3.107 \(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^4 (c i+d i x)^3} \, dx\)
Optimal. Leaf size=851 \[ -\frac {2 B^2 (c+d x)^3 b^5}{27 (b c-a d)^6 g^4 i^3 (a+b x)^3}-\frac {(c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 b^5}{3 (b c-a d)^6 g^4 i^3 (a+b x)^3}-\frac {2 B (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) b^5}{9 (b c-a d)^6 g^4 i^3 (a+b x)^3}+\frac {5 B^2 d (c+d x)^2 b^4}{4 (b c-a d)^6 g^4 i^3 (a+b x)^2}+\frac {5 d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 b^4}{2 (b c-a d)^6 g^4 i^3 (a+b x)^2}+\frac {5 B d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) b^4}{2 (b c-a d)^6 g^4 i^3 (a+b x)^2}-\frac {10 d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 b^3}{(b c-a d)^6 g^4 i^3 (a+b x)}-\frac {20 B^2 d^2 (c+d x) b^3}{(b c-a d)^6 g^4 i^3 (a+b x)}-\frac {20 B d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) b^3}{(b c-a d)^6 g^4 i^3 (a+b x)}-\frac {10 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^3 b^2}{3 B (b c-a d)^6 g^4 i^3}+\frac {5 d^4 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 b}{(b c-a d)^6 g^4 i^3 (c+d x)}-\frac {10 B^2 d^4 (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right ) b}{(b c-a d)^6 g^4 i^3 (c+d x)}+\frac {10 B^2 d^4 (a+b x) b}{(b c-a d)^6 g^4 i^3 (c+d x)}-\frac {10 A B d^4 (a+b x) b}{(b c-a d)^6 g^4 i^3 (c+d x)}-\frac {d^5 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^6 g^4 i^3 (c+d x)^2}+\frac {B d^5 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^6 g^4 i^3 (c+d x)^2}-\frac {B^2 d^5 (a+b x)^2}{4 (b c-a d)^6 g^4 i^3 (c+d x)^2} \]
[Out]
-1/4*B^2*d^5*(b*x+a)^2/(-a*d+b*c)^6/g^4/i^3/(d*x+c)^2-10*A*b*B*d^4*(b*x+a)/(-a*d+b*c)^6/g^4/i^3/(d*x+c)+10*b*B
^2*d^4*(b*x+a)/(-a*d+b*c)^6/g^4/i^3/(d*x+c)-20*b^3*B^2*d^2*(d*x+c)/(-a*d+b*c)^6/g^4/i^3/(b*x+a)+5/4*b^4*B^2*d*
(d*x+c)^2/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^2-2/27*b^5*B^2*(d*x+c)^3/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^3-10*b*B^2*d^4*(b
*x+a)*ln(e*(b*x+a)/(d*x+c))/(-a*d+b*c)^6/g^4/i^3/(d*x+c)+1/2*B*d^5*(b*x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d
+b*c)^6/g^4/i^3/(d*x+c)^2-20*b^3*B*d^2*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^6/g^4/i^3/(b*x+a)+5/2*b^
4*B*d*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^2-2/9*b^5*B*(d*x+c)^3*(A+B*ln(e*(b*x+
a)/(d*x+c)))/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^3-1/2*d^5*(b*x+a)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^6/g^4/i
^3/(d*x+c)^2+5*b*d^4*(b*x+a)*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^6/g^4/i^3/(d*x+c)-10*b^3*d^2*(d*x+c)*(A+
B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^6/g^4/i^3/(b*x+a)+5/2*b^4*d*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*
d+b*c)^6/g^4/i^3/(b*x+a)^2-1/3*b^5*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^2/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^3-10/3
*b^2*d^3*(A+B*ln(e*(b*x+a)/(d*x+c)))^3/B/(-a*d+b*c)^6/g^4/i^3
________________________________________________________________________________________
Rubi [C] time = 10.92, antiderivative size = 2454, normalized size of antiderivative = 2.88,
number of steps used = 207, number of rules used = 31, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.738, Rules used
= {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 6688, 6742, 2411, 2344, 2317,
2507, 2488, 2506, 6610, 2500, 2433, 2375, 2374, 6589, 2440, 2434, 2499, 2396, 2302, 30}
\[ \text {result too large to display} \]
Antiderivative was successfully verified.
[In]
Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/((a*g + b*g*x)^4*(c*i + d*i*x)^3),x]
[Out]
(-2*b^2*B^2)/(27*(b*c - a*d)^3*g^4*i^3*(a + b*x)^3) + (37*b^2*B^2*d)/(36*(b*c - a*d)^4*g^4*i^3*(a + b*x)^2) -
(319*b^2*B^2*d^2)/(18*(b*c - a*d)^5*g^4*i^3*(a + b*x)) - (B^2*d^3)/(4*(b*c - a*d)^4*g^4*i^3*(c + d*x)^2) - (19
*b*B^2*d^3)/(2*(b*c - a*d)^5*g^4*i^3*(c + d*x)) - (245*b^2*B^2*d^3*Log[a + b*x])/(9*(b*c - a*d)^6*g^4*i^3) + (
10*A*b^2*B*d^3*Log[a + b*x]^2)/((b*c - a*d)^6*g^4*i^3) + (10*b^2*B^2*d^3*Log[a + b*x]^2)/(3*(b*c - a*d)^6*g^4*
i^3) - (10*b^2*B^2*d^3*Log[a + b*x]*Log[(c + d*x)^(-1)]^2)/((b*c - a*d)^6*g^4*i^3) + (10*b^2*B^2*d^3*Log[-((d*
(a + b*x))/(b*c - a*d))]*Log[(c + d*x)^(-1)]^2)/((b*c - a*d)^6*g^4*i^3) + (10*b^2*B^2*d^3*Log[-((b*c - a*d)/(d
*(a + b*x)))]*Log[(e*(a + b*x))/(c + d*x)]^2)/((b*c - a*d)^6*g^4*i^3) + (10*b^2*B^2*d^3*Log[a + b*x]*Log[(e*(a
+ b*x))/(c + d*x)]^2)/((b*c - a*d)^6*g^4*i^3) - (2*b^2*B*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*(b*c - a*d)
^3*g^4*i^3*(a + b*x)^3) + (11*b^2*B*d*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(6*(b*c - a*d)^4*g^4*i^3*(a + b*x)
^2) - (47*b^2*B*d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^5*g^4*i^3*(a + b*x)) + (B*d^3*(A + B*
Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^4*g^4*i^3*(c + d*x)^2) + (9*b*B*d^3*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]))/((b*c - a*d)^5*g^4*i^3*(c + d*x)) - (20*b^2*B*d^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))
/(3*(b*c - a*d)^6*g^4*i^3) - (b^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(3*(b*c - a*d)^3*g^4*i^3*(a + b*x)^3
) + (3*b^2*d*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*(b*c - a*d)^4*g^4*i^3*(a + b*x)^2) - (6*b^2*d^2*(A + B
*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^5*g^4*i^3*(a + b*x)) - (d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]
)^2)/(2*(b*c - a*d)^4*g^4*i^3*(c + d*x)^2) - (4*b*d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^5*g
^4*i^3*(c + d*x)) - (10*b^2*d^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/((b*c - a*d)^6*g^4*i^3) +
(245*b^2*B^2*d^3*Log[c + d*x])/(9*(b*c - a*d)^6*g^4*i^3) - (10*b^2*B^2*d^3*Log[a + b*x]^2*Log[c + d*x])/((b*c
- a*d)^6*g^4*i^3) - (20*A*b^2*B*d^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/((b*c - a*d)^6*g^4*i^3) -
(20*b^2*B^2*d^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(3*(b*c - a*d)^6*g^4*i^3) - (20*b^2*B^2*d^3*L
og[a + b*x]*Log[(c + d*x)^(-1)]*Log[c + d*x])/((b*c - a*d)^6*g^4*i^3) + (20*b^2*B^2*d^3*Log[-((d*(a + b*x))/(b
*c - a*d))]*(Log[a + b*x] + Log[(c + d*x)^(-1)] - Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/((b*c - a*d)^6*g
