\(\int \frac {(c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))}{(a g+b g x)^6} \, dx\) [126]

Optimal result

Integrand size = 43, antiderivative size = 293 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^6} \, dx=-\frac {B d^2 i^2 n (c+d x)^3}{9 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b B d i^2 n (c+d x)^4}{8 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 B i^2 n (c+d x)^5}{25 (b c-a d)^3 g^6 (a+b x)^5}-\frac {d^2 i^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b d i^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 (b c-a d)^3 g^6 (a+b x)^5} \]

[Out]

-1/9*B*d^2*i^2*n*(d*x+c)^3/(-a*d+b*c)^3/g^6/(b*x+a)^3+1/8*b*B*d*i^2*n*(d*x+c)^4/(-a*d+b*c)^3/g^6/(b*x+a)^4-1/2
5*b^2*B*i^2*n*(d*x+c)^5/(-a*d+b*c)^3/g^6/(b*x+a)^5-1/3*d^2*i^2*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d
+b*c)^3/g^6/(b*x+a)^3+1/2*b*d*i^2*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/g^6/(b*x+a)^4-1/5*b^2
*i^2*(d*x+c)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^3/g^6/(b*x+a)^5

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {2561, 45, 2372, 12, 14} \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^6} \, dx=-\frac {b^2 i^2 (c+d x)^5 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{5 g^6 (a+b x)^5 (b c-a d)^3}-\frac {d^2 i^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^6 (a+b x)^3 (b c-a d)^3}+\frac {b d i^2 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^6 (a+b x)^4 (b c-a d)^3}-\frac {b^2 B i^2 n (c+d x)^5}{25 g^6 (a+b x)^5 (b c-a d)^3}-\frac {B d^2 i^2 n (c+d x)^3}{9 g^6 (a+b x)^3 (b c-a d)^3}+\frac {b B d i^2 n (c+d x)^4}{8 g^6 (a+b x)^4 (b c-a d)^3} \]

[In]

Int[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a*g + b*g*x)^6,x]

[Out]

-1/9*(B*d^2*i^2*n*(c + d*x)^3)/((b*c - a*d)^3*g^6*(a + b*x)^3) + (b*B*d*i^2*n*(c + d*x)^4)/(8*(b*c - a*d)^3*g^
6*(a + b*x)^4) - (b^2*B*i^2*n*(c + d*x)^5)/(25*(b*c - a*d)^3*g^6*(a + b*x)^5) - (d^2*i^2*(c + d*x)^3*(A + B*Lo
g[e*((a + b*x)/(c + d*x))^n]))/(3*(b*c - a*d)^3*g^6*(a + b*x)^3) + (b*d*i^2*(c + d*x)^4*(A + B*Log[e*((a + b*x
)/(c + d*x))^n]))/(2*(b*c - a*d)^3*g^6*(a + b*x)^4) - (b^2*i^2*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^
n]))/(5*(b*c - a*d)^3*g^6*(a + b*x)^5)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

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*} \text {integral}& = \frac {i^2 \text {Subst}\left (\int \frac {(b-d x)^2 \left (A+B \log \left (e x^n\right )\right )}{x^6} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^6} \\ & = -\frac {d^2 i^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b d 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 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 (b c-a d)^3 g^6 (a+b x)^5}-\frac {\left (B i^2 n\right ) \text {Subst}\left (\int \frac {-6 b^2+15 b d x-10 d^2 x^2}{30 x^6} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^3 g^6} \\ & = -\frac {d^2 i^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b d 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 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 (b c-a d)^3 g^6 (a+b x)^5}-\frac {\left (B i^2 n\right ) \text {Subst}\left (\int \frac {-6 b^2+15 b d x-10 d^2 x^2}{x^6} \, dx,x,\frac {a+b x}{c+d x}\right )}{30 (b c-a d)^3 g^6} \\ & = -\frac {d^2 i^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b d 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 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 (b c-a d)^3 g^6 (a+b x)^5}-\frac {\left (B i^2 n\right ) \text {Subst}\left (\int \left (-\frac {6 b^2}{x^6}+\frac {15 b d}{x^5}-\frac {10 d^2}{x^4}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{30 (b c-a d)^3 g^6} \\ & = -\frac {B d^2 i^2 n (c+d x)^3}{9 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b B d i^2 n (c+d x)^4}{8 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 B i^2 n (c+d x)^5}{25 (b c-a d)^3 g^6 (a+b x)^5}-\frac {d^2 i^2 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 (b c-a d)^3 g^6 (a+b x)^3}+\frac {b d 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 (b c-a d)^3 g^6 (a+b x)^4}-\frac {b^2 i^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{5 (b c-a d)^3 g^6 (a+b x)^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.56 (sec) , antiderivative size = 357, normalized size of antiderivative = 1.22 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^6} \, dx=\frac {i^2 \left (-\frac {360 A b^2 c^2}{(a+b x)^5}+\frac {720 a A b c d}{(a+b x)^5}-\frac {360 a^2 A d^2}{(a+b x)^5}-\frac {72 b^2 B c^2 n}{(a+b x)^5}+\frac {144 a b B c d n}{(a+b x)^5}-\frac {72 a^2 B d^2 n}{(a+b x)^5}-\frac {900 A b c d}{(a+b x)^4}+\frac {900 a A d^2}{(a+b x)^4}-\frac {135 b B c d n}{(a+b x)^4}+\frac {135 a B d^2 n}{(a+b x)^4}-\frac {600 A d^2}{(a+b x)^3}-\frac {20 B d^2 n}{(a+b x)^3}+\frac {30 B d^3 n}{(b c-a d) (a+b x)^2}-\frac {60 B d^4 n}{(b c-a d)^2 (a+b x)}-\frac {60 B d^5 n \log (a+b x)}{(b c-a d)^3}-\frac {60 B \left (a^2 d^2+a b d (3 c+5 d x)+b^2 \left (6 c^2+15 c d x+10 d^2 x^2\right )\right ) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^5}+\frac {60 B d^5 n \log (c+d x)}{(b c-a d)^3}\right )}{1800 b^3 g^6} \]

