3.2.76 \(\int \frac {(c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^5} \, dx\) [176]

Optimal. Leaf size=319 \[ \frac {2 B^2 d i^2 n^2 (c+d x)^3}{27 (b c-a d)^2 g^5 (a+b x)^3}-\frac {b B^2 i^2 n^2 (c+d x)^4}{32 (b c-a d)^2 g^5 (a+b x)^4}+\frac {2 B d i^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 (b c-a d)^2 g^5 (a+b x)^3}-\frac {b B i^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 (b c-a d)^2 g^5 (a+b x)^4}+\frac {d i^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 (b c-a d)^2 g^5 (a+b x)^3}-\frac {b i^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 (b c-a d)^2 g^5 (a+b x)^4} \]

[Out]

2/27*B^2*d*i^2*n^2*(d*x+c)^3/(-a*d+b*c)^2/g^5/(b*x+a)^3-1/32*b*B^2*i^2*n^2*(d*x+c)^4/(-a*d+b*c)^2/g^5/(b*x+a)^
4+2/9*B*d*i^2*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^2/g^5/(b*x+a)^3-1/8*b*B*i^2*n*(d*x+c)^4*(
A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^2/g^5/(b*x+a)^4+1/3*d*i^2*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^
2/(-a*d+b*c)^2/g^5/(b*x+a)^3-1/4*b*i^2*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^2/g^5/(b*x+a)^4

________________________________________________________________________________________

Rubi [A]
time = 0.21, antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {2561, 2395, 2342, 2341} \begin {gather*} -\frac {b i^2 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 g^5 (a+b x)^4 (b c-a d)^2}-\frac {b B i^2 n (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{8 g^5 (a+b x)^4 (b c-a d)^2}+\frac {d i^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 g^5 (a+b x)^3 (b c-a d)^2}+\frac {2 B d i^2 n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 g^5 (a+b x)^3 (b c-a d)^2}-\frac {b B^2 i^2 n^2 (c+d x)^4}{32 g^5 (a+b x)^4 (b c-a d)^2}+\frac {2 B^2 d i^2 n^2 (c+d x)^3}{27 g^5 (a+b x)^3 (b c-a d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a*g + b*g*x)^5,x]

[Out]

(2*B^2*d*i^2*n^2*(c + d*x)^3)/(27*(b*c - a*d)^2*g^5*(a + b*x)^3) - (b*B^2*i^2*n^2*(c + d*x)^4)/(32*(b*c - a*d)
^2*g^5*(a + b*x)^4) + (2*B*d*i^2*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*(b*c - a*d)^2*g^5*(a
+ b*x)^3) - (b*B*i^2*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(8*(b*c - a*d)^2*g^5*(a + b*x)^4)
+ (d*i^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*(b*c - a*d)^2*g^5*(a + b*x)^3) - (b*i^2*(c +
d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*(b*c - a*d)^2*g^5*(a + b*x)^4)