^4*i^3) + (20*b^2*B*d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(3*(b*c - a*d)^6*g^4*i^3) + (10*b^2
*d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2*Log[c + d*x])/((b*c - a*d)^6*g^4*i^3) + (10*A*b^2*B*d^3*Log[c + d*
x]^2)/((b*c - a*d)^6*g^4*i^3) + (10*b^2*B^2*d^3*Log[c + d*x]^2)/(3*(b*c - a*d)^6*g^4*i^3) - (10*b^2*B^2*d^3*Lo
g[a + b*x]*Log[c + d*x]^2)/((b*c - a*d)^6*g^4*i^3) + (10*b^2*B^2*d^3*Log[(e*(a + b*x))/(c + d*x)]*Log[c + d*x]
^2)/((b*c - a*d)^6*g^4*i^3) + (10*b^2*B^2*d^3*Log[c + d*x]^3)/(3*(b*c - a*d)^6*g^4*i^3) - (20*A*b^2*B*d^3*Log[
a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^6*g^4*i^3) - (20*b^2*B^2*d^3*Log[a + b*x]*Log[(b*(c + d*
x))/(b*c - a*d)])/(3*(b*c - a*d)^6*g^4*i^3) + (10*b^2*B^2*d^3*Log[a + b*x]^2*Log[(b*(c + d*x))/(b*c - a*d)])/(
(b*c - a*d)^6*g^4*i^3) - (20*A*b^2*B*d^3*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^6*g^4*i^3) - (
20*b^2*B^2*d^3*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(3*(b*c - a*d)^6*g^4*i^3) + (20*b^2*B^2*d^3*Log[a + b
*x]*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^6*g^4*i^3) - (20*A*b^2*B*d^3*PolyLog[2, (b*(c + d*x
))/(b*c - a*d)])/((b*c - a*d)^6*g^4*i^3) - (20*b^2*B^2*d^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(3*(b*c - a*
d)^6*g^4*i^3) - (20*b^2*B^2*d^3*Log[(c + d*x)^(-1)]*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^6*g^4*
i^3) + (20*b^2*B^2*d^3*(Log[a + b*x] + Log[(c + d*x)^(-1)] - Log[(e*(a + b*x))/(c + d*x)])*PolyLog[2, (b*(c +
d*x))/(b*c - a*d)])/((b*c - a*d)^6*g^4*i^3) - (20*b^2*B^2*d^3*Log[(e*(a + b*x))/(c + d*x)]*PolyLog[2, 1 + (b*c
- a*d)/(d*(a + b*x))])/((b*c - a*d)^6*g^4*i^3) - (20*b^2*B^2*d^3*PolyLog[3, -((d*(a + b*x))/(b*c - a*d))])/((
b*c - a*d)^6*g^4*i^3) - (20*b^2*B^2*d^3*PolyLog[3, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^6*g^4*i^3) - (20*b
^2*B^2*d^3*PolyLog[3, 1 + (b*c - a*d)/(d*(a + b*x))])/((b*c - a*d)^6*g^4*i^3)
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 30
Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]
Rule 44
Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])
Rule 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 2302
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 2317
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 2344
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((x_)*((d_) + (e_.)*(x_))), x_Symbol] :> Dist[1/d, Int[(a + b*
Log[c*x^n])^p/x, x], x] - Dist[e/d, Int[(a + b*Log[c*x^n])^p/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, n}, x]
&& IGtQ[p, 0]
Rule 2374
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :> -Sim
p[(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*Log
[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 2375
Int[(Log[(d_.)*((e_) + (f_.)*(x_)^(m_.))^(r_.)]*((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.))/(x_), x_Symbol] :
> Simp[(Log[d*(e + f*x^m)^r]*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1)), x] - Dist[(f*m*r)/(b*n*(p + 1)), Int[(
x^(m - 1)*(a + b*Log[c*x^n])^(p + 1))/(e + f*x^m), x], x] /; FreeQ[{a, b, c, d, e, f, r, m, n}, x] && IGtQ[p,
0] && NeQ[d*e, 1]
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 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 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 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 2396
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*
(f + g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n])^p)/g, x] - Dist[(b*e*n*p)/g, Int[(Log[(e*(f + g*x))/(e*f -
d*g)]*(a + b*Log[c*(d + e*x)^n])^(p - 1))/(d + e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*
f - d*g, 0] && IGtQ[p, 1]
Rule 2411
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + (g_.)*(x_))^(q_.)*((h_.) + (i_.)*(x_))
^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((g*x)/e)^q*((e*h - d*i)/e + (i*x)/e)^r*(a + b*Log[c*x^n])^p, x], x,
d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, n, p, q, r}, x] && EqQ[e*f - d*g, 0] && (IGtQ[p, 0] || IGtQ[
r, 0]) && IntegerQ[2*r]
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 2433
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*
(g_.))*((k_.) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/e, Subst[Int[((k*x)/d)^r*(a + b*Log[c*x^n])^p*(f + g*Lo
g[h*((e*i - d*j)/e + (j*x)/e)^m]), x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, l, n, p, r},
x] && EqQ[e*k - d*l, 0]
Rule 2434
Int[(((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.)
))/(x_), x_Symbol] :> Simp[Log[x]*(a + b*Log[c*(d + e*x)^n])*(f + g*Log[h*(i + j*x)^m]), x] + (-Dist[e*g*m, In
t[(Log[x]*(a + b*Log[c*(d + e*x)^n]))/(d + e*x), x], x] - Dist[b*j*n, Int[(Log[x]*(f + g*Log[h*(i + j*x)^m]))/
(i + j*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, m, n}, x] && EqQ[e*i - d*j, 0]
Rule 2440
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + Log[(h_.)*((i_.) + (j_.)*(x_))^(m_.)]*(g_.))
*((k_) + (l_.)*(x_))^(r_.), x_Symbol] :> Dist[1/l, Subst[Int[x^r*(a + b*Log[c*(-((e*k - d*l)/l) + (e*x)/l)^n])
*(f + g*Log[h*(-((j*k - i*l)/l) + (j*x)/l)^m]), x], x, k + l*x], x] /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k,
l, m, n}, x] && IntegerQ[r]
Rule 2488
Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)/((g_.) + (h_.)*(x_)),
x_Symbol] :> -Simp[(Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/h, x] + Dist[(p
*r*s*(b*c - a*d))/h, Int[(Log[-((b*c - a*d)/(d*(a + b*x)))]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a
+ b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q,
0] && EqQ[b*g - a*h, 0] && IGtQ[s, 0]
Rule 2499
Int[(Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((s_.) + Log[(i_.)*((g_.)
+ (h_.)*(x_))^(n_.)]*(t_.))^(m_.))/((j_.) + (k_.)*(x_)), x_Symbol] :> Simp[((s + t*Log[i*(g + h*x)^n])^(m + 1)
*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(k*n*t*(m + 1)), x] + (-Dist[(b*p*r)/(k*n*t*(m + 1)), Int[(s + t*Log[i*
(g + h*x)^n])^(m + 1)/(a + b*x), x], x] - Dist[(d*q*r)/(k*n*t*(m + 1)), Int[(s + t*Log[i*(g + h*x)^n])^(m + 1)
/(c + d*x), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, m, n, p, q, r}, x] && NeQ[b*c - a*d, 0] &
& EqQ[h*j - g*k, 0] && IGtQ[m, 0]
Rule 2500
Int[(Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((s_.) + Log[(i_.)*((g_.)