[In]

Integrate[((c*i + d*i*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a*g + b*g*x)^6,x]

[Out]

(i^2*((-360*A*b^2*c^2)/(a + b*x)^5 + (720*a*A*b*c*d)/(a + b*x)^5 - (360*a^2*A*d^2)/(a + b*x)^5 - (72*b^2*B*c^2
*n)/(a + b*x)^5 + (144*a*b*B*c*d*n)/(a + b*x)^5 - (72*a^2*B*d^2*n)/(a + b*x)^5 - (900*A*b*c*d)/(a + b*x)^4 + (
900*a*A*d^2)/(a + b*x)^4 - (135*b*B*c*d*n)/(a + b*x)^4 + (135*a*B*d^2*n)/(a + b*x)^4 - (600*A*d^2)/(a + b*x)^3
- (20*B*d^2*n)/(a + b*x)^3 + (30*B*d^3*n)/((b*c - a*d)*(a + b*x)^2) - (60*B*d^4*n)/((b*c - a*d)^2*(a + b*x))
- (60*B*d^5*n*Log[a + b*x])/(b*c - a*d)^3 - (60*B*(a^2*d^2 + a*b*d*(3*c + 5*d*x) + b^2*(6*c^2 + 15*c*d*x + 10*
d^2*x^2))*Log[e*((a + b*x)/(c + d*x))^n])/(a + b*x)^5 + (60*B*d^5*n*Log[c + d*x])/(b*c - a*d)^3))/(1800*b^3*g^
6)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1048\) vs. \(2(281)=562\).

Time = 57.56 (sec) , antiderivative size = 1049, normalized size of antiderivative = 3.58

method	result	size
parallelrisch	\(\text {Expression too large to display}\)	\(1049\)

[In]

int((d*i*x+c*i)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^6,x,method=_RETURNVERBOSE)

[Out]

-1/1800*(-1800*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^7*c*d^5*i^2*n+1800*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*a*b^8*
c^2*d^4*i^2*n-1800*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^7*c^2*d^4*i^2*n+2400*B*x*ln(e*((b*x+a)/(d*x+c))^n)*a*b^
8*c^3*d^3*i^2*n-300*B*x^4*ln(e*((b*x+a)/(d*x+c))^n)*a*b^8*d^6*i^2*n-600*B*x^3*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^
7*d^6*i^2*n-300*B*x^3*a*b^8*c*d^5*i^2*n^2-600*B*x^2*ln(e*((b*x+a)/(d*x+c))^n)*b^9*c^3*d^3*i^2*n-600*B*x^2*a^2*
b^7*c*d^5*i^2*n^2+150*B*x^2*a*b^8*c^2*d^4*i^2*n^2-1800*A*x^2*a^2*b^7*c*d^5*i^2*n+1800*A*x^2*a*b^8*c^2*d^4*i^2*
n-900*B*x*ln(e*((b*x+a)/(d*x+c))^n)*b^9*c^4*d^2*i^2*n-600*B*x*a^2*b^7*c^2*d^4*i^2*n^2+500*B*x*a*b^8*c^3*d^3*i^
2*n^2-1800*A*x*a^2*b^7*c^2*d^4*i^2*n+2400*A*x*a*b^8*c^3*d^3*i^2*n-600*B*ln(e*((b*x+a)/(d*x+c))^n)*a^2*b^7*c^3*
d^3*i^2*n+900*B*ln(e*((b*x+a)/(d*x+c))^n)*a*b^8*c^4*d^2*i^2*n+47*B*a^5*b^4*d^6*i^2*n^2-72*B*b^9*c^5*d*i^2*n^2+
60*A*a^5*b^4*d^6*i^2*n-360*A*b^9*c^5*d*i^2*n-60*B*x^5*ln(e*((b*x+a)/(d*x+c))^n)*b^9*d^6*i^2*n+60*B*x^4*a*b^8*d
^6*i^2*n^2-60*B*x^4*b^9*c*d^5*i^2*n^2+270*B*x^3*a^2*b^7*d^6*i^2*n^2+30*B*x^3*b^9*c^2*d^4*i^2*n^2+470*B*x^2*a^3
*b^6*d^6*i^2*n^2-20*B*x^2*b^9*c^3*d^3*i^2*n^2+600*A*x^2*a^3*b^6*d^6*i^2*n-600*A*x^2*b^9*c^3*d^3*i^2*n+235*B*x*
a^4*b^5*d^6*i^2*n^2-135*B*x*b^9*c^4*d^2*i^2*n^2+300*A*x*a^4*b^5*d^6*i^2*n-900*A*x*b^9*c^4*d^2*i^2*n-360*B*ln(e
*((b*x+a)/(d*x+c))^n)*b^9*c^5*d*i^2*n-200*B*a^2*b^7*c^3*d^3*i^2*n^2+225*B*a*b^8*c^4*d^2*i^2*n^2-600*A*a^2*b^7*
c^3*d^3*i^2*n+900*A*a*b^8*c^4*d^2*i^2*n)/g^6/(b*x+a)^5/n/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/b^7/d

