3.71 \(\int \frac{(c i+d i x)^2 (A+B \log (\frac{e (a+b x)}{c+d x}))^2}{(a g+b g x)^4} \, dx\)

Optimal. Leaf size=147 \[ -\frac{i^2 (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{3 g^4 (a+b x)^3 (b c-a d)}-\frac{2 B i^2 (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{9 g^4 (a+b x)^3 (b c-a d)}-\frac{2 B^2 i^2 (c+d x)^3}{27 g^4 (a+b x)^3 (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 3.13179, antiderivative size = 827, normalized size of antiderivative = 5.63, number of steps used = 92, 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} \[ \frac{B^2 i^2 \log ^2(a+b x) d^3}{3 b^3 (b c-a d) g^4}+\frac{B^2 i^2 \log ^2(c+d x) d^3}{3 b^3 (b c-a d) g^4}-\frac{2 B^2 i^2 \log (a+b x) d^3}{9 b^3 (b c-a d) g^4}-\frac{2 B i^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d^3}{3 b^3 (b c-a d) g^4}+\frac{2 B^2 i^2 \log (c+d x) d^3}{9 b^3 (b c-a d) g^4}-\frac{2 B^2 i^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) d^3}{3 b^3 (b c-a d) g^4}+\frac{2 B i^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x) d^3}{3 b^3 (b c-a d) g^4}-\frac{2 B^2 i^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) d^3}{3 b^3 (b c-a d) g^4}-\frac{2 B^2 i^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) d^3}{3 b^3 (b c-a d) g^4}-\frac{2 B^2 i^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) d^3}{3 b^3 (b c-a d) g^4}-\frac{i^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 d^2}{b^3 g^4 (a+b x)}-\frac{2 B i^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d^2}{3 b^3 g^4 (a+b x)}-\frac{2 B^2 i^2 d^2}{9 b^3 g^4 (a+b x)}-\frac{(b c-a d) i^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 d}{b^3 g^4 (a+b x)^2}-\frac{2 B (b c-a d) i^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d}{3 b^3 g^4 (a+b x)^2}-\frac{2 B^2 (b c-a d) i^2 d}{9 b^3 g^4 (a+b x)^2}-\frac{(b c-a d)^2 i^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac{2 B (b c-a d)^2 i^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac{2 B^2 (b c-a d)^2 i^2}{27 b^3 g^4 (a+b x)^3} \]

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)^4,x]

[Out]