+ (h_.)*(x_))^(n_.)]*(t_.)))/((j_.) + (k_.)*(x_)), x_Symbol] :> Dist[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r] - Lo
g[(a + b*x)^(p*r)] - Log[(c + d*x)^(q*r)], Int[(s + t*Log[i*(g + h*x)^n])/(j + k*x), x], x] + (Int[(Log[(a + b
*x)^(p*r)]*(s + t*Log[i*(g + h*x)^n]))/(j + k*x), x] + Int[(Log[(c + d*x)^(q*r)]*(s + t*Log[i*(g + h*x)^n]))/(
j + k*x), x]) /; FreeQ[{a, b, c, d, e, f, g, h, i, j, k, s, t, n, p, q, r}, x] && NeQ[b*c - a*d, 0]
Rule 2506
Int[Log[v_]*Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbo
l] :> With[{g = Simplify[((v - 1)*(c + d*x))/(a + b*x)], h = Simplify[u*(a + b*x)*(c + d*x)]}, -Simp[(h*PolyLo
g[2, 1 - v]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/(b*c - a*d), x] + Dist[h*p*r*s, Int[(PolyLog[2, 1 - v]*Log
[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s - 1))/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{g, h}, x]] /; FreeQ[{a, b,
c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && IGtQ[s, 0] && EqQ[p + q, 0]
Rule 2507
Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*Log[(i_.)*((j_.)*((g_
.) + (h_.)*(x_))^(t_.))^(u_.)]*(v_), x_Symbol] :> With[{k = Simplify[v*(a + b*x)*(c + d*x)]}, Simp[(k*Log[i*(j
*(g + h*x)^t)^u]*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1))/(p*r*(s + 1)*(b*c - a*d)), x] - Dist[(k*h*t*u)/
(p*r*(s + 1)*(b*c - a*d)), Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1)/(g + h*x), x], x] /; FreeQ[k, x]]
/; FreeQ[{a, b, c, d, e, f, g, h, i, j, p, q, r, s, t, u}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q, 0] && NeQ[s,
-1]
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 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 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 6589
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]
Rule 6610
Int[(u_)*PolyLog[n_, v_], x_Symbol] :> With[{w = DerivativeDivides[v, u*v, x]}, Simp[w*PolyLog[n + 1, v], x] /
; !FalseQ[w]] /; FreeQ[n, x]
Rule 6688
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
Rule 6742
Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]
Rubi steps
\begin {align*} \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(107 c+107 d x)^3 (a g+b g x)^4} \, dx &=-\frac {2 b^2 B^2}{33076161 (b c-a d)^3 g^4 (a+b x)^3}+\frac {37 b^2 B^2 d}{44101548 (b c-a d)^4 g^4 (a+b x)^2}-\frac {319 b^2 B^2 d^2}{22050774 (b c-a d)^5 g^4 (a+b x)}-\frac {B^2 d^3}{4900172 (b c-a d)^4 g^4 (c+d x)^2}-\frac {19 b B^2 d^3}{2450086 (b c-a d)^5 g^4 (c+d x)}-\frac {245 b^2 B^2 d^3 \log (a+b x)}{11025387 (b c-a d)^6 g^4}+\frac {10 A b^2 B d^3 \log ^2(a+b x)}{1225043 (b c-a d)^6 g^4}+\frac {10 b^2 B^2 d^3 \log ^2(a+b x)}{3675129 (b c-a d)^6 g^4}-\frac {10 b^2 B^2 d^3 \log (a+b x) \log ^2\left (\frac {1}{c+d x}\right )}{1225043 (b c-a d)^6 g^4}+\frac {10 b^2 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log ^2\left (\frac {1}{c+d x}\right )}{1225043 (b c-a d)^6 g^4}+\frac {10 b^2 B^2 d^3 \log \left (-\frac {b c-a d}{d (a+b x)}\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{1225043 (b c-a d)^6 g^4}+\frac {10 b^2 B^2 d^3 \log (a+b x) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )}{1225043 (b c-a d)^6 g^4}-\frac {2 b^2 B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{11025387 (b c-a d)^3 g^4 (a+b x)^3}+\frac {11 b^2 B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{7350258 (b c-a d)^4 g^4 (a+b x)^2}-\frac {47 b^2 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3675129 (b c-a d)^5 g^4 (a+b x)}+\frac {B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2450086 (b c-a d)^4 g^4 (c+d x)^2}+\frac {9 b B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1225043 (b c-a d)^5 g^4 (c+d x)}-\frac {20 b^2 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3675129 (b c-a d)^6 g^4}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3675129 (b c-a d)^3 g^4 (a+b x)^3}+\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2450086 (b c-a d)^4 g^4 (a+b x)^2}-\frac {6 b^2 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1225043 (b c-a d)^5 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2450086 (b c-a d)^4 g^4 (c+d x)^2}-\frac {4 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1225043 (b c-a d)^5 g^4 (c+d x)}-\frac {10 b^2 d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{1225043 (b c-a d)^6 g^4}+\frac {245 b^2 B^2 d^3 \log (c+d x)}{11025387 (b c-a d)^6 g^4}-\frac {10 b^2 B^2 d^3 \log ^2(a+b x) \log (c+d x)}{1225043 (b c-a d)^6 g^4}-\frac {20 A b^2 B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1225043 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3675129 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 \log (a+b x) \log \left (\frac {1}{c+d x}\right ) \log (c+d x)}{1225043 (b c-a d)^6 g^4}+\frac {20 b^2 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1225043 (b c-a d)^6 g^4}+\frac {20 b^2 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3675129 (b c-a d)^6 g^4}+\frac {10 b^2 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 \log (c+d x)}{1225043 (b c-a d)^6 g^4}+\frac {10 A b^2 B d^3 \log ^2(c+d x)}{1225043 (b c-a d)^6 g^4}+\frac {10 b^2 B^2 d^3 \log ^2(c+d x)}{3675129 (b c-a d)^6 g^4}-\frac {10 b^2 B^2 d^3 \log (a+b x) \log ^2(c+d x)}{1225043 (b c-a d)^6 g^4}+\frac {10 b^2 B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log ^2(c+d x)}{1225043 (b c-a d)^6 g^4}+\frac {10 b^2 B^2 d^3 \log ^3(c+d x)}{3675129 (b c-a d)^6 g^4}-\frac {20 A b^2 B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1225043 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3675129 (b c-a d)^6 g^4}+\frac {10 b^2 B^2 d^3 \log ^2(a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{1225043 (b c-a d)^6 g^4}-\frac {20 A b^2 B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{1225043 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3675129 (b c-a d)^6 g^4}+\frac {20 b^2 B^2 d^3 \log (a+b x) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{1225043 (b c-a d)^6 g^4}-\frac {20 A b^2 B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{1225043 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3675129 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 \log \left (\frac {1}{c+d x}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{1225043 (b c-a d)^6 g^4}+\frac {20 b^2 B^2 d^3 \left (\log (a+b x)+\log \left (\frac {1}{c+d x}\right )-\log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{1225043 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{1225043 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{1225043 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{1225043 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{1225043 (b c-a d)^6 g^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 3.05, size = 793, normalized size = 0.93 \[ -\frac {18 B \left (3 a^5 B d^5-15 a^4 b B d^4 (2 c+d x)+30 a^3 b^2 d^3 \left (2 A (c+d x)^2-B d x (4 c+3 d x)\right )+30 a^2 b^3 d^2 \left (6 A d x (c+d x)^2+B \left (2 c^3+6 c^2 d x-3 d^3 x^3\right )\right )+15 a b^4 d \left (12 A d^2 x^2 (c+d x)^2+B c \left (-c^3+4 c^2 d x+18 c d^2 x^2+12 d^3 x^3\right )\right )+b^5 \left (60 A d^3 x^3 (c+d x)^2+B \left (2 c^5-5 c^4 d x+20 c^3 d^2 x^2+110 c^2 d^3 x^3+100 c d^4 x^4+20 d^5 x^5\right )\right )\right ) \log ^2\left (\frac {e (a+b x)}{c+d x}\right )+60 b^2 d^3 \left (18 A^2+12 A B+49 B^2\right ) (a+b x)^3 (c+d x)^2 \log (a+b x)-60 b^2 d^3 \left (18 A^2+12 A B+49 B^2\right ) (a+b x)^3 (c+d x)^2 \log (c+d x)+6 b^2 d^2 \left (108 A^2+282 A B+319 B^2\right ) (a+b x)^2 (c+d x)^2 (b c-a d)+4 b^2 \left (9 A^2+6 A B+2 B^2\right ) (c+d x)^2 (b c-a d)^3-3 b^2 d \left (54 A^2+66 A B+37 B^2\right ) (a+b x) (c+d x)^2 (b c-a d)^2+27 d^3 \left (2 A^2-2 A B+B^2\right ) (a+b x)^3 (b c-a d)^2+54 b d^3 \left (8 A^2-18 A B+19 B^2\right ) (a+b x)^3 (c+d x) (b c-a d)+6 B (b c-a d) \left (6 b^2 d^2 (36 A+47 B) (a+b x)^2 (c+d x)^2-3 b^2 d (18 A+11 B) (a+b x) (c+d x)^2 (b c-a d)+4 b^2 (3 A+B) (c+d x)^2 (b c-a d)^2+9 d^3 (2 A-B) (a+b x)^3 (b c-a d)+18 b d^3 (8 A-9 B) (a+b x)^3 (c+d x)\right ) \log \left (\frac {e (a+b x)}{c+d x}\right )+360 b^2 B^2 d^3 (a+b x)^3 (c+d x)^2 \log ^3\left (\frac {e (a+b x)}{c+d x}\right )}{108 g^4 i^3 (a+b x)^3 (c+d x)^2 (b c-a d)^6} \]
Antiderivative was successfully verified.
