\(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx\) [105]

Optimal result

Integrand size = 42, antiderivative size = 525 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx=-\frac {B^2 d^3 (a+b x)^2}{4 (b c-a d)^4 g^2 i^3 (c+d x)^2}-\frac {6 A b B d^2 (a+b x)}{(b c-a d)^4 g^2 i^3 (c+d x)}+\frac {6 b B^2 d^2 (a+b x)}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {2 b^3 B^2 (c+d x)}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {6 b B^2 d^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d)^4 g^2 i^3 (c+d x)}+\frac {B d^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^4 g^2 i^3 (c+d x)^2}-\frac {2 b^3 B (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {d^3 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^4 g^2 i^3 (c+d x)^2}+\frac {3 b d^2 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^4 g^2 i^3 (c+d x)}-\frac {b^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^4 g^2 i^3 (a+b x)}-\frac {b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3}{B (b c-a d)^4 g^2 i^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 525, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2562, 2395, 2333, 2332, 2342, 2341, 2339, 30} \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx=-\frac {b^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g^2 i^3 (a+b x) (b c-a d)^4}-\frac {2 b^3 B (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (a+b x) (b c-a d)^4}-\frac {b^2 d \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^3}{B g^2 i^3 (b c-a d)^4}-\frac {d^3 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^2 i^3 (c+d x)^2 (b c-a d)^4}+\frac {B d^3 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^2 i^3 (c+d x)^2 (b c-a d)^4}+\frac {3 b d^2 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{g^2 i^3 (c+d x) (b c-a d)^4}-\frac {6 A b B d^2 (a+b x)}{g^2 i^3 (c+d x) (b c-a d)^4}-\frac {2 b^3 B^2 (c+d x)}{g^2 i^3 (a+b x) (b c-a d)^4}-\frac {B^2 d^3 (a+b x)^2}{4 g^2 i^3 (c+d x)^2 (b c-a d)^4}-\frac {6 b B^2 d^2 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{g^2 i^3 (c+d x) (b c-a d)^4}+\frac {6 b B^2 d^2 (a+b x)}{g^2 i^3 (c+d x) (b c-a d)^4} \]

[In]

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

[Out]

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

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 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2342

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

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 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]

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

Mathematica [A] (verified)

Time = 0.67 (sec) , antiderivative size = 453, normalized size of antiderivative = 0.86 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx=-\frac {\left (2 A^2-2 A B+B^2\right ) d (b c-a d)^2 (a+b x)+2 b \left (4 A^2-10 A B+11 B^2\right ) d (b c-a d) (a+b x) (c+d x)+4 b^2 \left (A^2+2 A B+2 B^2\right ) (b c-a d) (c+d x)^2+6 b^2 \left (2 A^2-2 A B+5 B^2\right ) d (a+b x) (c+d x)^2 \log (a+b x)+2 B (b c-a d) \left ((2 A-B) d (b c-a d) (a+b x)+2 b (4 A-5 B) d (a+b x) (c+d x)+4 b^2 (A+B) (c+d x)^2\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )+2 B \left (a^3 B d^3-3 a^2 b B d^2 (2 c+d x)+3 a b^2 d \left (2 A (c+d x)^2-B d x (4 c+3 d x)\right )+b^3 \left (6 A d x (c+d x)^2+B \left (2 c^3+6 c^2 d x-3 d^3 x^3\right )\right )\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+4 b^2 B^2 d (a+b x) (c+d x)^2 \log ^3\left (\frac {e (a+b x)}{c+d x}\right )-6 b^2 \left (2 A^2-2 A B+5 B^2\right ) d (a+b x) (c+d x)^2 \log (c+d x)}{4 (b c-a d)^4 g^2 i^3 (a+b x) (c+d x)^2} \]

[In]

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

[Out]