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

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{(71 c+71 d x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx &=\int \left (\frac{5041 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^4}+\frac{10082 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^3}+\frac{5041 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^2 g^4 (a+b x)^2}\right ) \, dx\\ &=\frac{\left (5041 d^2\right ) \int \frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a+b x)^2} \, dx}{b^2 g^4}+\frac{(10082 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)^3} \, dx}{b^2 g^4}+\frac{\left (5041 (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)^4} \, dx}{b^2 g^4}\\ &=-\frac{5041 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac{5041 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac{5041 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac{\left (10082 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)^2 (c+d x)} \, dx}{b^3 g^4}+\frac{(10082 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)^3 (c+d x)} \, dx}{b^3 g^4}+\frac{\left (10082 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)^4 (c+d x)} \, dx}{3 b^3 g^4}\\ &=-\frac{5041 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac{5041 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac{5041 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac{\left (10082 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)^2 (c+d x)} \, dx}{b^3 g^4}+\frac{\left (10082 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)^3 (c+d x)} \, dx}{b^3 g^4}+\frac{\left (10082 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)^4 (c+d x)} \, dx}{3 b^3 g^4}\\ &=-\frac{5041 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac{5041 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac{5041 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac{\left (10082 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)^2}-\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)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^4}+\frac{\left (10082 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)^3}-\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)^2}+\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)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^4}+\frac{\left (10082 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)^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^4}\\ &=-\frac{5041 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac{5041 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac{5041 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac{\left (10082 B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{3 b^2 g^4}-\frac{\left (10082 B d^3\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) g^4}+\frac{\left (10082 B d^4\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) g^4}-\frac{(10082 B d (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{3 b^2 g^4}+\frac{(10082 B d (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{b^2 g^4}+\frac{\left (10082 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)^4} \, dx}{3 b^2 g^4}\\ &=-\frac{10082 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac{10082 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac{10082 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac{10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac{5041 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac{5041 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac{5041 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac{10082 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac{\left (10082 B^2 d^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3 b^3 g^4}+\frac{\left (10082 B^2 d^3\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) g^4}-\frac{\left (10082 B^2 d^3\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) g^4}-\frac{\left (5041 B^2 d (b c-a d)\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{3 b^3 g^4}+\frac{\left (5041 B^2 d (b c-a d)\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac{\left (10082 B^2 (b c-a d)^2\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{9 b^3 g^4}\\ &=-\frac{10082 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac{10082 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac{10082 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac{10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac{5041 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac{5041 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac{5041 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac{10082 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac{\left (10082 B^2 d^2 (b c-a d)\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{3 b^3 g^4}-\frac{\left (5041 B^2 d (b c-a d)^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{3 b^3 g^4}+\frac{\left (5041 B^2 d (b c-a d)^2\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac{\left (10082 B^2 (b c-a d)^3\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{9 b^3 g^4}+\frac{\left (10082 B^2 d^3\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) e g^4}-\frac{\left (10082 B^2 d^3\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) e g^4}\\ &=-\frac{10082 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac{10082 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac{10082 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac{10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac{5041 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac{5041 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac{5041 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac{10082 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac{\left (10082 B^2 d^2 (b c-a d)\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3 b^3 g^4}-\frac{\left (5041 B^2 d (b c-a d)^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{3 b^3 g^4}+\frac{\left (5041 B^2 d (b c-a d)^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^4}+\frac{\left (10082 B^2 (b c-a d)^3\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^4}+\frac{\left (10082 B^2 d^3\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) e g^4}-\frac{\left (10082 B^2 d^3\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) e g^4}\\ &=-\frac{10082 B^2 (b c-a d)^2}{27 b^3 g^4 (a+b x)^3}-\frac{10082 B^2 d (b c-a d)}{9 b^3 g^4 (a+b x)^2}-\frac{10082 B^2 d^2}{9 b^3 g^4 (a+b x)}-\frac{10082 B^2 d^3 \log (a+b x)}{9 b^3 (b c-a d) g^4}-\frac{10082 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac{10082 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac{10082 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac{10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac{5041 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac{5041 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac{5041 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac{10082 B^2 d^3 \log (c+d x)}{9 b^3 (b c-a d) g^4}+\frac{10082 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac{\left (10082 B^2 d^3\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{3 b^2 (b c-a d) g^4}-\frac{\left (10082 B^2 d^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{3 b^2 (b c-a d) g^4}-\frac{\left (10082 B^2 d^4\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{3 b^3 (b c-a d) g^4}+\frac{\left (10082 B^2 d^4\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{3 b^3 (b c-a d) g^4}\\ &=-\frac{10082 B^2 (b c-a d)^2}{27 b^3 g^4 (a+b x)^3}-\frac{10082 B^2 d (b c-a d)}{9 b^3 g^4 (a+b x)^2}-\frac{10082 B^2 d^2}{9 b^3 g^4 (a+b x)}-\frac{10082 B^2 d^3 \log (a+b x)}{9 b^3 (b c-a d) g^4}-\frac{10082 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac{10082 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac{10082 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac{10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac{5041 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac{5041 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac{5041 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac{10082 B^2 d^3 \log (c+d x)}{9 b^3 (b c-a d) g^4}-\frac{10082 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac{10082 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}-\frac{10082 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}+\frac{\left (10082 B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{3 b^3 (b c-a d) g^4}+\frac{\left (10082 B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{3 b^3 (b c-a d) g^4}+\frac{\left (10082 B^2 d^3\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) g^4}+\frac{\left (10082 B^2 d^4\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) g^4}\\ &=-\frac{10082 B^2 (b c-a d)^2}{27 b^3 g^4 (a+b x)^3}-\frac{10082 B^2 d (b c-a d)}{9 b^3 g^4 (a+b x)^2}-\frac{10082 B^2 d^2}{9 b^3 g^4 (a+b x)}-\frac{10082 B^2 d^3 \log (a+b x)}{9 b^3 (b c-a d) g^4}+\frac{5041 B^2 d^3 \log ^2(a+b x)}{3 b^3 (b c-a d) g^4}-\frac{10082 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac{10082 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac{10082 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac{10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac{5041 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac{5041 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac{5041 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac{10082 B^2 d^3 \log (c+d x)}{9 b^3 (b c-a d) g^4}-\frac{10082 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac{10082 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac{5041 B^2 d^3 \log ^2(c+d x)}{3 b^3 (b c-a d) g^4}-\frac{10082 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}+\frac{\left (10082 B^2 d^3\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) g^4}+\frac{\left (10082 B^2 d^3\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) g^4}\\ &=-\frac{10082 B^2 (b c-a d)^2}{27 b^3 g^4 (a+b x)^3}-\frac{10082 B^2 d (b c-a d)}{9 b^3 g^4 (a+b x)^2}-\frac{10082 B^2 d^2}{9 b^3 g^4 (a+b x)}-\frac{10082 B^2 d^3 \log (a+b x)}{9 b^3 (b c-a d) g^4}+\frac{5041 B^2 d^3 \log ^2(a+b x)}{3 b^3 (b c-a d) g^4}-\frac{10082 B (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac{10082 B d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)^2}-\frac{10082 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 g^4 (a+b x)}-\frac{10082 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b^3 (b c-a d) g^4}-\frac{5041 (b c-a d)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac{5041 d (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)^2}-\frac{5041 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{b^3 g^4 (a+b x)}+\frac{10082 B^2 d^3 \log (c+d x)}{9 b^3 (b c-a d) g^4}-\frac{10082 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac{10082 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b^3 (b c-a d) g^4}+\frac{5041 B^2 d^3 \log ^2(c+d x)}{3 b^3 (b c-a d) g^4}-\frac{10082 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}-\frac{10082 B^2 d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}-\frac{10082 B^2 d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}\\ \end{align*}

