3.3.9 \(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^4 (c i+d i x)^3} \, dx\) [209]

Optimal. Leaf size=908 \[ -\frac {B^2 d^5 n^2 (a+b x)^2}{4 (b c-a d)^6 g^4 i^3 (c+d x)^2}-\frac {10 A b B d^4 n (a+b x)}{(b c-a d)^6 g^4 i^3 (c+d x)}+\frac {10 b B^2 d^4 n^2 (a+b x)}{(b c-a d)^6 g^4 i^3 (c+d x)}-\frac {20 b^3 B^2 d^2 n^2 (c+d x)}{(b c-a d)^6 g^4 i^3 (a+b x)}+\frac {5 b^4 B^2 d n^2 (c+d x)^2}{4 (b c-a d)^6 g^4 i^3 (a+b x)^2}-\frac {2 b^5 B^2 n^2 (c+d x)^3}{27 (b c-a d)^6 g^4 i^3 (a+b x)^3}-\frac {10 b B^2 d^4 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c-a d)^6 g^4 i^3 (c+d x)}+\frac {B d^5 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^6 g^4 i^3 (c+d x)^2}-\frac {20 b^3 B d^2 n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^6 g^4 i^3 (a+b x)}+\frac {5 b^4 B d n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^6 g^4 i^3 (a+b x)^2}-\frac {2 b^5 B n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 (b c-a d)^6 g^4 i^3 (a+b x)^3}-\frac {d^5 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^6 g^4 i^3 (c+d x)^2}+\frac {5 b d^4 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^6 g^4 i^3 (c+d x)}-\frac {10 b^3 d^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(b c-a d)^6 g^4 i^3 (a+b x)}+\frac {5 b^4 d (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^6 g^4 i^3 (a+b x)^2}-\frac {b^5 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 (b c-a d)^6 g^4 i^3 (a+b x)^3}-\frac {10 b^2 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3}{3 B (b c-a d)^6 g^4 i^3 n} \]

[Out]

-1/4*B^2*d^5*n^2*(b*x+a)^2/(-a*d+b*c)^6/g^4/i^3/(d*x+c)^2-10*A*b*B*d^4*n*(b*x+a)/(-a*d+b*c)^6/g^4/i^3/(d*x+c)+
10*b*B^2*d^4*n^2*(b*x+a)/(-a*d+b*c)^6/g^4/i^3/(d*x+c)-20*b^3*B^2*d^2*n^2*(d*x+c)/(-a*d+b*c)^6/g^4/i^3/(b*x+a)+
5/4*b^4*B^2*d*n^2*(d*x+c)^2/(-a*d+b*c)^6/g^4/i^3/(b*x+a)^2-2/27*b^5*B^2*n^2*(d*x+c)^3/(-a*d+b*c)^6/g^4/i^3/(b*
x+a)^3-10*b*B^2*d^4*n*(b*x+a)*ln(e*((b*x+a)/(d*x+c))^n)/(-a*d+b*c)^6/g^4/i^3/(d*x+c)+1/2*B*d^5*n*(b*x+a)^2*(A+
B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^6/g^4/i^3/(d*x+c)^2-20*b^3*B*d^2*n*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))
^n))/(-a*d+b*c)^6/g^4/i^3/(b*x+a)+5/2*b^4*B*d*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^6/g^4/i^3
/(b*x+a)^2-2/9*b^5*B*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-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))^n))^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
))^n))^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))^n))^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))^n))^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))^n))^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))^n
))^3/B/(-a*d+b*c)^6/g^4/i^3/n

________________________________________________________________________________________

Rubi [A]
time = 0.45, antiderivative size = 908, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 8, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.178, Rules used = {2561, 2395, 2333, 2332, 2342, 2341, 2339, 30} \begin {gather*} -\frac {2 B^2 n^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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 b^5}{3 (b c-a d)^6 g^4 i^3 (a+b x)^3}-\frac {2 B n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) b^5}{9 (b c-a d)^6 g^4 i^3 (a+b x)^3}+\frac {5 B^2 d n^2 (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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 b^4}{2 (b c-a d)^6 g^4 i^3 (a+b x)^2}+\frac {5 B d n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 b^3}{(b c-a d)^6 g^4 i^3 (a+b x)}-\frac {20 B^2 d^2 n^2 (c+d x) b^3}{(b c-a d)^6 g^4 i^3 (a+b x)}-\frac {20 B d^2 n (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^3 b^2}{3 B (b c-a d)^6 g^4 i^3 n}+\frac {5 d^4 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 b}{(b c-a d)^6 g^4 i^3 (c+d x)}-\frac {10 B^2 d^4 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) b}{(b c-a d)^6 g^4 i^3 (c+d x)}+\frac {10 B^2 d^4 n^2 (a+b x) b}{(b c-a d)^6 g^4 i^3 (c+d x)}-\frac {10 A B d^4 n (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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{2 (b c-a d)^6 g^4 i^3 (c+d x)^2}+\frac {B d^5 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^6 g^4 i^3 (c+d x)^2}-\frac {B^2 d^5 n^2 (a+b x)^2}{4 (b c-a d)^6 g^4 i^3 (c+d x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*(B^2*d^5*n^2*(a + b*x)^2)/((b*c - a*d)^6*g^4*i^3*(c + d*x)^2) - (10*A*b*B*d^4*n*(a + b*x))/((b*c - a*d)^6
*g^4*i^3*(c + d*x)) + (10*b*B^2*d^4*n^2*(a + b*x))/((b*c - a*d)^6*g^4*i^3*(c + d*x)) - (20*b^3*B^2*d^2*n^2*(c
+ d*x))/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*B^2*d*n^2*(c + d*x)^2)/(4*(b*c - a*d)^6*g^4*i^3*(a + b*x)^2
) - (2*b^5*B^2*n^2*(c + d*x)^3)/(27*(b*c - a*d)^6*g^4*i^3*(a + b*x)^3) - (10*b*B^2*d^4*n*(a + b*x)*Log[e*((a +
b*x)/(c + d*x))^n])/((b*c - a*d)^6*g^4*i^3*(c + d*x)) + (B*d^5*n*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x
))^n]))/(2*(b*c - a*d)^6*g^4*i^3*(c + d*x)^2) - (20*b^3*B*d^2*n*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n
]))/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*B*d*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*(b
*c - a*d)^6*g^4*i^3*(a + b*x)^2) - (2*b^5*B*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(b*c - a*
d)^6*g^4*i^3*(a + b*x)^3) - (d^5*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^6*g^4*i^
3*(c + d*x)^2) + (5*b*d^4*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^6*g^4*i^3*(c + d*x)
) - (10*b^3*d^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4
*d*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(2*(b*c - a*d)^6*g^4*i^3*(a + b*x)^2) - (b^5*(c + d*x
)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*(b*c - a*d)^6*g^4*i^3*(a + b*x)^3) - (10*b^2*d^3*(A + B*Log[e
*((a + b*x)/(c + d*x))^n])^3)/(3*B*(b*c - a*d)^6*g^4*i^3*n)

