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

Optimal. Leaf size=463 \[ -\frac {b^2 i^2 (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{5 g^6 (a+b x)^5 (b c-a d)^3}-\frac {2 b^2 B i^2 (c+d x)^5 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{25 g^6 (a+b x)^5 (b c-a d)^3}-\frac {d^2 i^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{3 g^6 (a+b x)^3 (b c-a d)^3}-\frac {2 B d^2 i^2 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{9 g^6 (a+b x)^3 (b c-a d)^3}+\frac {b d i^2 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 g^6 (a+b x)^4 (b c-a d)^3}+\frac {b B d i^2 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{4 g^6 (a+b x)^4 (b c-a d)^3}-\frac {2 b^2 B^2 i^2 (c+d x)^5}{125 g^6 (a+b x)^5 (b c-a d)^3}-\frac {2 B^2 d^2 i^2 (c+d x)^3}{27 g^6 (a+b x)^3 (b c-a d)^3}+\frac {b B^2 d i^2 (c+d x)^4}{16 g^6 (a+b x)^4 (b c-a d)^3} \]

[Out]

-2/27*B^2*d^2*i^2*(d*x+c)^3/(-a*d+b*c)^3/g^6/(b*x+a)^3+1/16*b*B^2*d*i^2*(d*x+c)^4/(-a*d+b*c)^3/g^6/(b*x+a)^4-2
/125*b^2*B^2*i^2*(d*x+c)^5/(-a*d+b*c)^3/g^6/(b*x+a)^5-2/9*B*d^2*i^2*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*
d+b*c)^3/g^6/(b*x+a)^3+1/4*b*B*d*i^2*(d*x+c)^4*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^3/g^6/(b*x+a)^4-2/25*b^2
*B*i^2*(d*x+c)^5*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-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)))^2/(-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)))^2/(-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)))^2/(-a*d+b*c)^3/g^6/(b*x+a)^5

________________________________________________________________________________________

Rubi [C]  time = 4.22, antiderivative size = 1009, normalized size of antiderivative = 2.18, number of steps used = 116, number of rules used = 11, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.262, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {B^2 i^2 \log ^2(a+b x) d^5}{30 b^3 (b c-a d)^3 g^6}+\frac {B^2 i^2 \log ^2(c+d x) d^5}{30 b^3 (b c-a d)^3 g^6}-\frac {47 B^2 i^2 \log (a+b x) d^5}{900 b^3 (b c-a d)^3 g^6}-\frac {B i^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) d^5}{15 b^3 (b c-a d)^3 g^6}+\frac {47 B^2 i^2 \log (c+d x) d^5}{900 b^3 (b c-a d)^3 g^6}-\frac {B^2 i^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) d^5}{15 b^3 (b c-a d)^3 g^6}+\frac {B i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x) d^5}{15 b^3 (b c-a d)^3 g^6}-\frac {B^2 i^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) d^5}{15 b^3 (b c-a d)^3 g^6}-\frac {B^2 i^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) d^5}{15 b^3 (b c-a d)^3 g^6}-\frac {B^2 i^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) d^5}{15 b^3 (b c-a d)^3 g^6}-\frac {B i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) d^4}{15 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {47 B^2 i^2 d^4}{900 b^3 (b c-a d)^2 g^6 (a+b x)}+\frac {B i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) d^3}{30 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {13 B^2 i^2 d^3}{1800 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 d^2}{3 b^3 g^6 (a+b x)^3}-\frac {B i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) d^2}{45 b^3 g^6 (a+b x)^3}+\frac {43 B^2 i^2 d^2}{2700 b^3 g^6 (a+b x)^3}-\frac {(b c-a d) i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2 d}{2 b^3 g^6 (a+b x)^4}-\frac {3 B (b c-a d) i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) d}{20 b^3 g^6 (a+b x)^4}-\frac {7 B^2 (b c-a d) i^2 d}{400 b^3 g^6 (a+b x)^4}-\frac {(b c-a d)^2 i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {2 B (b c-a d)^2 i^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac {2 B^2 (b c-a d)^2 i^2}{125 b^3 g^6 (a+b x)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*B^2*(b*c - a*d)^2*i^2)/(125*b^3*g^6*(a + b*x)^5) - (7*B^2*d*(b*c - a*d)*i^2)/(400*b^3*g^6*(a + b*x)^4) + (
43*B^2*d^2*i^2)/(2700*b^3*g^6*(a + b*x)^3) - (13*B^2*d^3*i^2)/(1800*b^3*(b*c - a*d)*g^6*(a + b*x)^2) - (47*B^2
*d^4*i^2)/(900*b^3*(b*c - a*d)^2*g^6*(a + b*x)) - (47*B^2*d^5*i^2*Log[a + b*x])/(900*b^3*(b*c - a*d)^3*g^6) +
(B^2*d^5*i^2*Log[a + b*x]^2)/(30*b^3*(b*c - a*d)^3*g^6) - (2*B*(b*c - a*d)^2*i^2*(A + B*Log[(e*(a + b*x))/(c +
d*x)]))/(25*b^3*g^6*(a + b*x)^5) - (3*B*d*(b*c - a*d)*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(20*b^3*g^6*(
a + b*x)^4) - (B*d^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(45*b^3*g^6*(a + b*x)^3) + (B*d^3*i^2*(A + B*Lo
g[(e*(a + b*x))/(c + d*x)]))/(30*b^3*(b*c - a*d)*g^6*(a + b*x)^2) - (B*d^4*i^2*(A + B*Log[(e*(a + b*x))/(c + d
*x)]))/(15*b^3*(b*c - a*d)^2*g^6*(a + b*x)) - (B*d^5*i^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(1
5*b^3*(b*c - a*d)^3*g^6) - ((b*c - a*d)^2*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(5*b^3*g^6*(a + b*x)^5)
- (d*(b*c - a*d)*i^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2)/(2*b^3*g^6*(a + b*x)^4) - (d^2*i^2*(A + B*Log[(e*
(a + b*x))/(c + d*x)])^2)/(3*b^3*g^6*(a + b*x)^3) + (47*B^2*d^5*i^2*Log[c + d*x])/(900*b^3*(b*c - a*d)^3*g^6)
- (B^2*d^5*i^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(15*b^3*(b*c - a*d)^3*g^6) + (B*d^5*i^2*(A + B*
Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(15*b^3*(b*c - a*d)^3*g^6) + (B^2*d^5*i^2*Log[c + d*x]^2)/(30*b^3*
(b*c - a*d)^3*g^6) - (B^2*d^5*i^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(15*b^3*(b*c - a*d)^3*g^6) - (B
^2*d^5*i^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(15*b^3*(b*c - a*d)^3*g^6) - (B^2*d^5*i^2*PolyLog[2, (b*(
c + d*x))/(b*c - a*d)])/(15*b^3*(b*c - a*d)^3*g^6)

Rule 12

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2524

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[d + e*x]*(a + b
*Log[c*RFx^p])^n)/e, x] - Dist[(b*n*p)/e, Int[(Log[d + e*x]*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x
] /; FreeQ[{a, b, c, d, e, p}, x] && RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 2525

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[((d + e*x)^(m
+ 1)*(a + b*Log[c*RFx^p])^n)/(e*(m + 1)), x] - Dist[(b*n*p)/(e*(m + 1)), Int[SimplifyIntegrand[((d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*D[RFx, x])/RFx, x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 2528