-1/4*((2*A^2 - 2*A*B + B^2)*d*(b*c - a*d)^2*(a + b*x) + 2*b*(4*A^2 - 10*A*B + 11*B^2)*d*(b*c - a*d)*(a + b*x)*
(c + d*x) + 4*b^2*(A^2 + 2*A*B + 2*B^2)*(b*c - a*d)*(c + d*x)^2 + 6*b^2*(2*A^2 - 2*A*B + 5*B^2)*d*(a + b*x)*(c
+ d*x)^2*Log[a + b*x] + 2*B*(b*c - a*d)*((2*A - B)*d*(b*c - a*d)*(a + b*x) + 2*b*(4*A - 5*B)*d*(a + b*x)*(c +
d*x) + 4*b^2*(A + B)*(c + d*x)^2)*Log[(e*(a + b*x))/(c + d*x)] + 2*B*(a^3*B*d^3 - 3*a^2*b*B*d^2*(2*c + d*x) +
3*a*b^2*d*(2*A*(c + d*x)^2 - B*d*x*(4*c + 3*d*x)) + b^3*(6*A*d*x*(c + d*x)^2 + B*(2*c^3 + 6*c^2*d*x - 3*d^3*x
^3)))*Log[(e*(a + b*x))/(c + d*x)]^2 + 4*b^2*B^2*d*(a + b*x)*(c + d*x)^2*Log[(e*(a + b*x))/(c + d*x)]^3 - 6*b^
2*(2*A^2 - 2*A*B + 5*B^2)*d*(a + b*x)*(c + d*x)^2*Log[c + d*x])/((b*c - a*d)^4*g^2*i^3*(a + b*x)*(c + d*x)^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1079\) vs. \(2(519)=1038\).

Time = 2.78 (sec) , antiderivative size = 1080, normalized size of antiderivative = 2.06

method	result	size
parts	\(\text {Expression too large to display}\)	\(1080\)
derivativedivides	\(\text {Expression too large to display}\)	\(1233\)
default	\(\text {Expression too large to display}\)	\(1233\)
parallelrisch	\(\text {Expression too large to display}\)	\(1762\)
norman	\(\text {Expression too large to display}\)	\(1852\)
risch	\(\text {Expression too large to display}\)	\(2114\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 1008, normalized size of antiderivative = 1.92 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx=-\frac {4 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b^{3} c^{3} + 3 \, {\left (2 \, A^{2} - 10 \, A B + 5 \, B^{2}\right )} a b^{2} c^{2} d - 12 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a^{2} b c d^{2} + {\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} a^{3} d^{3} + 4 \, {\left (B^{2} b^{3} d^{3} x^{3} + B^{2} a b^{2} c^{2} d + {\left (2 \, B^{2} b^{3} c d^{2} + B^{2} a b^{2} d^{3}\right )} x^{2} + {\left (B^{2} b^{3} c^{2} d + 2 \, B^{2} a b^{2} c d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{3} + 6 \, {\left ({\left (2 \, A^{2} - 2 \, A B + 5 \, B^{2}\right )} b^{3} c d^{2} - {\left (2 \, A^{2} - 2 \, A B + 5 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} + 2 \, {\left (3 \, {\left (2 \, A B - B^{2}\right )} b^{3} d^{3} x^{3} + 2 \, B^{2} b^{3} c^{3} + 6 \, A B a b^{2} c^{2} d - 6 \, B^{2} a^{2} b c d^{2} + B^{2} a^{3} d^{3} + 3 \, {\left (4 \, A B b^{3} c d^{2} + {\left (2 \, A B - 3 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (B^{2} a^{2} b d^{3} - 2 \, {\left (A B + B^{2}\right )} b^{3} c^{2} d - 4 \, {\left (A B - B^{2}\right )} a b^{2} c d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 3 \, {\left ({\left (6 \, A^{2} - 2 \, A B + 13 \, B^{2}\right )} b^{3} c^{2} d - 2 \, {\left (2 \, A^{2} + 2 \, A B + 3 \, B^{2}\right )} a b^{2} c d^{2} - {\left (2 \, A^{2} - 6 \, A B + 7 \, B^{2}\right )} a^{2} b d^{3}\right )} x + 2 \, {\left (3 \, {\left (2 \, A^{2} - 2 \, A B + 5 \, B^{2}\right )} b^{3} d^{3} x^{3} + 6 \, A^{2} a b^{2} c^{2} d + 4 \, {\left (A B + B^{2}\right )} b^{3} c^{3} - 12 \, {\left (A B - B^{2}\right )} a^{2} b c d^{2} + {\left (2 \, A B - B^{2}\right )} a^{3} d^{3} + 3 \, {\left (4 \, {\left (A^{2} + 2 \, B^{2}\right )} b^{3} c d^{2} + {\left (2 \, A^{2} - 6 \, A B + 7 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} + 3 \, {\left (2 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} b^{3} c^{2} d + 4 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a b^{2} c d^{2} - {\left (2 \, A B - 3 \, B^{2}\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} g^{2} i^{3} x^{3} + {\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} g^{2} i^{3} x^{2} + {\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} g^{2} i^{3} x + {\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4}\right )} g^{2} i^{3}\right )}} \]