Mathematica [C]  time = 2.20987, size = 1355, normalized size = 9.22 \[ -\frac{i^2 \left (18 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)^3+54 d (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2 (b c-a d)^2-54 d^2 (a d-b c) (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2+54 B d^2 (a+b x)^2 \left (2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )+2 d (a+b x) \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )-2 d (a+b x) \log (c+d x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )+2 B (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+B d (a+b x) \left (\left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right )\right )+27 B d (a+b x) \left (2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)^2+4 d (a d-b c) (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )-4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )+4 d^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-4 B d (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+B \left ((b c-a d)^2+2 d (a d-b c) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B d^2 (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-2 B d^2 (a+b x)^2 \left (\left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right )\right )+B \left (12 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)^3-18 d (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)^2+36 d^2 (a+b x)^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) (b c-a d)+36 d^3 (a+b x)^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )-36 d^3 (a+b x)^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \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) \left ((b c-a d)^2+2 d (a d-b c) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )+2 B \left (2 (b c-a d)^3-3 d (a+b x) (b c-a d)^2+6 d^2 (a+b x)^2 (b c-a d)+6 d^3 (a+b x)^3 \log (a+b x)-6 d^3 (a+b x)^3 \log (c+d x)\right )-18 B d^3 (a+b x)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+18 B d^3 (a+b x)^3 \left (\left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right ) \log (c+d x)+2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )\right )\right )\right )}{54 b^3 (b c-a d) g^4 (a+b x)^3} \]

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)^4,x]

[Out]

-(i^2*(18*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 54*d*(b*c - a*d)^2*(a + b*x)*(A + B*Log[(e*(a
+ b*x))/(c + d*x)])^2 - 54*d^2*(-(b*c) + a*d)*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + 54*B*d^2*(
a + b*x)^2*(2*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 2*d*(a + b*x)*Log[a + b*x]*(A + B*Log[(e*(a +
b*x))/(c + d*x)]) - 2*d*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] + 2*B*(b*c - a*d + d*(a +
b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) - B*d*(a + b*x)*(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)]) + B*d*(a + b*x)*((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)])) + 27*B*d*(a + b*x)*(2*(b*c - a
*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 4*d*(-(b*c) + a*d)*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])
- 4*d^2*(a + b*x)^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 4*d^2*(a + b*x)^2*(A + B*Log[(e*(a +
b*x))/(c + d*x)])*Log[c + d*x] - 4*B*d*(a + b*x)*(b*c - a*d + d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d
*x]) + B*((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*Lo
g[c + d*x]) + 2*B*d^2*(a + b*x)^2*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[
2, (d*(a + b*x))/(-(b*c) + a*d)]) - 2*B*d^2*(a + b*x)^2*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*
Log[c + d*x] + 2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])) + B*(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)]))))/(54*b^3*(b*c - a*d)*g^4*(a + b*x)^3)

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Maple [B]  time = 0.055, size = 890, normalized size = 6.1 \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)^4,x)

[Out]

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

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Maxima [B]  time = 3.36555, size = 7468, normalized size = 50.8 \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)^4,x, algorithm="maxima")

[Out]

