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

________________________________________________________________________________________

Rubi [C]  time = 4.21895, 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 2528

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

Rule 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 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 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 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 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 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 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 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 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]

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*}

Mathematica [C]  time = 4.23366, 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]

-(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)]) + 3
6*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]*(L
og[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) + 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)])*Log[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*Log[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 - 36
00*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] + 900*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)] - Lo
g[c + d*x])*Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]))))/(54000*b^3*(b*c - a*d)^3*g^6*(a + b*x)^
5)

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Maple [B]  time = 0.056, size = 2761, normalized size = 6. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 7.22908, size = 14688, normalized size = 31.72 \begin{align*} \text{result too large to display} \end{align*}

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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Fricas [B]  time = 0.632417, size = 2804, normalized size = 6.06 \begin{align*} \text{result too large to display} \end{align*}

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

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (d i x + c i\right )}^{2}{\left (B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A\right )}^{2}}{{\left (b g x + a g\right )}^{6}}\,{d x} \end{align*}

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]

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