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {(73 c+73 d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^6} \, dx &=\int \left (\frac {5329 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^6 (a+b x)^6}+\frac {10658 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^6 (a+b x)^5}+\frac {5329 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^6 (a+b x)^4}\right ) \, dx\\ &=\frac {\left (5329 d^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^4} \, dx}{b^2 g^6}+\frac {(10658 d (b c-a d)) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^5} \, dx}{b^2 g^6}+\frac {\left (5329 (b c-a d)^2\right ) \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^6} \, dx}{b^2 g^6}\\ &=-\frac {5329 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {5329 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {5329 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^6 (a+b x)^3}+\frac {\left (10658 B d^2\right ) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^6}+\frac {(5329 B d (b c-a d)) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^6}+\frac {\left (10658 B (b c-a d)^2\right ) \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^6 (c+d x)} \, dx}{5 b^3 g^6}\\ &=-\frac {5329 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {5329 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {5329 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^6 (a+b x)^3}+\frac {\left (10658 B d^2 (b c-a d)\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^6}+\frac {\left (5329 B d (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^6}+\frac {\left (10658 B (b c-a d)^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^6 (c+d x)} \, dx}{5 b^3 g^6}\\ &=-\frac {5329 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {5329 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {5329 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^6 (a+b x)^3}+\frac {\left (10658 B d^2 (b c-a d)\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^4}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b^3 g^6}+\frac {\left (5329 B d (b c-a d)^2\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^5}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 (a+b x)}-\frac {d^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{b^3 g^6}+\frac {\left (10658 B (b c-a d)^3\right ) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^6}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)^5}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (a+b x)^4}-\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 (a+b x)^3}+\frac {b d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 (a+b x)^2}-\frac {b d^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^6 (a+b x)}+\frac {d^6 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^6 (c+d x)}\right ) \, dx}{5 b^3 g^6}\\ &=-\frac {5329 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {5329 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {5329 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^6 (a+b x)^3}+\frac {\left (10658 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{5 b^2 g^6}+\frac {\left (10658 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{3 b^2 g^6}-\frac {\left (5329 B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{b^2 g^6}-\frac {\left (10658 B d^5\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{5 b^2 (b c-a d)^3 g^6}-\frac {\left (10658 B d^5\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{3 b^2 (b c-a d)^3 g^6}+\frac {\left (5329 B d^5\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}+\frac {\left (10658 B d^6\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B d^6\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{3 b^3 (b c-a d)^3 g^6}-\frac {\left (5329 B d^6\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{5 b^2 (b c-a d)^2 g^6}+\frac {\left (10658 B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{3 b^2 (b c-a d)^2 g^6}-\frac {\left (5329 B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{b^2 (b c-a d)^2 g^6}-\frac {\left (10658 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{5 b^2 (b c-a d) g^6}-\frac {\left (10658 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{3 b^2 (b c-a d) g^6}+\frac {\left (5329 B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^2 (b c-a d) g^6}-\frac {(10658 B d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{5 b^2 g^6}+\frac {(5329 B d (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{b^2 g^6}+\frac {\left (10658 B (b c-a d)^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^6} \, dx}{5 b^2 g^6}\\ &=-\frac {10658 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac {15987 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac {5329 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b^3 g^6 (a+b x)^3}+\frac {5329 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {5329 B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {5329 B d^5 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 (b c-a d)^3 g^6}-\frac {5329 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {5329 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {5329 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^6 (a+b x)^3}+\frac {5329 B d^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{15 b^3 g^6}+\frac {\left (10658 B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{9 b^3 g^6}-\frac {\left (5329 B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^6}+\frac {\left (10658 B^2 d^5\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{5 b^3 (b c-a d)^3 g^6}-\frac {\left (10658 B^2 d^5\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^5\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{3 b^3 (b c-a d)^3 g^6}-\frac {\left (10658 B^2 d^5\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{3 b^3 (b c-a d)^3 g^6}-\frac {\left (5329 B^2 d^5\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{b^3 (b c-a d)^3 g^6}+\frac {\left (5329 B^2 d^5\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^4\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{5 b^3 (b c-a d)^2 g^6}+\frac {\left (10658 B^2 d^4\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3 b^3 (b c-a d)^2 g^6}-\frac {\left (5329 B^2 d^4\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d)^2 g^6}-\frac {\left (5329 B^2 d^3\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{5 b^3 (b c-a d) g^6}-\frac {\left (5329 B^2 d^3\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{3 b^3 (b c-a d) g^6}+\frac {\left (5329 B^2 d^3\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2 b^3 (b c-a d) g^6}-\frac {\left (5329 B^2 d (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{10 b^3 g^6}+\frac {\left (5329 B^2 d (b c-a d)\right ) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{4 b^3 g^6}+\frac {\left (10658 B^2 (b c-a d)^2\right ) \int \frac {b c-a d}{(a+b x)^6 (c+d x)} \, dx}{25 b^3 g^6}\\ &=-\frac {10658 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac {15987 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac {5329 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b^3 g^6 (a+b x)^3}+\frac {5329 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {5329 B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {5329 B d^5 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 (b c-a d)^3 g^6}-\frac {5329 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {5329 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {5329 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^6 (a+b x)^3}+\frac {5329 B d^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 (b c-a d)^3 g^6}-\frac {\left (5329 B^2 d^3\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{5 b^3 g^6}-\frac {\left (5329 B^2 d^3\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{3 b^3 g^6}+\frac {\left (5329 B^2 d^3\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2 b^3 g^6}+\frac {\left (10658 B^2 d^4\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{5 b^3 (b c-a d) g^6}+\frac {\left (10658 B^2 d^4\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3 b^3 (b c-a d) g^6}-\frac {\left (5329 B^2 d^4\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^6}+\frac {\left (10658 B^2 d^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{15 b^3 g^6}+\frac {\left (10658 B^2 d^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{9 b^3 g^6}-\frac {\left (5329 B^2 d^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^6}-\frac {\left (5329 B^2 d (b c-a d)^2\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{10 b^3 g^6}+\frac {\left (5329 B^2 d (b c-a d)^2\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{4 b^3 g^6}+\frac {\left (10658 B^2 (b c-a d)^3\right ) \int \frac {1}{(a+b x)^6 (c+d x)} \, dx}{25 b^3 g^6}+\frac {\left (10658 B^2 d^5\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{5 b^3 (b c-a d)^3 e g^6}-\frac {\left (10658 B^2 d^5\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{5 b^3 (b c-a d)^3 e g^6}+\frac {\left (10658 B^2 d^5\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{3 b^3 (b c-a d)^3 e g^6}-\frac {\left (10658 B^2 d^5\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{3 b^3 (b c-a d)^3 e g^6}-\frac {\left (5329 B^2 d^5\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{b^3 (b c-a d)^3 e g^6}+\frac {\left (5329 B^2 d^5\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{b^3 (b c-a d)^3 e g^6}\\ &=-\frac {10658 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac {15987 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac {5329 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b^3 g^6 (a+b x)^3}+\frac {5329 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {5329 B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {5329 B d^5 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 (b c-a d)^3 g^6}-\frac {5329 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {5329 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {5329 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^6 (a+b x)^3}+\frac {5329 B d^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 (b c-a d)^3 g^6}-\frac {\left (5329 B^2 d^3\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}{5 b^3 g^6}-\frac {\left (5329 B^2 d^3\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^6}+\frac {\left (5329 B^2 d^3\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}{2 b^3 g^6}+\frac {\left (10658 B^2 d^4\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}{5 b^3 (b c-a d) g^6}+\frac {\left (10658 B^2 d^4\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 (b c-a d) g^6}-\frac {\left (5329 B^2 d^4\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 (b c-a d) g^6}+\frac {\left (10658 B^2 d^2 (b c-a d)\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}{15 b^3 g^6}+\frac {\left (10658 B^2 d^2 (b c-a d)\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^6}-\frac {\left (5329 B^2 d^2 (b c-a d)\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^6}-\frac {\left (5329 B^2 d (b c-a d)^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}{10 b^3 g^6}+\frac {\left (5329 B^2 d (b c-a d)^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}{4 b^3 g^6}+\frac {\left (10658 B^2 (b c-a d)^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^6}-\frac {b d}{(b c-a d)^2 (a+b x)^5}+\frac {b d^2}{(b c-a d)^3 (a+b x)^4}-\frac {b d^3}{(b c-a d)^4 (a+b x)^3}+\frac {b d^4}{(b c-a d)^5 (a+b x)^2}-\frac {b d^5}{(b c-a d)^6 (a+b x)}+\frac {d^6}{(b c-a d)^6 (c+d x)}\right ) \, dx}{25 b^3 g^6}+\frac {\left (10658 B^2 d^5\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{5 b^3 (b c-a d)^3 e g^6}-\frac {\left (10658 B^2 d^5\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{5 b^3 (b c-a d)^3 e g^6}+\frac {\left (10658 B^2 d^5\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{3 b^3 (b c-a d)^3 e g^6}-\frac {\left (10658 B^2 d^5\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{3 b^3 (b c-a d)^3 e g^6}-\frac {\left (5329 B^2 d^5\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^3 e g^6}+\frac {\left (5329 B^2 d^5\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{b^3 (b c-a d)^3 e g^6}\\ &=-\frac {10658 B^2 (b c-a d)^2}{125 b^3 g^6 (a+b x)^5}-\frac {37303 B^2 d (b c-a d)}{400 b^3 g^6 (a+b x)^4}+\frac {229147 B^2 d^2}{2700 b^3 g^6 (a+b x)^3}-\frac {69277 B^2 d^3}{1800 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {250463 B^2 d^4}{900 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {250463 B^2 d^5 \log (a+b x)}{900 b^3 (b c-a d)^3 g^6}-\frac {10658 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac {15987 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac {5329 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b^3 g^6 (a+b x)^3}+\frac {5329 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {5329 B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {5329 B d^5 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 (b c-a d)^3 g^6}-\frac {5329 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {5329 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {5329 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^6 (a+b x)^3}+\frac {250463 B^2 d^5 \log (c+d x)}{900 b^3 (b c-a d)^3 g^6}+\frac {5329 B d^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^5\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{5 b^2 (b c-a d)^3 g^6}-\frac {\left (10658 B^2 d^5\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{5 b^2 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^5\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3 b^2 (b c-a d)^3 g^6}-\frac {\left (10658 B^2 d^5\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 b^2 (b c-a d)^3 g^6}-\frac {\left (5329 B^2 d^5\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}+\frac {\left (5329 B^2 d^5\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^3 g^6}-\frac {\left (10658 B^2 d^6\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^6\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{5 b^3 (b c-a d)^3 g^6}-\frac {\left (10658 B^2 d^6\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^6\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b^3 (b c-a d)^3 g^6}+\frac {\left (5329 B^2 d^6\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}-\frac {\left (5329 B^2 d^6\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^3 g^6}\\ &=-\frac {10658 B^2 (b c-a d)^2}{125 b^3 g^6 (a+b x)^5}-\frac {37303 B^2 d (b c-a d)}{400 b^3 g^6 (a+b x)^4}+\frac {229147 B^2 d^2}{2700 b^3 g^6 (a+b x)^3}-\frac {69277 B^2 d^3}{1800 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {250463 B^2 d^4}{900 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {250463 B^2 d^5 \log (a+b x)}{900 b^3 (b c-a d)^3 g^6}-\frac {10658 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac {15987 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac {5329 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b^3 g^6 (a+b x)^3}+\frac {5329 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {5329 B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {5329 B d^5 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 (b c-a d)^3 g^6}-\frac {5329 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {5329 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {5329 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^6 (a+b x)^3}+\frac {250463 B^2 d^5 \log (c+d x)}{900 b^3 (b c-a d)^3 g^6}-\frac {5329 B^2 d^5 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{15 b^3 (b c-a d)^3 g^6}+\frac {5329 B d^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 (b c-a d)^3 g^6}-\frac {5329 B^2 d^5 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{15 b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^5\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^5\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^5\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^5\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b^3 (b c-a d)^3 g^6}-\frac {\left (5329 B^2 d^5\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^3 g^6}-\frac {\left (5329 B^2 d^5\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^5\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{5 b^2 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^5\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)^3 g^6}-\frac {\left (5329 B^2 d^5\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)^3 g^6}+\frac {\left (10658 B^2 d^6\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^6\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)^3 g^6}-\frac {\left (5329 B^2 d^6\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)^3 g^6}\\ &=-\frac {10658 B^2 (b c-a d)^2}{125 b^3 g^6 (a+b x)^5}-\frac {37303 B^2 d (b c-a d)}{400 b^3 g^6 (a+b x)^4}+\frac {229147 B^2 d^2}{2700 b^3 g^6 (a+b x)^3}-\frac {69277 B^2 d^3}{1800 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {250463 B^2 d^4}{900 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {250463 B^2 d^5 \log (a+b x)}{900 b^3 (b c-a d)^3 g^6}+\frac {5329 B^2 d^5 \log ^2(a+b x)}{30 b^3 (b c-a d)^3 g^6}-\frac {10658 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac {15987 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac {5329 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b^3 g^6 (a+b x)^3}+\frac {5329 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {5329 B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {5329 B d^5 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 (b c-a d)^3 g^6}-\frac {5329 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {5329 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {5329 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^6 (a+b x)^3}+\frac {250463 B^2 d^5 \log (c+d x)}{900 b^3 (b c-a d)^3 g^6}-\frac {5329 B^2 d^5 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{15 b^3 (b c-a d)^3 g^6}+\frac {5329 B d^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 (b c-a d)^3 g^6}+\frac {5329 B^2 d^5 \log ^2(c+d x)}{30 b^3 (b c-a d)^3 g^6}-\frac {5329 B^2 d^5 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{15 b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^5\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^5\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{5 b^3 (b c-a d)^3 g^6}+\frac {\left (10658 B^2 d^5\right ) \operatorname {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)^3 g^6}+\frac {\left (10658 B^2 d^5\right ) \operatorname {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)^3 g^6}-\frac {\left (5329 B^2 d^5\right ) \operatorname {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)^3 g^6}-\frac {\left (5329 B^2 d^5\right ) \operatorname {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)^3 g^6}\\ &=-\frac {10658 B^2 (b c-a d)^2}{125 b^3 g^6 (a+b x)^5}-\frac {37303 B^2 d (b c-a d)}{400 b^3 g^6 (a+b x)^4}+\frac {229147 B^2 d^2}{2700 b^3 g^6 (a+b x)^3}-\frac {69277 B^2 d^3}{1800 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {250463 B^2 d^4}{900 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {250463 B^2 d^5 \log (a+b x)}{900 b^3 (b c-a d)^3 g^6}+\frac {5329 B^2 d^5 \log ^2(a+b x)}{30 b^3 (b c-a d)^3 g^6}-\frac {10658 B (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{25 b^3 g^6 (a+b x)^5}-\frac {15987 B d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{20 b^3 g^6 (a+b x)^4}-\frac {5329 B d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{45 b^3 g^6 (a+b x)^3}+\frac {5329 B d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{30 b^3 (b c-a d) g^6 (a+b x)^2}-\frac {5329 B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 (b c-a d)^2 g^6 (a+b x)}-\frac {5329 B d^5 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{15 b^3 (b c-a d)^3 g^6}-\frac {5329 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{5 b^3 g^6 (a+b x)^5}-\frac {5329 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 b^3 g^6 (a+b x)^4}-\frac {5329 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^6 (a+b x)^3}+\frac {250463 B^2 d^5 \log (c+d x)}{900 b^3 (b c-a d)^3 g^6}-\frac {5329 B^2 d^5 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{15 b^3 (b c-a d)^3 g^6}+\frac {5329 B d^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{15 b^3 (b c-a d)^3 g^6}+\frac {5329 B^2 d^5 \log ^2(c+d x)}{30 b^3 (b c-a d)^3 g^6}-\frac {5329 B^2 d^5 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{15 b^3 (b c-a d)^3 g^6}-\frac {5329 B^2 d^5 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{15 b^3 (b c-a d)^3 g^6}-\frac {5329 B^2 d^5 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{15 b^3 (b c-a d)^3 g^6}\\ \end {align*}