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1087 vs. \(2 (281) = 562\).

Time = 0.37 (sec) , antiderivative size = 1087, normalized size of antiderivative = 3.71 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^6} \, dx=-\frac {60 \, {\left (B b^{5} c d^{4} - B a b^{4} d^{5}\right )} i^{2} n x^{4} - 30 \, {\left (B b^{5} c^{2} d^{3} - 10 \, B a b^{4} c d^{4} + 9 \, B a^{2} b^{3} d^{5}\right )} i^{2} n x^{3} + {\left (72 \, B b^{5} c^{5} - 225 \, B a b^{4} c^{4} d + 200 \, B a^{2} b^{3} c^{3} d^{2} - 47 \, B a^{5} d^{5}\right )} i^{2} n + 60 \, {\left (6 \, A b^{5} c^{5} - 15 \, A a b^{4} c^{4} d + 10 \, A a^{2} b^{3} c^{3} d^{2} - A a^{5} d^{5}\right )} i^{2} + 10 \, {\left ({\left (2 \, B b^{5} c^{3} d^{2} - 15 \, B a b^{4} c^{2} d^{3} + 60 \, B a^{2} b^{3} c d^{4} - 47 \, B a^{3} b^{2} d^{5}\right )} i^{2} n + 60 \, {\left (A b^{5} c^{3} d^{2} - 3 \, A a b^{4} c^{2} d^{3} + 3 \, A a^{2} b^{3} c d^{4} - A a^{3} b^{2} d^{5}\right )} i^{2}\right )} x^{2} + 5 \, {\left ({\left (27 \, B b^{5} c^{4} d - 100 \, B a b^{4} c^{3} d^{2} + 120 \, B a^{2} b^{3} c^{2} d^{3} - 47 \, B a^{4} b d^{5}\right )} i^{2} n + 60 \, {\left (3 \, A b^{5} c^{4} d - 8 \, A a b^{4} c^{3} d^{2} + 6 \, A a^{2} b^{3} c^{2} d^{3} - A a^{4} b d^{5}\right )} i^{2}\right )} x + 60 \, {\left (10 \, {\left (B b^{5} c^{3} d^{2} - 3 \, B a b^{4} c^{2} d^{3} + 3 \, B a^{2} b^{3} c d^{4} - B a^{3} b^{2} d^{5}\right )} i^{2} x^{2} + 5 \, {\left (3 \, B b^{5} c^{4} d - 8 \, B a b^{4} c^{3} d^{2} + 6 \, B a^{2} b^{3} c^{2} d^{3} - B a^{4} b d^{5}\right )} i^{2} x + {\left (6 \, B b^{5} c^{5} - 15 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2} - B a^{5} d^{5}\right )} i^{2}\right )} \log \left (e\right ) + 60 \, {\left (B b^{5} d^{5} i^{2} n x^{5} + 5 \, B a b^{4} d^{5} i^{2} n x^{4} + 10 \, B a^{2} b^{3} d^{5} i^{2} n x^{3} + 10 \, {\left (B b^{5} c^{3} d^{2} - 3 \, B a b^{4} c^{2} d^{3} + 3 \, B a^{2} b^{3} c d^{4}\right )} i^{2} n x^{2} + 5 \, {\left (3 \, B b^{5} c^{4} d - 8 \, B a b^{4} c^{3} d^{2} + 6 \, B a^{2} b^{3} c^{2} d^{3}\right )} i^{2} n x + {\left (6 \, B b^{5} c^{5} - 15 \, B a b^{4} c^{4} d + 10 \, B a^{2} b^{3} c^{3} d^{2}\right )} i^{2} n\right )} \log \left (\frac {b x + a}{d x + c}\right )}{1800 \, {\left ({\left (b^{11} c^{3} - 3 \, a b^{10} c^{2} d + 3 \, a^{2} b^{9} c d^{2} - a^{3} b^{8} d^{3}\right )} g^{6} x^{5} + 5 \, {\left (a b^{10} c^{3} - 3 \, a^{2} b^{9} c^{2} d + 3 \, a^{3} b^{8} c d^{2} - a^{4} b^{7} d^{3}\right )} g^{6} x^{4} + 10 \, {\left (a^{2} b^{9} c^{3} - 3 \, a^{3} b^{8} c^{2} d + 3 \, a^{4} b^{7} c d^{2} - a^{5} b^{6} d^{3}\right )} g^{6} x^{3} + 10 \, {\left (a^{3} b^{8} c^{3} - 3 \, a^{4} b^{7} c^{2} d + 3 \, a^{5} b^{6} c d^{2} - a^{6} b^{5} d^{3}\right )} g^{6} x^{2} + 5 \, {\left (a^{4} b^{7} c^{3} - 3 \, a^{5} b^{6} c^{2} d + 3 \, a^{6} b^{5} c d^{2} - a^{7} b^{4} d^{3}\right )} g^{6} x + {\left (a^{5} b^{6} c^{3} - 3 \, a^{6} b^{5} c^{2} d + 3 \, a^{7} b^{4} c d^{2} - a^{8} b^{3} d^{3}\right )} g^{6}\right )}} \]