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 {(176 c+176 d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx &=\int \left (\frac {30976 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^5 (a+b x)^5}+\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^5 (a+b x)^4}+\frac {30976 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^5 (a+b x)^3}\right ) \, dx\\ &=\frac {\left (30976 d^2\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^3} \, dx}{b^2 g^5}+\frac {(61952 d (b c-a d)) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^4} \, dx}{b^2 g^5}+\frac {\left (30976 (b c-a d)^2\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^5} \, dx}{b^2 g^5}\\ &=-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}+\frac {\left (30976 B d^2 n\right ) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac {(123904 B d (b c-a d) n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^5}+\frac {\left (15488 B (b c-a d)^2 n\right ) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}+\frac {\left (30976 B d^2 (b c-a d) n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (123904 B d (b c-a d)^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^5}+\frac {\left (15488 B (b c-a d)^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}+\frac {\left (30976 B d^2 (b c-a d) n\right ) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^3}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^5}+\frac {\left (123904 B d (b c-a d)^2 n\right ) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^4}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b^3 g^5}+\frac {\left (15488 B (b c-a d)^3 n\right ) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^5}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 (a+b x)}-\frac {d^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{b^3 g^5}\\ &=-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}+\frac {\left (15488 B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{b^2 g^5}+\frac {\left (30976 B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{b^2 g^5}-\frac {\left (123904 B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{3 b^2 g^5}+\frac {\left (15488 B d^4 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac {\left (30976 B d^4 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac {\left (123904 B d^4 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{3 b^2 (b c-a d)^2 g^5}-\frac {\left (15488 B d^5 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 B d^5 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (123904 B d^5 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (15488 B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^2 (b c-a d) g^5}-\frac {\left (30976 B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{b^2 (b c-a d) g^5}+\frac {\left (123904 B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{3 b^2 (b c-a d) g^5}-\frac {(15488 B d (b c-a d) n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{b^2 g^5}+\frac {(123904 B d (b c-a d) n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{3 b^2 g^5}+\frac {\left (15488 B (b c-a d)^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^5} \, dx}{b^2 g^5}\\ &=-\frac {3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac {7744 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac {15488 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac {15488 B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d)^2 g^5}-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {15488 B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}+\frac {\left (7744 B^2 d^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (15488 B^2 d^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}-\frac {\left (61952 B^2 d^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{3 b^3 g^5}-\frac {\left (15488 B^2 d^4 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (15488 B^2 d^4 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 B^2 d^4 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (30976 B^2 d^4 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (123904 B^2 d^4 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (123904 B^2 d^4 n^2\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (15488 B^2 d^3 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^5}-\frac {\left (30976 B^2 d^3 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^5}+\frac {\left (123904 B^2 d^3 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3 b^3 (b c-a d) g^5}-\frac {\left (15488 B^2 d (b c-a d) n^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^5}+\frac {\left (123904 B^2 d (b c-a d) n^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{9 b^3 g^5}+\frac {\left (3872 B^2 (b c-a d)^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac {3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac {7744 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac {15488 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac {15488 B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d)^2 g^5}-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {15488 B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (15488 B^2 d^3 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^5}-\frac {\left (30976 B^2 d^3 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (123904 B^2 d^3 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3 b^3 g^5}-\frac {\left (15488 B^2 d^4 n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (15488 B^2 d^4 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 B^2 d^4 n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (30976 B^2 d^4 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (123904 B^2 d^4 n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (123904 B^2 d^4 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{3 b^3 (b c-a d)^2 g^5}+\frac {\left (7744 B^2 d^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (15488 B^2 d^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}-\frac {\left (61952 B^2 d^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{3 b^3 g^5}-\frac {\left (15488 B^2 d (b c-a d)^2 n^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^5}+\frac {\left (123904 B^2 d (b c-a d)^2 n^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{9 b^3 g^5}+\frac {\left (3872 B^2 (b c-a d)^3 n^2\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac {3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac {7744 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac {15488 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac {15488 B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d)^2 g^5}-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {15488 B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (15488 B^2 d^3 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^5}-\frac {\left (30976 B^2 d^3 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^5}+\frac {\left (123904 B^2 d^3 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3 b^3 g^5}-\frac {\left (15488 B^2 d^4 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac {\left (15488 B^2 d^4 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac {\left (30976 B^2 d^4 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac {\left (30976 B^2 d^4 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac {\left (123904 B^2 d^4 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3 b^2 (b c-a d)^2 g^5}-\frac {\left (123904 B^2 d^4 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 b^2 (b c-a d)^2 g^5}+\frac {\left (15488 B^2 d^5 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (15488 B^2 d^5 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (30976 B^2 d^5 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 B^2 d^5 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (123904 B^2 d^5 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 b^3 (b c-a d)^2 g^5}+\frac {\left (123904 B^2 d^5 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b^3 (b c-a d)^2 g^5}+\frac {\left (7744 B^2 d^2 (b c-a d) n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^5}+\frac {\left (15488 B^2 d^2 (b c-a d) n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^5}-\frac {\left (61952 B^2 d^2 (b c-a d) n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{3 b^3 g^5}-\frac {\left (15488 B^2 d (b c-a d)^2 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b^3 g^5}+\frac {\left (123904 B^2 d (b c-a d)^2 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{9 b^3 g^5}+\frac {\left (3872 B^2 (b c-a d)^3 n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{b^3 g^5}\\ &=-\frac {968 B^2 (b c-a d)^2 n^2}{b^3 g^5 (a+b x)^4}-\frac {42592 B^2 d (b c-a d) n^2}{27 b^3 g^5 (a+b x)^3}+\frac {9680 B^2 d^2 n^2}{9 b^3 g^5 (a+b x)^2}+\frac {27104 B^2 d^3 n^2}{9 b^3 (b c-a d) g^5 (a+b x)}+\frac {27104 B^2 d^4 n^2 \log (a+b x)}{9 b^3 (b c-a d)^2 g^5}-\frac {3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac {7744 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac {15488 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac {15488 B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d)^2 g^5}-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {27104 B^2 d^4 n^2 \log (c+d x)}{9 b^3 (b c-a d)^2 g^5}+\frac {15488 B^2 d^4 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac {15488 B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}+\frac {15488 B^2 d^4 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (15488 B^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (15488 B^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 B^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 B^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}+\frac {\left (123904 B^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 b^3 (b c-a d)^2 g^5}+\frac {\left (123904 B^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (15488 B^2 d^4 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac {\left (30976 B^2 d^4 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac {\left (123904 B^2 d^4 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 b^2 (b c-a d)^2 g^5}-\frac {\left (15488 B^2 d^5 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 B^2 d^5 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (123904 B^2 d^5 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b^3 (b c-a d)^2 g^5}\\ &=-\frac {968 B^2 (b c-a d)^2 n^2}{b^3 g^5 (a+b x)^4}-\frac {42592 B^2 d (b c-a d) n^2}{27 b^3 g^5 (a+b x)^3}+\frac {9680 B^2 d^2 n^2}{9 b^3 g^5 (a+b x)^2}+\frac {27104 B^2 d^3 n^2}{9 b^3 (b c-a d) g^5 (a+b x)}+\frac {27104 B^2 d^4 n^2 \log (a+b x)}{9 b^3 (b c-a d)^2 g^5}-\frac {7744 B^2 d^4 n^2 \log ^2(a+b x)}{3 b^3 (b c-a d)^2 g^5}-\frac {3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac {7744 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac {15488 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac {15488 B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d)^2 g^5}-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {27104 B^2 d^4 n^2 \log (c+d x)}{9 b^3 (b c-a d)^2 g^5}+\frac {15488 B^2 d^4 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac {15488 B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac {7744 B^2 d^4 n^2 \log ^2(c+d x)}{3 b^3 (b c-a d)^2 g^5}+\frac {15488 B^2 d^4 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (15488 B^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (15488 B^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 B^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 B^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}+\frac {\left (123904 B^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3 b^3 (b c-a d)^2 g^5}+\frac {\left (123904 B^2 d^4 n^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b^3 (b c-a d)^2 g^5}\\ &=-\frac {968 B^2 (b c-a d)^2 n^2}{b^3 g^5 (a+b x)^4}-\frac {42592 B^2 d (b c-a d) n^2}{27 b^3 g^5 (a+b x)^3}+\frac {9680 B^2 d^2 n^2}{9 b^3 g^5 (a+b x)^2}+\frac {27104 B^2 d^3 n^2}{9 b^3 (b c-a d) g^5 (a+b x)}+\frac {27104 B^2 d^4 n^2 \log (a+b x)}{9 b^3 (b c-a d)^2 g^5}-\frac {7744 B^2 d^4 n^2 \log ^2(a+b x)}{3 b^3 (b c-a d)^2 g^5}-\frac {3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac {7744 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac {15488 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac {15488 B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d)^2 g^5}-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {27104 B^2 d^4 n^2 \log (c+d x)}{9 b^3 (b c-a d)^2 g^5}+\frac {15488 B^2 d^4 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac {15488 B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac {7744 B^2 d^4 n^2 \log ^2(c+d x)}{3 b^3 (b c-a d)^2 g^5}+\frac {15488 B^2 d^4 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d)^2 g^5}+\frac {15488 B^2 d^4 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3 b^3 (b c-a d)^2 g^5}+\frac {15488 B^2 d^4 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d)^2 g^5}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 1.85, size = 1860, normalized size = 5.83 \begin {gather*} -\frac {i^2 \left (216 (b c-a d)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2-576 d (-b c+a d)^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+432 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2+216 B d^2 n (a+b x)^2 \left (2 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+4 d (-b c+a d) (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+4 d^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-4 B d n (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+B n \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B d^2 n (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )-2 B d^2 n (a+b x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )+32 B d n (a+b x) \left (12 (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-18 d (b c-a d)^2 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+36 d^2 (b c-a d) (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+36 d^3 (a+b x)^3 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-36 d^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)+36 B d^2 n (a+b x)^2 (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-9 B d n (a+b x) \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B n \left (2 (b c-a d)^3-3 d (b c-a d)^2 (a+b x)+6 d^2 (b c-a d) (a+b x)^2+6 d^3 (a+b x)^3 \log (a+b x)-6 d^3 (a+b x)^3 \log (c+d x)\right )-18 B d^3 n (a+b x)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )+18 B d^3 n (a+b x)^3 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )+3 B n \left (36 (b c-a d)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+48 d (-b c+a d)^3 (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+72 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+144 d^3 (-b c+a d) (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )-144 d^4 (a+b x)^4 \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )+144 d^4 (a+b x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)-144 B d^3 n (a+b x)^3 (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+36 B d^2 n (a+b x)^2 \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )-8 B d n (a+b x) \left (2 (b c-a d)^3-3 d (b c-a d)^2 (a+b x)+6 d^2 (b c-a d) (a+b x)^2+6 d^3 (a+b x)^3 \log (a+b x)-6 d^3 (a+b x)^3 \log (c+d x)\right )+3 B n \left (3 (b c-a d)^4+4 d (-b c+a d)^3 (a+b x)+6 d^2 (b c-a d)^2 (a+b x)^2+12 d^3 (-b c+a d) (a+b x)^3-12 d^4 (a+b x)^4 \log (a+b x)+12 d^4 (a+b x)^4 \log (c+d x)\right )+72 B d^4 n (a+b x)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )\right )-72 B d^4 n (a+b x)^4 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )\right )}{864 b^3 (b c-a d)^2 g^5 (a+b x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(a*g + b*g*x)^5,x]