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Mathematica [C]  time = 4.70, size = 2220, normalized size = 4.79 \[ \text {Result too large to show} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/54000*(i^2*(10800*(b*c - a*d)^5*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 27000*d*(b*c - a*d)^4*(a + b*x)*(A
+ B*Log[(e*(a + b*x))/(c + d*x)])^2 - 18000*d^2*(-(b*c) + a*d)^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*
x)])^2 + 1000*B*d^2*(a + b*x)^2*(12*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 18*d*(b*c - a*d)^2*(a
+ b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 36*d^2*(b*c - a*d)*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*
x)]) + 36*d^3*(a + b*x)^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 36*d^3*(a + b*x)^3*(A + B*Log[(e
*(a + b*x))/(c + d*x)])*Log[c + d*x] + 36*B*d^2*(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*(a + b*x)*((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*(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*(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*(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)])) + 375*B*d*(a + b*x)*(36*(b*c - a*d)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 48*d*(-(b*c) + a*
d)^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 72*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[(e*(a + b*x)
)/(c + d*x)]) + 144*d^3*(-(b*c) + a*d)*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 144*d^4*(a + b*x)^4*
Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 144*d^4*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])*L
og[c + d*x] - 144*B*d^3*(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*(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*(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*(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*L
og[a + b*x] + 12*d^4*(a + b*x)^4*Log[c + d*x]) + 72*B*d^4*(a + b*x)^4*(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)]) - 72*B*d^4*(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)])) + 6*B*(-225*a*B*
d*(b*c - a*d)^4 + 144*B*(b*c - a*d)^5 - 225*b*B*d*(b*c - a*d)^4*x + 300*a*B*d^2*(b*c - a*d)^3*(a + b*x) - 180*
B*d*(b*c - a*d)^4*(a + b*x) + 300*b*B*d^2*(b*c - a*d)^3*x*(a + b*x) - 450*a*B*d^3*(b*c - a*d)^2*(a + b*x)^2 +
640*B*d^2*(b*c - a*d)^3*(a + b*x)^2 - 450*b*B*d^3*(b*c - a*d)^2*x*(a + b*x)^2 + 900*a*B*d^4*(b*c - a*d)*(a + b
*x)^3 - 1860*B*d^3*(b*c - a*d)^2*(a + b*x)^3 + 900*b*B*d^4*(b*c - a*d)*x*(a + b*x)^3 + 3600*b*B*c*d^4*(a + b*x
)^4 - 3600*a*B*d^5*(a + b*x)^4 + 3720*B*d^4*(b*c - a*d)*(a + b*x)^4 + 900*a*B*d^5*(a + b*x)^4*Log[a + b*x] + 9
00*b*B*d^5*x*(a + b*x)^4*Log[a + b*x] + 7320*B*d^5*(a + b*x)^5*Log[a + b*x] + 720*(b*c - a*d)^5*(A + B*Log[(e*
(a + b*x))/(c + d*x)]) - 900*d*(b*c - a*d)^4*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 1200*d^2*(b*c -
a*d)^3*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 1800*d^3*(b*c - a*d)^2*(a + b*x)^3*(A + B*Log[(e*(a
+ b*x))/(c + d*x)]) + 3600*d^4*(b*c - a*d)*(a + b*x)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 3600*d^5*(a + b*
x)^5*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 900*a*B*d^5*(a + b*x)^4*Log[c + d*x] - 900*b*B*d^5*x*
(a + b*x)^4*Log[c + d*x] - 7320*B*d^5*(a + b*x)^5*Log[c + d*x] - 3600*d^5*(a + b*x)^5*(A + B*Log[(e*(a + b*x))
/(c + d*x)])*Log[c + d*x] - 1800*B*d^5*(a + b*x)^5*(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)]) + 1800*B*d^5*(a + b*x)^5*((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)^3*g^6*(a + b*x
)^5)

