = 4443, normalized size of antiderivative = 6.15 \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\text {Result too large to show} \]

[In]

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

[Out]

(g^3*(-4*A^2*b^2*d*(2*b*c - 3*a*d)*x + 2*A^2*b^3*d^2*x^2 + (4*A^2*(b*c - a*d)^3)/(c + d*x) + 12*A^2*b*(b*c - a
*d)^2*Log[c + d*x] + (8*a^3*A*B*d^3*(b*c - a*d + b*(c + d*x)*Log[a/b + x] + (-(b*c) + a*d)*Log[(e*(a + b*x))/(
c + d*x)] - b*c*Log[(b*(c + d*x))/(b*c - a*d)] - b*d*x*Log[(b*(c + d*x))/(b*c - a*d)]))/((b*c - a*d)*(c + d*x)
) + 12*a^2*A*b*B*d^2*(-Log[c/d + x]^2 + 2*Log[c/d + x]*Log[c + d*x] + 2*(-(c/(c + d*x)) + (b*c*Log[a + b*x])/(
-(b*c) + a*d) + (b*c*Log[c + d*x])/(b*c - a*d) - Log[a/b + x]*Log[c + d*x] + Log[(e*(a + b*x))/(c + d*x)]*(c/(
c + d*x) + Log[c + d*x]) + Log[a/b + x]*Log[(b*(c + d*x))/(b*c - a*d)]) + 2*PolyLog[2, (d*(a + b*x))/(-(b*c) +
a*d)]) + 4*A*b^3*B*(-4*c^2 + (4*a*c*d)/b - c*d*x + (a*d^2*x)/b - (2*c^3)/(c + d*x) + 4*c^2*Log[c/d + x] - 3*c
^2*Log[c/d + x]^2 - (a^2*d^2*Log[a + b*x])/b^2 + (2*b*c^3*Log[a + b*x])/(-(b*c) + a*d) - 4*c*d*x*Log[(e*(a + b
*x))/(c + d*x)] + d^2*x^2*Log[(e*(a + b*x))/(c + d*x)] + (2*c^3*Log[(e*(a + b*x))/(c + d*x)])/(c + d*x) + c^2*
Log[c + d*x] + (2*b*c^3*Log[c + d*x])/(b*c - a*d) + 6*c^2*Log[c/d + x]*Log[c + d*x] + 6*c^2*Log[(e*(a + b*x))/
(c + d*x)]*Log[c + d*x] - (2*c*Log[a/b + x]*(2*a*d + 3*b*c*Log[c + d*x] - 3*b*c*Log[(b*(c + d*x))/(b*c - a*d)]
))/b + 6*c^2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) + 24*a*A*b^2*B*d*(d*(a/b + x)*(-1 + Log[a/b + x]) - (c^
2*Log[a/b + x])/(c + d*x) - (c + d*x)*(-1 + Log[c/d + x]) + c*Log[c/d + x]^2 + (c^2*(1 + Log[c/d + x]))/(c + d
*x) + (b*c^2*(Log[a + b*x] - Log[c + d*x]))/(b*c - a*d) + (-Log[a/b + x] + Log[c/d + x] + Log[(e*(a + b*x))/(c
+ d*x)])*(d*x - c^2/(c + d*x) - 2*c*Log[c + d*x]) - 2*c*(Log[a/b + x]*Log[(b*(c + d*x))/(b*c - a*d)] + PolyLo
g[2, (d*(a + b*x))/(-(b*c) + a*d)])) - (4*a^3*B^2*d^3*(2*b*c - 2*a*d + 2*b*(c + d*x)*Log[a + b*x] - 2*(b*c - a
*d)*Log[(e*(a + b*x))/(c + d*x)] - 2*b*(c + d*x)*Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)] + (b*c - a*d)*Log[(