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 2561

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

Rubi steps

\begin {align*} \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(209 c+209 d x)^3 (a g+b g x)^4} \, dx &=-\frac {2 b^2 B^2 n^2}{246491883 (b c-a d)^3 g^4 (a+b x)^3}+\frac {37 b^2 B^2 d n^2}{328655844 (b c-a d)^4 g^4 (a+b x)^2}-\frac {29 b^2 B^2 d^2 n^2}{14938902 (b c-a d)^5 g^4 (a+b x)}-\frac {B^2 d^3 n^2}{36517316 (b c-a d)^4 g^4 (c+d x)^2}-\frac {b B^2 d^3 n^2}{960982 (b c-a d)^5 g^4 (c+d x)}-\frac {245 b^2 B^2 d^3 n^2 \log (a+b x)}{82163961 (b c-a d)^6 g^4}+\frac {10 A b^2 B d^3 n \log ^2(a+b x)}{9129329 (b c-a d)^6 g^4}+\frac {10 b^2 B^2 d^3 n^2 \log ^2(a+b x)}{27387987 (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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{9129329 (b c-a d)^6 g^4}+\frac {10 b^2 B^2 d^3 \log (a+b x) \log ^2\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{9129329 (b c-a d)^6 g^4}-\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{82163961 (b c-a d)^3 g^4 (a+b x)^3}+\frac {b^2 B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4979634 (b c-a d)^4 g^4 (a+b x)^2}-\frac {47 b^2 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{27387987 (b c-a d)^5 g^4 (a+b x)}+\frac {B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{18258658 (b c-a d)^4 g^4 (c+d x)^2}+\frac {9 b B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9129329 (b c-a d)^5 g^4 (c+d x)}-\frac {20 b^2 B d^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{27387987 (b c-a d)^6 g^4}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{27387987 (b c-a d)^3 g^4 (a+b x)^3}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{18258658 (b c-a d)^4 g^4 (a+b x)^2}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{9129329 (b c-a d)^5 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{18258658 (b c-a d)^4 g^4 (c+d x)^2}-\frac {4 b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{9129329 (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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{9129329 (b c-a d)^6 g^4}+\frac {245 b^2 B^2 d^3 n^2 \log (c+d x)}{82163961 (b c-a d)^6 g^4}-\frac {20 A b^2 B d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9129329 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{27387987 (b c-a d)^6 g^4}-\frac {10 b^2 B^2 d^3 \log ^2\left ((a+b x)^n\right ) \log (c+d x)}{9129329 (b c-a d)^6 g^4}+\frac {20 b^2 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{27387987 (b c-a d)^6 g^4}+\frac {10 b^2 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \log (c+d x)}{9129329 (b c-a d)^6 g^4}+\frac {10 A b^2 B d^3 n \log ^2(c+d x)}{9129329 (b c-a d)^6 g^4}+\frac {10 b^2 B^2 d^3 n^2 \log ^2(c+d x)}{27387987 (b c-a d)^6 g^4}-\frac {10 b^2 B^2 d^3 n^2 \log (a+b x) \log ^2(c+d x)}{9129329 (b c-a d)^6 g^4}+\frac {10 b^2 B^2 d^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log ^2(c+d x)}{9129329 (b c-a d)^6 g^4}+\frac {10 b^2 B^2 d^3 n^2 \log ^3(c+d x)}{27387987 (b c-a d)^6 g^4}-\frac {20 A b^2 B d^3 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9129329 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{27387987 (b c-a d)^6 g^4}+\frac {10 b^2 B^2 d^3 \log ^2\left ((a+b x)^n\right ) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{9129329 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 n \log (a+b x) \log (c+d x) \log \left ((c+d x)^{-n}\right )}{9129329 (b c-a d)^6 g^4}-\frac {10 b^2 B^2 d^3 \log (a+b x) \log ^2\left ((c+d x)^{-n}\right )}{9129329 (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 ((c+d x)^{-n}\right )}{9129329 (b c-a d)^6 g^4}+\frac {20 b^2 B^2 d^3 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right )}{9129329 (b c-a d)^6 g^4}-\frac {20 A b^2 B d^3 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{9129329 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{27387987 (b c-a d)^6 g^4}+\frac {20 b^2 B^2 d^3 n \log \left ((a+b x)^n\right ) \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{9129329 (b c-a d)^6 g^4}-\frac {20 A b^2 B d^3 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9129329 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{27387987 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 n \log \left ((c+d x)^{-n}\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9129329 (b c-a d)^6 g^4}+\frac {20 b^2 B^2 d^3 n \left (\log \left ((a+b x)^n\right )-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+\log \left ((c+d x)^{-n}\right )\right ) \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{9129329 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 n \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \text {Li}_2\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9129329 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 n^2 \text {Li}_3\left (-\frac {d (a+b x)}{b c-a d}\right )}{9129329 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 n^2 \text {Li}_3\left (\frac {b (c+d x)}{b c-a d}\right )}{9129329 (b c-a d)^6 g^4}-\frac {20 b^2 B^2 d^3 n^2 \text {Li}_3\left (1+\frac {b c-a d}{d (a+b x)}\right )}{9129329 (b c-a d)^6 g^4}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(2138\) vs. \(2(908)=1816\).