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^6,x, algorithm="fricas")

[Out]

-1/1800*(60*(B*b^5*c*d^4 - B*a*b^4*d^5)*i^2*n*x^4 - 30*(B*b^5*c^2*d^3 - 10*B*a*b^4*c*d^4 + 9*B*a^2*b^3*d^5)*i^
2*n*x^3 + (72*B*b^5*c^5 - 225*B*a*b^4*c^4*d + 200*B*a^2*b^3*c^3*d^2 - 47*B*a^5*d^5)*i^2*n + 60*(6*A*b^5*c^5 -
15*A*a*b^4*c^4*d + 10*A*a^2*b^3*c^3*d^2 - A*a^5*d^5)*i^2 + 10*((2*B*b^5*c^3*d^2 - 15*B*a*b^4*c^2*d^3 + 60*B*a^
2*b^3*c*d^4 - 47*B*a^3*b^2*d^5)*i^2*n + 60*(A*b^5*c^3*d^2 - 3*A*a*b^4*c^2*d^3 + 3*A*a^2*b^3*c*d^4 - A*a^3*b^2*
d^5)*i^2)*x^2 + 5*((27*B*b^5*c^4*d - 100*B*a*b^4*c^3*d^2 + 120*B*a^2*b^3*c^2*d^3 - 47*B*a^4*b*d^5)*i^2*n + 60*
(3*A*b^5*c^4*d - 8*A*a*b^4*c^3*d^2 + 6*A*a^2*b^3*c^2*d^3 - A*a^4*b*d^5)*i^2)*x + 60*(10*(B*b^5*c^3*d^2 - 3*B*a
*b^4*c^2*d^3 + 3*B*a^2*b^3*c*d^4 - B*a^3*b^2*d^5)*i^2*x^2 + 5*(3*B*b^5*c^4*d - 8*B*a*b^4*c^3*d^2 + 6*B*a^2*b^3
*c^2*d^3 - B*a^4*b*d^5)*i^2*x + (6*B*b^5*c^5 - 15*B*a*b^4*c^4*d + 10*B*a^2*b^3*c^3*d^2 - B*a^5*d^5)*i^2)*log(e
) + 60*(B*b^5*d^5*i^2*n*x^5 + 5*B*a*b^4*d^5*i^2*n*x^4 + 10*B*a^2*b^3*d^5*i^2*n*x^3 + 10*(B*b^5*c^3*d^2 - 3*B*a
*b^4*c^2*d^3 + 3*B*a^2*b^3*c*d^4)*i^2*n*x^2 + 5*(3*B*b^5*c^4*d - 8*B*a*b^4*c^3*d^2 + 6*B*a^2*b^3*c^2*d^3)*i^2*
n*x + (6*B*b^5*c^5 - 15*B*a*b^4*c^4*d + 10*B*a^2*b^3*c^3*d^2)*i^2*n)*log((b*x + a)/(d*x + c)))/((b^11*c^3 - 3*
a*b^10*c^2*d + 3*a^2*b^9*c*d^2 - a^3*b^8*d^3)*g^6*x^5 + 5*(a*b^10*c^3 - 3*a^2*b^9*c^2*d + 3*a^3*b^8*c*d^2 - a^
4*b^7*d^3)*g^6*x^4 + 10*(a^2*b^9*c^3 - 3*a^3*b^8*c^2*d + 3*a^4*b^7*c*d^2 - a^5*b^6*d^3)*g^6*x^3 + 10*(a^3*b^8*
c^3 - 3*a^4*b^7*c^2*d + 3*a^5*b^6*c*d^2 - a^6*b^5*d^3)*g^6*x^2 + 5*(a^4*b^7*c^3 - 3*a^5*b^6*c^2*d + 3*a^6*b^5*
c*d^2 - a^7*b^4*d^3)*g^6*x + (a^5*b^6*c^3 - 3*a^6*b^5*c^2*d + 3*a^7*b^4*c*d^2 - a^8*b^3*d^3)*g^6)

Sympy [F(-1)]

Timed out. \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^6} \, dx=\text {Timed out} \]

[In]

integrate((d*i*x+c*i)**2*(A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**6,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 3058 vs. \(2 (281) = 562\).