e*(a + b*x))/(c + d*x)]^2 - 2*b*(c + d*x)*Log[c + d*x] - 2*b*(c + d*x)*Log[(e*(a + b*x))/(c + d*x)]*Log[(b*c -
a*d)/(b*c + b*d*x)] + b*(c + d*x)*(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*(c + d*x)*(Log[(b*c - a*d)/(b*c + b*d*x)]*(2*Log[(d*(a + b*x))/(-(b*c)
+ a*d)] + Log[(b*c - a*d)/(b*c + b*d*x)]) - 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])))/((b*c - a*d)*(c + d*x))
+ 4*a*b^2*B^2*d*(6*d*x + (3*d*(2*b*x - 2*(a + b*x)*Log[a/b + x] + (a + b*x)*Log[a/b + x]^2))/b - 6*(c + d*x)*
Log[c/d + x] + 3*(c + d*x)*Log[c/d + x]^2 - 2*c*Log[c/d + x]^3 - (3*c^2*(2 + 2*Log[c/d + x] + Log[c/d + x]^2))
/(c + d*x) + 3*(-Log[a/b + x] + Log[c/d + x] + Log[(e*(a + b*x))/(c + d*x)])^2*(d*x - c^2/(c + d*x) - 2*c*Log[
c + d*x]) - (6*(a*d + 2*b*d*x - b*d*x*Log[c/d + x] - b*c*Log[c + d*x] + Log[a/b + x]*(-(d*(a + b*x)) + d*(a +
b*x)*Log[c/d + x] + (b*c - a*d)*Log[(b*(c + d*x))/(b*c - a*d)]) + (b*c - a*d)*PolyLog[2, (d*(a + b*x))/(-(b*c)
+ a*d)]))/b + (3*c^2*(-(d*(a + b*x)*Log[a/b + x]^2) + 2*b*(c + d*x)*Log[a/b + x]*Log[(b*(c + d*x))/(b*c - a*d
)] + 2*b*(c + d*x)*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]))/((-(b*c) + a*d)*(c + d*x)) + 6*(-Log[a/b + x] +
Log[c/d + x] + Log[(e*(a + b*x))/(c + d*x)])*(d*(a/b + x)*(-1 + Log[a/b + x]) - (c^2*Log[a/b + x])/(c + d*x) -
(c + d*x)*(-1 + Log[c/d + x]) + c*Log[c/d + x]^2 + (c^2*(1 + Log[c/d + x]))/(c + d*x) + (b*c^2*(Log[a + b*x]
- Log[c + d*x]))/(b*c - a*d) - 2*c*(Log[a/b + x]*Log[(b*(c + d*x))/(b*c - a*d)] + PolyLog[2, (d*(a + b*x))/(-(
b*c) + a*d)])) + 3*c^2*((2*Log[a/b + x]*(1 + Log[c/d + x]))/(c + d*x) + (b*(Log[c/d + x]^2 - 2*Log[a + b*x] -
2*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + 2*Log[c + d*x]))/(b*c - a*d) - (2*b*PolyLog[2, (b*(c + d*x)
)/(b*c - a*d)])/(b*c - a*d)) - 6*c*(Log[a/b + x]^2*Log[(b*(c + d*x))/(b*c - a*d)] + 2*Log[a/b + x]*PolyLog[2,
(d*(a + b*x))/(-(b*c) + a*d)] - 2*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)]) + 6*c*(Log[c/d + x]^2*(Log[a/b + x