time = 2.50, size = 2138, normalized size = 2.35 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/108*(360*b^2*B^2*d^3*n^2*(a + b*x)^3*(c + d*x)^2*Log[(a + b*x)/(c + d*x)]^3 + 18*B*n*Log[(a + b*x)/(c + d*x
)]^2*(60*a^3*A*b^2*c^2*d^3 + 2*b^5*B*c^5*n - 15*a*b^4*B*c^4*d*n + 60*a^2*b^3*B*c^3*d^2*n - 30*a^4*b*B*c*d^4*n
+ 3*a^5*B*d^5*n + 180*a^2*A*b^3*c^2*d^3*x + 120*a^3*A*b^2*c*d^4*x - 5*b^5*B*c^4*d*n*x + 60*a*b^4*B*c^3*d^2*n*x
+ 180*a^2*b^3*B*c^2*d^3*n*x - 120*a^3*b^2*B*c*d^4*n*x - 15*a^4*b*B*d^5*n*x + 180*a*A*b^4*c^2*d^3*x^2 + 360*a^
2*A*b^3*c*d^4*x^2 + 60*a^3*A*b^2*d^5*x^2 + 20*b^5*B*c^3*d^2*n*x^2 + 270*a*b^4*B*c^2*d^3*n*x^2 - 90*a^3*b^2*B*d
^5*n*x^2 + 60*A*b^5*c^2*d^3*x^3 + 360*a*A*b^4*c*d^4*x^3 + 180*a^2*A*b^3*d^5*x^3 + 110*b^5*B*c^2*d^3*n*x^3 + 18
0*a*b^4*B*c*d^4*n*x^3 - 90*a^2*b^3*B*d^5*n*x^3 + 120*A*b^5*c*d^4*x^4 + 180*a*A*b^4*d^5*x^4 + 100*b^5*B*c*d^4*n
*x^4 + 60*A*b^5*d^5*x^5 + 20*b^5*B*d^5*n*x^5 + 60*b^2*B*d^3*(a + b*x)^3*(c + d*x)^2*Log[e*((a + b*x)/(c + d*x)
)^n] - 60*b^2*B*d^3*n*(a + b*x)^3*(c + d*x)^2*Log[(a + b*x)/(c + d*x)]) + 6*b^2*d^2*(b*c - a*d)*(a + b*x)^2*(c
+ d*x)^2*(108*A^2 + 282*A*B*n + 319*B^2*n^2 + 108*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 6*B*n*(36*A + 47*B*n
)*Log[(a + b*x)/(c + d*x)] + 108*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 6*B*Log[e*((a + b*x)/(c + d*x))^n]*(36*A
+ 47*B*n - 36*B*n*Log[(a + b*x)/(c + d*x)])) - 3*b^2*d*(b*c - a*d)^2*(a + b*x)*(c + d*x)^2*(54*A^2 + 66*A*B*n
+ 37*B^2*n^2 + 54*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 6*B*n*(18*A + 11*B*n)*Log[(a + b*x)/(c + d*x)] + 54*
B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 6*B*Log[e*((a + b*x)/(c + d*x))^n]*(18*A + 11*B*n - 18*B*n*Log[(a + b*x)/
(c + d*x)])) + 4*b^2*(b*c - a*d)^3*(c + d*x)^2*(9*A^2 + 6*A*B*n + 2*B^2*n^2 + 9*B^2*Log[e*((a + b*x)/(c + d*x)
)^n]^2 - 6*B*n*(3*A + B*n)*Log[(a + b*x)/(c + d*x)] + 9*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 6*B*Log[e*((a + b
*x)/(c + d*x))^n]*(3*A + B*n - 3*B*n*Log[(a + b*x)/(c + d*x)])) + 60*b^2*d^3*(a + b*x)^3*(c + d*x)^2*Log[a + b
*x]*(18*A^2 + 12*A*B*n + 49*B^2*n^2 + 18*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 - 12*B*n*(3*A + B*n)*Log[(a + b*
x)/(c + d*x)] + 18*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 + 12*B*Log[e*((a + b*x)/(c + d*x))^n]*(3*A + B*n - 3*B*n
*Log[(a + b*x)/(c + d*x)])) + 27*d^3*(b*c - a*d)^2*(a + b*x)^3*(2*A^2 - 2*A*B*n + B^2*n^2 + 2*B^2*Log[e*((a +
b*x)/(c + d*x))^n]^2 + 2*B*n*(-2*A + B*n)*Log[(a + b*x)/(c + d*x)] + 2*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 - 2*
B*Log[e*((a + b*x)/(c + d*x))^n]*(-2*A + B*n + 2*B*n*Log[(a + b*x)/(c + d*x)])) + 54*b*d^3*(b*c - a*d)*(a + b*
x)^3*(c + d*x)*(8*A^2 - 18*A*B*n + 19*B^2*n^2 + 8*B^2*Log[e*((a + b*x)/(c + d*x))^n]^2 + 2*B*n*(-8*A + 9*B*n)*
Log[(a + b*x)/(c + d*x)] + 8*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 - 2*B*Log[e*((a + b*x)/(c + d*x))^n]*(-8*A + 9
*B*n + 8*B*n*Log[(a + b*x)/(c + d*x)])) + 6*B*(b*c - a*d)*n*Log[(a + b*x)/(c + d*x)]*(18*b*d^3*(a + b*x)^3*(c
+ d*x)*(8*A - 9*B*n + 8*B*Log[e*((a + b*x)/(c + d*x))^n] - 8*B*n*Log[(a + b*x)/(c + d*x)]) + 4*b^2*(b*c - a*d)
^2*(c + d*x)^2*(3*A + B*n + 3*B*Log[e*((a + b*x)/(c + d*x))^n] - 3*B*n*Log[(a + b*x)/(c + d*x)]) + 9*d^3*(b*c
- a*d)*(a + b*x)^3*(2*A - B*n + 2*B*Log[e*((a + b*x)/(c + d*x))^n] - 2*B*n*Log[(a + b*x)/(c + d*x)]) - 3*b^2*d
*(b*c - a*d)*(a + b*x)*(c + d*x)^2*(18*A + 11*B*n + 18*B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a + b*x)/(c
+ d*x)])) + 6*b^2*d^2*(a + b*x)^2*(c + d*x)^2*(36*A + 47*B*n + 36*B*(Log[e*((a + b*x)/(c + d*x))^n] - n*Log[(a
+ b*x)/(c + d*x)]))) - 60*b^2*d^3*(a + b*x)^3*(c + d*x)^2*(18*A^2 + 12*A*B*n + 49*B^2*n^2 + 18*B^2*Log[e*((a
+ b*x)/(c + d*x))^n]^2 - 12*B*n*(3*A + B*n)*Log[(a + b*x)/(c + d*x)] + 18*B^2*n^2*Log[(a + b*x)/(c + d*x)]^2 +
12*B*Log[e*((a + b*x)/(c + d*x))^n]*(3*A + B*n - 3*B*n*Log[(a + b*x)/(c + d*x)]))*Log[c + d*x])/((b*c - a*d)^
6*g^4*i^3*(a + b*x)^3*(c + d*x)^2)