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fricas [B]  time = 0.68, size = 1323, normalized size = 2.86 \[ \text {result too large to display} \]

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)))^2/(b*g*x+a*g)^6,x, algorithm="fricas")

[Out]

-1/54000*(60*((60*A*B + 47*B^2)*b^5*c*d^4 - (60*A*B + 47*B^2)*a*b^4*d^5)*i^2*x^4 - 30*((60*A*B - 13*B^2)*b^5*c
^2*d^3 - 50*(12*A*B + 7*B^2)*a*b^4*c*d^4 + 3*(180*A*B + 121*B^2)*a^2*b^3*d^5)*i^2*x^3 + 10*(2*(900*A^2 + 60*A*
B - 43*B^2)*b^5*c^3*d^2 - 75*(72*A^2 + 12*A*B - 5*B^2)*a*b^4*c^2*d^3 + 600*(9*A^2 + 6*A*B + 2*B^2)*a^2*b^3*c*d
^4 - (1800*A^2 + 2820*A*B + 1489*B^2)*a^3*b^2*d^5)*i^2*x^2 + 5*(27*(200*A^2 + 60*A*B + 7*B^2)*b^5*c^4*d - 100*
(144*A^2 + 60*A*B + 11*B^2)*a*b^4*c^3*d^2 + 1200*(9*A^2 + 6*A*B + 2*B^2)*a^2*b^3*c^2*d^3 - (1800*A^2 + 2820*A*
B + 1489*B^2)*a^4*b*d^5)*i^2*x + (432*(25*A^2 + 10*A*B + 2*B^2)*b^5*c^5 - 3375*(8*A^2 + 4*A*B + B^2)*a*b^4*c^4
*d + 2000*(9*A^2 + 6*A*B + 2*B^2)*a^2*b^3*c^3*d^2 - (1800*A^2 + 2820*A*B + 1489*B^2)*a^5*d^5)*i^2 + 1800*(B^2*
b^5*d^5*i^2*x^5 + 5*B^2*a*b^4*d^5*i^2*x^4 + 10*B^2*a^2*b^3*d^5*i^2*x^3 + 10*(B^2*b^5*c^3*d^2 - 3*B^2*a*b^4*c^2
*d^3 + 3*B^2*a^2*b^3*c*d^4)*i^2*x^2 + 5*(3*B^2*b^5*c^4*d - 8*B^2*a*b^4*c^3*d^2 + 6*B^2*a^2*b^3*c^2*d^3)*i^2*x
+ (6*B^2*b^5*c^5 - 15*B^2*a*b^4*c^4*d + 10*B^2*a^2*b^3*c^3*d^2)*i^2)*log((b*e*x + a*e)/(d*x + c))^2 + 60*((60*
A*B + 47*B^2)*b^5*d^5*i^2*x^5 + 5*(12*B^2*b^5*c*d^4 + 5*(12*A*B + 7*B^2)*a*b^4*d^5)*i^2*x^4 - 10*(3*B^2*b^5*c^
2*d^3 - 30*B^2*a*b^4*c*d^4 - 20*(3*A*B + B^2)*a^2*b^3*d^5)*i^2*x^3 + 10*(2*(30*A*B + B^2)*b^5*c^3*d^2 - 15*(12
*A*B + B^2)*a*b^4*c^2*d^3 + 60*(3*A*B + B^2)*a^2*b^3*c*d^4)*i^2*x^2 + 5*(9*(20*A*B + 3*B^2)*b^5*c^4*d - 20*(24
*A*B + 5*B^2)*a*b^4*c^3*d^2 + 120*(3*A*B + B^2)*a^2*b^3*c^2*d^3)*i^2*x + (72*(5*A*B + B^2)*b^5*c^5 - 225*(4*A*
B + B^2)*a*b^4*c^4*d + 200*(3*A*B + B^2)*a^2*b^3*c^3*d^2)*i^2)*log((b*e*x + a*e)/(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)

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giac [A]  time = 3.09, size = 709, normalized size = 1.53 \[ \frac {{\left (10800 \, B^{2} b^{2} e^{6} \log \left (\frac {b x e + a e}{d x + c}\right )^{2} - \frac {27000 \, {\left (b x e + a e\right )} B^{2} b d e^{5} \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{d x + c} + \frac {18000 \, {\left (b x e + a e\right )}^{2} B^{2} d^{2} e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{{\left (d x + c\right )}^{2}} + 21600 \, A B b^{2} e^{6} \log \left (\frac {b x e + a e}{d x + c}\right ) + 4320 \, B^{2} b^{2} e^{6} \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {54000 \, {\left (b x e + a e\right )} A B b d e^{5} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} - \frac {13500 \, {\left (b x e + a e\right )} B^{2} b d e^{5} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + \frac {36000 \, {\left (b x e + a e\right )}^{2} A B d^{2} e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + \frac {12000 \, {\left (b x e + a e\right )}^{2} B^{2} d^{2} e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + 10800 \, A^{2} b^{2} e^{6} + 4320 \, A B b^{2} e^{6} + 864 \, B^{2} b^{2} e^{6} - \frac {27000 \, {\left (b x e + a e\right )} A^{2} b d e^{5}}{d x + c} - \frac {13500 \, {\left (b x e + a e\right )} A B b d e^{5}}{d x + c} - \frac {3375 \, {\left (b x e + a e\right )} B^{2} b d e^{5}}{d x + c} + \frac {18000 \, {\left (b x e + a e\right )}^{2} A^{2} d^{2} e^{4}}{{\left (d x + c\right )}^{2}} + \frac {12000 \, {\left (b x e + a e\right )}^{2} A B d^{2} e^{4}}{{\left (d x + c\right )}^{2}} + \frac {4000 \, {\left (b x e + a e\right )}^{2} B^{2} d^{2} e^{4}}{{\left (d x + c\right )}^{2}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{54000 \, {\left (\frac {{\left (b x e + a e\right )}^{5} b^{2} c^{2} g^{6}}{{\left (d x + c\right )}^{5}} - \frac {2 \, {\left (b x e + a e\right )}^{5} a b c d g^{6}}{{\left (d x + c\right )}^{5}} + \frac {{\left (b x e + a e\right )}^{5} a^{2} d^{2} g^{6}}{{\left (d x + c\right )}^{5}}\right )}} \]

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)))^2/(b*g*x+a*g)^6,x, algorithm="giac")

[Out]

1/54000*(10800*B^2*b^2*e^6*log((b*x*e + a*e)/(d*x + c))^2 - 27000*(b*x*e + a*e)*B^2*b*d*e^5*log((b*x*e + a*e)/
(d*x + c))^2/(d*x + c) + 18000*(b*x*e + a*e)^2*B^2*d^2*e^4*log((b*x*e + a*e)/(d*x + c))^2/(d*x + c)^2 + 21600*
A*B*b^2*e^6*log((b*x*e + a*e)/(d*x + c)) + 4320*B^2*b^2*e^6*log((b*x*e + a*e)/(d*x + c)) - 54000*(b*x*e + a*e)
*A*B*b*d*e^5*log((b*x*e + a*e)/(d*x + c))/(d*x + c) - 13500*(b*x*e + a*e)*B^2*b*d*e^5*log((b*x*e + a*e)/(d*x +
c))/(d*x + c) + 36000*(b*x*e + a*e)^2*A*B*d^2*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 12000*(b*x*e + a
*e)^2*B^2*d^2*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 10800*A^2*b^2*e^6 + 4320*A*B*b^2*e^6 + 864*B^2*b^
2*e^6 - 27000*(b*x*e + a*e)*A^2*b*d*e^5/(d*x + c) - 13500*(b*x*e + a*e)*A*B*b*d*e^5/(d*x + c) - 3375*(b*x*e +
a*e)*B^2*b*d*e^5/(d*x + c) + 18000*(b*x*e + a*e)^2*A^2*d^2*e^4/(d*x + c)^2 + 12000*(b*x*e + a*e)^2*A*B*d^2*e^4
/(d*x + c)^2 + 4000*(b*x*e + a*e)^2*B^2*d^2*e^4/(d*x + c)^2)*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e
- a*d*e)*(b*c - a*d)))/((b*x*e + a*e)^5*b^2*c^2*g^6/(d*x + c)^5 - 2*(b*x*e + a*e)^5*a*b*c*d*g^6/(d*x + c)^5 +
(b*x*e + a*e)^5*a^2*d^2*g^6/(d*x + c)^5)