[In]
Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/((a*g + b*g*x)^4*(c*i + d*i*x)^3),x]
[Out]
-1/108*(27*(2*A^2 - 2*A*B + B^2)*d^3*(b*c - a*d)^2*(a + b*x)^3 + 54*b*(8*A^2 - 18*A*B + 19*B^2)*d^3*(b*c - a*d
)*(a + b*x)^3*(c + d*x) + 4*b^2*(9*A^2 + 6*A*B + 2*B^2)*(b*c - a*d)^3*(c + d*x)^2 - 3*b^2*(54*A^2 + 66*A*B + 3
7*B^2)*d*(b*c - a*d)^2*(a + b*x)*(c + d*x)^2 + 6*b^2*(108*A^2 + 282*A*B + 319*B^2)*d^2*(b*c - a*d)*(a + b*x)^2
*(c + d*x)^2 + 60*b^2*(18*A^2 + 12*A*B + 49*B^2)*d^3*(a + b*x)^3*(c + d*x)^2*Log[a + b*x] + 6*B*(b*c - a*d)*(9
*(2*A - B)*d^3*(b*c - a*d)*(a + b*x)^3 + 18*b*(8*A - 9*B)*d^3*(a + b*x)^3*(c + d*x) + 4*b^2*(3*A + B)*(b*c - a
*d)^2*(c + d*x)^2 - 3*b^2*(18*A + 11*B)*d*(b*c - a*d)*(a + b*x)*(c + d*x)^2 + 6*b^2*(36*A + 47*B)*d^2*(a + b*x
)^2*(c + d*x)^2)*Log[(e*(a + b*x))/(c + d*x)] + 18*B*(3*a^5*B*d^5 - 15*a^4*b*B*d^4*(2*c + d*x) + 30*a^3*b^2*d^
3*(2*A*(c + d*x)^2 - B*d*x*(4*c + 3*d*x)) + 30*a^2*b^3*d^2*(6*A*d*x*(c + d*x)^2 + B*(2*c^3 + 6*c^2*d*x - 3*d^3
*x^3)) + 15*a*b^4*d*(12*A*d^2*x^2*(c + d*x)^2 + B*c*(-c^3 + 4*c^2*d*x + 18*c*d^2*x^2 + 12*d^3*x^3)) + b^5*(60*
A*d^3*x^3*(c + d*x)^2 + B*(2*c^5 - 5*c^4*d*x + 20*c^3*d^2*x^2 + 110*c^2*d^3*x^3 + 100*c*d^4*x^4 + 20*d^5*x^5))
)*Log[(e*(a + b*x))/(c + d*x)]^2 + 360*b^2*B^2*d^3*(a + b*x)^3*(c + d*x)^2*Log[(e*(a + b*x))/(c + d*x)]^3 - 60
*b^2*(18*A^2 + 12*A*B + 49*B^2)*d^3*(a + b*x)^3*(c + d*x)^2*Log[c + d*x])/((b*c - a*d)^6*g^4*i^3*(a + b*x)^3*(
c + d*x)^2)
________________________________________________________________________________________
fricas [B] time = 1.07, size = 2257, normalized size = 2.65 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x, algorithm="fricas")
[Out]
-1/108*(4*(9*A^2 + 6*A*B + 2*B^2)*b^5*c^5 - 135*(2*A^2 + 2*A*B + B^2)*a*b^4*c^4*d + 1080*(A^2 + 2*A*B + 2*B^2)
*a^2*b^3*c^3*d^2 - 20*(18*A^2 + 147*A*B + 49*B^2)*a^3*b^2*c^2*d^3 - 540*(A^2 - 2*A*B + 2*B^2)*a^4*b*c*d^4 + 27
*(2*A^2 - 2*A*B + B^2)*a^5*d^5 + 60*((18*A^2 + 12*A*B + 49*B^2)*b^5*c*d^4 - (18*A^2 + 12*A*B + 49*B^2)*a*b^4*d
^5)*x^4 + 30*(3*(18*A^2 + 24*A*B + 53*B^2)*b^5*c^2*d^3 + 2*(18*A^2 - 24*A*B + 37*B^2)*a*b^4*c*d^4 - (90*A^2 +
24*A*B + 233*B^2)*a^2*b^3*d^5)*x^3 + 360*(B^2*b^5*d^5*x^5 + B^2*a^3*b^2*c^2*d^3 + (2*B^2*b^5*c*d^4 + 3*B^2*a*b
^4*d^5)*x^4 + (B^2*b^5*c^2*d^3 + 6*B^2*a*b^4*c*d^4 + 3*B^2*a^2*b^3*d^5)*x^3 + (3*B^2*a*b^4*c^2*d^3 + 6*B^2*a^2
*b^3*c*d^4 + B^2*a^3*b^2*d^5)*x^2 + (3*B^2*a^2*b^3*c^2*d^3 + 2*B^2*a^3*b^2*c*d^4)*x)*log((b*e*x + a*e)/(d*x +
c))^3 + 10*(2*(18*A^2 + 66*A*B + 85*B^2)*b^5*c^3*d^2 + 3*(126*A^2 + 84*A*B + 307*B^2)*a*b^4*c^2*d^3 - 12*(18*A
^2 + 39*A*B + 49*B^2)*a^2*b^3*c*d^4 - (198*A^2 - 84*A*B + 503*B^2)*a^3*b^2*d^5)*x^2 + 18*(20*(3*A*B + B^2)*b^5
*d^5*x^5 + 2*B^2*b^5*c^5 - 15*B^2*a*b^4*c^4*d + 60*B^2*a^2*b^3*c^3*d^2 + 60*A*B*a^3*b^2*c^2*d^3 - 30*B^2*a^4*b
*c*d^4 + 3*B^2*a^5*d^5 + 20*(9*A*B*a*b^4*d^5 + (6*A*B + 5*B^2)*b^5*c*d^4)*x^4 + 10*((6*A*B + 11*B^2)*b^5*c^2*d
^3 + 18*(2*A*B + B^2)*a*b^4*c*d^4 + 9*(2*A*B - B^2)*a^2*b^3*d^5)*x^3 + 10*(2*B^2*b^5*c^3*d^2 + 36*A*B*a^2*b^3*
c*d^4 + 9*(2*A*B + 3*B^2)*a*b^4*c^2*d^3 + 3*(2*A*B - 3*B^2)*a^3*b^2*d^5)*x^2 - 5*(B^2*b^5*c^4*d - 12*B^2*a*b^4
*c^3*d^2 + 3*B^2*a^4*b*d^5 - 36*(A*B + B^2)*a^2*b^3*c^2*d^3 - 24*(A*B - B^2)*a^3*b^2*c*d^4)*x)*log((b*e*x + a*
e)/(d*x + c))^2 - 5*((18*A^2 + 30*A*B + 19*B^2)*b^5*c^4*d - 108*(2*A^2 + 6*A*B + 7*B^2)*a*b^4*c^3*d^2 - 12*(36
*A^2 - 39*A*B + 59*B^2)*a^2*b^3*c^2*d^3 + 8*(72*A^2 + 39*A*B + 157*B^2)*a^3*b^2*c*d^4 + 27*(2*A^2 - 6*A*B + 7*
B^2)*a^4*b*d^5)*x + 6*(10*(18*A^2 + 12*A*B + 49*B^2)*b^5*d^5*x^5 + 180*A^2*a^3*b^2*c^2*d^3 + 4*(3*A*B + B^2)*b
^5*c^5 - 45*(2*A*B + B^2)*a*b^4*c^4*d + 360*(A*B + B^2)*a^2*b^3*c^3*d^2 - 180*(A*B - B^2)*a^4*b*c*d^4 + 9*(2*A
*B - B^2)*a^5*d^5 + 10*(2*(18*A^2 + 30*A*B + 55*B^2)*b^5*c*d^4 + 27*(2*A^2 + 5*B^2)*a*b^4*d^5)*x^4 + 10*((18*A
^2 + 66*A*B + 85*B^2)*b^5*c^2*d^3 + 54*(2*A^2 + 2*A*B + 5*B^2)*a*b^4*c*d^4 + 27*(2*A^2 - 2*A*B + 5*B^2)*a^2*b^
3*d^5)*x^3 + 10*(2*(6*A*B + 11*B^2)*b^5*c^3*d^2 + 27*(2*A^2 + 6*A*B + 7*B^2)*a*b^4*c^2*d^3 + 108*(A^2 + 2*B^2)
*a^2*b^3*c*d^4 + 9*(2*A^2 - 6*A*B + 7*B^2)*a^3*b^2*d^5)*x^2 - 5*((6*A*B + 5*B^2)*b^5*c^4*d - 36*(2*A*B + 3*B^2
)*a*b^4*c^3*d^2 - 108*(A^2 + 2*A*B + 2*B^2)*a^2*b^3*c^2*d^3 - 72*(A^2 - 2*A*B + 2*B^2)*a^3*b^2*c*d^4 + 9*(2*A*
B - 3*B^2)*a^4*b*d^5)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^9*c^6*d^2 - 6*a*b^8*c^5*d^3 + 15*a^2*b^7*c^4*d^4 -
20*a^3*b^6*c^3*d^5 + 15*a^4*b^5*c^2*d^6 - 6*a^5*b^4*c*d^7 + a^6*b^3*d^8)*g^4*i^3*x^5 + (2*b^9*c^7*d - 9*a*b^8*
c^6*d^2 + 12*a^2*b^7*c^5*d^3 + 5*a^3*b^6*c^4*d^4 - 30*a^4*b^5*c^3*d^5 + 33*a^5*b^4*c^2*d^6 - 16*a^6*b^3*c*d^7
+ 3*a^7*b^2*d^8)*g^4*i^3*x^4 + (b^9*c^8 - 18*a^2*b^7*c^6*d^2 + 52*a^3*b^6*c^5*d^3 - 60*a^4*b^5*c^4*d^4 + 24*a^
5*b^4*c^3*d^5 + 10*a^6*b^3*c^2*d^6 - 12*a^7*b^2*c*d^7 + 3*a^8*b*d^8)*g^4*i^3*x^3 + (3*a*b^8*c^8 - 12*a^2*b^7*c
^7*d + 10*a^3*b^6*c^6*d^2 + 24*a^4*b^5*c^5*d^3 - 60*a^5*b^4*c^4*d^4 + 52*a^6*b^3*c^3*d^5 - 18*a^7*b^2*c^2*d^6