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Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {\left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{\left (b g x +a g \right )^{4} \left (d i x +c i \right )^{3}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 8914 vs. \(2 (848) = 1696\).
time = 2.28, size = 8914, normalized size = 9.82 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x, algorithm="maxima")

[Out]

1/6*B^2*(60*b^2*d^3*log(b*x + a)/((I*b^6*c^6 - 6*I*a*b^5*c^5*d + 15*I*a^2*b^4*c^4*d^2 - 20*I*a^3*b^3*c^3*d^3 +
15*I*a^4*b^2*c^2*d^4 - 6*I*a^5*b*c*d^5 + I*a^6*d^6)*g^4) - 60*b^2*d^3*log(d*x + c)/((I*b^6*c^6 - 6*I*a*b^5*c^
5*d + 15*I*a^2*b^4*c^4*d^2 - 20*I*a^3*b^3*c^3*d^3 + 15*I*a^4*b^2*c^2*d^4 - 6*I*a^5*b*c*d^5 + I*a^6*d^6)*g^4) +
(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)/((I*b^8*c^5*d^2 - 5*I*a*b^7*c^4*d^3 + 10*I*a^2*b^6*c^3*d^4 - 10*I*a
^3*b^5*c^2*d^5 + 5*I*a^4*b^4*c*d^6 - I*a^5*b^3*d^7)*g^4*x^5 + (2*I*b^8*c^6*d - 7*I*a*b^7*c^5*d^2 + 5*I*a^2*b^6
*c^4*d^3 + 10*I*a^3*b^5*c^3*d^4 - 20*I*a^4*b^4*c^2*d^5 + 13*I*a^5*b^3*c*d^6 - 3*I*a^6*b^2*d^7)*g^4*x^4 + (I*b^
8*c^7 + I*a*b^7*c^6*d - 17*I*a^2*b^6*c^5*d^2 + 35*I*a^3*b^5*c^4*d^3 - 25*I*a^4*b^4*c^3*d^4 - I*a^5*b^3*c^2*d^5
+ 9*I*a^6*b^2*c*d^6 - 3*I*a^7*b*d^7)*g^4*x^3 + (3*I*a*b^7*c^7 - 9*I*a^2*b^6*c^6*d + I*a^3*b^5*c^5*d^2 + 25*I*
a^4*b^4*c^4*d^3 - 35*I*a^5*b^3*c^3*d^4 + 17*I*a^6*b^2*c^2*d^5 - I*a^7*b*c*d^6 - I*a^8*d^7)*g^4*x^2 + (3*I*a^2*
b^6*c^7 - 13*I*a^3*b^5*c^6*d + 20*I*a^4*b^4*c^5*d^2 - 10*I*a^5*b^3*c^4*d^3 - 5*I*a^6*b^2*c^3*d^4 + 7*I*a^7*b*c
^2*d^5 - 2*I*a^8*c*d^6)*g^4*x + (I*a^3*b^5*c^7 - 5*I*a^4*b^4*c^6*d + 10*I*a^5*b^3*c^5*d^2 - 10*I*a^6*b^2*c^4*d
^3 + 5*I*a^7*b*c^3*d^4 - I*a^8*c^2*d^5)*g^4))*log((b*x/(d*x + c) + a/(d*x + c))^n*e)^2 + 1/3*A*B*(60*b^2*d^3*l
og(b*x + a)/((I*b^6*c^6 - 6*I*a*b^5*c^5*d + 15*I*a^2*b^4*c^4*d^2 - 20*I*a^3*b^3*c^3*d^3 + 15*I*a^4*b^2*c^2*d^4
- 6*I*a^5*b*c*d^5 + I*a^6*d^6)*g^4) - 60*b^2*d^3*log(d*x + c)/((I*b^6*c^6 - 6*I*a*b^5*c^5*d + 15*I*a^2*b^4*c^
4*d^2 - 20*I*a^3*b^3*c^3*d^3 + 15*I*a^4*b^2*c^2*d^4 - 6*I*a^5*b*c*d^5 + I*a^6*d^6)*g^4) + (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)/((I*b^8*c^5*d^2 - 5*I*a*b^7*c^4*d^3 + 10*I*a^2*b^6*c^3*d^4 - 10*I*a^3*b^5*c^2*d^5 + 5*I*
a^4*b^4*c*d^6 - I*a^5*b^3*d^7)*g^4*x^5 + (2*I*b^8*c^6*d - 7*I*a*b^7*c^5*d^2 + 5*I*a^2*b^6*c^4*d^3 + 10*I*a^3*b
^5*c^3*d^4 - 20*I*a^4*b^4*c^2*d^5 + 13*I*a^5*b^3*c*d^6 - 3*I*a^6*b^2*d^7)*g^4*x^4 + (I*b^8*c^7 + I*a*b^7*c^6*d
- 17*I*a^2*b^6*c^5*d^2 + 35*I*a^3*b^5*c^4*d^3 - 25*I*a^4*b^4*c^3*d^4 - I*a^5*b^3*c^2*d^5 + 9*I*a^6*b^2*c*d^6
- 3*I*a^7*b*d^7)*g^4*x^3 + (3*I*a*b^7*c^7 - 9*I*a^2*b^6*c^6*d + I*a^3*b^5*c^5*d^2 + 25*I*a^4*b^4*c^4*d^3 - 35*
I*a^5*b^3*c^3*d^4 + 17*I*a^6*b^2*c^2*d^5 - I*a^7*b*c*d^6 - I*a^8*d^7)*g^4*x^2 + (3*I*a^2*b^6*c^7 - 13*I*a^3*b^
5*c^6*d + 20*I*a^4*b^4*c^5*d^2 - 10*I*a^5*b^3*c^4*d^3 - 5*I*a^6*b^2*c^3*d^4 + 7*I*a^7*b*c^2*d^5 - 2*I*a^8*c*d^
6)*g^4*x + (I*a^3*b^5*c^7 - 5*I*a^4*b^4*c^6*d + 10*I*a^5*b^3*c^5*d^2 - 10*I*a^6*b^2*c^4*d^3 + 5*I*a^7*b*c^3*d^
4 - I*a^8*c^2*d^5)*g^4))*log((b*x/(d*x + c) + a/(d*x + c))^n*e) + 1/108*((-8*I*b^5*c^5 + 135*I*a*b^4*c^4*d - 2
160*I*a^2*b^3*c^3*d^2 + 980*I*a^3*b^2*c^2*d^3 + 1080*I*a^4*b*c*d^4 - 27*I*a^5*d^5 - 2940*(I*b^5*c*d^4 - I*a*b^
4*d^5)*x^4 - 30*(159*I*b^5*c^2*d^3 + 74*I*a*b^4*c*d^4 - 233*I*a^2*b^3*d^5)*x^3 - 360*(I*b^5*d^5*x^5 + I*a^3*b^
2*c^2*d^3 + (2*I*b^5*c*d^4 + 3*I*a*b^4*d^5)*x^4 + (I*b^5*c^2*d^3 + 6*I*a*b^4*c*d^4 + 3*I*a^2*b^3*d^5)*x^3 + (3
*I*a*b^4*c^2*d^3 + 6*I*a^2*b^3*c*d^4 + I*a^3*b^2*d^5)*x^2 + (3*I*a^2*b^3*c^2*d^3 + 2*I*a^3*b^2*c*d^4)*x)*log(b
*x + a)^3 - 360*(-I*b^5*d^5*x^5 - I*a^3*b^2*c^2*d^3 + (-2*I*b^5*c*d^4 - 3*I*a*b^4*d^5)*x^4 + (-I*b^5*c^2*d^3 -
6*I*a*b^4*c*d^4 - 3*I*a^2*b^3*d^5)*x^3 + (-3*I*a*b^4*c^2*d^3 - 6*I*a^2*b^3*c*d^4 - I*a^3*b^2*d^5)*x^2 + (-3*I
*a^2*b^3*c^2*d^3 - 2*I*a^3*b^2*c*d^4)*x)*log(d*x + c)^3 - 10*(170*I*b^5*c^3*d^2 + 921*I*a*b^4*c^2*d^3 - 588*I*
a^2*b^3*c*d^4 - 503*I*a^3*b^2*d^5)*x^2 - 360*(-I*b^5*d^5*x^5 - I*a^3*b^2*c^2*d^3 + (-2*I*b^5*c*d^4 - 3*I*a*b^4
*d^5)*x^4 + (-I*b^5*c^2*d^3 - 6*I*a*b^4*c*d^4 - 3*I*a^2*b^3*d^5)*x^3 + (-3*I*a*b^4*c^2*d^3 - 6*I*a^2*b^3*c*d^4
- I*a^3*b^2*d^5)*x^2 + (-3*I*a^2*b^3*c^2*d^3 - 2*I*a^3*b^2*c*d^4)*x)*log(b*x + a)^2 - 360*(-I*b^5*d^5*x^5 - I
*a^3*b^2*c^2*d^3 + (-2*I*b^5*c*d^4 - 3*I*a*b^4*d^5)*x^4 + (-I*b^5*c^2*d^3 - 6*I*a*b^4*c*d^4 - 3*I*a^2*b^3*d^5)
*x^3 + (-3*I*a*b^4*c^2*d^3 - 6*I*a^2*b^3*c*d^4 - I*a^3*b^2*d^5)*x^2 + (-3*I*a^2*b^3*c^2*d^3 - 2*I*a^3*b^2*c*d^
4)*x + 3*(I*b^5*d^5*x^5 + I*a^3*b^2*c^2*d^3 + (2*I*b^5*c*d^4 + 3*I*a*b^4*d^5)*x^4 + (I*b^5*c^2*d^3 + 6*I*a*b^4
*c*d^4 + 3*I*a^2*b^3*d^5)*x^3 + (3*I*a*b^4*c^2*d^3 + 6*I*a^2*b^3*c*d^4 + I*a^3*b^2*d^5)*x^2 + (3*I*a^2*b^3*c^2
*d^3 + 2*I*a^3*b^2*c*d^4)*x)*log(b*x + a))*log(d*x + c)^2 - 5*(-19*I*b^5*c^4*d + 756*I*a*b^4*c^3*d^2 + 708*I*a
^2*b^3*c^2*d^3 - 1256*I*a^3*b^2*c*d^4 - 189*I*a^4*b*d^5)*x - 2940*(I*b^5*d^5*x^5 + I*a^3*b^2*c^2*d^3 + (2*I*b^
5*c*d^4 + 3*I*a*b^4*d^5)*x^4 + (I*b^5*c^2*d^3 + 6*I*a*b^4*c*d^4 + 3*I*a^2*b^3*d^5)*x^3 + (3*I*a*b^4*c^2*d^3 +
6*I*a^2*b^3*c*d^4 + I*a^3*b^2*d^5)*x^2 + (3*I*a...