[In]

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

[Out]

-1/4*(4*(A^2 + 2*A*B + 2*B^2)*b^3*c^3 + 3*(2*A^2 - 10*A*B + 5*B^2)*a*b^2*c^2*d - 12*(A^2 - 2*A*B + 2*B^2)*a^2*
b*c*d^2 + (2*A^2 - 2*A*B + B^2)*a^3*d^3 + 4*(B^2*b^3*d^3*x^3 + B^2*a*b^2*c^2*d + (2*B^2*b^3*c*d^2 + B^2*a*b^2*
d^3)*x^2 + (B^2*b^3*c^2*d + 2*B^2*a*b^2*c*d^2)*x)*log((b*e*x + a*e)/(d*x + c))^3 + 6*((2*A^2 - 2*A*B + 5*B^2)*
b^3*c*d^2 - (2*A^2 - 2*A*B + 5*B^2)*a*b^2*d^3)*x^2 + 2*(3*(2*A*B - B^2)*b^3*d^3*x^3 + 2*B^2*b^3*c^3 + 6*A*B*a*
b^2*c^2*d - 6*B^2*a^2*b*c*d^2 + B^2*a^3*d^3 + 3*(4*A*B*b^3*c*d^2 + (2*A*B - 3*B^2)*a*b^2*d^3)*x^2 - 3*(B^2*a^2
*b*d^3 - 2*(A*B + B^2)*b^3*c^2*d - 4*(A*B - B^2)*a*b^2*c*d^2)*x)*log((b*e*x + a*e)/(d*x + c))^2 + 3*((6*A^2 -
2*A*B + 13*B^2)*b^3*c^2*d - 2*(2*A^2 + 2*A*B + 3*B^2)*a*b^2*c*d^2 - (2*A^2 - 6*A*B + 7*B^2)*a^2*b*d^3)*x + 2*(
3*(2*A^2 - 2*A*B + 5*B^2)*b^3*d^3*x^3 + 6*A^2*a*b^2*c^2*d + 4*(A*B + B^2)*b^3*c^3 - 12*(A*B - B^2)*a^2*b*c*d^2
+ (2*A*B - B^2)*a^3*d^3 + 3*(4*(A^2 + 2*B^2)*b^3*c*d^2 + (2*A^2 - 6*A*B + 7*B^2)*a*b^2*d^3)*x^2 + 3*(2*(A^2 +
2*A*B + 2*B^2)*b^3*c^2*d + 4*(A^2 - 2*A*B + 2*B^2)*a*b^2*c*d^2 - (2*A*B - 3*B^2)*a^2*b*d^3)*x)*log((b*e*x + a
*e)/(d*x + c)))/((b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4 - 4*a^3*b^2*c*d^5 + a^4*b*d^6)*g^2*i^3*x^3
+ (2*b^5*c^5*d - 7*a*b^4*c^4*d^2 + 8*a^2*b^3*c^3*d^3 - 2*a^3*b^2*c^2*d^4 - 2*a^4*b*c*d^5 + a^5*d^6)*g^2*i^3*x
^2 + (b^5*c^6 - 2*a*b^4*c^5*d - 2*a^2*b^3*c^4*d^2 + 8*a^3*b^2*c^3*d^3 - 7*a^4*b*c^2*d^4 + 2*a^5*c*d^5)*g^2*i^3
*x + (a*b^4*c^6 - 4*a^2*b^3*c^5*d + 6*a^3*b^2*c^4*d^2 - 4*a^4*b*c^3*d^3 + a^5*c^2*d^4)*g^2*i^3)

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4188 vs. \(2 (519) = 1038\).