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maple [B]  time = 0.05, size = 2761, normalized size = 5.96 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*d^2*e^3*i^2/(a*d-b*c)^4/g^6*A*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)
*b*c-d^2*e^4*i^2/(a*d-b*c)^4/g^6*A*B*b/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^4*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*
e)*a+d*e^4*i^2/(a*d-b*c)^4/g^6*A*B*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^4*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*
e)*c+2/5*d*e^5*i^2/(a*d-b*c)^4/g^6*A*B*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^5*ln(b/d*e+(a*d-b*c)/(d*x+c
)/d*e)*a+1/3*d^3*e^3*i^2/(a*d-b*c)^4/g^6*A^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*a+2/27*d^3*e^3*i^2/(a*d
-b*c)^4/g^6*B^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*a-2/125*e^5*i^2/(a*d-b*c)^4/g^6*B^2*b^3/(1/(d*x+c)*a
*e-1/(d*x+c)*b*c/d*e+b/d*e)^5*c-1/5*e^5*i^2/(a*d-b*c)^4/g^6*A^2*b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^5*
c-1/3*d^2*e^3*i^2/(a*d-b*c)^4/g^6*B^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e
)^2*b*c+1/3*d^3*e^3*i^2/(a*d-b*c)^4/g^6*B^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*ln(b/d*e+(a*d-b*c)/(d*x+
c)/d*e)^2*a+1/2*d*e^4*i^2/(a*d-b*c)^4/g^6*A^2*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^4*c+1/5*d*e^5*i^2/(a
*d-b*c)^4/g^6*A^2*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^5*a+2/9*d^3*e^3*i^2/(a*d-b*c)^4/g^6*A*B/(1/(d*x+
c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*a-2/25*e^5*i^2/(a*d-b*c)^4/g^6*B^2*b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*
e)^5*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*c+2/25*d*e^5*i^2/(a*d-b*c)^4/g^6*B^2*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e
+b/d*e)^5*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*a+2/3*d^3*e^3*i^2/(a*d-b*c)^4/g^6*A*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d
*e+b/d*e)^3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*a-2/9*d^2*e^3*i^2/(a*d-b*c)^4/g^6*A*B/(1/(d*x+c)*a*e-1/(d*x+c)*b*c
/d*e+b/d*e)^3*b*c-1/4*d^2*e^4*i^2/(a*d-b*c)^4/g^6*A*B*b/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^4*a-1/2*d^2*e^
4*i^2/(a*d-b*c)^4/g^6*B^2*b/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^4*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*a+1/2*
d*e^4*i^2/(a*d-b*c)^4/g^6*B^2*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^4*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*
c-1/4*d^2*e^4*i^2/(a*d-b*c)^4/g^6*B^2*b/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^4*ln(b/d*e+(a*d-b*c)/(d*x+c)/d
*e)*a+1/4*d*e^4*i^2/(a*d-b*c)^4/g^6*B^2*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^4*ln(b/d*e+(a*d-b*c)/(d*x+
c)/d*e)*c+1/5*d*e^5*i^2/(a*d-b*c)^4/g^6*B^2*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^5*ln(b/d*e+(a*d-b*c)/(
d*x+c)/d*e)^2*a-2/25*e^5*i^2/(a*d-b*c)^4/g^6*A*B*b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^5*c-2/27*d^2*e^3*
i^2/(a*d-b*c)^4/g^6*B^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*b*c+2/125*d*e^5*i^2/(a*d-b*c)^4/g^6*B^2*b^2/
(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^5*a-1/16*d^2*e^4*i^2/(a*d-b*c)^4/g^6*B^2*b/(1/(d*x+c)*a*e-1/(d*x+c)*b*
c/d*e+b/d*e)^4*a+1/16*d*e^4*i^2/(a*d-b*c)^4/g^6*B^2*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^4*c-1/3*d^2*e^
3*i^2/(a*d-b*c)^4/g^6*A^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*b*c-1/2*d^2*e^4*i^2/(a*d-b*c)^4/g^6*A^2*b/
(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^4*a+2/9*d^3*e^3*i^2/(a*d-b*c)^4/g^6*B^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d
*e+b/d*e)^3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*a-1/5*e^5*i^2/(a*d-b*c)^4/g^6*B^2*b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c
/d*e+b/d*e)^5*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)^2*c+1/4*d*e^4*i^2/(a*d-b*c)^4/g^6*A*B*b^2/(1/(d*x+c)*a*e-1/(d*x+
c)*b*c/d*e+b/d*e)^4*c+2/25*d*e^5*i^2/(a*d-b*c)^4/g^6*A*B*b^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^5*a-2/5*e
^5*i^2/(a*d-b*c)^4/g^6*A*B*b^3/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^5*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*c-2/9
*d^2*e^3*i^2/(a*d-b*c)^4/g^6*B^2/(1/(d*x+c)*a*e-1/(d*x+c)*b*c/d*e+b/d*e)^3*ln(b/d*e+(a*d-b*c)/(d*x+c)/d*e)*b*c

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maxima [B]  time = 12.03, size = 10880, normalized size = 23.50 \[ \text {result too large to display} \]

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)))^2/(b*g*x+a*g)^6,x, algorithm="maxima")

[Out]