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 3747 vs. \(2 (848) = 1696\).
time = 0.51, size = 3747, normalized size = 4.13 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x, algorithm="fricas")

[Out]

-1/108*(36*(I*A^2 + 2*I*A*B + I*B^2)*b^5*c^5 + 270*(-I*A^2 - 2*I*A*B - I*B^2)*a*b^4*c^4*d + 1080*(I*A^2 + 2*I*
A*B + I*B^2)*a^2*b^3*c^3*d^2 + 360*(-I*A^2 - 2*I*A*B - I*B^2)*a^3*b^2*c^2*d^3 + 540*(-I*A^2 - 2*I*A*B - I*B^2)
*a^4*b*c*d^4 + 54*(I*A^2 + 2*I*A*B + I*B^2)*a^5*d^5 + 60*(18*(I*A^2 + 2*I*A*B + I*B^2)*b^5*c*d^4 + 18*(-I*A^2
- 2*I*A*B - I*B^2)*a*b^4*d^5 + 49*(I*B^2*b^5*c*d^4 - I*B^2*a*b^4*d^5)*n^2 + 12*((I*A*B + I*B^2)*b^5*c*d^4 + (-
I*A*B - I*B^2)*a*b^4*d^5)*n)*x^4 + 30*(54*(I*A^2 + 2*I*A*B + I*B^2)*b^5*c^2*d^3 + 36*(I*A^2 + 2*I*A*B + I*B^2)
*a*b^4*c*d^4 + 90*(-I*A^2 - 2*I*A*B - I*B^2)*a^2*b^3*d^5 + (159*I*B^2*b^5*c^2*d^3 + 74*I*B^2*a*b^4*c*d^4 - 233
*I*B^2*a^2*b^3*d^5)*n^2 + 24*(3*(I*A*B + I*B^2)*b^5*c^2*d^3 + 2*(-I*A*B - I*B^2)*a*b^4*c*d^4 + (-I*A*B - I*B^2
)*a^2*b^3*d^5)*n)*x^3 + 360*(I*B^2*b^5*d^5*n^2*x^5 + I*B^2*a^3*b^2*c^2*d^3*n^2 + (2*I*B^2*b^5*c*d^4 + 3*I*B^2*
a*b^4*d^5)*n^2*x^4 + (I*B^2*b^5*c^2*d^3 + 6*I*B^2*a*b^4*c*d^4 + 3*I*B^2*a^2*b^3*d^5)*n^2*x^3 + (3*I*B^2*a*b^4*
c^2*d^3 + 6*I*B^2*a^2*b^3*c*d^4 + I*B^2*a^3*b^2*d^5)*n^2*x^2 + (3*I*B^2*a^2*b^3*c^2*d^3 + 2*I*B^2*a^3*b^2*c*d^
4)*n^2*x)*log((b*x + a)/(d*x + c))^3 - (-8*I*B^2*b^5*c^5 + 135*I*B^2*a*b^4*c^4*d - 2160*I*B^2*a^2*b^3*c^3*d^2
+ 980*I*B^2*a^3*b^2*c^2*d^3 + 1080*I*B^2*a^4*b*c*d^4 - 27*I*B^2*a^5*d^5)*n^2 + 10*(36*(I*A^2 + 2*I*A*B + I*B^2
)*b^5*c^3*d^2 + 378*(I*A^2 + 2*I*A*B + I*B^2)*a*b^4*c^2*d^3 + 216*(-I*A^2 - 2*I*A*B - I*B^2)*a^2*b^3*c*d^4 + 1
98*(-I*A^2 - 2*I*A*B - I*B^2)*a^3*b^2*d^5 + (170*I*B^2*b^5*c^3*d^2 + 921*I*B^2*a*b^4*c^2*d^3 - 588*I*B^2*a^2*b
^3*c*d^4 - 503*I*B^2*a^3*b^2*d^5)*n^2 + 12*(11*(I*A*B + I*B^2)*b^5*c^3*d^2 + 21*(I*A*B + I*B^2)*a*b^4*c^2*d^3
+ 39*(-I*A*B - I*B^2)*a^2*b^3*c*d^4 + 7*(I*A*B + I*B^2)*a^3*b^2*d^5)*n)*x^2 + 18*(60*(I*A*B + I*B^2)*a^3*b^2*c
^2*d^3*n + 20*(I*B^2*b^5*d^5*n^2 + 3*(I*A*B + I*B^2)*b^5*d^5*n)*x^5 + 20*(5*I*B^2*b^5*c*d^4*n^2 + 3*(2*(I*A*B
+ I*B^2)*b^5*c*d^4 + 3*(I*A*B + I*B^2)*a*b^4*d^5)*n)*x^4 + 10*((11*I*B^2*b^5*c^2*d^3 + 18*I*B^2*a*b^4*c*d^4 -
9*I*B^2*a^2*b^3*d^5)*n^2 + 6*((I*A*B + I*B^2)*b^5*c^2*d^3 + 6*(I*A*B + I*B^2)*a*b^4*c*d^4 + 3*(I*A*B + I*B^2)*
a^2*b^3*d^5)*n)*x^3 + (2*I*B^2*b^5*c^5 - 15*I*B^2*a*b^4*c^4*d + 60*I*B^2*a^2*b^3*c^3*d^2 - 30*I*B^2*a^4*b*c*d^
4 + 3*I*B^2*a^5*d^5)*n^2 + 10*((2*I*B^2*b^5*c^3*d^2 + 27*I*B^2*a*b^4*c^2*d^3 - 9*I*B^2*a^3*b^2*d^5)*n^2 + 6*(3
*(I*A*B + I*B^2)*a*b^4*c^2*d^3 + 6*(I*A*B + I*B^2)*a^2*b^3*c*d^4 + (I*A*B + I*B^2)*a^3*b^2*d^5)*n)*x^2 + 5*((-