[Out]

-1/864*(i^2*(216*(b*c - a*d)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - 576*d*(-(b*c) + a*d)^3*(a + b*x)*(A
+ B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 432*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n
])^2 + 216*B*d^2*n*(a + b*x)^2*(2*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 4*d*(-(b*c) + a*d)*(a
+ b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 4*d^2*(a + b*x)^2*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d
*x))^n]) + 4*d^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - 4*B*d*n*(a + b*x)*(b*c - a*
d + d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) + B*n*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x) -
2*d^2*(a + b*x)^2*Log[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d*x]) + 2*B*d^2*n*(a + b*x)^2*(Log[a + b*x]*(Log[a
+ b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) - 2*B*d^2*n*(a + b*x
)^2*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d
)])) + 32*B*d*n*(a + b*x)*(12*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 18*d*(b*c - a*d)^2*(a + b
*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 36*d^2*(b*c - a*d)*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))
^n]) + 36*d^3*(a + b*x)^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 36*d^3*(a + b*x)^3*(A + B*Log[
e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 36*B*d^2*n*(a + b*x)^2*(b*c - a*d + d*(a + b*x)*Log[a + b*x] - d*(a
+ b*x)*Log[c + d*x]) - 9*B*d*n*(a + b*x)*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x) - 2*d^2*(a + b*x)^2*Lo
g[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d*x]) + 2*B*n*(2*(b*c - a*d)^3 - 3*d*(b*c - a*d)^2*(a + b*x) + 6*d^2*(b
*c - a*d)*(a + b*x)^2 + 6*d^3*(a + b*x)^3*Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[c + d*x]) - 18*B*d^3*n*(a + b*x
)^3*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d
)]) + 18*B*d^3*n*(a + b*x)^3*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2,
(b*(c + d*x))/(b*c - a*d)])) + 3*B*n*(36*(b*c - a*d)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 48*d*(-(b*c)
+ a*d)^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 72*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[e*((a
+ b*x)/(c + d*x))^n]) + 144*d^3*(-(b*c) + a*d)*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 144*d^4*(a
+ b*x)^4*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 144*d^4*(a + b*x)^4*(A + B*Log[e*((a + b*x)/(c
+ d*x))^n])*Log[c + d*x] - 144*B*d^3*n*(a + b*x)^3*(b*c - a*d + d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c
+ d*x]) + 36*B*d^2*n*(a + b*x)^2*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x) - 2*d^2*(a + b*x)^2*Log[a + b*x
] + 2*d^2*(a + b*x)^2*Log[c + d*x]) - 8*B*d*n*(a + b*x)*(2*(b*c - a*d)^3 - 3*d*(b*c - a*d)^2*(a + b*x) + 6*d^2
*(b*c - a*d)*(a + b*x)^2 + 6*d^3*(a + b*x)^3*Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[c + d*x]) + 3*B*n*(3*(b*c -
a*d)^4 + 4*d*(-(b*c) + a*d)^3*(a + b*x) + 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 12*d^3*(-(b*c) + a*d)*(a + b*x)^3
- 12*d^4*(a + b*x)^4*Log[a + b*x] + 12*d^4*(a + b*x)^4*Log[c + d*x]) + 72*B*d^4*n*(a + b*x)^4*(Log[a + b*x]*(L
og[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) - 72*B*d^4*n*(a
+ b*x)^4*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c
- a*d)]))))/(b^3*(b*c - a*d)^2*g^5*(a + b*x)^4)

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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\left (d i x +c i \right )^{2} \left (A +B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )\right )^{2}}{\left (b g x +a g \right )^{5}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^5,x)

[Out]

int((d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^5,x)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 8051 vs. \(2 (293) = 586\).
time = 1.06, size = 8051, normalized size = 25.24 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^5,x, algorithm="maxima")

[Out]