+ a^9*d^8)*g^4*i^3*x^2 + (3*a^2*b^7*c^8 - 16*a^3*b^6*c^7*d + 33*a^4*b^5*c^6*d^2 - 30*a^5*b^4*c^5*d^3 + 5*a^6*b
^3*c^4*d^4 + 12*a^7*b^2*c^3*d^5 - 9*a^8*b*c^2*d^6 + 2*a^9*c*d^7)*g^4*i^3*x + (a^3*b^6*c^8 - 6*a^4*b^5*c^7*d +
15*a^5*b^4*c^6*d^2 - 20*a^6*b^3*c^5*d^3 + 15*a^7*b^2*c^4*d^4 - 6*a^8*b*c^3*d^5 + a^9*c^2*d^6)*g^4*i^3)
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x, algorithm="giac")
[Out]
Timed out
________________________________________________________________________________________
maple [B] time = 0.06, size = 5731, normalized size = 6.73 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((B*ln((b*x+a)/(d*x+c)*e)+A)^2/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x)
[Out]
result too large to display
________________________________________________________________________________________
maxima [B] time = 12.47, size = 9282, normalized size = 10.91 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x, algorithm="maxima")
[Out]
-1/6*B^2*((60*b^4*d^4*x^4 + 2*b^4*c^4 - 13*a*b^3*c^3*d + 47*a^2*b^2*c^2*d^2 + 27*a^3*b*c*d^3 - 3*a^4*d^4 + 30*
(3*b^4*c*d^3 + 5*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 + 23*a*b^3*c*d^3 + 11*a^2*b^2*d^4)*x^2 - 5*(b^4*c^3*d - 11
*a*b^3*c^2*d^2 - 35*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x)/((b^8*c^5*d^2 - 5*a*b^7*c^4*d^3 + 10*a^2*b^6*c^3*d^4 - 10*
a^3*b^5*c^2*d^5 + 5*a^4*b^4*c*d^6 - a^5*b^3*d^7)*g^4*i^3*x^5 + (2*b^8*c^6*d - 7*a*b^7*c^5*d^2 + 5*a^2*b^6*c^4*
d^3 + 10*a^3*b^5*c^3*d^4 - 20*a^4*b^4*c^2*d^5 + 13*a^5*b^3*c*d^6 - 3*a^6*b^2*d^7)*g^4*i^3*x^4 + (b^8*c^7 + a*b
^7*c^6*d - 17*a^2*b^6*c^5*d^2 + 35*a^3*b^5*c^4*d^3 - 25*a^4*b^4*c^3*d^4 - a^5*b^3*c^2*d^5 + 9*a^6*b^2*c*d^6 -
3*a^7*b*d^7)*g^4*i^3*x^3 + (3*a*b^7*c^7 - 9*a^2*b^6*c^6*d + a^3*b^5*c^5*d^2 + 25*a^4*b^4*c^4*d^3 - 35*a^5*b^3*
c^3*d^4 + 17*a^6*b^2*c^2*d^5 - a^7*b*c*d^6 - a^8*d^7)*g^4*i^3*x^2 + (3*a^2*b^6*c^7 - 13*a^3*b^5*c^6*d + 20*a^4
*b^4*c^5*d^2 - 10*a^5*b^3*c^4*d^3 - 5*a^6*b^2*c^3*d^4 + 7*a^7*b*c^2*d^5 - 2*a^8*c*d^6)*g^4*i^3*x + (a^3*b^5*c^
7 - 5*a^4*b^4*c^6*d + 10*a^5*b^3*c^5*d^2 - 10*a^6*b^2*c^4*d^3 + 5*a^7*b*c^3*d^4 - a^8*c^2*d^5)*g^4*i^3) + 60*b
^2*d^3*log(b*x + a)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 -
6*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3) - 60*b^2*d^3*log(d*x + c)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 -
20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3))*log(b*e*x/(d*x + c) + a*e/(d*x +
c))^2 - 1/3*A*B*((60*b^4*d^4*x^4 + 2*b^4*c^4 - 13*a*b^3*c^3*d + 47*a^2*b^2*c^2*d^2 + 27*a^3*b*c*d^3 - 3*a^4*d^
4 + 30*(3*b^4*c*d^3 + 5*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 + 23*a*b^3*c*d^3 + 11*a^2*b^2*d^4)*x^2 - 5*(b^4*c^3
*d - 11*a*b^3*c^2*d^2 - 35*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x)/((b^8*c^5*d^2 - 5*a*b^7*c^4*d^3 + 10*a^2*b^6*c^3*d^
4 - 10*a^3*b^5*c^2*d^5 + 5*a^4*b^4*c*d^6 - a^5*b^3*d^7)*g^4*i^3*x^5 + (2*b^8*c^6*d - 7*a*b^7*c^5*d^2 + 5*a^2*b
^6*c^4*d^3 + 10*a^3*b^5*c^3*d^4 - 20*a^4*b^4*c^2*d^5 + 13*a^5*b^3*c*d^6 - 3*a^6*b^2*d^7)*g^4*i^3*x^4 + (b^8*c^
7 + a*b^7*c^6*d - 17*a^2*b^6*c^5*d^2 + 35*a^3*b^5*c^4*d^3 - 25*a^4*b^4*c^3*d^4 - a^5*b^3*c^2*d^5 + 9*a^6*b^2*c
*d^6 - 3*a^7*b*d^7)*g^4*i^3*x^3 + (3*a*b^7*c^7 - 9*a^2*b^6*c^6*d + a^3*b^5*c^5*d^2 + 25*a^4*b^4*c^4*d^3 - 35*a
^5*b^3*c^3*d^4 + 17*a^6*b^2*c^2*d^5 - a^7*b*c*d^6 - a^8*d^7)*g^4*i^3*x^2 + (3*a^2*b^6*c^7 - 13*a^3*b^5*c^6*d +
20*a^4*b^4*c^5*d^2 - 10*a^5*b^3*c^4*d^3 - 5*a^6*b^2*c^3*d^4 + 7*a^7*b*c^2*d^5 - 2*a^8*c*d^6)*g^4*i^3*x + (a^3
*b^5*c^7 - 5*a^4*b^4*c^6*d + 10*a^5*b^3*c^5*d^2 - 10*a^6*b^2*c^4*d^3 + 5*a^7*b*c^3*d^4 - a^8*c^2*d^5)*g^4*i^3)
+ 60*b^2*d^3*log(b*x + a)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^
2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3) - 60*b^2*d^3*log(d*x + c)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4
*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3))*log(b*e*x/(d*x + c) + a*e/
(d*x + c)) - 1/108*B^2*(6*(4*b^5*c^5 - 45*a*b^4*c^4*d + 360*a^2*b^3*c^3*d^2 - 490*a^3*b^2*c^2*d^3 + 180*a^4*b*
c*d^4 - 9*a^5*d^5 + 120*(b^5*c*d^4 - a*b^4*d^5)*x^4 + 120*(3*b^5*c^2*d^3 - 2*a*b^4*c*d^4 - a^2*b^3*d^5)*x^3 +
20*(11*b^5*c^3*d^2 + 21*a*b^4*c^2*d^3 - 39*a^2*b^3*c*d^4 + 7*a^3*b^2*d^5)*x^2 - 180*(b^5*d^5*x^5 + a^3*b^2*c^2
*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3
+ 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(b*x + a)^2 - 180*(b^5*d^5*
x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 +
(3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(d*x + c)^
2 - 5*(5*b^5*c^4*d - 108*a*b^4*c^3*d^2 + 78*a^2*b^3*c^2*d^3 + 52*a^3*b^2*c*d^4 - 27*a^4*b*d^5)*x + 120*(b^5*d^
5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3
+ (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(b*x + a
) - 120*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^
2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)
*x - 3*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2