I*B^2*b^5*c^4*d + 12*I*B^2*a*b^4*c^3*d^2 + 36*I*B^2*a^2*b^3*c^2*d^3 - 24*I*B^2*a^3*b^2*c*d^4 - 3*I*B^2*a^4*b*d
^5)*n^2 + 12*(3*(I*A*B + I*B^2)*a^2*b^3*c^2*d^3 + 2*(I*A*B + I*B^2)*a^3*b^2*c*d^4)*n)*x)*log((b*x + a)/(d*x +
c))^2 + 6*(4*(I*A*B + I*B^2)*b^5*c^5 + 45*(-I*A*B - I*B^2)*a*b^4*c^4*d + 360*(I*A*B + I*B^2)*a^2*b^3*c^3*d^2 +
490*(-I*A*B - I*B^2)*a^3*b^2*c^2*d^3 + 180*(I*A*B + I*B^2)*a^4*b*c*d^4 + 9*(-I*A*B - I*B^2)*a^5*d^5)*n + 5*(1
8*(-I*A^2 - 2*I*A*B - I*B^2)*b^5*c^4*d + 216*(I*A^2 + 2*I*A*B + I*B^2)*a*b^4*c^3*d^2 + 432*(I*A^2 + 2*I*A*B +
I*B^2)*a^2*b^3*c^2*d^3 + 576*(-I*A^2 - 2*I*A*B - I*B^2)*a^3*b^2*c*d^4 + 54*(-I*A^2 - 2*I*A*B - I*B^2)*a^4*b*d^
5 + (-19*I*B^2*b^5*c^4*d + 756*I*B^2*a*b^4*c^3*d^2 + 708*I*B^2*a^2*b^3*c^2*d^3 - 1256*I*B^2*a^3*b^2*c*d^4 - 18
9*I*B^2*a^4*b*d^5)*n^2 + 6*(5*(-I*A*B - I*B^2)*b^5*c^4*d + 108*(I*A*B + I*B^2)*a*b^4*c^3*d^2 + 78*(-I*A*B - I*
B^2)*a^2*b^3*c^2*d^3 + 52*(-I*A*B - I*B^2)*a^3*b^2*c*d^4 + 27*(I*A*B + I*B^2)*a^4*b*d^5)*n)*x + 6*(180*(I*A^2
+ 2*I*A*B + I*B^2)*a^3*b^2*c^2*d^3 + 10*(49*I*B^2*b^5*d^5*n^2 + 12*(I*A*B + I*B^2)*b^5*d^5*n + 18*(I*A^2 + 2*I
*A*B + I*B^2)*b^5*d^5)*x^5 + 10*(60*(I*A*B + I*B^2)*b^5*c*d^4*n + 36*(I*A^2 + 2*I*A*B + I*B^2)*b^5*c*d^4 + 54*
(I*A^2 + 2*I*A*B + I*B^2)*a*b^4*d^5 + 5*(22*I*B^2*b^5*c*d^4 + 27*I*B^2*a*b^4*d^5)*n^2)*x^4 + 10*(18*(I*A^2 + 2
*I*A*B + I*B^2)*b^5*c^2*d^3 + 108*(I*A^2 + 2*I*A*B + I*B^2)*a*b^4*c*d^4 + 54*(I*A^2 + 2*I*A*B + I*B^2)*a^2*b^3
*d^5 + 5*(17*I*B^2*b^5*c^2*d^3 + 54*I*B^2*a*b^4*c*d^4 + 27*I*B^2*a^2*b^3*d^5)*n^2 + 6*(11*(I*A*B + I*B^2)*b^5*
c^2*d^3 + 18*(I*A*B + I*B^2)*a*b^4*c*d^4 + 9*(-I*A*B - I*B^2)*a^2*b^3*d^5)*n)*x^3 + (4*I*B^2*b^5*c^5 - 45*I*B^
2*a*b^4*c^4*d + 360*I*B^2*a^2*b^3*c^3*d^2 + 180*I*B^2*a^4*b*c*d^4 - 9*I*B^2*a^5*d^5)*n^2 + 10*(54*(I*A^2 + 2*I
*A*B + I*B^2)*a*b^4*c^2*d^3 + 108*(I*A^2 + 2*I*A*B + I*B^2)*a^2*b^3*c*d^4 + 18*(I*A^2 + 2*I*A*B + I*B^2)*a^3*b
^2*d^5 + (22*I*B^2*b^5*c^3*d^2 + 189*I*B^2*a*b^4*c^2*d^3 + 216*I*B^2*a^2*b^3*c*d^4 + 63*I*B^2*a^3*b^2*d^5)*n^2
+ 6*(2*(I*A*B + I*B^2)*b^5*c^3*d^2 + 27*(I*A*B + I*B^2)*a*b^4*c^2*d^3 + 9*(-I*A*B - I*B^2)*a^3*b^2*d^5)*n)*x^
2 + 6*(2*(I*A*B + I*B^2)*b^5*c^5 + 15*(-I*A*B - I*B^2)*a*b^4*c^4*d + 60*(I*A*B + I*B^2)*a^2*b^3*c^3*d^2 + 30*(
-I*A*B - I*B^2)*a^4*b*c*d^4 + 3*(I*A*B + I*B^2)*a^5*d^5)*n + 5*(108*(I*A^2 + 2*I*A*B + I*B^2)*a^2*b^3*c^2*d^3
+ 72*(I*A^2 + 2*I*A*B + I*B^2)*a^3*b^2*c*d^4 + (-5*I*B^2*b^5*c^4*d + 108*I*B^2*a*b^4*c^3*d^2 + 216*I*B^2*a^2*b
^3*c^2*d^3 + 144*I*B^2*a^3*b^2*c*d^4 + 27*I*B^2*a^4*b*d^5)*n^2 + 6*((-I*A*B - I*B^2)*b^5*c^4*d + 12*(I*A*B + I
*B^2)*a*b^4*c^3*d^2 + 36*(I*A*B + I*B^2)*a^2*b^...