-1/24*A*B*c^2*n*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7
*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2
- a^3*b^5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*
c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*
c*d^2 - a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) + 12*d^4*log(b
*x + a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c
)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5)) + 1/72*A*B*d^2*n*((13*a^2
*b^3*c^3 - 75*a^3*b^2*c^2*d + 33*a^4*b*c*d^2 - 7*a^5*d^3 - 12*(6*b^5*c^2*d - 4*a*b^4*c*d^2 + a^2*b^3*d^3)*x^3
+ 6*(6*b^5*c^3 - 46*a*b^4*c^2*d + 29*a^2*b^3*c*d^2 - 7*a^3*b^2*d^3)*x^2 + 4*(10*a*b^4*c^3 - 63*a^2*b^3*c^2*d +
33*a^3*b^2*c*d^2 - 7*a^4*b*d^3)*x)/((b^10*c^3 - 3*a*b^9*c^2*d + 3*a^2*b^8*c*d^2 - a^3*b^7*d^3)*g^5*x^4 + 4*(a
*b^9*c^3 - 3*a^2*b^8*c^2*d + 3*a^3*b^7*c*d^2 - a^4*b^6*d^3)*g^5*x^3 + 6*(a^2*b^8*c^3 - 3*a^3*b^7*c^2*d + 3*a^4
*b^6*c*d^2 - a^5*b^5*d^3)*g^5*x^2 + 4*(a^3*b^7*c^3 - 3*a^4*b^6*c^2*d + 3*a^5*b^5*c*d^2 - a^6*b^4*d^3)*g^5*x +
(a^4*b^6*c^3 - 3*a^5*b^5*c^2*d + 3*a^6*b^4*c*d^2 - a^7*b^3*d^3)*g^5) - 12*(6*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d
^4)*log(b*x + a)/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^5) + 12*(6*b
^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(d*x + c)/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d
^3 + a^4*b^3*d^4)*g^5)) + 1/36*A*B*c*d*n*((7*a*b^3*c^3 - 33*a^2*b^2*c^2*d + 75*a^3*b*c*d^2 - 13*a^4*d^3 + 12*(
4*b^4*c*d^2 - a*b^3*d^3)*x^3 - 6*(4*b^4*c^2*d - 29*a*b^3*c*d^2 + 7*a^2*b^2*d^3)*x^2 + 4*(4*b^4*c^3 - 21*a*b^3*
c^2*d + 57*a^2*b^2*c*d^2 - 13*a^3*b*d^3)*x)/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4
+ 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d
+ 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g
^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*g^5) + 12*(4*b*c*d^3 - a*d^4)*log(b*x +
a)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5) - 12*(4*b*c*d^3 - a*d^
4)*log(d*x + c)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5)) + 1/6*(4*
b*x + a)*B^2*c*d*log((b*x/(d*x + c) + a/(d*x + c))^n*e)^2/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2 +
4*a^3*b^3*g^5*x + a^4*b^2*g^5) + 1/12*(6*b^2*x^2 + 4*a*b*x + a^2)*B^2*d^2*log((b*x/(d*x + c) + a/(d*x + c))^n
*e)^2/(b^7*g^5*x^4 + 4*a*b^6*g^5*x^3 + 6*a^2*b^5*g^5*x^2 + 4*a^3*b^4*g^5*x + a^4*b^3*g^5) - 1/288*(12*n*((12*b
^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b
^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*g^5*x^4
+ 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d
+ 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*g
^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 4*
a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*c^4 - 4*a*b^4*
c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5))*log((b*x/(d*x + c) + a/(d*x + c))^n*e) - (9*b^4
*c^4 - 64*a*b^3*c^3*d + 216*a^2*b^2*c^2*d^2 - 576*a^3*b*c*d^3 + 415*a^4*d^4 - 300*(b^4*c*d^3 - a*b^3*d^4)*x^3
+ 6*(13*b^4*c^2*d^2 - 176*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d
^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a)^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a
^3*b*d^4*x + a^4*d^4)*log(d*x + c)^2 - 4*(7*b^4*c^3*d - 60*a*b^3*c^2*d^2 + 324*a^2*b^2*c*d^3 - 271*a^3*b*d^4)*
x - 300*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a) + 12*(25*b^
4*d^4*x^4 + 100*a*b^3*d^4*x^3 + 150*a^2*b^2*d^4*x^2 + 100*a^3*b*d^4*x + 25*a^4*d^4 - 12*(b^4*d^4*x^4 + 4*a*b^3
*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a))*log(d*x + c))*n^2/(a^4*b^5*c^4*g^5 - 4*a
^5*b^4*c^3*d*g^5 + 6*a^6*b^3*c^2*d^2*g^5 - 4*a^7*b^2*c*d^3*g^5 + a^8*b*d^4*g^5 + (b^9*c^4*g^5 - 4*a*b^8*c^3*d*
g^5 + 6*a^2*b^7*c^2*d^2*g^5 - 4*a^3*b^6*c*d^3*g^5 + a^4*b^5*d^4*g^5)*x^4 + 4*(a*b^8*c^4*g^5 - 4*a^2*b^7*c^3*d*
g^5 + 6*a^3*b^6*c^2*d^2*g^5 - 4*a^4*b^5*c*d^3*g^5 + a^5*b^4*d^4*g^5)*x^3 + 6*(a^2*b^7*c^4*g^5 - 4*a^3*b^6*c^3*
d*g^5 + 6*a^4*b^5*c^2*d^2*g^5 - 4*a^5*b^4*c*d^3*g^5 + a^6*b^3*d^4*g^5)*x^2 + 4*(a^3*b^6*c^4*g^5 - 4*a^4*b^5*c^
3*d*g^5 + 6*a^5*b^4*c^2*d^2*g^5 - 4*a^6*b^3*c*d^3*g^5 + a^7*b^2*d^4*g^5)*x))*B^2*c^2 + 1/432*(12*n*((7*a*b^3*c
^3 - 33*a^2*b^2*c^2*d + 75*a^3*b*c*d^2 - 13*a^4...