Time = 0.50 (sec) , antiderivative size = 4188, normalized size of antiderivative = 7.98 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

-1/2*B^2*((6*b^2*d^2*x^2 + 2*b^2*c^2 + 5*a*b*c*d - a^2*d^2 + 3*(3*b^2*c*d + a*b*d^2)*x)/((b^4*c^3*d^2 - 3*a*b^
3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*g^2*i^3*x^3 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^
3*b*c*d^4 - a^4*d^5)*g^2*i^3*x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)
*g^2*i^3*x + (a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3)*g^2*i^3) + 6*b^2*d*log(b*x + a)/((b
^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3) - 6*b^2*d*log(d*x + c)/((b^4*c^
4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3))*log(b*e*x/(d*x + c) + a*e/(d*x + c)
)^2 - A*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 + 5*a*b*c*d - a^2*d^2 + 3*(3*b^2*c*d + a*b*d^2)*x)/((b^4*c^3*d^2 - 3*a*b
^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*g^2*i^3*x^3 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a
^3*b*c*d^4 - a^4*d^5)*g^2*i^3*x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4
)*g^2*i^3*x + (a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3)*g^2*i^3) + 6*b^2*d*log(b*x + a)/((
b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3) - 6*b^2*d*log(d*x + c)/((b^4*c
^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3))*log(b*e*x/(d*x + c) + a*e/(d*x + c
)) - 1/4*B^2*(2*(4*b^3*c^3 - 15*a*b^2*c^2*d + 12*a^2*b*c*d^2 - a^3*d^3 - 6*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 6*(b^
3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x + a)^2 - 6*(b
^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(d*x + c)^2 - 3*(
b^3*c^2*d + 2*a*b^2*c*d^2 - 3*a^2*b*d^3)*x - 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b
^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x + a) + 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^
3*c^2*d + 2*a*b^2*c*d^2)*x + 2*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b
^2*c*d^2)*x)*log(b*x + a))*log(d*x + c))*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(a*b^4*c^6*g^2*i^3 - 4*a^2*b^3*c
^5*d*g^2*i^3 + 6*a^3*b^2*c^4*d^2*g^2*i^3 - 4*a^4*b*c^3*d^3*g^2*i^3 + a^5*c^2*d^4*g^2*i^3 + (b^5*c^4*d^2*g^2*i^
3 - 4*a*b^4*c^3*d^3*g^2*i^3 + 6*a^2*b^3*c^2*d^4*g^2*i^3 - 4*a^3*b^2*c*d^5*g^2*i^3 + a^4*b*d^6*g^2*i^3)*x^3 + (
2*b^5*c^5*d*g^2*i^3 - 7*a*b^4*c^4*d^2*g^2*i^3 + 8*a^2*b^3*c^3*d^3*g^2*i^3 - 2*a^3*b^2*c^2*d^4*g^2*i^3 - 2*a^4*
b*c*d^5*g^2*i^3 + a^5*d^6*g^2*i^3)*x^2 + (b^5*c^6*g^2*i^3 - 2*a*b^4*c^5*d*g^2*i^3 - 2*a^2*b^3*c^4*d^2*g^2*i^3
+ 8*a^3*b^2*c^3*d^3*g^2*i^3 - 7*a^4*b*c^2*d^4*g^2*i^3 + 2*a^5*c*d^5*g^2*i^3)*x) + (8*b^3*c^3 + 15*a*b^2*c^2*d
- 24*a^2*b*c*d^2 + a^3*d^3 + 4*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b
^2*c*d^2)*x)*log(b*x + a)^3 - 4*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*
b^2*c*d^2)*x)*log(d*x + c)^3 + 30*(b^3*c*d^2 - a*b^2*d^3)*x^2 + 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 +