-1/10*(5*b*x + a)*B^2*c*d*i^2*log(b*e*x/(d*x + c) + a*e/(d*x + c))^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)*B^2*d^2*i
^2*log(b*e*x/(d*x + c) + a*e/(d*x + c))^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) - 1/9000*(60*((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 + 137*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))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + (144*b^5*c^5 - 1125*a*b^4*c^4*d + 4000*
a^2*b^3*c^3*d^2 - 9000*a^3*b^2*c^2*d^3 + 18000*a^4*b*c*d^4 - 12019*a^5*d^5 + 8220*(b^5*c*d^4 - a*b^4*d^5)*x^4
- 30*(77*b^5*c^2*d^3 - 1250*a*b^4*c*d^4 + 1173*a^2*b^3*d^5)*x^3 + 10*(94*b^5*c^3*d^2 - 975*a*b^4*c^2*d^3 + 660
0*a^2*b^3*c*d^4 - 5719*a^3*b^2*d^5)*x^2 - 1800*(b^5*d^5*x^5 + 5*a*b^4*d^5*x^4 + 10*a^2*b^3*d^5*x^3 + 10*a^3*b^
2*d^5*x^2 + 5*a^4*b*d^5*x + a^5*d^5)*log(b*x + a)^2 - 1800*(b^5*d^5*x^5 + 5*a*b^4*d^5*x^4 + 10*a^2*b^3*d^5*x^3
+ 10*a^3*b^2*d^5*x^2 + 5*a^4*b*d^5*x + a^5*d^5)*log(d*x + c)^2 - 5*(81*b^5*c^4*d - 700*a*b^4*c^3*d^2 + 3000*a
^2*b^3*c^2*d^3 - 10800*a^3*b^2*c*d^4 + 8419*a^4*b*d^5)*x + 8220*(b^5*d^5*x^5 + 5*a*b^4*d^5*x^4 + 10*a^2*b^3*d^
5*x^3 + 10*a^3*b^2*d^5*x^2 + 5*a^4*b*d^5*x + a^5*d^5)*log(b*x + a) - 60*(137*b^5*d^5*x^5 + 685*a*b^4*d^5*x^4 +
1370*a^2*b^3*d^5*x^3 + 1370*a^3*b^2*d^5*x^2 + 685*a^4*b*d^5*x + 137*a^5*d^5 - 60*(b^5*d^5*x^5 + 5*a*b^4*d^5*x
^4 + 10*a^2*b^3*d^5*x^3 + 10*a^3*b^2*d^5*x^2 + 5*a^4*b*d^5*x + a^5*d^5)*log(b*x + a))*log(d*x + c))/(a^5*b^6*c
^5*g^6 - 5*a^6*b^5*c^4*d*g^6 + 10*a^7*b^4*c^3*d^2*g^6 - 10*a^8*b^3*c^2*d^3*g^6 + 5*a^9*b^2*c*d^4*g^6 - a^10*b*
d^5*g^6 + (b^11*c^5*g^6 - 5*a*b^10*c^4*d*g^6 + 10*a^2*b^9*c^3*d^2*g^6 - 10*a^3*b^8*c^2*d^3*g^6 + 5*a^4*b^7*c*d
^4*g^6 - a^5*b^6*d^5*g^6)*x^5 + 5*(a*b^10*c^5*g^6 - 5*a^2*b^9*c^4*d*g^6 + 10*a^3*b^8*c^3*d^2*g^6 - 10*a^4*b^7*
c^2*d^3*g^6 + 5*a^5*b^6*c*d^4*g^6 - a^6*b^5*d^5*g^6)*x^4 + 10*(a^2*b^9*c^5*g^6 - 5*a^3*b^8*c^4*d*g^6 + 10*a^4*
b^7*c^3*d^2*g^6 - 10*a^5*b^6*c^2*d^3*g^6 + 5*a^6*b^5*c*d^4*g^6 - a^7*b^4*d^5*g^6)*x^3 + 10*(a^3*b^8*c^5*g^6 -
5*a^4*b^7*c^4*d*g^6 + 10*a^5*b^6*c^3*d^2*g^6 - 10*a^6*b^5*c^2*d^3*g^6 + 5*a^7*b^4*c*d^4*g^6 - a^8*b^3*d^5*g^6)
*x^2 + 5*(a^4*b^7*c^5*g^6 - 5*a^5*b^6*c^4*d*g^6 + 10*a^6*b^5*c^3*d^2*g^6 - 10*a^7*b^4*c^2*d^3*g^6 + 5*a^8*b^3*
c*d^4*g^6 - a^9*b^2*d^5*g^6)*x))*B^2*c^2*i^2 - 1/18000*(60*((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))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + (549*a*b^5*c^5
- 4625*a^2*b^4*c^4*d + 19000*a^3*b^3*c^3*d^2 - 63000*a^4*b^2*c^2*d^3 + 51875*a^5*b*c*d^4 - 3799*a^6*d^5 - 60*(
625*b^6*c^2*d^3 - 702*a*b^5*c*d^4 + 77*a^2*b^4*d^5)*x^4 + 30*(325*b^6*c^3*d^2 - 5667*a*b^5*c^2*d^3 + 5975*a^2*
b^4*c*d^4 - 633*a^3*b^3*d^5)*x^3 - 10*(350*b^6*c^4*d - 3949*a*b^5*c^3*d^2 + 29475*a^2*b^4*c^2*d^3 - 28775*a^3*
b^3*c*d^4 + 2899*a^4*b^2*d^5)*x^2 + 1800*(5*a^5*b*c*d^4 - a^6*d^5 + (5*b^6*c*d^4 - a*b^5*d^5)*x^5 + 5*(5*a*b^5
*c*d^4 - a^2*b^4*d^5)*x^4 + 10*(5*a^2*b^4*c*d^4 - a^3*b^3*d^5)*x^3 + 10*(5*a^3*b^3*c*d^4 - a^4*b^2*d^5)*x^2 +
5*(5*a^4*b^2*c*d^4 - a^5*b*d^5)*x)*log(b*x + a)^2 + 1800*(5*a^5*b*c*d^4 - a^6*d^5 + (5*b^6*c*d^4 - a*b^5*d^5)*
x^5 + 5*(5*a*b^5*c*d^4 - a^2*b^4*d^5)*x^4 + 10*(5*a^2*b^4*c*d^4 - a^3*b^3*d^5)*x^3 + 10*(5*a^3*b^3*c*d^4 - a^4
*b^2*d^5)*x^2 + 5*(5*a^4*b^2*c*d^4 - a^5*b*d^5)*x)*log(d*x + c)^2 + 5*(225*b^6*c^5 - 2201*a*b^5*c^4*d + 10900*
a^2*b^4*c^3*d^2 - 46200*a^3*b^3*c^2*d^3 + 41075*a^4*b^2*c*d^4 - 3799*a^5*b*d^5)*x - 60*(625*a^5*b*c*d^4 - 77*a
^6*d^5 + (625*b^6*c*d^4 - 77*a*b^5*d^5)*x^5 + 5*(625*a*b^5*c*d^4 - 77*a^2*b^4*d^5)*x^4 + 10*(625*a^2*b^4*c*d^4
- 77*a^3*b^3*d^5)*x^3 + 10*(625*a^3*b^3*c*d^4 - 77*a^4*b^2*d^5)*x^2 + 5*(625*a^4*b^2*c*d^4 - 77*a^5*b*d^5)*x)
*log(b*x + a) + 60*(625*a^5*b*c*d^4 - 77*a^6*d^5 + (625*b^6*c*d^4 - 77*a*b^5*d^5)*x^5 + 5*(625*a*b^5*c*d^4 - 7
7*a^2*b^4*d^5)*x^4 + 10*(625*a^2*b^4*c*d^4 - 77*a^3*b^3*d^5)*x^3 + 10*(625*a^3*b^3*c*d^4 - 77*a^4*b^2*d^5)*x^2
+ 5*(625*a^4*b^2*c*d^4 - 77*a^5*b*d^5)*x - 60*(5*a^5*b*c*d^4 - a^6*d^5 + (5*b^6*c*d^4 - a*b^5*d^5)*x^5 + 5*(5
*a*b^5*c*d^4 - a^2*b^4*d^5)*x^4 + 10*(5*a^2*b^4*c*d^4 - a^3*b^3*d^5)*x^3 + 10*(5*a^3*b^3*c*d^4 - a^4*b^2*d^5)*
x^2 + 5*(5*a^4*b^2*c*d^4 - a^5*b*d^5)*x)*log(b*x + a))*log(d*x + c))/(a^5*b^7*c^5*g^6 - 5*a^6*b^6*c^4*d*g^6 +
10*a^7*b^5*c^3*d^2*g^6 - 10*a^8*b^4*c^2*d^3*g^6 + 5*a^9*b^3*c*d^4*g^6 - a^10*b^2*d^5*g^6 + (b^12*c^5*g^6 - 5*a
*b^11*c^4*d*g^6 + 10*a^2*b^10*c^3*d^2*g^6 - 10*a^3*b^9*c^2*d^3*g^6 + 5*a^4*b^8*c*d^4*g^6 - a^5*b^7*d^5*g^6)*x^
5 + 5*(a*b^11*c^5*g^6 - 5*a^2*b^10*c^4*d*g^6 + 10*a^3*b^9*c^3*d^2*g^6 - 10*a^4*b^8*c^2*d^3*g^6 + 5*a^5*b^7*c*d
^4*g^6 - a^6*b^6*d^5*g^6)*x^4 + 10*(a^2*b^10*c^5*g^6 - 5*a^3*b^9*c^4*d*g^6 + 10*a^4*b^8*c^3*d^2*g^6 - 10*a^5*b
^7*c^2*d^3*g^6 + 5*a^6*b^6*c*d^4*g^6 - a^7*b^5*d^5*g^6)*x^3 + 10*(a^3*b^9*c^5*g^6 - 5*a^4*b^8*c^4*d*g^6 + 10*a