*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*
x)*log(b*x + a))*log(d*x + c))*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(a^3*b^6*c^8*g^4*i^3 - 6*a^4*b^5*c^7*d*g^4
*i^3 + 15*a^5*b^4*c^6*d^2*g^4*i^3 - 20*a^6*b^3*c^5*d^3*g^4*i^3 + 15*a^7*b^2*c^4*d^4*g^4*i^3 - 6*a^8*b*c^3*d^5*
g^4*i^3 + a^9*c^2*d^6*g^4*i^3 + (b^9*c^6*d^2*g^4*i^3 - 6*a*b^8*c^5*d^3*g^4*i^3 + 15*a^2*b^7*c^4*d^4*g^4*i^3 -
20*a^3*b^6*c^3*d^5*g^4*i^3 + 15*a^4*b^5*c^2*d^6*g^4*i^3 - 6*a^5*b^4*c*d^7*g^4*i^3 + a^6*b^3*d^8*g^4*i^3)*x^5 +
(2*b^9*c^7*d*g^4*i^3 - 9*a*b^8*c^6*d^2*g^4*i^3 + 12*a^2*b^7*c^5*d^3*g^4*i^3 + 5*a^3*b^6*c^4*d^4*g^4*i^3 - 30*
a^4*b^5*c^3*d^5*g^4*i^3 + 33*a^5*b^4*c^2*d^6*g^4*i^3 - 16*a^6*b^3*c*d^7*g^4*i^3 + 3*a^7*b^2*d^8*g^4*i^3)*x^4 +
(b^9*c^8*g^4*i^3 - 18*a^2*b^7*c^6*d^2*g^4*i^3 + 52*a^3*b^6*c^5*d^3*g^4*i^3 - 60*a^4*b^5*c^4*d^4*g^4*i^3 + 24*
a^5*b^4*c^3*d^5*g^4*i^3 + 10*a^6*b^3*c^2*d^6*g^4*i^3 - 12*a^7*b^2*c*d^7*g^4*i^3 + 3*a^8*b*d^8*g^4*i^3)*x^3 + (
3*a*b^8*c^8*g^4*i^3 - 12*a^2*b^7*c^7*d*g^4*i^3 + 10*a^3*b^6*c^6*d^2*g^4*i^3 + 24*a^4*b^5*c^5*d^3*g^4*i^3 - 60*
a^5*b^4*c^4*d^4*g^4*i^3 + 52*a^6*b^3*c^3*d^5*g^4*i^3 - 18*a^7*b^2*c^2*d^6*g^4*i^3 + a^9*d^8*g^4*i^3)*x^2 + (3*
a^2*b^7*c^8*g^4*i^3 - 16*a^3*b^6*c^7*d*g^4*i^3 + 33*a^4*b^5*c^6*d^2*g^4*i^3 - 30*a^5*b^4*c^5*d^3*g^4*i^3 + 5*a
^6*b^3*c^4*d^4*g^4*i^3 + 12*a^7*b^2*c^3*d^5*g^4*i^3 - 9*a^8*b*c^2*d^6*g^4*i^3 + 2*a^9*c*d^7*g^4*i^3)*x) + (8*b
^5*c^5 - 135*a*b^4*c^4*d + 2160*a^2*b^3*c^3*d^2 - 980*a^3*b^2*c^2*d^3 - 1080*a^4*b*c*d^4 + 27*a^5*d^5 + 2940*(
b^5*c*d^4 - a*b^4*d^5)*x^4 + 30*(159*b^5*c^2*d^3 + 74*a*b^4*c*d^4 - 233*a^2*b^3*d^5)*x^3 + 360*(b^5*d^5*x^5 +
a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b
^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(b*x + a)^3 - 36
0*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*
d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*lo
g(d*x + c)^3 + 10*(170*b^5*c^3*d^2 + 921*a*b^4*c^2*d^3 - 588*a^2*b^3*c*d^4 - 503*a^3*b^2*d^5)*x^2 - 360*(b^5*d
^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3
+ (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(b*x +
a)^2 - 360*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3
*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d
^4)*x - 3*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*
a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^
4)*x)*log(b*x + a))*log(d*x + c)^2 - 5*(19*b^5*c^4*d - 756*a*b^4*c^3*d^2 - 708*a^2*b^3*c^2*d^3 + 1256*a^3*b^2*
c*d^4 + 189*a^4*b*d^5)*x + 2940*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^
3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^
2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(b*x + a) - 60*(49*b^5*d^5*x^5 + 49*a^3*b^2*c^2*d^3 + 49*(2*b^5*c*d^4 + 3*a*b^4
*d^5)*x^4 + 49*(b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + 49*(3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3
*b^2*d^5)*x^2 + 18*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c
*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3
*b^2*c*d^4)*x)*log(b*x + a)^2 + 49*(3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x - 12*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3
+ (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*
a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(b*x + a))*log(d*x + c))/(a^3*b
^6*c^8*g^4*i^3 - 6*a^4*b^5*c^7*d*g^4*i^3 + 15*a^5*b^4*c^6*d^2*g^4*i^3 - 20*a^6*b^3*c^5*d^3*g^4*i^3 + 15*a^7*b^
2*c^4*d^4*g^4*i^3 - 6*a^8*b*c^3*d^5*g^4*i^3 + a^9*c^2*d^6*g^4*i^3 + (b^9*c^6*d^2*g^4*i^3 - 6*a*b^8*c^5*d^3*g^4
*i^3 + 15*a^2*b^7*c^4*d^4*g^4*i^3 - 20*a^3*b^6*c^3*d^5*g^4*i^3 + 15*a^4*b^5*c^2*d^6*g^4*i^3 - 6*a^5*b^4*c*d^7*
g^4*i^3 + a^6*b^3*d^8*g^4*i^3)*x^5 + (2*b^9*c^7*d*g^4*i^3 - 9*a*b^8*c^6*d^2*g^4*i^3 + 12*a^2*b^7*c^5*d^3*g^4*i
^3 + 5*a^3*b^6*c^4*d^4*g^4*i^3 - 30*a^4*b^5*c^3*d^5*g^4*i^3 + 33*a^5*b^4*c^2*d^6*g^4*i^3 - 16*a^6*b^3*c*d^7*g^
4*i^3 + 3*a^7*b^2*d^8*g^4*i^3)*x^4 + (b^9*c^8*g^4*i^3 - 18*a^2*b^7*c^6*d^2*g^4*i^3 + 52*a^3*b^6*c^5*d^3*g^4*i^
3 - 60*a^4*b^5*c^4*d^4*g^4*i^3 + 24*a^5*b^4*c^3*d^5*g^4*i^3 + 10*a^6*b^3*c^2*d^6*g^4*i^3 - 12*a^7*b^2*c*d^7*g^
4*i^3 + 3*a^8*b*d^8*g^4*i^3)*x^3 + (3*a*b^8*c^8*g^4*i^3 - 12*a^2*b^7*c^7*d*g^4*i^3 + 10*a^3*b^6*c^6*d^2*g^4*i^
3 + 24*a^4*b^5*c^5*d^3*g^4*i^3 - 60*a^5*b^4*c^4*d^4*g^4*i^3 + 52*a^6*b^3*c^3*d^5*g^4*i^3 - 18*a^7*b^2*c^2*d^6*
g^4*i^3 + a^9*d^8*g^4*i^3)*x^2 + (3*a^2*b^7*c^8*g^4*i^3 - 16*a^3*b^6*c^7*d*g^4*i^3 + 33*a^4*b^5*c^6*d^2*g^4*i^
3 - 30*a^5*b^4*c^5*d^3*g^4*i^3 + 5*a^6*b^3*c^4*d^4*g^4*i^3 + 12*a^7*b^2*c^3*d^5*g^4*i^3 - 9*a^8*b*c^2*d^6*g^4*
i^3 + 2*a^9*c*d^7*g^4*i^3)*x)) - 1/6*A^2*((60*b^4*d^4*x^4 + 2*b^4*c^4 - 13*a*b^3*c^3*d + 47*a^2*b^2*c^2*d^2 +
27*a^3*b*c*d^3 - 3*a^4*d^4 + 30*(3*b^4*c*d^3 + 5*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 + 23*a*b^3*c*d^3 + 11*a^2*
b^2*d^4)*x^2 - 5*(b^4*c^3*d - 11*a*b^3*c^2*d^2 - 35*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x)/((b^8*c^5*d^2 - 5*a*b^7*c^