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(b*g*x+a*g)**4/(d*i*x+c*i)**3,x)

[Out]

Timed out

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^4/(d*i*x+c*i)^3,x, algorithm="giac")

[Out]

integrate((B*log(((b*x + a)/(d*x + c))^n*e) + A)^2/((b*g*x + a*g)^4*(I*d*x + I*c)^3), x)

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Mupad [B]
time = 13.70, size = 2500, normalized size = 2.75 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log(e*((a + b*x)/(c + d*x))^n)*((x*((a*d + b*c)*(20*A*B*a*b*d^2 + 10*A*B*b^2*c*d - (70*B^2*a*b*d^2*n)/3 + (10*
B^2*b^2*c*d*n)/3) + a*c*(30*A*B*b^2*d^2 - 20*B^2*b^2*d^2*n) + (5*B^2*a^2*b*d^3*n)/6 + (5*B^2*b^3*c^2*d*n)/6 -
5*A*B*a^2*b*d^3 - 5*A*B*b^3*c^2*d + 10*A*B*a*b^2*c*d^2 - (5*B^2*a*b^2*c*d^2*n)/3) + x^2*((a*d + b*c)*(30*A*B*b
^2*d^2 - 20*B^2*b^2*d^2*n) + b*d*(20*A*B*a*b*d^2 + 10*A*B*b^2*c*d - (70*B^2*a*b*d^2*n)/3 + (10*B^2*b^2*c*d*n)/
3)) + a*c*(20*A*B*a*b*d^2 + 10*A*B*b^2*c*d - (70*B^2*a*b*d^2*n)/3 + (10*B^2*b^2*c*d*n)/3) - 3*A*B*a^3*d^3 - 2*
A*B*b^3*c^3 + b*d*x^3*(30*A*B*b^2*d^2 - 20*B^2*b^2*d^2*n) + (3*B^2*a^3*d^3*n)/2 - (2*B^2*b^3*c^3*n)/3 + A*B*a*
b^2*c^2*d + 4*A*B*a^2*b*c*d^2 + (17*B^2*a*b^2*c^2*d*n)/6 - (11*B^2*a^2*b*c*d^2*n)/3)/(x^5*(3*a^4*b^3*d^6*g^4*i
^3 + 3*b^7*c^4*d^2*g^4*i^3 - 12*a*b^6*c^3*d^3*g^4*i^3 - 12*a^3*b^4*c*d^5*g^4*i^3 + 18*a^2*b^5*c^2*d^4*g^4*i^3)
+ x*(9*a^2*b^5*c^6*g^4*i^3 + 6*a^7*c*d^5*g^4*i^3 - 30*a^3*b^4*c^5*d*g^4*i^3 - 15*a^6*b*c^2*d^4*g^4*i^3 + 30*a
^4*b^3*c^4*d^2*g^4*i^3) + x^2*(3*a^7*d^6*g^4*i^3 + 9*a*b^6*c^6*g^4*i^3 + 6*a^6*b*c*d^5*g^4*i^3 - 18*a^2*b^5*c^
5*d*g^4*i^3 - 15*a^3*b^4*c^4*d^2*g^4*i^3 + 60*a^4*b^3*c^3*d^3*g^4*i^3 - 45*a^5*b^2*c^2*d^4*g^4*i^3) + x^3*(3*b
^7*c^6*g^4*i^3 + 9*a^6*b*d^6*g^4*i^3 + 6*a*b^6*c^5*d*g^4*i^3 - 18*a^5*b^2*c*d^5*g^4*i^3 - 45*a^2*b^5*c^4*d^2*g
^4*i^3 + 60*a^3*b^4*c^3*d^3*g^4*i^3 - 15*a^4*b^3*c^2*d^4*g^4*i^3) + x^4*(9*a^5*b^2*d^6*g^4*i^3 + 6*b^7*c^5*d*g
^4*i^3 - 15*a*b^6*c^4*d^2*g^4*i^3 - 30*a^4*b^3*c*d^5*g^4*i^3 + 30*a^3*b^4*c^2*d^4*g^4*i^3) + 3*a^3*b^4*c^6*g^4
*i^3 + 3*a^7*c^2*d^4*g^4*i^3 - 12*a^4*b^3*c^5*d*g^4*i^3 - 12*a^6*b*c^3*d^3*g^4*i^3 + 18*a^5*b^2*c^4*d^2*g^4*i^
3) + (20*B*b^2*d^3*(3*A + B*n)*(x^2*((3*g^4*i^3*n*(a*d + b*c)^2*(a*d - b*c)^5)/d + 6*a*b*c*g^4*i^3*n*(a*d - b*
c)^5) + 6*b*g^4*i^3*n*x^3*(a*d + b*c)*(a*d - b*c)^5 + 3*b^2*d*g^4*i^3*n*x^4*(a*d - b*c)^5 + (3*a^2*c^2*g^4*i^3
*n*(a*d - b*c)^5)/d + (6*a*c*g^4*i^3*n*x*(a*d + b*c)*(a*d - b*c)^5)/d))/(3*g^4*i^3*n*(a*d - b*c)^6*(x^5*(3*a^4
*b^3*d^6*g^4*i^3 + 3*b^7*c^4*d^2*g^4*i^3 - 12*a*b^6*c^3*d^3*g^4*i^3 - 12*a^3*b^4*c*d^5*g^4*i^3 + 18*a^2*b^5*c^
2*d^4*g^4*i^3) + x*(9*a^2*b^5*c^6*g^4*i^3 + 6*a^7*c*d^5*g^4*i^3 - 30*a^3*b^4*c^5*d*g^4*i^3 - 15*a^6*b*c^2*d^4*
g^4*i^3 + 30*a^4*b^3*c^4*d^2*g^4*i^3) + x^2*(3*a^7*d^6*g^4*i^3 + 9*a*b^6*c^6*g^4*i^3 + 6*a^6*b*c*d^5*g^4*i^3 -