] - Log[(d*(a + b*x))/(-(b*c) + a*d)]) - 2*Log[c/d + x]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] + 2*PolyLog[3, (
b*(c + d*x))/(b*c - a*d)])) + b^3*B^2*(d*x*(-6*c + d*x) - (8*c*d*(2*b*x - 2*(a + b*x)*Log[a/b + x] + (a + b*x)
*Log[a/b + x]^2))/b + (d^2*(b*x*(-6*a + b*x) + (6*a^2 + 4*a*b*x - 2*b^2*x^2)*Log[a/b + x] - 2*(a^2 - b^2*x^2)*
Log[a/b + x]^2))/b^2 + (6*c^2 + 4*c*d*x - 2*d^2*x^2)*Log[c/d + x] - 2*(c^2 - d^2*x^2)*Log[c/d + x]^2 + 4*c^2*L
og[c/d + x]^3 + (4*c^3*(2 + 2*Log[c/d + x] + Log[c/d + x]^2))/(c + d*x) - 8*c*(2*d*x - 2*(c + d*x)*Log[c/d + x
] + (c + d*x)*Log[c/d + x]^2) + 2*(-Log[a/b + x] + Log[c/d + x] + Log[(e*(a + b*x))/(c + d*x)])^2*(-4*c*d*x +
d^2*x^2 + (2*c^3)/(c + d*x) + 6*c^2*Log[c + d*x]) + 4*(-Log[a/b + x] + Log[c/d + x] + Log[(e*(a + b*x))/(c + d
*x)])*(-4*c^2 + (4*a*c*d)/b - c*d*x + (a*d^2*x)/b - (2*c^3)/(c + d*x) + 4*c^2*Log[c/d + x] + 4*c*d*x*Log[c/d +
x] - d^2*x^2*Log[c/d + x] - (2*c^3*Log[c/d + x])/(c + d*x) - 3*c^2*Log[c/d + x]^2 - (a^2*d^2*Log[a + b*x])/b^
2 + (2*b*c^3*Log[a + b*x])/(-(b*c) + a*d) + c^2*Log[c + d*x] + (2*b*c^3*Log[c + d*x])/(b*c - a*d) + Log[a/b +
x]*((-4*a*c*d)/b - 4*c*d*x + d^2*x^2 + (2*c^3)/(c + d*x) + 6*c^2*Log[(b*(c + d*x))/(b*c - a*d)]) + 6*c^2*PolyL
og[2, (d*(a + b*x))/(-(b*c) + a*d)]) + (4*c^3*(-(d*(a + b*x)*Log[a/b + x]^2) + 2*b*(c + d*x)*Log[a/b + x]*Log[
(b*(c + d*x))/(b*c - a*d)] + 2*b*(c + d*x)*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]))/((b*c - a*d)*(c + d*x))
+ 12*c^2*(Log[a/b + x]^2*Log[(b*(c + d*x))/(b*c - a*d)] + 2*Log[a/b + x]*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*
d)] - 2*PolyLog[3, (d*(a + b*x))/(-(b*c) + a*d)]) + 2*((8*c*(a*d + 2*b*d*x - b*d*x*Log[c/d + x] - b*c*Log[c +
d*x] + Log[a/b + x]*(-(d*(a + b*x)) + d*(a + b*x)*Log[c/d + x] + (b*c - a*d)*Log[(b*(c + d*x))/(b*c - a*d)]) +
(b*c - a*d)*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]))/b - (-2*a*b*c*d - 3*b^2*c*d*x - 3*a*b*d^2*x + b^2*d^2*
x^2 + 2*a*b*d^2*x*Log[c/d + x] - b^2*d^2*x^2*Log[c/d + x] + a^2*d^2*Log[a + b*x] + b^2*c^2*Log[c + d*x] + 2*a*