4*d^3 + 10*a^2*b^6*c^3*d^4 - 10*a^3*b^5*c^2*d^5 + 5*a^4*b^4*c*d^6 - a^5*b^3*d^7)*g^4*i^3*x^5 + (2*b^8*c^6*d -
7*a*b^7*c^5*d^2 + 5*a^2*b^6*c^4*d^3 + 10*a^3*b^5*c^3*d^4 - 20*a^4*b^4*c^2*d^5 + 13*a^5*b^3*c*d^6 - 3*a^6*b^2*d
^7)*g^4*i^3*x^4 + (b^8*c^7 + a*b^7*c^6*d - 17*a^2*b^6*c^5*d^2 + 35*a^3*b^5*c^4*d^3 - 25*a^4*b^4*c^3*d^4 - a^5*
b^3*c^2*d^5 + 9*a^6*b^2*c*d^6 - 3*a^7*b*d^7)*g^4*i^3*x^3 + (3*a*b^7*c^7 - 9*a^2*b^6*c^6*d + a^3*b^5*c^5*d^2 +
25*a^4*b^4*c^4*d^3 - 35*a^5*b^3*c^3*d^4 + 17*a^6*b^2*c^2*d^5 - a^7*b*c*d^6 - a^8*d^7)*g^4*i^3*x^2 + (3*a^2*b^6
*c^7 - 13*a^3*b^5*c^6*d + 20*a^4*b^4*c^5*d^2 - 10*a^5*b^3*c^4*d^3 - 5*a^6*b^2*c^3*d^4 + 7*a^7*b*c^2*d^5 - 2*a^
8*c*d^6)*g^4*i^3*x + (a^3*b^5*c^7 - 5*a^4*b^4*c^6*d + 10*a^5*b^3*c^5*d^2 - 10*a^6*b^2*c^4*d^3 + 5*a^7*b*c^3*d^
4 - a^8*c^2*d^5)*g^4*i^3) + 60*b^2*d^3*log(b*x + a)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^
3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3) - 60*b^2*d^3*log(d*x + c)/((b^6*c^6 - 6*a*b
^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3)) -
1/18*(4*b^5*c^5 - 45*a*b^4*c^4*d + 360*a^2*b^3*c^3*d^2 - 490*a^3*b^2*c^2*d^3 + 180*a^4*b*c*d^4 - 9*a^5*d^5 +
120*(b^5*c*d^4 - a*b^4*d^5)*x^4 + 120*(3*b^5*c^2*d^3 - 2*a*b^4*c*d^4 - a^2*b^3*d^5)*x^3 + 20*(11*b^5*c^3*d^2 +
21*a*b^4*c^2*d^3 - 39*a^2*b^3*c*d^4 + 7*a^3*b^2*d^5)*x^2 - 180*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4
+ 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 +
a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(b*x + a)^2 - 180*(b^5*d^5*x^5 + a^3*b^2*c^2*d^
3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6
*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(d*x + c)^2 - 5*(5*b^5*c^4*d -
108*a*b^4*c^3*d^2 + 78*a^2*b^3*c^2*d^3 + 52*a^3*b^2*c*d^4 - 27*a^4*b*d^5)*x + 120*(b^5*d^5*x^5 + a^3*b^2*c^2*
d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 +
6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(b*x + a) - 120*(b^5*d^5*x^5
+ a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*
a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x - 3*(b^5*d^5*x^5
+ a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a
*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(b*x + a))*log
(d*x + c))*A*B/(a^3*b^6*c^8*g^4*i^3 - 6*a^4*b^5*c^7*d*g^4*i^3 + 15*a^5*b^4*c^6*d^2*g^4*i^3 - 20*a^6*b^3*c^5*d^
3*g^4*i^3 + 15*a^7*b^2*c^4*d^4*g^4*i^3 - 6*a^8*b*c^3*d^5*g^4*i^3 + a^9*c^2*d^6*g^4*i^3 + (b^9*c^6*d^2*g^4*i^3
- 6*a*b^8*c^5*d^3*g^4*i^3 + 15*a^2*b^7*c^4*d^4*g^4*i^3 - 20*a^3*b^6*c^3*d^5*g^4*i^3 + 15*a^4*b^5*c^2*d^6*g^4*i
^3 - 6*a^5*b^4*c*d^7*g^4*i^3 + a^6*b^3*d^8*g^4*i^3)*x^5 + (2*b^9*c^7*d*g^4*i^3 - 9*a*b^8*c^6*d^2*g^4*i^3 + 12*
a^2*b^7*c^5*d^3*g^4*i^3 + 5*a^3*b^6*c^4*d^4*g^4*i^3 - 30*a^4*b^5*c^3*d^5*g^4*i^3 + 33*a^5*b^4*c^2*d^6*g^4*i^3
- 16*a^6*b^3*c*d^7*g^4*i^3 + 3*a^7*b^2*d^8*g^4*i^3)*x^4 + (b^9*c^8*g^4*i^3 - 18*a^2*b^7*c^6*d^2*g^4*i^3 + 52*a
^3*b^6*c^5*d^3*g^4*i^3 - 60*a^4*b^5*c^4*d^4*g^4*i^3 + 24*a^5*b^4*c^3*d^5*g^4*i^3 + 10*a^6*b^3*c^2*d^6*g^4*i^3
- 12*a^7*b^2*c*d^7*g^4*i^3 + 3*a^8*b*d^8*g^4*i^3)*x^3 + (3*a*b^8*c^8*g^4*i^3 - 12*a^2*b^7*c^7*d*g^4*i^3 + 10*a
^3*b^6*c^6*d^2*g^4*i^3 + 24*a^4*b^5*c^5*d^3*g^4*i^3 - 60*a^5*b^4*c^4*d^4*g^4*i^3 + 52*a^6*b^3*c^3*d^5*g^4*i^3
- 18*a^7*b^2*c^2*d^6*g^4*i^3 + a^9*d^8*g^4*i^3)*x^2 + (3*a^2*b^7*c^8*g^4*i^3 - 16*a^3*b^6*c^7*d*g^4*i^3 + 33*a
^4*b^5*c^6*d^2*g^4*i^3 - 30*a^5*b^4*c^5*d^3*g^4*i^3 + 5*a^6*b^3*c^4*d^4*g^4*i^3 + 12*a^7*b^2*c^3*d^5*g^4*i^3 -
9*a^8*b*c^2*d^6*g^4*i^3 + 2*a^9*c*d^7*g^4*i^3)*x)
________________________________________________________________________________________
mupad [B] time = 18.95, size = 3550, normalized size = 4.17 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((A + B*log((e*(a + b*x))/(c + d*x)))^2/((a*g + b*g*x)^4*(c*i + d*i*x)^3),x)
[Out]
((36*A^2*b^4*c^4 - 54*A^2*a^4*d^4 - 27*B^2*a^4*d^4 + 8*B^2*b^4*c^4 + 54*A*B*a^4*d^4 + 24*A*B*b^4*c^4 + 846*A^2
*a^2*b^2*c^2*d^2 + 2033*B^2*a^2*b^2*c^2*d^2 - 234*A^2*a*b^3*c^3*d + 486*A^2*a^3*b*c*d^3 - 127*B^2*a*b^3*c^3*d
+ 1053*B^2*a^3*b*c*d^3 - 246*A*B*a*b^3*c^3*d - 1026*A*B*a^3*b*c*d^3 + 1914*A*B*a^2*b^2*c^2*d^2)/(6*(a*d - b*c)
) + (10*x^4*(18*A^2*b^4*d^4 + 49*B^2*b^4*d^4 + 12*A*B*b^4*d^4))/(a*d - b*c) + (5*x*(54*A^2*a^3*b*d^4 + 189*B^2
*a^3*b*d^4 - 18*A^2*b^4*c^3*d - 19*B^2*b^4*c^3*d + 198*A^2*a*b^3*c^2*d^2 + 630*A^2*a^2*b^2*c*d^3 + 737*B^2*a*b
^3*c^2*d^2 + 1445*B^2*a^2*b^2*c*d^3 - 162*A*B*a^3*b*d^4 - 30*A*B*b^4*c^3*d + 618*A*B*a*b^3*c^2*d^2 + 150*A*B*a
^2*b^2*c*d^3))/(6*(a*d - b*c)) + (5*x^2*(198*A^2*a^2*b^2*d^4 + 503*B^2*a^2*b^2*d^4 + 36*A^2*b^4*c^2*d^2 + 170*
B^2*b^4*c^2*d^2 - 84*A*B*a^2*b^2*d^4 + 132*A*B*b^4*c^2*d^2 + 414*A^2*a*b^3*c*d^3 + 1091*B^2*a*b^3*c*d^3 + 384*
A*B*a*b^3*c*d^3))/(3*(a*d - b*c)) + (5*x^3*(90*A^2*a*b^3*d^4 + 233*B^2*a*b^3*d^4 + 54*A^2*b^4*c*d^3 + 159*B^2*
b^4*c*d^3 + 24*A*B*a*b^3*d^4 + 72*A*B*b^4*c*d^3))/(a*d - b*c))/(x^5*(18*a^4*b^3*d^6*g^4*i^3 + 18*b^7*c^4*d^2*g
^4*i^3 - 72*a*b^6*c^3*d^3*g^4*i^3 - 72*a^3*b^4*c*d^5*g^4*i^3 + 108*a^2*b^5*c^2*d^4*g^4*i^3) + x*(54*a^2*b^5*c^