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1219 vs. \(2 (293) = 586\).
time = 0.45, size = 1219, normalized size = 3.82 \begin {gather*} \frac {216 \, {\left (A^{2} + 2 \, A B + B^{2}\right )} b^{4} c^{4} - 288 \, {\left (A^{2} + 2 \, A B + B^{2}\right )} a b^{3} c^{3} d + 72 \, {\left (A^{2} + 2 \, A B + B^{2}\right )} a^{4} d^{4} - 12 \, {\left (7 \, {\left (B^{2} b^{4} c d^{3} - B^{2} a b^{3} d^{4}\right )} n^{2} + 12 \, {\left ({\left (A B + B^{2}\right )} b^{4} c d^{3} - {\left (A B + B^{2}\right )} a b^{3} d^{4}\right )} n\right )} x^{3} + {\left (27 \, B^{2} b^{4} c^{4} - 64 \, B^{2} a b^{3} c^{3} d + 37 \, B^{2} a^{4} d^{4}\right )} n^{2} + 6 \, {\left (72 \, {\left (A^{2} + 2 \, A B + B^{2}\right )} b^{4} c^{2} d^{2} - 144 \, {\left (A^{2} + 2 \, A B + B^{2}\right )} a b^{3} c d^{3} + 72 \, {\left (A^{2} + 2 \, A B + B^{2}\right )} a^{2} b^{2} d^{4} - {\left (5 \, B^{2} b^{4} c^{2} d^{2} + 32 \, B^{2} a b^{3} c d^{3} - 37 \, B^{2} a^{2} b^{2} d^{4}\right )} n^{2} + 12 \, {\left ({\left (A B + B^{2}\right )} b^{4} c^{2} d^{2} - 8 \, {\left (A B + B^{2}\right )} a b^{3} c d^{3} + 7 \, {\left (A B + B^{2}\right )} a^{2} b^{2} d^{4}\right )} n\right )} x^{2} - 72 \, {\left (B^{2} b^{4} d^{4} n^{2} x^{4} + 4 \, B^{2} a b^{3} d^{4} n^{2} x^{3} - 6 \, {\left (B^{2} b^{4} c^{2} d^{2} - 2 \, B^{2} a b^{3} c d^{3}\right )} n^{2} x^{2} - 4 \, {\left (2 \, B^{2} b^{4} c^{3} d - 3 \, B^{2} a b^{3} c^{2} d^{2}\right )} n^{2} x - {\left (3 \, B^{2} b^{4} c^{4} - 4 \, B^{2} a b^{3} c^{3} d\right )} n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + 12 \, {\left (9 \, {\left (A B + B^{2}\right )} b^{4} c^{4} - 16 \, {\left (A B + B^{2}\right )} a b^{3} c^{3} d + 7 \, {\left (A B + B^{2}\right )} a^{4} d^{4}\right )} n + 4 \, {\left (144 \, {\left (A^{2} + 2 \, A B + B^{2}\right )} b^{4} c^{3} d - 216 \, {\left (A^{2} + 2 \, A B + B^{2}\right )} a b^{3} c^{2} d^{2} + 72 \, {\left (A^{2} + 2 \, A B + B^{2}\right )} a^{3} b d^{4} + {\left (11 \, B^{2} b^{4} c^{3} d - 48 \, B^{2} a b^{3} c^{2} d^{2} + 37 \, B^{2} a^{3} b d^{4}\right )} n^{2} + 12 \, {\left (5 \, {\left (A B + B^{2}\right )} b^{4} c^{3} d - 12 \, {\left (A B + B^{2}\right )} a b^{3} c^{2} d^{2} + 7 \, {\left (A B + B^{2}\right )} a^{3} b d^{4}\right )} n\right )} x - 12 \, {\left ({\left (7 \, B^{2} b^{4} d^{4} n^{2} + 12 \, {\left (A B + B^{2}\right )} b^{4} d^{4} n\right )} x^{4} + 4 \, {\left (12 \, {\left (A B + B^{2}\right )} a b^{3} d^{4} n + {\left (3 \, B^{2} b^{4} c d^{3} + 4 \, B^{2} a b^{3} d^{4}\right )} n^{2}\right )} x^{3} - {\left (9 \, B^{2} b^{4} c^{4} - 16 \, B^{2} a b^{3} c^{3} d\right )} n^{2} - 6 \, {\left ({\left (B^{2} b^{4} c^{2} d^{2} - 8 \, B^{2} a b^{3} c d^{3}\right )} n^{2} + 12 \, {\left ({\left (A B + B^{2}\right )} b^{4} c^{2} d^{2} - 2 \, {\left (A B + B^{2}\right )} a b^{3} c d^{3}\right )} n\right )} x^{2} - 12 \, {\left (3 \, {\left (A B + B^{2}\right )} b^{4} c^{4} - 4 \, {\left (A B + B^{2}\right )} a b^{3} c^{3} d\right )} n - 4 \, {\left ({\left (5 \, B^{2} b^{4} c^{3} d - 12 \, B^{2} a b^{3} c^{2} d^{2}\right )} n^{2} + 12 \, {\left (2 \, {\left (A B + B^{2}\right )} b^{4} c^{3} d - 3 \, {\left (A B + B^{2}\right )} a b^{3} c^{2} d^{2}\right )} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{864 \, {\left ({\left (b^{9} c^{2} - 2 \, a b^{8} c d + a^{2} b^{7} d^{2}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{2} - 2 \, a^{2} b^{7} c d + a^{3} b^{6} d^{2}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{2} - 2 \, a^{3} b^{6} c d + a^{4} b^{5} d^{2}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{2} - 2 \, a^{4} b^{5} c d + a^{5} b^{4} d^{2}\right )} g^{5} x + {\left (a^{4} b^{5} c^{2} - 2 \, a^{5} b^{4} c d + a^{6} b^{3} d^{2}\right )} g^{5}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^5,x, algorithm="fricas")

[Out]

1/864*(216*(A^2 + 2*A*B + B^2)*b^4*c^4 - 288*(A^2 + 2*A*B + B^2)*a*b^3*c^3*d + 72*(A^2 + 2*A*B + B^2)*a^4*d^4
- 12*(7*(B^2*b^4*c*d^3 - B^2*a*b^3*d^4)*n^2 + 12*((A*B + B^2)*b^4*c*d^3 - (A*B + B^2)*a*b^3*d^4)*n)*x^3 + (27*
B^2*b^4*c^4 - 64*B^2*a*b^3*c^3*d + 37*B^2*a^4*d^4)*n^2 + 6*(72*(A^2 + 2*A*B + B^2)*b^4*c^2*d^2 - 144*(A^2 + 2*
A*B + B^2)*a*b^3*c*d^3 + 72*(A^2 + 2*A*B + B^2)*a^2*b^2*d^4 - (5*B^2*b^4*c^2*d^2 + 32*B^2*a*b^3*c*d^3 - 37*B^2
*a^2*b^2*d^4)*n^2 + 12*((A*B + B^2)*b^4*c^2*d^2 - 8*(A*B + B^2)*a*b^3*c*d^3 + 7*(A*B + B^2)*a^2*b^2*d^4)*n)*x^
2 - 72*(B^2*b^4*d^4*n^2*x^4 + 4*B^2*a*b^3*d^4*n^2*x^3 - 6*(B^2*b^4*c^2*d^2 - 2*B^2*a*b^3*c*d^3)*n^2*x^2 - 4*(2
*B^2*b^4*c^3*d - 3*B^2*a*b^3*c^2*d^2)*n^2*x - (3*B^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d)*n^2)*log((b*x + a)/(d*x + c)
)^2 + 12*(9*(A*B + B^2)*b^4*c^4 - 16*(A*B + B^2)*a*b^3*c^3*d + 7*(A*B + B^2)*a^4*d^4)*n + 4*(144*(A^2 + 2*A*B
+ B^2)*b^4*c^3*d - 216*(A^2 + 2*A*B + B^2)*a*b^3*c^2*d^2 + 72*(A^2 + 2*A*B + B^2)*a^3*b*d^4 + (11*B^2*b^4*c^3*
d - 48*B^2*a*b^3*c^2*d^2 + 37*B^2*a^3*b*d^4)*n^2 + 12*(5*(A*B + B^2)*b^4*c^3*d - 12*(A*B + B^2)*a*b^3*c^2*d^2
+ 7*(A*B + B^2)*a^3*b*d^4)*n)*x - 12*((7*B^2*b^4*d^4*n^2 + 12*(A*B + B^2)*b^4*d^4*n)*x^4 + 4*(12*(A*B + B^2)*a
*b^3*d^4*n + (3*B^2*b^4*c*d^3 + 4*B^2*a*b^3*d^4)*n^2)*x^3 - (9*B^2*b^4*c^4 - 16*B^2*a*b^3*c^3*d)*n^2 - 6*((B^2
*b^4*c^2*d^2 - 8*B^2*a*b^3*c*d^3)*n^2 + 12*((A*B + B^2)*b^4*c^2*d^2 - 2*(A*B + B^2)*a*b^3*c*d^3)*n)*x^2 - 12*(
3*(A*B + B^2)*b^4*c^4 - 4*(A*B + B^2)*a*b^3*c^3*d)*n - 4*((5*B^2*b^4*c^3*d - 12*B^2*a*b^3*c^2*d^2)*n^2 + 12*(2
*(A*B + B^2)*b^4*c^3*d - 3*(A*B + B^2)*a*b^3*c^2*d^2)*n)*x)*log((b*x + a)/(d*x + c)))/((b^9*c^2 - 2*a*b^8*c*d
+ a^2*b^7*d^2)*g^5*x^4 + 4*(a*b^8*c^2 - 2*a^2*b^7*c*d + a^3*b^6*d^2)*g^5*x^3 + 6*(a^2*b^7*c^2 - 2*a^3*b^6*c*d
+ a^4*b^5*d^2)*g^5*x^2 + 4*(a^3*b^6*c^2 - 2*a^4*b^5*c*d + a^5*b^4*d^2)*g^5*x + (a^4*b^5*c^2 - 2*a^5*b^4*c*d +
a^6*b^3*d^2)*g^5)