b*c*d*Log[c + d*x] + Log[a/b + x]*(b*d*(2*a*c + b*x*(2*c - d*x)) - 2*d^2*(a^2 - b^2*x^2)*Log[c/d + x] + (-2*b^
2*c^2 + 2*a^2*d^2)*Log[(b*(c + d*x))/(b*c - a*d)]) + (-2*b^2*c^2 + 2*a^2*d^2)*PolyLog[2, (d*(a + b*x))/(-(b*c)
+ a*d)])/b^2 + (2*c^3*(2*(b*c - a*d)*Log[a/b + x]*(1 + Log[c/d + x]) + b*(c + d*x)*(Log[c/d + x]^2 - 2*Log[a
+ b*x] - 2*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + 2*Log[c + d*x]) - 2*b*(c + d*x)*PolyLog[2, (b*(c +
d*x))/(b*c - a*d)]))/((-(b*c) + a*d)*(c + d*x)) - 6*c^2*(Log[c/d + x]^2*(Log[a/b + x] - Log[(d*(a + b*x))/(-(
b*c) + a*d)]) - 2*Log[c/d + x]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] + 2*PolyLog[3, (b*(c + d*x))/(b*c - a*d)]
))) + (4*a^2*b*B^2*d^2*((b*c - a*d)*(c + d*x)*Log[c/d + x]^3 + 3*c*(b*c - a*d)*(2 + 2*Log[c/d + x] + Log[c/d +
x]^2) + 3*(b*c - a*d)*(-Log[a/b + x] + Log[c/d + x] + Log[(e*(a + b*x))/(c + d*x)])^2*(c + (c + d*x)*Log[c +
d*x]) + 3*c*Log[a/b + x]*(-(d*(a + b*x)*Log[a/b + x]) + 2*b*(c + d*x)*Log[(b*(c + d*x))/(b*c - a*d)]) + 6*b*c*
(c + d*x)*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] + 3*(Log[a/b + x] - Log[c/d + x] - Log[(e*(a + b*x))/(c + d
*x)])*(-2*c*(b*c - a*d)*Log[a/b + x] + (b*c - a*d)*(c + d*x)*Log[c/d + x]^2 + 2*c*(b*c - a*d)*(1 + Log[c/d + x
]) + 2*b*c*(c + d*x)*(Log[a + b*x] - Log[c + d*x]) - 2*(b*c - a*d)*(c + d*x)*(Log[a/b + x]*Log[(b*(c + d*x))/(
b*c - a*d)] + PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])) + 3*(b*c - a*d)*(c + d*x)*(Log[a/b + x]^2*Log[(b*(c +
d*x))/(b*c - a*d)] + 2*Log[a/b + x]*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] - 2*PolyLog[3, (d*(a + b*x))/(-(
b*c) + a*d)]) - 3*(c*(2*(b*c - a*d)*Log[a/b + x]*(1 + Log[c/d + x]) + b*(c + d*x)*(Log[c/d + x]^2 - 2*Log[a +
b*x] - 2*Log[c/d + x]*Log[(d*(a + b*x))/(-(b*c) + a*d)] + 2*Log[c + d*x]) - 2*b*(c + d*x)*PolyLog[2, (b*(c + d
*x))/(b*c - a*d)]) + (b*c - a*d)*(c + d*x)*(Log[c/d + x]^2*(Log[a/b + x] - Log[(d*(a + b*x))/(-(b*c) + a*d)])
- 2*Log[c/d + x]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)] + 2*PolyLog[3, (b*(c + d*x))/(b*c - a*d)]))))/((b*c - a
*d)*(c + d*x))))/(4*d^4*i^2)