6*g^4*i^3 + 36*a^7*c*d^5*g^4*i^3 - 180*a^3*b^4*c^5*d*g^4*i^3 - 90*a^6*b*c^2*d^4*g^4*i^3 + 180*a^4*b^3*c^4*d^2*
g^4*i^3) + x^2*(18*a^7*d^6*g^4*i^3 + 54*a*b^6*c^6*g^4*i^3 + 36*a^6*b*c*d^5*g^4*i^3 - 108*a^2*b^5*c^5*d*g^4*i^3
- 90*a^3*b^4*c^4*d^2*g^4*i^3 + 360*a^4*b^3*c^3*d^3*g^4*i^3 - 270*a^5*b^2*c^2*d^4*g^4*i^3) + x^3*(18*b^7*c^6*g
^4*i^3 + 54*a^6*b*d^6*g^4*i^3 + 36*a*b^6*c^5*d*g^4*i^3 - 108*a^5*b^2*c*d^5*g^4*i^3 - 270*a^2*b^5*c^4*d^2*g^4*i
^3 + 360*a^3*b^4*c^3*d^3*g^4*i^3 - 90*a^4*b^3*c^2*d^4*g^4*i^3) + x^4*(54*a^5*b^2*d^6*g^4*i^3 + 36*b^7*c^5*d*g^
4*i^3 - 90*a*b^6*c^4*d^2*g^4*i^3 - 180*a^4*b^3*c*d^5*g^4*i^3 + 180*a^3*b^4*c^2*d^4*g^4*i^3) + 18*a^3*b^4*c^6*g
^4*i^3 + 18*a^7*c^2*d^4*g^4*i^3 - 72*a^4*b^3*c^5*d*g^4*i^3 - 72*a^6*b*c^3*d^3*g^4*i^3 + 108*a^5*b^2*c^4*d^2*g^
4*i^3) + log((e*(a + b*x))/(c + d*x))^2*((x*((5*B^2*(a*d + b*c)*(2*a*d + b*c))/(3*g^4*i^3*(a^2*d^2 + b^2*c^2 -
2*a*b*c*d)^2) - (5*B^2)/(6*g^4*i^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (5*B^2*a*b*c*d)/(g^4*i^3*(a^2*d^2 + b^2
*c^2 - 2*a*b*c*d)^2) + (20*B^2*a*b*c*d*(a*d + b*c))/(g^4*i^3*(a*d - b*c)^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) +
x^3*((5*B^2*b^2*d^2)/(g^4*i^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) + (20*B^2*b^2*d^2*(a*d + b*c))/(g^4*i^3*(a*d
- b*c)^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) + x^2*((5*B^2*b*d*(a*d + b*c))/(g^4*i^3*(a^2*d^2 + b^2*c^2 - 2*a*b
*c*d)^2) + (5*B^2*b*d*(2*a*d + b*c))/(3*g^4*i^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) + (10*B^2*b^2*d^3*((2*a*c*(
a*d - b*c))/d + ((a*d + b*c)^2*(a*d - b*c))/(b*d^2)))/(g^4*i^3*(a*d - b*c)^4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))
- (B^2*(3*a*d + 2*b*c))/(6*g^4*i^3*(a^2*b*d^3 + b^3*c^2*d - 2*a*b^2*c*d^2)) + (5*B^2*a*c*(2*a*d + b*c))/(3*g^
4*i^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) + (10*B^2*b^3*d^3*x^4)/(g^4*i^3*(a*d - b*c)^3*(a^2*d^2 + b^2*c^2 - 2*
a*b*c*d)) + (10*B^2*a^2*b*c^2*d)/(g^4*i^3*(a*d - b*c)^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(b^2*d*x^5 + (x^4*(3
*a*b^2*d^2 + 2*b^3*c*d))/(b*d) + (a^3*c^2)/(b*d) + (x^2*(a^3*d^2 + 3*a*b^2*c^2 + 6*a^2*b*c*d))/(b*d) + (x^3*(b
^3*c^2 + 3*a^2*b*d^2 + 6*a*b^2*c*d))/(b*d) + (x*(3*a^2*b*c^2 + 2*a^3*c*d))/(b*d)) - (10*B*b^2*d^3*(3*A + B))/(
3*g^4*i^3*(a*d - b*c)^4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) + (log((e*(a + b*x))/(c + d*x))*(x^2*((10*b*d*(B^2*b
*c - 7*B^2*a*d + 6*A*B*a*d + 3*A*B*b*c))/(9*g^4*i^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) - (10*(a*d + b*c)*(2*B^
2*b*d - 3*A*B*b*d))/(3*g^4*i^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) + (20*B*b^2*d^3*(3*A + B)*((2*a*c*(a*d - b*c
))/d + ((a*d + b*c)^2*(a*d - b*c))/(b*d^2)))/(3*g^4*i^3*(a*d - b*c)^4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) - x^3*
((10*b*d*(2*B^2*b*d - 3*A*B*b*d))/(3*g^4*i^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) - (40*B*b^2*d^2*(3*A + B)*(a*d
+ b*c))/(3*g^4*i^3*(a*d - b*c)^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) + x*((5*(B^2 - 6*A*B))/(18*g^4*i^3*(a^2*d^
2 + b^2*c^2 - 2*a*b*c*d)) + (10*(a*d + b*c)*(B^2*b*c - 7*B^2*a*d + 6*A*B*a*d + 3*A*B*b*c))/(9*g^4*i^3*(a^2*d^2
+ b^2*c^2 - 2*a*b*c*d)^2) - (10*a*c*(2*B^2*b*d - 3*A*B*b*d))/(3*g^4*i^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) +
(40*B*a*b*c*d*(3*A + B)*(a*d + b*c))/(3*g^4*i^3*(a*d - b*c)^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) - (4*B^2*b*c -
9*B^2*a*d + 18*A*B*a*d + 12*A*B*b*c)/(18*g^4*i^3*(a^2*b*d^3 + b^3*c^2*d - 2*a*b^2*c*d^2)) + (10*a*c*(B^2*b*c
- 7*B^2*a*d + 6*A*B*a*d + 3*A*B*b*c))/(9*g^4*i^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) + (20*B*b^3*d^3*x^4*(3*A +
B))/(3*g^4*i^3*(a*d - b*c)^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (20*B*a^2*b*c^2*d*(3*A + B))/(3*g^4*i^3*(a*d
- b*c)^3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))))/(b^2*d*x^5 + (x^4*(3*a*b^2*d^2 + 2*b^3*c*d))/(b*d) + (a^3*c^2)/(b*
d) + (x^2*(a^3*d^2 + 3*a*b^2*c^2 + 6*a^2*b*c*d))/(b*d) + (x^3*(b^3*c^2 + 3*a^2*b*d^2 + 6*a*b^2*c*d))/(b*d) + (
x*(3*a^2*b*c^2 + 2*a^3*c*d))/(b*d)) + (b^2*d^3*atan((b^2*d^3*(18*A^2 + 49*B^2 + 12*A*B)*(9*a^6*d^6*g^4*i^3 - 9
*b^6*c^6*g^4*i^3 + 36*a*b^5*c^5*d*g^4*i^3 - 36*a^5*b*c*d^5*g^4*i^3 - 45*a^2*b^4*c^4*d^2*g^4*i^3 + 45*a^4*b^2*c
^2*d^4*g^4*i^3)*5i)/(9*g^4*i^3*(a*d - b*c)^6*(90*A^2*b^2*d^3 + 245*B^2*b^2*d^3 + 60*A*B*b^2*d^3)) + (b^3*d^4*x
*(18*A^2 + 49*B^2 + 12*A*B)*(a^5*d^5*g^4*i^3 - b^5*c^5*g^4*i^3 + 5*a*b^4*c^4*d*g^4*i^3 - 5*a^4*b*c*d^4*g^4*i^3
- 10*a^2*b^3*c^3*d^2*g^4*i^3 + 10*a^3*b^2*c^2*d^3*g^4*i^3)*10i)/(g^4*i^3*(a*d - b*c)^6*(90*A^2*b^2*d^3 + 245*
B^2*b^2*d^3 + 60*A*B*b^2*d^3)))*(18*A^2 + 49*B^2 + 12*A*B)*10i)/(9*g^4*i^3*(a*d - b*c)^6) - (10*B^2*b^2*d^3*lo
g((e*(a + b*x))/(c + d*x))^3)/(3*g^4*i^3*(a*d - b*c)^4*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**4/(d*i*x+c*i)**3,x)
[Out]
Timed out
________________________________________________________________________________________