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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((d*i*x+c*i)**2*(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(b*g*x+a*g)**5,x)

[Out]

Timed out

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Giac [A]
time = 10.64, size = 461, normalized size = 1.45 \begin {gather*} \frac {1}{864} \, {\left (\frac {72 \, {\left (3 \, B^{2} b n^{2} - \frac {4 \, {\left (b x + a\right )} B^{2} d n^{2}}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{\frac {{\left (b x + a\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x + a\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {12 \, {\left (9 \, B^{2} b n^{2} - \frac {16 \, {\left (b x + a\right )} B^{2} d n^{2}}{d x + c} + 36 \, A B b n + 36 \, B^{2} b n - \frac {48 \, {\left (b x + a\right )} A B d n}{d x + c} - \frac {48 \, {\left (b x + a\right )} B^{2} d n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x + a\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {27 \, B^{2} b n^{2} - \frac {64 \, {\left (b x + a\right )} B^{2} d n^{2}}{d x + c} + 108 \, A B b n + 108 \, B^{2} b n - \frac {192 \, {\left (b x + a\right )} A B d n}{d x + c} - \frac {192 \, {\left (b x + a\right )} B^{2} d n}{d x + c} + 216 \, A^{2} b + 432 \, A B b + 216 \, B^{2} b - \frac {288 \, {\left (b x + a\right )} A^{2} d}{d x + c} - \frac {576 \, {\left (b x + a\right )} A B d}{d x + c} - \frac {288 \, {\left (b x + a\right )} B^{2} d}{d x + c}}{\frac {{\left (b x + a\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x + a\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^5,x, algorithm="giac")

[Out]

1/864*(72*(3*B^2*b*n^2 - 4*(b*x + a)*B^2*d*n^2/(d*x + c))*log((b*x + a)/(d*x + c))^2/((b*x + a)^4*b*c*g^5/(d*x
+ c)^4 - (b*x + a)^4*a*d*g^5/(d*x + c)^4) + 12*(9*B^2*b*n^2 - 16*(b*x + a)*B^2*d*n^2/(d*x + c) + 36*A*B*b*n +
36*B^2*b*n - 48*(b*x + a)*A*B*d*n/(d*x + c) - 48*(b*x + a)*B^2*d*n/(d*x + c))*log((b*x + a)/(d*x + c))/((b*x
+ a)^4*b*c*g^5/(d*x + c)^4 - (b*x + a)^4*a*d*g^5/(d*x + c)^4) + (27*B^2*b*n^2 - 64*(b*x + a)*B^2*d*n^2/(d*x +
c) + 108*A*B*b*n + 108*B^2*b*n - 192*(b*x + a)*A*B*d*n/(d*x + c) - 192*(b*x + a)*B^2*d*n/(d*x + c) + 216*A^2*b
+ 432*A*B*b + 216*B^2*b - 288*(b*x + a)*A^2*d/(d*x + c) - 576*(b*x + a)*A*B*d/(d*x + c) - 288*(b*x + a)*B^2*d
/(d*x + c))/((b*x + a)^4*b*c*g^5/(d*x + c)^4 - (b*x + a)^4*a*d*g^5/(d*x + c)^4))*(b*c/(b*c - a*d)^2 - a*d/(b*c
- a*d)^2)