Maple [F]

\[\int \frac {\left (b g x +a g \right )^{3} \left (A +B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )\right )^{2}}{\left (d i x +c i \right )^{2}}d x\]

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\int { \frac {{\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (d i x + c i\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-1)]

Timed out. \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\int { \frac {{\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (d i x + c i\right )}^{2}} \,d x } \]

[In]

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

[Out]

1/2*(2*c^3/(d^5*i^2*x + c*d^4*i^2) + 6*c^2*log(d*x + c)/(d^4*i^2) + (d*x^2 - 4*c*x)/(d^3*i^2))*A^2*b^3*g^3 - 3
*A^2*a*b^2*(c^2/(d^4*i^2*x + c*d^3*i^2) - x/(d^2*i^2) + 2*c*log(d*x + c)/(d^3*i^2))*g^3 + 3*A^2*a^2*b*g^3*(c/(
d^3*i^2*x + c*d^2*i^2) + log(d*x + c)/(d^2*i^2)) - 2*A*B*a^3*g^3*(log(b*e*x/(d*x + c) + a*e/(d*x + c))/(d^2*i^
2*x + c*d*i^2) - 1/(d^2*i^2*x + c*d*i^2) - b*log(b*x + a)/((b*c*d - a*d^2)*i^2) + b*log(d*x + c)/((b*c*d - a*d
^2)*i^2)) - A^2*a^3*g^3/(d^2*i^2*x + c*d*i^2) + 1/2*(2*((b^3*c^2*d*g^3 - 2*a*b^2*c*d^2*g^3 + a^2*b*d^3*g^3)*B^
2*x + (b^3*c^3*g^3 - 2*a*b^2*c^2*d*g^3 + a^2*b*c*d^2*g^3)*B^2)*log(d*x + c)^3 + (B^2*b^3*d^3*g^3*x^3 - 3*(b^3*
c*d^2*g^3 - 2*a*b^2*d^3*g^3)*B^2*x^2 - 2*(2*b^3*c^2*d*g^3 - 3*a*b^2*c*d^2*g^3)*B^2*x + 2*(b^3*c^3*g^3 - 3*a*b^
2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^3)*B^2)*log(d*x + c)^2)/(d^5*i^2*x + c*d^4*i^2) - integrate(-(B^2*
a^3*d^3*g^3*log(e)^2 + (B^2*b^3*d^3*g^3*log(e)^2 + 2*A*B*b^3*d^3*g^3*log(e))*x^3 + 3*(B^2*a*b^2*d^3*g^3*log(e)
^2 + 2*A*B*a*b^2*d^3*g^3*log(e))*x^2 + (B^2*b^3*d^3*g^3*x^3 + 3*B^2*a*b^2*d^3*g^3*x^2 + 3*B^2*a^2*b*d^3*g^3*x
+ B^2*a^3*d^3*g^3)*log(b*x + a)^2 + 3*(B^2*a^2*b*d^3*g^3*log(e)^2 + 2*A*B*a^2*b*d^3*g^3*log(e))*x + 2*(B^2*a^3
*d^3*g^3*log(e) + (B^2*b^3*d^3*g^3*log(e) + A*B*b^3*d^3*g^3)*x^3 + 3*(B^2*a*b^2*d^3*g^3*log(e) + A*B*a*b^2*d^3
*g^3)*x^2 + 3*(B^2*a^2*b*d^3*g^3*log(e) + A*B*a^2*b*d^3*g^3)*x)*log(b*x + a) - ((2*A*B*b^3*d^3*g^3 + (2*g^3*lo
g(e) + g^3)*B^2*b^3*d^3)*x^3 + 2*(b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 + (g^3*log(e) - g^3)*a^3
*d^3)*B^2 + 3*(2*A*B*a*b^2*d^3*g^3 - (b^3*c*d^2*g^3 - 2*(g^3*log(e) + g^3)*a*b^2*d^3)*B^2)*x^2 + 2*(3*A*B*a^2*
b*d^3*g^3 + (3*a^2*b*d^3*g^3*log(e) - 2*b^3*c^2*d*g^3 + 3*a*b^2*c*d^2*g^3)*B^2)*x + 2*(B^2*b^3*d^3*g^3*x^3 + 3
*B^2*a*b^2*d^3*g^3*x^2 + 3*B^2*a^2*b*d^3*g^3*x + B^2*a^3*d^3*g^3)*log(b*x + a))*log(d*x + c))/(d^5*i^2*x^2 + 2
*c*d^4*i^2*x + c^2*d^3*i^2), x)

Giac [F]

\[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\int { \frac {{\left (b g x + a g\right )}^{3} {\left (B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (d i x + c i\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(a g+b g x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(c i+d i x)^2} \, dx=\int \frac {{\left (a\,g+b\,g\,x\right )}^3\,{\left (A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\right )}^2}{{\left (c\,i+d\,i\,x\right )}^2} \,d x \]

[In]

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

[Out]

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