Time = 0.35 (sec) , antiderivative size = 3058, normalized size of antiderivative = 10.44 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^6} \, dx=\text {Too large to display} \]

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^6,x, algorithm="maxima")

[Out]

-1/300*B*c^2*i^2*n*((60*b^4*d^4*x^4 + 12*b^4*c^4 - 63*a*b^3*c^3*d + 137*a^2*b^2*c^2*d^2 - 163*a^3*b*c*d^3 + 13
7*a^4*d^4 - 30*(b^4*c*d^3 - 9*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 - 13*a*b^3*c*d^3 + 47*a^2*b^2*d^4)*x^2 - 5*(3
*b^4*c^3*d - 17*a*b^3*c^2*d^2 + 43*a^2*b^2*c*d^3 - 77*a^3*b*d^4)*x)/((b^10*c^4 - 4*a*b^9*c^3*d + 6*a^2*b^8*c^2
*d^2 - 4*a^3*b^7*c*d^3 + a^4*b^6*d^4)*g^6*x^5 + 5*(a*b^9*c^4 - 4*a^2*b^8*c^3*d + 6*a^3*b^7*c^2*d^2 - 4*a^4*b^6
*c*d^3 + a^5*b^5*d^4)*g^6*x^4 + 10*(a^2*b^8*c^4 - 4*a^3*b^7*c^3*d + 6*a^4*b^6*c^2*d^2 - 4*a^5*b^5*c*d^3 + a^6*
b^4*d^4)*g^6*x^3 + 10*(a^3*b^7*c^4 - 4*a^4*b^6*c^3*d + 6*a^5*b^5*c^2*d^2 - 4*a^6*b^4*c*d^3 + a^7*b^3*d^4)*g^6*
x^2 + 5*(a^4*b^6*c^4 - 4*a^5*b^5*c^3*d + 6*a^6*b^4*c^2*d^2 - 4*a^7*b^3*c*d^3 + a^8*b^2*d^4)*g^6*x + (a^5*b^5*c
^4 - 4*a^6*b^4*c^3*d + 6*a^7*b^3*c^2*d^2 - 4*a^8*b^2*c*d^3 + a^9*b*d^4)*g^6) + 60*d^5*log(b*x + a)/((b^6*c^5 -
5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^6) - 60*d^5*log(d*x
+ c)/((b^6*c^5 - 5*a*b^5*c^4*d + 10*a^2*b^4*c^3*d^2 - 10*a^3*b^3*c^2*d^3 + 5*a^4*b^2*c*d^4 - a^5*b*d^5)*g^6))
- 1/1800*B*d^2*i^2*n*((47*a^2*b^4*c^4 - 278*a^3*b^3*c^3*d + 822*a^4*b^2*c^2*d^2 - 278*a^5*b*c*d^3 + 47*a^6*d^4
+ 60*(10*b^6*c^2*d^2 - 5*a*b^5*c*d^3 + a^2*b^4*d^4)*x^4 - 30*(10*b^6*c^3*d - 95*a*b^5*c^2*d^2 + 46*a^2*b^4*c*
d^3 - 9*a^3*b^3*d^4)*x^3 + 10*(20*b^6*c^4 - 140*a*b^5*c^3*d + 537*a^2*b^4*c^2*d^2 - 248*a^3*b^3*c*d^3 + 47*a^4
*b^2*d^4)*x^2 + 5*(35*a*b^5*c^4 - 218*a^2*b^4*c^3*d + 702*a^3*b^3*c^2*d^2 - 278*a^4*b^2*c*d^3 + 47*a^5*b*d^4)*
x)/((b^12*c^4 - 4*a*b^11*c^3*d + 6*a^2*b^10*c^2*d^2 - 4*a^3*b^9*c*d^3 + a^4*b^8*d^4)*g^6*x^5 + 5*(a*b^11*c^4 -
4*a^2*b^10*c^3*d + 6*a^3*b^9*c^2*d^2 - 4*a^4*b^8*c*d^3 + a^5*b^7*d^4)*g^6*x^4 + 10*(a^2*b^10*c^4 - 4*a^3*b^9*
c^3*d + 6*a^4*b^8*c^2*d^2 - 4*a^5*b^7*c*d^3 + a^6*b^6*d^4)*g^6*x^3 + 10*(a^3*b^9*c^4 - 4*a^4*b^8*c^3*d + 6*a^5
*b^7*c^2*d^2 - 4*a^6*b^6*c*d^3 + a^7*b^5*d^4)*g^6*x^2 + 5*(a^4*b^8*c^4 - 4*a^5*b^7*c^3*d + 6*a^6*b^6*c^2*d^2 -
4*a^7*b^5*c*d^3 + a^8*b^4*d^4)*g^6*x + (a^5*b^7*c^4 - 4*a^6*b^6*c^3*d + 6*a^7*b^5*c^2*d^2 - 4*a^8*b^4*c*d^3 +
a^9*b^3*d^4)*g^6) + 60*(10*b^2*c^2*d^3 - 5*a*b*c*d^4 + a^2*d^5)*log(b*x + a)/((b^8*c^5 - 5*a*b^7*c^4*d + 10*a