a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x + a)^2 + 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 +
a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x + 2*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2
+ (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x + a))*log(d*x + c)^2 + 3*(13*b^3*c^2*d - 6*a*b^2*c*d^2 - 7*a^2*b*d^3
)*x + 30*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x +
a) - 6*(5*b^3*d^3*x^3 + 5*a*b^2*c^2*d + 5*(2*b^3*c*d^2 + a*b^2*d^3)*x^2 + 2*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b
^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x + a)^2 + 5*(b^3*c^2*d + 2*a*b^2*c*d^2)*x +
2*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x + a))*lo
g(d*x + c))/(a*b^4*c^6*g^2*i^3 - 4*a^2*b^3*c^5*d*g^2*i^3 + 6*a^3*b^2*c^4*d^2*g^2*i^3 - 4*a^4*b*c^3*d^3*g^2*i^3
+ a^5*c^2*d^4*g^2*i^3 + (b^5*c^4*d^2*g^2*i^3 - 4*a*b^4*c^3*d^3*g^2*i^3 + 6*a^2*b^3*c^2*d^4*g^2*i^3 - 4*a^3*b^
2*c*d^5*g^2*i^3 + a^4*b*d^6*g^2*i^3)*x^3 + (2*b^5*c^5*d*g^2*i^3 - 7*a*b^4*c^4*d^2*g^2*i^3 + 8*a^2*b^3*c^3*d^3*
g^2*i^3 - 2*a^3*b^2*c^2*d^4*g^2*i^3 - 2*a^4*b*c*d^5*g^2*i^3 + a^5*d^6*g^2*i^3)*x^2 + (b^5*c^6*g^2*i^3 - 2*a*b^
4*c^5*d*g^2*i^3 - 2*a^2*b^3*c^4*d^2*g^2*i^3 + 8*a^3*b^2*c^3*d^3*g^2*i^3 - 7*a^4*b*c^2*d^4*g^2*i^3 + 2*a^5*c*d^
5*g^2*i^3)*x)) - 1/2*A^2*((6*b^2*d^2*x^2 + 2*b^2*c^2 + 5*a*b*c*d - a^2*d^2 + 3*(3*b^2*c*d + a*b*d^2)*x)/((b^4*
c^3*d^2 - 3*a*b^3*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*g^2*i^3*x^3 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*
b^2*c^2*d^3 + a^3*b*c*d^4 - a^4*d^5)*g^2*i^3*x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^
3 - 2*a^4*c*d^4)*g^2*i^3*x + (a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3)*g^2*i^3) + 6*b^2*d*
log(b*x + a)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3) - 6*b^2*d*log(d
*x + c)/((b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3)) - 1/2*(4*b^3*c^3 -
15*a*b^2*c^2*d + 12*a^2*b*c*d^2 - a^3*d^3 - 6*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*
b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x + a)^2 - 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2
*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(d*x + c)^2 - 3*(b^3*c^2*d + 2*a*b^2*c*d^2 - 3
*a^2*b*d^3)*x - 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*
log(b*x + a) + 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x +
2*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x + a))*lo
g(d*x + c))*A*B/(a*b^4*c^6*g^2*i^3 - 4*a^2*b^3*c^5*d*g^2*i^3 + 6*a^3*b^2*c^4*d^2*g^2*i^3 - 4*a^4*b*c^3*d^3*g^2
*i^3 + a^5*c^2*d^4*g^2*i^3 + (b^5*c^4*d^2*g^2*i^3 - 4*a*b^4*c^3*d^3*g^2*i^3 + 6*a^2*b^3*c^2*d^4*g^2*i^3 - 4*a^
3*b^2*c*d^5*g^2*i^3 + a^4*b*d^6*g^2*i^3)*x^3 + (2*b^5*c^5*d*g^2*i^3 - 7*a*b^4*c^4*d^2*g^2*i^3 + 8*a^2*b^3*c^3*
d^3*g^2*i^3 - 2*a^3*b^2*c^2*d^4*g^2*i^3 - 2*a^4*b*c*d^5*g^2*i^3 + a^5*d^6*g^2*i^3)*x^2 + (b^5*c^6*g^2*i^3 - 2*
a*b^4*c^5*d*g^2*i^3 - 2*a^2*b^3*c^4*d^2*g^2*i^3 + 8*a^3*b^2*c^3*d^3*g^2*i^3 - 7*a^4*b*c^2*d^4*g^2*i^3 + 2*a^5*
c*d^5*g^2*i^3)*x)