18*a^2*b^5*c^5*d*g^4*i^3 - 15*a^3*b^4*c^4*d^2*g^4*i^3 + 60*a^4*b^3*c^3*d^3*g^4*i^3 - 45*a^5*b^2*c^2*d^4*g^4*i
^3) + x^3*(3*b^7*c^6*g^4*i^3 + 9*a^6*b*d^6*g^4*i^3 + 6*a*b^6*c^5*d*g^4*i^3 - 18*a^5*b^2*c*d^5*g^4*i^3 - 45*a^2
*b^5*c^4*d^2*g^4*i^3 + 60*a^3*b^4*c^3*d^3*g^4*i^3 - 15*a^4*b^3*c^2*d^4*g^4*i^3) + x^4*(9*a^5*b^2*d^6*g^4*i^3 +
6*b^7*c^5*d*g^4*i^3 - 15*a*b^6*c^4*d^2*g^4*i^3 - 30*a^4*b^3*c*d^5*g^4*i^3 + 30*a^3*b^4*c^2*d^4*g^4*i^3) + 3*a
^3*b^4*c^6*g^4*i^3 + 3*a^7*c^2*d^4*g^4*i^3 - 12*a^4*b^3*c^5*d*g^4*i^3 - 12*a^6*b*c^3*d^3*g^4*i^3 + 18*a^5*b^2*
c^4*d^2*g^4*i^3))) + log(e*((a + b*x)/(c + d*x))^n)^2*((x*((5*B^2*(2*a*b*d^2 + b^2*c*d)*(a*d + b*c))/(3*(a^2*d
^2 + b^2*c^2 - 2*a*b*c*d)^2) - (5*B^2*b*d)/(6*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (5*B^2*a*b^2*c*d^2)/(a^2*d^2
+ b^2*c^2 - 2*a*b*c*d)^2) + x^2*((5*B^2*b*d*(2*a*b*d^2 + b^2*c*d))/(3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) + (5*
B^2*b^2*d^2*(a*d + b*c))/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) - (B^2*(3*a*d + 2*b*c))/(6*(a^2*d^2 + b^2*c^2 - 2*
a*b*c*d)) + (5*B^2*a*c*(2*a*b*d^2 + b^2*c*d))/(3*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) + (5*B^2*b^3*d^3*x^3)/(a^2
*d^2 + b^2*c^2 - 2*a*b*c*d)^2)/(x*(2*a^3*c*d*g^4*i^3 + 3*a^2*b*c^2*g^4*i^3) + x^2*(a^3*d^2*g^4*i^3 + 3*a*b^2*c
^2*g^4*i^3 + 6*a^2*b*c*d*g^4*i^3) + x^3*(b^3*c^2*g^4*i^3 + 3*a^2*b*d^2*g^4*i^3 + 6*a*b^2*c*d*g^4*i^3) + x^4*(2
*b^3*c*d*g^4*i^3 + 3*a*b^2*d^2*g^4*i^3) + a^3*c^2*g^4*i^3 + b^3*d^2*g^4*i^3*x^5) - (10*B*b^2*d^3*(3*A + B*n))/
(3*g^4*i^3*n*(a*d - b*c)^6) + (10*B^2*b^2*d^3*(x^2*((g^4*i^3*n*(a*d + b*c)^2*(a*d - b*c))/d + 2*a*b*c*g^4*i^3*
n*(a*d - b*c)) + b^2*d*g^4*i^3*n*x^4*(a*d - b*c) + (a^2*c^2*g^4*i^3*n*(a*d - b*c))/d + 2*b*g^4*i^3*n*x^3*(a*d
+ b*c)*(a*d - b*c) + (2*a*c*g^4*i^3*n*x*(a*d + b*c)*(a*d - b*c))/d))/(g^4*i^3*n*(a*d - b*c)^6*(x*(2*a^3*c*d*g^
4*i^3 + 3*a^2*b*c^2*g^4*i^3) + x^2*(a^3*d^2*g^4*i^3 + 3*a*b^2*c^2*g^4*i^3 + 6*a^2*b*c*d*g^4*i^3) + x^3*(b^3*c^
2*g^4*i^3 + 3*a^2*b*d^2*g^4*i^3 + 6*a*b^2*c*d*g^4*i^3) + x^4*(2*b^3*c*d*g^4*i^3 + 3*a*b^2*d^2*g^4*i^3) + a^3*c
^2*g^4*i^3 + b^3*d^2*g^4*i^3*x^5))) + ((36*A^2*b^4*c^4 - 54*A^2*a^4*d^4 - 27*B^2*a^4*d^4*n^2 + 8*B^2*b^4*c^4*n
^2 + 846*A^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 + 54*A*B*a^4*d^4*n + 24*A*B*b^4*c^4*n
- 127*B^2*a*b^3*c^3*d*n^2 + 1053*B^2*a^3*b*c*d^3*n^2 + 2033*B^2*a^2*b^2*c^2*d^2*n^2 + 1914*A*B*a^2*b^2*c^2*d^
2*n - 246*A*B*a*b^3*c^3*d*n - 1026*A*B*a^3*b*c*d^3*n)/(6*(a*d - b*c)) + (5*x*(54*A^2*a^3*b*d^4 - 18*A^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 + 189*B^2*a^3*b*d^4*n^2 - 19*B^2*b^4*c^3*d*n^2 - 162*A*B*a
^3*b*d^4*n - 30*A*B*b^4*c^3*d*n + 737*B^2*a*b^3*c^2*d^2*n^2 + 1445*B^2*a^2*b^2*c*d^3*n^2 + 618*A*B*a*b^3*c^2*d
^2*n + 150*A*B*a^2*b^2*c*d^3*n))/(6*(a*d - b*c)...

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