^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 - a^5*b^3*d^5)*g^6) - 60*(10*b^2*c^2*d^3 - 5*a*b*c*d^4 +
a^2*d^5)*log(d*x + c)/((b^8*c^5 - 5*a*b^7*c^4*d + 10*a^2*b^6*c^3*d^2 - 10*a^3*b^5*c^2*d^3 + 5*a^4*b^4*c*d^4 -
a^5*b^3*d^5)*g^6)) - 1/600*B*c*d*i^2*n*((27*a*b^4*c^4 - 148*a^2*b^3*c^3*d + 352*a^3*b^2*c^2*d^2 - 548*a^4*b*c
*d^3 + 77*a^5*d^4 - 60*(5*b^5*c*d^3 - a*b^4*d^4)*x^4 + 30*(5*b^5*c^2*d^2 - 46*a*b^4*c*d^3 + 9*a^2*b^3*d^4)*x^3
- 10*(10*b^5*c^3*d - 67*a*b^4*c^2*d^2 + 248*a^2*b^3*c*d^3 - 47*a^3*b^2*d^4)*x^2 + 5*(15*b^5*c^4 - 88*a*b^4*c^
3*d + 232*a^2*b^3*c^2*d^2 - 428*a^3*b^2*c*d^3 + 77*a^4*b*d^4)*x)/((b^11*c^4 - 4*a*b^10*c^3*d + 6*a^2*b^9*c^2*d
^2 - 4*a^3*b^8*c*d^3 + a^4*b^7*d^4)*g^6*x^5 + 5*(a*b^10*c^4 - 4*a^2*b^9*c^3*d + 6*a^3*b^8*c^2*d^2 - 4*a^4*b^7*
c*d^3 + a^5*b^6*d^4)*g^6*x^4 + 10*(a^2*b^9*c^4 - 4*a^3*b^8*c^3*d + 6*a^4*b^7*c^2*d^2 - 4*a^5*b^6*c*d^3 + a^6*b
^5*d^4)*g^6*x^3 + 10*(a^3*b^8*c^4 - 4*a^4*b^7*c^3*d + 6*a^5*b^6*c^2*d^2 - 4*a^6*b^5*c*d^3 + a^7*b^4*d^4)*g^6*x
^2 + 5*(a^4*b^7*c^4 - 4*a^5*b^6*c^3*d + 6*a^6*b^5*c^2*d^2 - 4*a^7*b^4*c*d^3 + a^8*b^3*d^4)*g^6*x + (a^5*b^6*c^
4 - 4*a^6*b^5*c^3*d + 6*a^7*b^4*c^2*d^2 - 4*a^8*b^3*c*d^3 + a^9*b^2*d^4)*g^6) - 60*(5*b*c*d^4 - a*d^5)*log(b*x
+ a)/((b^7*c^5 - 5*a*b^6*c^4*d + 10*a^2*b^5*c^3*d^2 - 10*a^3*b^4*c^2*d^3 + 5*a^4*b^3*c*d^4 - a^5*b^2*d^5)*g^6
) + 60*(5*b*c*d^4 - a*d^5)*log(d*x + c)/((b^7*c^5 - 5*a*b^6*c^4*d + 10*a^2*b^5*c^3*d^2 - 10*a^3*b^4*c^2*d^3 +
5*a^4*b^3*c*d^4 - a^5*b^2*d^5)*g^6)) - 1/10*(5*b*x + a)*B*c*d*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^7*
g^6*x^5 + 5*a*b^6*g^6*x^4 + 10*a^2*b^5*g^6*x^3 + 10*a^3*b^4*g^6*x^2 + 5*a^4*b^3*g^6*x + a^5*b^2*g^6) - 1/30*(1
0*b^2*x^2 + 5*a*b*x + a^2)*B*d^2*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^8*g^6*x^5 + 5*a*b^7*g^6*x^4 + 1
0*a^2*b^6*g^6*x^3 + 10*a^3*b^5*g^6*x^2 + 5*a^4*b^4*g^6*x + a^5*b^3*g^6) - 1/10*(5*b*x + a)*A*c*d*i^2/(b^7*g^6*
x^5 + 5*a*b^6*g^6*x^4 + 10*a^2*b^5*g^6*x^3 + 10*a^3*b^4*g^6*x^2 + 5*a^4*b^3*g^6*x + a^5*b^2*g^6) - 1/30*(10*b^
2*x^2 + 5*a*b*x + a^2)*A*d^2*i^2/(b^8*g^6*x^5 + 5*a*b^7*g^6*x^4 + 10*a^2*b^6*g^6*x^3 + 10*a^3*b^5*g^6*x^2 + 5*
a^4*b^4*g^6*x + a^5*b^3*g^6) - 1/5*B*c^2*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^6*g^6*x^5 + 5*a*b^5*g^6
*x^4 + 10*a^2*b^4*g^6*x^3 + 10*a^3*b^3*g^6*x^2 + 5*a^4*b^2*g^6*x + a^5*b*g^6) - 1/5*A*c^2*i^2/(b^6*g^6*x^5 + 5
*a*b^5*g^6*x^4 + 10*a^2*b^4*g^6*x^3 + 10*a^3*b^3*g^6*x^2 + 5*a^4*b^2*g^6*x + a^5*b*g^6)