Giac [F]

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

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 7.47 (sec) , antiderivative size = 1505, normalized size of antiderivative = 2.87 \[ \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^2 (c i+d i x)^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

((4*A^2*b^2*c^2 - 2*A^2*a^2*d^2 - B^2*a^2*d^2 + 8*B^2*b^2*c^2 + 2*A*B*a^2*d^2 + 8*A*B*b^2*c^2 + 10*A^2*a*b*c*d
+ 23*B^2*a*b*c*d - 22*A*B*a*b*c*d)/(2*(a*d - b*c)) + (3*x^2*(2*A^2*b^2*d^2 + 5*B^2*b^2*d^2 - 2*A*B*b^2*d^2))/
(a*d - b*c) + (3*x*(2*A^2*a*b*d^2 + 7*B^2*a*b*d^2 + 6*A^2*b^2*c*d + 13*B^2*b^2*c*d - 6*A*B*a*b*d^2 - 2*A*B*b^2
*c*d))/(2*(a*d - b*c)))/(x*(2*b^3*c^4*g^2*i^3 + 4*a^3*c*d^3*g^2*i^3 - 6*a^2*b*c^2*d^2*g^2*i^3) + x^2*(2*a^3*d^
4*g^2*i^3 + 4*b^3*c^3*d*g^2*i^3 - 6*a*b^2*c^2*d^2*g^2*i^3) + x^3*(2*b^3*c^2*d^2*g^2*i^3 + 2*a^2*b*d^4*g^2*i^3
- 4*a*b^2*c*d^3*g^2*i^3) + 2*a^3*c^2*d^2*g^2*i^3 + 2*a*b^2*c^4*g^2*i^3 - 4*a^2*b*c^3*d*g^2*i^3) - log((e*(a +
b*x))/(c + d*x))^2*((x*((3*B^2)/(2*g^2*i^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (3*B^2*(a*d + b*c))/(g^2*i^3*(a*
d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) + (B^2*(a*d + 2*b*c))/(2*g^2*i^3*(a^2*b*d^3 + b^3*c^2*d - 2*a*b^2*c
*d^2)) - (3*B^2*a*c)/(g^2*i^3*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (3*B^2*b*d*x^2)/(g^2*i^3*(a*d - b
*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(d*x^3 + (a*c^2)/(b*d) + (x^2*(a*d^2 + 2*b*c*d))/(b*d) + (x*(b*c^2 + 2*a
*c*d))/(b*d)) + (3*B*b^2*d*(2*A - B))/(2*g^2*i^3*(a*d - b*c)^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) - (log((e*(a
+ b*x))/(c + d*x))*(x*((3*(B^2 + 2*A*B))/(2*g^2*i^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) - (3*B*(2*A - B)*(a*d + b
*c))/(g^2*i^3*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) + (4*B^2*b*c - B^2*a*d + 2*A*B*a*d + 4*A*B*b*c)/(2
*g^2*i^3*(a^2*b*d^3 + b^3*c^2*d - 2*a*b^2*c*d^2)) - (3*B*a*c*(2*A - B))/(g^2*i^3*(a*d - b*c)*(a^2*d^2 + b^2*c^
2 - 2*a*b*c*d)) - (3*B*b*d*x^2*(2*A - B))/(g^2*i^3*(a*d - b*c)*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))))/(d*x^3 + (a*
c^2)/(b*d) + (x^2*(a*d^2 + 2*b*c*d))/(b*d) + (x*(b*c^2 + 2*a*c*d))/(b*d)) + (b^2*d*atan((b^2*d*(2*A^2 + 5*B^2
- 2*A*B)*(2*a^4*d^4*g^2*i^3 - 2*b^4*c^4*g^2*i^3 + 4*a*b^3*c^3*d*g^2*i^3 - 4*a^3*b*c*d^3*g^2*i^3)*3i)/(2*g^2*i^
3*(a*d - b*c)^4*(6*A^2*b^2*d + 15*B^2*b^2*d - 6*A*B*b^2*d)) + (b^3*d^2*x*(2*A^2 + 5*B^2 - 2*A*B)*(a^3*d^3*g^2*
i^3 - b^3*c^3*g^2*i^3 + 3*a*b^2*c^2*d*g^2*i^3 - 3*a^2*b*c*d^2*g^2*i^3)*6i)/(g^2*i^3*(a*d - b*c)^4*(6*A^2*b^2*d
+ 15*B^2*b^2*d - 6*A*B*b^2*d)))*(2*A^2 + 5*B^2 - 2*A*B)*3i)/(g^2*i^3*(a*d - b*c)^4) - (B^2*b^2*d*log((e*(a +
b*x))/(c + d*x))^3)/(g^2*i^3*(a*d - b*c)^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))