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Mupad [B]
time = 9.42, size = 1934, normalized size = 6.06 \begin {gather*} -\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {a\,\left (-\frac {a\,n\,B^2\,d^2\,i^2}{2}+\frac {b\,c\,n\,B^2\,d\,i^2}{2}+A\,a\,B\,d^2\,i^2+2\,A\,b\,c\,B\,d\,i^2\right )+x\,\left (b\,\left (-\frac {a\,n\,B^2\,d^2\,i^2}{2}+\frac {b\,c\,n\,B^2\,d\,i^2}{2}+A\,a\,B\,d^2\,i^2+2\,A\,b\,c\,B\,d\,i^2\right )+3\,A\,B\,a\,b\,d^2\,i^2+6\,A\,B\,b^2\,c\,d\,i^2-\frac {3\,B^2\,a\,b\,d^2\,i^2\,n}{2}+\frac {3\,B^2\,b^2\,c\,d\,i^2\,n}{2}\right )+3\,A\,B\,b^2\,c^2\,i^2-B^2\,a^2\,d^2\,i^2\,n+\frac {B^2\,b^2\,c^2\,i^2\,n}{2}+6\,A\,B\,b^2\,d^2\,i^2\,x^2+\frac {B^2\,a\,b\,c\,d\,i^2\,n}{2}}{6\,a^4\,b^3\,g^5+24\,a^3\,b^4\,g^5\,x+36\,a^2\,b^5\,g^5\,x^2+24\,a\,b^6\,g^5\,x^3+6\,b^7\,g^5\,x^4}+\frac {B^2\,d^4\,i^2\,\left (x^2\,\left (b\,\left (b\,\left (\frac {3\,a\,b^3\,g^5\,n\,\left (a\,d-b\,c\right )}{2\,d}+\frac {b^3\,g^5\,n\,\left (a\,d-b\,c\right )\,\left (4\,a\,d-b\,c\right )}{2\,d^2}\right )+\frac {3\,a\,b^4\,g^5\,n\,\left (a\,d-b\,c\right )}{d}+\frac {b^4\,g^5\,n\,\left (a\,d-b\,c\right )\,\left (4\,a\,d-b\,c\right )}{d^2}\right )+\frac {9\,a\,b^5\,g^5\,n\,\left (a\,d-b\,c\right )}{2\,d}+\frac {3\,b^5\,g^5\,n\,\left (a\,d-b\,c\right )\,\left (4\,a\,d-b\,c\right )}{2\,d^2}\right )+a\,\left (a\,\left (\frac {3\,a\,b^3\,g^5\,n\,\left (a\,d-b\,c\right )}{2\,d}+\frac {b^3\,g^5\,n\,\left (a\,d-b\,c\right )\,\left (4\,a\,d-b\,c\right )}{2\,d^2}\right )+\frac {b^3\,g^5\,n\,\left (a\,d-b\,c\right )\,\left (6\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2\right )}{2\,d^3}\right )+x\,\left (a\,\left (b\,\left (\frac {3\,a\,b^3\,g^5\,n\,\left (a\,d-b\,c\right )}{2\,d}+\frac {b^3\,g^5\,n\,\left (a\,d-b\,c\right )\,\left (4\,a\,d-b\,c\right )}{2\,d^2}\right )+\frac {3\,a\,b^4\,g^5\,n\,\left (a\,d-b\,c\right )}{d}+\frac {b^4\,g^5\,n\,\left (a\,d-b\,c\right )\,\left (4\,a\,d-b\,c\right )}{d^2}\right )+b\,\left (a\,\left (\frac {3\,a\,b^3\,g^5\,n\,\left (a\,d-b\,c\right )}{2\,d}+\frac {b^3\,g^5\,n\,\left (a\,d-b\,c\right )\,\left (4\,a\,d-b\,c\right )}{2\,d^2}\right )+\frac {b^3\,g^5\,n\,\left (a\,d-b\,c\right )\,\left (6\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2\right )}{2\,d^3}\right )+\frac {3\,b^4\,g^5\,n\,\left (a\,d-b\,c\right )\,\left (6\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2\right )}{2\,d^3}\right )+\frac {3\,b^3\,g^5\,n\,\left (a\,d-b\,c\right )\,\left (4\,a^3\,d^3-6\,a^2\,b\,c\,d^2+4\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{2\,d^4}+\frac {6\,b^6\,g^5\,n\,x^3\,\left (a\,d-b\,c\right )}{d}\right )}{6\,b^3\,g^5\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )\,\left (6\,a^4\,b^3\,g^5+24\,a^3\,b^4\,g^5\,x+36\,a^2\,b^5\,g^5\,x^2+24\,a\,b^6\,g^5\,x^3+6\,b^7\,g^5\,x^4\right )}\right )-\frac {\frac {72\,A^2\,a^3\,d^3\,i^2+72\,A^2\,a^2\,b\,c\,d^2\,i^2+72\,A^2\,a\,b^2\,c^2\,d\,i^2-216\,A^2\,b^3\,c^3\,i^2+84\,A\,B\,a^3\,d^3\,i^2\,n+84\,A\,B\,a^2\,b\,c\,d^2\,i^2\,n+84\,A\,B\,a\,b^2\,c^2\,d\,i^2\,n-108\,A\,B\,b^3\,c^3\,i^2\,n+37\,B^2\,a^3\,d^3\,i^2\,n^2+37\,B^2\,a^2\,b\,c\,d^2\,i^2\,n^2+37\,B^2\,a\,b^2\,c^2\,d\,i^2\,n^2-27\,B^2\,b^3\,c^3\,i^2\,n^2}{12\,\left (a\,d-b\,c\right )}+\frac {x^3\,\left (7\,B^2\,b^3\,d^3\,i^2\,n^2+12\,A\,B\,b^3\,d^3\,i^2\,n\right )}{a\,d-b\,c}+\frac {x\,\left (72\,A^2\,a^2\,b\,d^3\,i^2+72\,A^2\,a\,b^2\,c\,d^2\,i^2-144\,A^2\,b^3\,c^2\,d\,i^2+84\,A\,B\,a^2\,b\,d^3\,i^2\,n+84\,A\,B\,a\,b^2\,c\,d^2\,i^2\,n-60\,A\,B\,b^3\,c^2\,d\,i^2\,n+37\,B^2\,a^2\,b\,d^3\,i^2\,n^2+37\,B^2\,a\,b^2\,c\,d^2\,i^2\,n^2-11\,B^2\,b^3\,c^2\,d\,i^2\,n^2\right )}{3\,\left (a\,d-b\,c\right )}+\frac {x^2\,\left (-72\,c\,A^2\,b^3\,d^2\,i^2+72\,a\,A^2\,b^2\,d^3\,i^2-12\,c\,A\,B\,b^3\,d^2\,i^2\,n+84\,a\,A\,B\,b^2\,d^3\,i^2\,n+5\,c\,B^2\,b^3\,d^2\,i^2\,n^2+37\,a\,B^2\,b^2\,d^3\,i^2\,n^2\right )}{2\,\left (a\,d-b\,c\right )}}{72\,a^4\,b^3\,g^5+288\,a^3\,b^4\,g^5\,x+432\,a^2\,b^5\,g^5\,x^2+288\,a\,b^6\,g^5\,x^3+72\,b^7\,g^5\,x^4}-{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {a\,\left (\frac {B^2\,c\,d\,i^2}{6\,b^2}+\frac {B^2\,a\,d^2\,i^2}{12\,b^3}\right )+x\,\left (b\,\left (\frac {B^2\,c\,d\,i^2}{6\,b^2}+\frac {B^2\,a\,d^2\,i^2}{12\,b^3}\right )+\frac {B^2\,c\,d\,i^2}{2\,b}+\frac {B^2\,a\,d^2\,i^2}{4\,b^2}\right )+\frac {B^2\,c^2\,i^2}{4\,b}+\frac {B^2\,d^2\,i^2\,x^2}{2\,b}}{a^4\,g^5+4\,a^3\,b\,g^5\,x+6\,a^2\,b^2\,g^5\,x^2+4\,a\,b^3\,g^5\,x^3+b^4\,g^5\,x^4}-\frac {B^2\,d^4\,i^2}{12\,b^3\,g^5\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}\right )-\frac {B\,d^4\,i^2\,n\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x-\frac {72\,b^5\,c^2\,g^5-72\,a^2\,b^3\,d^2\,g^5}{72\,b^3\,g^5\,\left (a\,d-b\,c\right )}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (12\,A+7\,B\,n\right )\,1{}\mathrm {i}}{36\,b^3\,g^5\,{\left (a\,d-b\,c\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*i + d*i*x)^2*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2)/(a*g + b*g*x)^5,x)