Giac [A] (verification not implemented)

none

Time = 2.58 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.43 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^6} \, dx=-\frac {1}{1800} \, {\left (\frac {60 \, {\left (6 \, B b^{2} i^{2} n - \frac {15 \, {\left (b x + a\right )} B b d i^{2} n}{d x + c} + \frac {10 \, {\left (b x + a\right )}^{2} B d^{2} i^{2} n}{{\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b x + a\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b x + a\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}} + \frac {72 \, B b^{2} i^{2} n - \frac {225 \, {\left (b x + a\right )} B b d i^{2} n}{d x + c} + \frac {200 \, {\left (b x + a\right )}^{2} B d^{2} i^{2} n}{{\left (d x + c\right )}^{2}} + 360 \, B b^{2} i^{2} \log \left (e\right ) - \frac {900 \, {\left (b x + a\right )} B b d i^{2} \log \left (e\right )}{d x + c} + \frac {600 \, {\left (b x + a\right )}^{2} B d^{2} i^{2} \log \left (e\right )}{{\left (d x + c\right )}^{2}} + 360 \, A b^{2} i^{2} - \frac {900 \, {\left (b x + a\right )} A b d i^{2}}{d x + c} + \frac {600 \, {\left (b x + a\right )}^{2} A d^{2} i^{2}}{{\left (d x + c\right )}^{2}}}{\frac {{\left (b x + a\right )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b x + a\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b x + a\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]

[In]

integrate((d*i*x+c*i)^2*(A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^6,x, algorithm="giac")

[Out]

-1/1800*(60*(6*B*b^2*i^2*n - 15*(b*x + a)*B*b*d*i^2*n/(d*x + c) + 10*(b*x + a)^2*B*d^2*i^2*n/(d*x + c)^2)*log(
(b*x + a)/(d*x + c))/((b*x + a)^5*b^2*c^2*g^6/(d*x + c)^5 - 2*(b*x + a)^5*a*b*c*d*g^6/(d*x + c)^5 + (b*x + a)^
5*a^2*d^2*g^6/(d*x + c)^5) + (72*B*b^2*i^2*n - 225*(b*x + a)*B*b*d*i^2*n/(d*x + c) + 200*(b*x + a)^2*B*d^2*i^2
*n/(d*x + c)^2 + 360*B*b^2*i^2*log(e) - 900*(b*x + a)*B*b*d*i^2*log(e)/(d*x + c) + 600*(b*x + a)^2*B*d^2*i^2*l
og(e)/(d*x + c)^2 + 360*A*b^2*i^2 - 900*(b*x + a)*A*b*d*i^2/(d*x + c) + 600*(b*x + a)^2*A*d^2*i^2/(d*x + c)^2)
/((b*x + a)^5*b^2*c^2*g^6/(d*x + c)^5 - 2*(b*x + a)^5*a*b*c*d*g^6/(d*x + c)^5 + (b*x + a)^5*a^2*d^2*g^6/(d*x +
c)^5))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

Mupad [B] (verification not implemented)

Time = 3.37 (sec) , antiderivative size = 954, normalized size of antiderivative = 3.26 \[ \int \frac {(c i+d i x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a g+b g x)^6} \, dx=\frac {B\,d^5\,i^2\,n\,\mathrm {atanh}\left (\frac {30\,a^3\,b^3\,d^3\,g^6-30\,a^2\,b^4\,c\,d^2\,g^6-30\,a\,b^5\,c^2\,d\,g^6+30\,b^6\,c^3\,g^6}{30\,b^3\,g^6\,{\left (a\,d-b\,c\right )}^3}+\frac {2\,b\,d\,x\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}{{\left (a\,d-b\,c\right )}^3}\right )}{15\,b^3\,g^6\,{\left (a\,d-b\,c\right )}^3}-\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (a\,\left (\frac {B\,a\,d^2\,i^2}{30\,b^3}+\frac {B\,c\,d\,i^2}{10\,b^2}\right )+x\,\left (b\,\left (\frac {B\,a\,d^2\,i^2}{30\,b^3}+\frac {B\,c\,d\,i^2}{10\,b^2}\right )+\frac {2\,B\,a\,d^2\,i^2}{15\,b^2}+\frac {2\,B\,c\,d\,i^2}{5\,b}\right )+\frac {B\,c^2\,i^2}{5\,b}+\frac {B\,d^2\,i^2\,x^2}{3\,b}\right )}{a^5\,g^6+5\,a^4\,b\,g^6\,x+10\,a^3\,b^2\,g^6\,x^2+10\,a^2\,b^3\,g^6\,x^3+5\,a\,b^4\,g^6\,x^4+b^5\,g^6\,x^5}-\frac {\frac {60\,A\,a^4\,d^4\,i^2+360\,A\,b^4\,c^4\,i^2+47\,B\,a^4\,d^4\,i^2\,n+72\,B\,b^4\,c^4\,i^2\,n+60\,A\,a^2\,b^2\,c^2\,d^2\,i^2-540\,A\,a\,b^3\,c^3\,d\,i^2+60\,A\,a^3\,b\,c\,d^3\,i^2-153\,B\,a\,b^3\,c^3\,d\,i^2\,n+47\,B\,a^3\,b\,c\,d^3\,i^2\,n+47\,B\,a^2\,b^2\,c^2\,d^2\,i^2\,n}{60\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x^2\,\left (60\,A\,a^2\,b^2\,d^4\,i^2+60\,A\,b^4\,c^2\,d^2\,i^2+47\,B\,a^2\,b^2\,d^4\,i^2\,n+2\,B\,b^4\,c^2\,d^2\,i^2\,n-120\,A\,a\,b^3\,c\,d^3\,i^2-13\,B\,a\,b^3\,c\,d^3\,i^2\,n\right )}{6\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {x\,\left (60\,A\,a^3\,b\,d^4\,i^2+180\,A\,b^4\,c^3\,d\,i^2-300\,A\,a\,b^3\,c^2\,d^2\,i^2+60\,A\,a^2\,b^2\,c\,d^3\,i^2+47\,B\,a^3\,b\,d^4\,i^2\,n+27\,B\,b^4\,c^3\,d\,i^2\,n-73\,B\,a\,b^3\,c^2\,d^2\,i^2\,n+47\,B\,a^2\,b^2\,c\,d^3\,i^2\,n\right )}{12\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {d\,x^3\,\left (9\,B\,a\,b^3\,d^3\,i^2\,n-B\,b^4\,c\,d^2\,i^2\,n\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {B\,b^4\,d^4\,i^2\,n\,x^4}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}}{30\,a^5\,b^3\,g^6+150\,a^4\,b^4\,g^6\,x+300\,a^3\,b^5\,g^6\,x^2+300\,a^2\,b^6\,g^6\,x^3+150\,a\,b^7\,g^6\,x^4+30\,b^8\,g^6\,x^5} \]