^5*b^7*c^3*d^2*g^6 - 10*a^6*b^6*c^2*d^3*g^6 + 5*a^7*b^5*c*d^4*g^6 - a^8*b^4*d^5*g^6)*x^2 + 5*(a^4*b^8*c^5*g^6
- 5*a^5*b^7*c^4*d*g^6 + 10*a^6*b^6*c^3*d^2*g^6 - 10*a^7*b^5*c^2*d^3*g^6 + 5*a^8*b^4*c*d^4*g^6 - a^9*b^3*d^5*g^
6)*x))*B^2*c*d*i^2 - 1/54000*(60*((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 + 4
6*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))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + (1489*a^2*b^5*c^5 - 14375*a^3*b^4*c^4*d
+ 85000*a^4*b^3*c^3*d^2 - 85000*a^5*b^2*c^2*d^3 + 14375*a^6*b*c*d^4 - 1489*a^7*d^5 + 60*(1100*b^7*c^3*d^2 - 1
425*a*b^6*c^2*d^3 + 372*a^2*b^5*c*d^4 - 47*a^3*b^4*d^5)*x^4 - 30*(500*b^7*c^4*d - 9825*a*b^6*c^3*d^2 + 11937*a
^2*b^5*c^2*d^3 - 2975*a^3*b^4*c*d^4 + 363*a^4*b^3*d^5)*x^3 + 10*(400*b^7*c^5 - 5450*a*b^6*c^4*d + 49189*a^2*b^
5*c^3*d^2 - 55525*a^3*b^4*c^2*d^3 + 12875*a^4*b^3*c*d^4 - 1489*a^5*b^2*d^5)*x^2 - 1800*(10*a^5*b^2*c^2*d^3 - 5
*a^6*b*c*d^4 + a^7*d^5 + (10*b^7*c^2*d^3 - 5*a*b^6*c*d^4 + a^2*b^5*d^5)*x^5 + 5*(10*a*b^6*c^2*d^3 - 5*a^2*b^5*
c*d^4 + a^3*b^4*d^5)*x^4 + 10*(10*a^2*b^5*c^2*d^3 - 5*a^3*b^4*c*d^4 + a^4*b^3*d^5)*x^3 + 10*(10*a^3*b^4*c^2*d^
3 - 5*a^4*b^3*c*d^4 + a^5*b^2*d^5)*x^2 + 5*(10*a^4*b^3*c^2*d^3 - 5*a^5*b^2*c*d^4 + a^6*b*d^5)*x)*log(b*x + a)^
2 - 1800*(10*a^5*b^2*c^2*d^3 - 5*a^6*b*c*d^4 + a^7*d^5 + (10*b^7*c^2*d^3 - 5*a*b^6*c*d^4 + a^2*b^5*d^5)*x^5 +
5*(10*a*b^6*c^2*d^3 - 5*a^2*b^5*c*d^4 + a^3*b^4*d^5)*x^4 + 10*(10*a^2*b^5*c^2*d^3 - 5*a^3*b^4*c*d^4 + a^4*b^3*
d^5)*x^3 + 10*(10*a^3*b^4*c^2*d^3 - 5*a^4*b^3*c*d^4 + a^5*b^2*d^5)*x^2 + 5*(10*a^4*b^3*c^2*d^3 - 5*a^5*b^2*c*d
^4 + a^6*b*d^5)*x)*log(d*x + c)^2 + 5*(925*a*b^6*c^5 - 9911*a^2*b^5*c^4*d + 67900*a^3*b^4*c^3*d^2 - 71800*a^4*
b^3*c^2*d^3 + 14375*a^5*b^2*c*d^4 - 1489*a^6*b*d^5)*x + 60*(1100*a^5*b^2*c^2*d^3 - 325*a^6*b*c*d^4 + 47*a^7*d^
5 + (1100*b^7*c^2*d^3 - 325*a*b^6*c*d^4 + 47*a^2*b^5*d^5)*x^5 + 5*(1100*a*b^6*c^2*d^3 - 325*a^2*b^5*c*d^4 + 47
*a^3*b^4*d^5)*x^4 + 10*(1100*a^2*b^5*c^2*d^3 - 325*a^3*b^4*c*d^4 + 47*a^4*b^3*d^5)*x^3 + 10*(1100*a^3*b^4*c^2*
d^3 - 325*a^4*b^3*c*d^4 + 47*a^5*b^2*d^5)*x^2 + 5*(1100*a^4*b^3*c^2*d^3 - 325*a^5*b^2*c*d^4 + 47*a^6*b*d^5)*x)
*log(b*x + a) - 60*(1100*a^5*b^2*c^2*d^3 - 325*a^6*b*c*d^4 + 47*a^7*d^5 + (1100*b^7*c^2*d^3 - 325*a*b^6*c*d^4
+ 47*a^2*b^5*d^5)*x^5 + 5*(1100*a*b^6*c^2*d^3 - 325*a^2*b^5*c*d^4 + 47*a^3*b^4*d^5)*x^4 + 10*(1100*a^2*b^5*c^2
*d^3 - 325*a^3*b^4*c*d^4 + 47*a^4*b^3*d^5)*x^3 + 10*(1100*a^3*b^4*c^2*d^3 - 325*a^4*b^3*c*d^4 + 47*a^5*b^2*d^5
)*x^2 + 5*(1100*a^4*b^3*c^2*d^3 - 325*a^5*b^2*c*d^4 + 47*a^6*b*d^5)*x - 60*(10*a^5*b^2*c^2*d^3 - 5*a^6*b*c*d^4
+ a^7*d^5 + (10*b^7*c^2*d^3 - 5*a*b^6*c*d^4 + a^2*b^5*d^5)*x^5 + 5*(10*a*b^6*c^2*d^3 - 5*a^2*b^5*c*d^4 + a^3*
b^4*d^5)*x^4 + 10*(10*a^2*b^5*c^2*d^3 - 5*a^3*b^4*c*d^4 + a^4*b^3*d^5)*x^3 + 10*(10*a^3*b^4*c^2*d^3 - 5*a^4*b^
3*c*d^4 + a^5*b^2*d^5)*x^2 + 5*(10*a^4*b^3*c^2*d^3 - 5*a^5*b^2*c*d^4 + a^6*b*d^5)*x)*log(b*x + a))*log(d*x + c
))/(a^5*b^8*c^5*g^6 - 5*a^6*b^7*c^4*d*g^6 + 10*a^7*b^6*c^3*d^2*g^6 - 10*a^8*b^5*c^2*d^3*g^6 + 5*a^9*b^4*c*d^4*
g^6 - a^10*b^3*d^5*g^6 + (b^13*c^5*g^6 - 5*a*b^12*c^4*d*g^6 + 10*a^2*b^11*c^3*d^2*g^6 - 10*a^3*b^10*c^2*d^3*g^
6 + 5*a^4*b^9*c*d^4*g^6 - a^5*b^8*d^5*g^6)*x^5 + 5*(a*b^12*c^5*g^6 - 5*a^2*b^11*c^4*d*g^6 + 10*a^3*b^10*c^3*d^
2*g^6 - 10*a^4*b^9*c^2*d^3*g^6 + 5*a^5*b^8*c*d^4*g^6 - a^6*b^7*d^5*g^6)*x^4 + 10*(a^2*b^11*c^5*g^6 - 5*a^3*b^1
0*c^4*d*g^6 + 10*a^4*b^9*c^3*d^2*g^6 - 10*a^5*b^8*c^2*d^3*g^6 + 5*a^6*b^7*c*d^4*g^6 - a^7*b^6*d^5*g^6)*x^3 + 1
0*(a^3*b^10*c^5*g^6 - 5*a^4*b^9*c^4*d*g^6 + 10*a^5*b^8*c^3*d^2*g^6 - 10*a^6*b^7*c^2*d^3*g^6 + 5*a^7*b^6*c*d^4*
g^6 - a^8*b^5*d^5*g^6)*x^2 + 5*(a^4*b^9*c^5*g^6 - 5*a^5*b^8*c^4*d*g^6 + 10*a^6*b^7*c^3*d^2*g^6 - 10*a^7*b^6*c^
2*d^3*g^6 + 5*a^8*b^5*c*d^4*g^6 - a^9*b^4*d^5*g^6)*x))*B^2*d^2*i^2 - 1/900*A*B*d^2*i^2*(60*(10*b^2*x^2 + 5*a*b
*x + a^2)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(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) + (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)) - 1/300*A*B*c*d*i^2*(60*(5*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x
+ c))/(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
) + (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))
- 1/150*A*B*c^2*i^2*((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 + 1
37*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*log(b*e*x/(d*x + c) + a*e/(
d*x + c))/(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) + 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)) - 1/5*B^2*c^2*i^2*log(b*e*x/(d*x + c) + a*e/(d*x + c))^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) - 1/10*(5*b*x + a)*A^2
*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^2*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*A^2*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)