[Out]

- log(e*((a + b*x)/(c + d*x))^n)*((a*(A*B*a*d^2*i^2 - (B^2*a*d^2*i^2*n)/2 + (B^2*b*c*d*i^2*n)/2 + 2*A*B*b*c*d*
i^2) + x*(b*(A*B*a*d^2*i^2 - (B^2*a*d^2*i^2*n)/2 + (B^2*b*c*d*i^2*n)/2 + 2*A*B*b*c*d*i^2) + 3*A*B*a*b*d^2*i^2
+ 6*A*B*b^2*c*d*i^2 - (3*B^2*a*b*d^2*i^2*n)/2 + (3*B^2*b^2*c*d*i^2*n)/2) + 3*A*B*b^2*c^2*i^2 - B^2*a^2*d^2*i^2
*n + (B^2*b^2*c^2*i^2*n)/2 + 6*A*B*b^2*d^2*i^2*x^2 + (B^2*a*b*c*d*i^2*n)/2)/(6*a^4*b^3*g^5 + 6*b^7*g^5*x^4 + 2
4*a^3*b^4*g^5*x + 24*a*b^6*g^5*x^3 + 36*a^2*b^5*g^5*x^2) + (B^2*d^4*i^2*(x^2*(b*(b*((3*a*b^3*g^5*n*(a*d - b*c)
)/(2*d) + (b^3*g^5*n*(a*d - b*c)*(4*a*d - b*c))/(2*d^2)) + (3*a*b^4*g^5*n*(a*d - b*c))/d + (b^4*g^5*n*(a*d - b
*c)*(4*a*d - b*c))/d^2) + (9*a*b^5*g^5*n*(a*d - b*c))/(2*d) + (3*b^5*g^5*n*(a*d - b*c)*(4*a*d - b*c))/(2*d^2))
+ a*(a*((3*a*b^3*g^5*n*(a*d - b*c))/(2*d) + (b^3*g^5*n*(a*d - b*c)*(4*a*d - b*c))/(2*d^2)) + (b^3*g^5*n*(a*d
- b*c)*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(2*d^3)) + x*(a*(b*((3*a*b^3*g^5*n*(a*d - b*c))/(2*d) + (b^3*g^5*n*(
a*d - b*c)*(4*a*d - b*c))/(2*d^2)) + (3*a*b^4*g^5*n*(a*d - b*c))/d + (b^4*g^5*n*(a*d - b*c)*(4*a*d - b*c))/d^2
) + b*(a*((3*a*b^3*g^5*n*(a*d - b*c))/(2*d) + (b^3*g^5*n*(a*d - b*c)*(4*a*d - b*c))/(2*d^2)) + (b^3*g^5*n*(a*d
- b*c)*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(2*d^3)) + (3*b^4*g^5*n*(a*d - b*c)*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*
d))/(2*d^3)) + (3*b^3*g^5*n*(a*d - b*c)*(4*a^3*d^3 - b^3*c^3 + 4*a*b^2*c^2*d - 6*a^2*b*c*d^2))/(2*d^4) + (6*b^
6*g^5*n*x^3*(a*d - b*c))/d))/(6*b^3*g^5*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(6*a^4*b^3*g^5 + 6*b^7*g^5*x^4 + 24*a^
3*b^4*g^5*x + 24*a*b^6*g^5*x^3 + 36*a^2*b^5*g^5*x^2))) - ((72*A^2*a^3*d^3*i^2 - 216*A^2*b^3*c^3*i^2 + 37*B^2*a
^3*d^3*i^2*n^2 - 27*B^2*b^3*c^3*i^2*n^2 + 72*A^2*a*b^2*c^2*d*i^2 + 72*A^2*a^2*b*c*d^2*i^2 + 84*A*B*a^3*d^3*i^2
*n - 108*A*B*b^3*c^3*i^2*n + 37*B^2*a*b^2*c^2*d*i^2*n^2 + 37*B^2*a^2*b*c*d^2*i^2*n^2 + 84*A*B*a*b^2*c^2*d*i^2*
n + 84*A*B*a^2*b*c*d^2*i^2*n)/(12*(a*d - b*c)) + (x^3*(7*B^2*b^3*d^3*i^2*n^2 + 12*A*B*b^3*d^3*i^2*n))/(a*d - b
*c) + (x*(72*A^2*a^2*b*d^3*i^2 - 144*A^2*b^3*c^2*d*i^2 + 72*A^2*a*b^2*c*d^2*i^2 + 37*B^2*a^2*b*d^3*i^2*n^2 - 1
1*B^2*b^3*c^2*d*i^2*n^2 - 60*A*B*b^3*c^2*d*i^2*n + 37*B^2*a*b^2*c*d^2*i^2*n^2 + 84*A*B*a^2*b*d^3*i^2*n + 84*A*
B*a*b^2*c*d^2*i^2*n))/(3*(a*d - b*c)) + (x^2*(72*A^2*a*b^2*d^3*i^2 - 72*A^2*b^3*c*d^2*i^2 + 37*B^2*a*b^2*d^3*i
^2*n^2 + 5*B^2*b^3*c*d^2*i^2*n^2 - 12*A*B*b^3*c*d^2*i^2*n + 84*A*B*a*b^2*d^3*i^2*n))/(2*(a*d - b*c)))/(72*a^4*
b^3*g^5 + 72*b^7*g^5*x^4 + 288*a^3*b^4*g^5*x + 288*a*b^6*g^5*x^3 + 432*a^2*b^5*g^5*x^2) - log(e*((a + b*x)/(c
+ d*x))^n)^2*((a*((B^2*c*d*i^2)/(6*b^2) + (B^2*a*d^2*i^2)/(12*b^3)) + x*(b*((B^2*c*d*i^2)/(6*b^2) + (B^2*a*d^2
*i^2)/(12*b^3)) + (B^2*c*d*i^2)/(2*b) + (B^2*a*d^2*i^2)/(4*b^2)) + (B^2*c^2*i^2)/(4*b) + (B^2*d^2*i^2*x^2)/(2*
b))/(a^4*g^5 + b^4*g^5*x^4 + 4*a*b^3*g^5*x^3 + 6*a^2*b^2*g^5*x^2 + 4*a^3*b*g^5*x) - (B^2*d^4*i^2)/(12*b^3*g^5*
(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) - (B*d^4*i^2*n*atan(((2*b*d*x - (72*b^5*c^2*g^5 - 72*a^2*b^3*d^2*g^5)/(72*b^
3*g^5*(a*d - b*c)))*1i)/(a*d - b*c))*(12*A + 7*B*n)*1i)/(36*b^3*g^5*(a*d - b*c)^2)

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