[In]

int(((c*i + d*i*x)^2*(A + B*log(e*((a + b*x)/(c + d*x))^n)))/(a*g + b*g*x)^6,x)

[Out]

(B*d^5*i^2*n*atanh((30*b^6*c^3*g^6 + 30*a^3*b^3*d^3*g^6 - 30*a*b^5*c^2*d*g^6 - 30*a^2*b^4*c*d^2*g^6)/(30*b^3*g
^6*(a*d - b*c)^3) + (2*b*d*x*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(a*d - b*c)^3))/(15*b^3*g^6*(a*d - b*c)^3) - (lo
g(e*((a + b*x)/(c + d*x))^n)*(a*((B*a*d^2*i^2)/(30*b^3) + (B*c*d*i^2)/(10*b^2)) + x*(b*((B*a*d^2*i^2)/(30*b^3)
+ (B*c*d*i^2)/(10*b^2)) + (2*B*a*d^2*i^2)/(15*b^2) + (2*B*c*d*i^2)/(5*b)) + (B*c^2*i^2)/(5*b) + (B*d^2*i^2*x^
2)/(3*b)))/(a^5*g^6 + b^5*g^6*x^5 + 5*a*b^4*g^6*x^4 + 10*a^3*b^2*g^6*x^2 + 10*a^2*b^3*g^6*x^3 + 5*a^4*b*g^6*x)
- ((60*A*a^4*d^4*i^2 + 360*A*b^4*c^4*i^2 + 47*B*a^4*d^4*i^2*n + 72*B*b^4*c^4*i^2*n + 60*A*a^2*b^2*c^2*d^2*i^2
- 540*A*a*b^3*c^3*d*i^2 + 60*A*a^3*b*c*d^3*i^2 - 153*B*a*b^3*c^3*d*i^2*n + 47*B*a^3*b*c*d^3*i^2*n + 47*B*a^2*
b^2*c^2*d^2*i^2*n)/(60*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (x^2*(60*A*a^2*b^2*d^4*i^2 + 60*A*b^4*c^2*d^2*i^2 +
47*B*a^2*b^2*d^4*i^2*n + 2*B*b^4*c^2*d^2*i^2*n - 120*A*a*b^3*c*d^3*i^2 - 13*B*a*b^3*c*d^3*i^2*n))/(6*(a^2*d^2
+ b^2*c^2 - 2*a*b*c*d)) + (x*(60*A*a^3*b*d^4*i^2 + 180*A*b^4*c^3*d*i^2 - 300*A*a*b^3*c^2*d^2*i^2 + 60*A*a^2*b^
2*c*d^3*i^2 + 47*B*a^3*b*d^4*i^2*n + 27*B*b^4*c^3*d*i^2*n - 73*B*a*b^3*c^2*d^2*i^2*n + 47*B*a^2*b^2*c*d^3*i^2*
n))/(12*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (d*x^3*(9*B*a*b^3*d^3*i^2*n - B*b^4*c*d^2*i^2*n))/(2*(a^2*d^2 + b^2
*c^2 - 2*a*b*c*d)) + (B*b^4*d^4*i^2*n*x^4)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))/(30*a^5*b^3*g^6 + 30*b^8*g^6*x^5 +
150*a^4*b^4*g^6*x + 150*a*b^7*g^6*x^4 + 300*a^3*b^5*g^6*x^2 + 300*a^2*b^6*g^6*x^3)