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mupad [B]  time = 12.77, size = 3434, normalized size = 7.42 \[ \text {result too large to display} \]

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)))^2)/(a*g + b*g*x)^6,x)

[Out]

((1800*A^2*a^4*d^4*i^2 + 10800*A^2*b^4*c^4*i^2 + 1489*B^2*a^4*d^4*i^2 + 864*B^2*b^4*c^4*i^2 + 2820*A*B*a^4*d^4
*i^2 + 4320*A*B*b^4*c^4*i^2 - 16200*A^2*a*b^3*c^3*d*i^2 + 1800*A^2*a^3*b*c*d^3*i^2 - 2511*B^2*a*b^3*c^3*d*i^2
+ 1489*B^2*a^3*b*c*d^3*i^2 + 1800*A^2*a^2*b^2*c^2*d^2*i^2 + 1489*B^2*a^2*b^2*c^2*d^2*i^2 + 2820*A*B*a^2*b^2*c^
2*d^2*i^2 - 9180*A*B*a*b^3*c^3*d*i^2 + 2820*A*B*a^3*b*c*d^3*i^2)/(60*(a*d - b*c)) + (x^3*(363*B^2*a*b^3*d^4*i^
2 + 13*B^2*b^4*c*d^3*i^2 + 540*A*B*a*b^3*d^4*i^2 - 60*A*B*b^4*c*d^3*i^2))/(2*(a*d - b*c)) + (x*(1800*A^2*a^3*b
*d^4*i^2 + 1489*B^2*a^3*b*d^4*i^2 + 5400*A^2*b^4*c^3*d*i^2 + 189*B^2*b^4*c^3*d*i^2 - 9000*A^2*a*b^3*c^2*d^2*i^
2 + 1800*A^2*a^2*b^2*c*d^3*i^2 - 911*B^2*a*b^3*c^2*d^2*i^2 + 1489*B^2*a^2*b^2*c*d^3*i^2 + 2820*A*B*a^3*b*d^4*i
^2 + 1620*A*B*b^4*c^3*d*i^2 - 4380*A*B*a*b^3*c^2*d^2*i^2 + 2820*A*B*a^2*b^2*c*d^3*i^2))/(12*(a*d - b*c)) + (x^
2*(1800*A^2*a^2*b^2*d^4*i^2 + 1489*B^2*a^2*b^2*d^4*i^2 + 1800*A^2*b^4*c^2*d^2*i^2 - 86*B^2*b^4*c^2*d^2*i^2 - 3
600*A^2*a*b^3*c*d^3*i^2 + 289*B^2*a*b^3*c*d^3*i^2 + 2820*A*B*a^2*b^2*d^4*i^2 + 120*A*B*b^4*c^2*d^2*i^2 - 780*A
*B*a*b^3*c*d^3*i^2))/(6*(a*d - b*c)) + (d*x^4*(47*B^2*b^4*d^3*i^2 + 60*A*B*b^4*d^3*i^2))/(a*d - b*c))/(x*(4500
*a^4*b^5*c*g^6 - 4500*a^5*b^4*d*g^6) - x^4*(4500*a^2*b^7*d*g^6 - 4500*a*b^8*c*g^6) + x^5*(900*b^9*c*g^6 - 900*
a*b^8*d*g^6) + x^2*(9000*a^3*b^6*c*g^6 - 9000*a^4*b^5*d*g^6) + x^3*(9000*a^2*b^7*c*g^6 - 9000*a^3*b^6*d*g^6) +
900*a^5*b^4*c*g^6 - 900*a^6*b^3*d*g^6) - log((e*(a + b*x))/(c + d*x))^2*((x*(b*((B^2*c*d*i^2)/(10*b^3*g^6) +
(B^2*a*d^2*i^2)/(30*b^4*g^6)) + (2*B^2*c*d*i^2)/(5*b^2*g^6) + (2*B^2*a*d^2*i^2)/(15*b^3*g^6)) + a*((B^2*c*d*i^
2)/(10*b^3*g^6) + (B^2*a*d^2*i^2)/(30*b^4*g^6)) + (B^2*c^2*i^2)/(5*b^2*g^6) + (B^2*d^2*i^2*x^2)/(3*b^2*g^6))/(
5*a^4*x + a^5/b + b^4*x^5 + 10*a^3*b*x^2 + 5*a*b^3*x^4 + 10*a^2*b^2*x^3) - (B^2*d^5*i^2)/(30*b^3*g^6*(a^3*d^3
- b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) - (log((e*(a + b*x))/(c + d*x))*(a*((B*i^2*(6*A*b*c - B*a*d + B*b
*c))/(30*b^4*g^6) + (A*B*a*d*i^2)/(15*b^4*g^6)) + x*(b*((B*i^2*(6*A*b*c - B*a*d + B*b*c))/(30*b^4*g^6) + (A*B*
a*d*i^2)/(15*b^4*g^6)) + (2*B*i^2*(6*A*b*c - B*a*d + B*b*c))/(15*b^3*g^6) + (B^2*d^5*i^2*((10*a^4*d^4 + b^4*c^
4 + 15*a^2*b^2*c^2*d^2 - 6*a*b^3*c^3*d - 20*a^3*b*c*d^3)/(5*d^5) + b*(a*(a*((5*a^2*d^2 + b^2*c^2 - 6*a*b*c*d)/
(20*b*d^3) + (a*(a*d - b*c))/(5*b*d^2)) + (10*a^3*d^3 - b^3*c^3 + 6*a*b^2*c^2*d - 15*a^2*b*c*d^2)/(30*b*d^4))
+ (10*a^4*d^4 + b^4*c^4 + 15*a^2*b^2*c^2*d^2 - 6*a*b^3*c^3*d - 20*a^3*b*c*d^3)/(20*b*d^5)) + a*(b*(a*((5*a^2*d
^2 + b^2*c^2 - 6*a*b*c*d)/(20*b*d^3) + (a*(a*d - b*c))/(5*b*d^2)) + (10*a^3*d^3 - b^3*c^3 + 6*a*b^2*c^2*d - 15
*a^2*b*c*d^2)/(30*b*d^4)) + a*(b*((5*a^2*d^2 + b^2*c^2 - 6*a*b*c*d)/(20*b*d^3) + (a*(a*d - b*c))/(5*b*d^2)) +
(5*a^2*d^2 + b^2*c^2 - 6*a*b*c*d)/(10*d^3) + (2*a*(a*d - b*c))/(5*d^2)) + (10*a^3*d^3 - b^3*c^3 + 6*a*b^2*c^2*
d - 15*a^2*b*c*d^2)/(10*d^4))))/(15*b^3*g^6*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (4*A*B*a*d*
i^2)/(15*b^3*g^6)) + x^2*((2*A*B*d*i^2)/(3*b^2*g^6) + (B^2*d^5*i^2*(a*(b*(b*((5*a^2*d^2 + b^2*c^2 - 6*a*b*c*d)
/(20*b*d^3) + (a*(a*d - b*c))/(5*b*d^2)) + (5*a^2*d^2 + b^2*c^2 - 6*a*b*c*d)/(10*d^3) + (2*a*(a*d - b*c))/(5*d
^2)) - a*((b^2*c - a*b*d)/(5*d^2) - (2*b*(a*d - b*c))/(5*d^2)) + (3*(b^3*c^2 + 5*a^2*b*d^2 - 6*a*b^2*c*d))/(20
*d^3)) - (b^4*c^3 - 10*a^3*b*d^3 + 15*a^2*b^2*c*d^2 - 6*a*b^3*c^2*d)/(5*d^4) + b*(b*(a*((5*a^2*d^2 + b^2*c^2 -
6*a*b*c*d)/(20*b*d^3) + (a*(a*d - b*c))/(5*b*d^2)) + (10*a^3*d^3 - b^3*c^3 + 6*a*b^2*c^2*d - 15*a^2*b*c*d^2)/
(30*b*d^4)) + a*(b*((5*a^2*d^2 + b^2*c^2 - 6*a*b*c*d)/(20*b*d^3) + (a*(a*d - b*c))/(5*b*d^2)) + (5*a^2*d^2 + b
^2*c^2 - 6*a*b*c*d)/(10*d^3) + (2*a*(a*d - b*c))/(5*d^2)) + (10*a^3*d^3 - b^3*c^3 + 6*a*b^2*c^2*d - 15*a^2*b*c
*d^2)/(10*d^4))))/(15*b^3*g^6*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) + (B*i^2*(6*A*b^2*c^2 - B*
a^2*d^2 + B*b^2*c^2))/(15*b^4*d*g^6) + (B^2*d^5*i^2*(a*(a*(a*((5*a^2*d^2 + b^2*c^2 - 6*a*b*c*d)/(20*b*d^3) + (
a*(a*d - b*c))/(5*b*d^2)) + (10*a^3*d^3 - b^3*c^3 + 6*a*b^2*c^2*d - 15*a^2*b*c*d^2)/(30*b*d^4)) + (10*a^4*d^4
+ b^4*c^4 + 15*a^2*b^2*c^2*d^2 - 6*a*b^3*c^3*d - 20*a^3*b*c*d^3)/(20*b*d^5)) + (5*a^5*d^5 - b^5*c^5 - 15*a^2*b
^3*c^3*d^2 + 20*a^3*b^2*c^2*d^3 + 6*a*b^4*c^4*d - 15*a^4*b*c*d^4)/(5*b*d^6)))/(15*b^3*g^6*(a^3*d^3 - b^3*c^3 +
3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (B^2*d^5*i^2*x^3*((b^4*c^2 + 5*a^2*b^2*d^2 - 6*a*b^3*c*d)/(5*d^3) + b*(b*(b
*((5*a^2*d^2 + b^2*c^2 - 6*a*b*c*d)/(20*b*d^3) + (a*(a*d - b*c))/(5*b*d^2)) + (5*a^2*d^2 + b^2*c^2 - 6*a*b*c*d
)/(10*d^3) + (2*a*(a*d - b*c))/(5*d^2)) - a*((b^2*c - a*b*d)/(5*d^2) - (2*b*(a*d - b*c))/(5*d^2)) + (3*(b^3*c^
2 + 5*a^2*b*d^2 - 6*a*b^2*c*d))/(20*d^3)) - a*(b*((b^2*c - a*b*d)/(5*d^2) - (2*b*(a*d - b*c))/(5*d^2)) + (b^3*
c - a*b^2*d)/(5*d^2))))/(15*b^3*g^6*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (B^2*d^5*i^2*x^4*(b
*(b*((b^2*c - a*b*d)/(5*d^2) - (2*b*(a*d - b*c))/(5*d^2)) + (b^3*c - a*b^2*d)/(5*d^2)) + (b^4*c - a*b^3*d)/(5*
d^2)))/(15*b^3*g^6*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))))/((5*a^4*x)/d + a^5/(b*d) + (b^4*x^5)
/d + (10*a^3*b*x^2)/d + (5*a*b^3*x^4)/d + (10*a^2*b^2*x^3)/d) - (B*d^5*i^2*atan((B*d^5*i^2*(60*A + 47*B)*(900*
b^6*c^3*g^6 + 900*a^3*b^3*d^3*g^6 - 900*a*b^5*c^2*d*g^6 - 900*a^2*b^4*c*d^2*g^6)*1i)/(900*b^3*g^6*(47*B^2*d^5*
i^2 + 60*A*B*d^5*i^2)*(a*d - b*c)^3) + (B*d^6*i^2*x*(60*A + 47*B)*(b^5*c^2*g^6 + a^2*b^3*d^2*g^6 - 2*a*b^4*c*d
*g^6)*2i)/(b^2*g^6*(47*B^2*d^5*i^2 + 60*A*B*d^5*i^2)*(a*d - b*c)^3))*(60*A + 47*B)*1i)/(450*b^3*g^6*(a*d - b*c
)^3)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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