-1/3*(3*b*x + a)*B^2*c*d*i^2*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3
*g^4*x + a^3*b^2*g^4) - 1/3*(3*b^2*x^2 + 3*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^
6*g^4*x^3 + 3*a*b^5*g^4*x^2 + 3*a^2*b^4*g^4*x + a^3*b^3*g^4) - 1/54*(6*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d
+ 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a
^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4
*b^2*c*d + a^5*b*d^2)*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4)
- 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4))*log(b*e*x/(d*x + c) + a*e/
(d*x + c)) + (4*b^3*c^3 - 27*a*b^2*c^2*d + 108*a^2*b*c*d^2 - 85*a^3*d^3 + 66*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 18*
(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a)^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 +
3*a^2*b*d^3*x + a^3*d^3)*log(d*x + c)^2 - 3*(5*b^3*c^2*d - 54*a*b^2*c*d^2 + 49*a^2*b*d^3)*x + 66*(b^3*d^3*x^3
+ 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a) - 6*(11*b^3*d^3*x^3 + 33*a*b^2*d^3*x^2 + 33*a^2*b*d
^3*x + 11*a^3*d^3 - 6*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a))*log(d*x + c))/(a
^3*b^4*c^3*g^4 - 3*a^4*b^3*c^2*d*g^4 + 3*a^5*b^2*c*d^2*g^4 - a^6*b*d^3*g^4 + (b^7*c^3*g^4 - 3*a*b^6*c^2*d*g^4
+ 3*a^2*b^5*c*d^2*g^4 - a^3*b^4*d^3*g^4)*x^3 + 3*(a*b^6*c^3*g^4 - 3*a^2*b^5*c^2*d*g^4 + 3*a^3*b^4*c*d^2*g^4 -
a^4*b^3*d^3*g^4)*x^2 + 3*(a^2*b^5*c^3*g^4 - 3*a^3*b^4*c^2*d*g^4 + 3*a^4*b^3*c*d^2*g^4 - a^5*b^2*d^3*g^4)*x))*B
^2*c^2*i^2 - 1/54*(6*((5*a*b^2*c^2 - 22*a^2*b*c*d + 5*a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 -
16*a*b^2*c*d + 5*a^2*b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d
+ a^3*b^4*d^2)*g^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d +
a^5*b^2*d^2)*g^4) - 6*(3*b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d
^3)*g^4) + 6*(3*b*c*d^2 - a*d^3)*log(d*x + c)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4))
*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + (19*a*b^3*c^3 - 189*a^2*b^2*c^2*d + 189*a^3*b*c*d^2 - 19*a^4*d^3 - 6*(
27*b^4*c^2*d - 32*a*b^3*c*d^2 + 5*a^2*b^2*d^3)*x^2 + 18*(3*a^3*b*c*d^2 - a^4*d^3 + (3*b^4*c*d^2 - a*b^3*d^3)*x
^3 + 3*(3*a*b^3*c*d^2 - a^2*b^2*d^3)*x^2 + 3*(3*a^2*b^2*c*d^2 - a^3*b*d^3)*x)*log(b*x + a)^2 + 18*(3*a^3*b*c*d
^2 - a^4*d^3 + (3*b^4*c*d^2 - a*b^3*d^3)*x^3 + 3*(3*a*b^3*c*d^2 - a^2*b^2*d^3)*x^2 + 3*(3*a^2*b^2*c*d^2 - a^3*
b*d^3)*x)*log(d*x + c)^2 + 3*(9*b^4*c^3 - 125*a*b^3*c^2*d + 135*a^2*b^2*c*d^2 - 19*a^3*b*d^3)*x - 6*(27*a^3*b*
c*d^2 - 5*a^4*d^3 + (27*b^4*c*d^2 - 5*a*b^3*d^3)*x^3 + 3*(27*a*b^3*c*d^2 - 5*a^2*b^2*d^3)*x^2 + 3*(27*a^2*b^2*
c*d^2 - 5*a^3*b*d^3)*x)*log(b*x + a) + 6*(27*a^3*b*c*d^2 - 5*a^4*d^3 + (27*b^4*c*d^2 - 5*a*b^3*d^3)*x^3 + 3*(2
7*a*b^3*c*d^2 - 5*a^2*b^2*d^3)*x^2 + 3*(27*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x - 6*(3*a^3*b*c*d^2 - a^4*d^3 + (3*b^
4*c*d^2 - a*b^3*d^3)*x^3 + 3*(3*a*b^3*c*d^2 - a^2*b^2*d^3)*x^2 + 3*(3*a^2*b^2*c*d^2 - a^3*b*d^3)*x)*log(b*x +
a))*log(d*x + c))/(a^3*b^5*c^3*g^4 - 3*a^4*b^4*c^2*d*g^4 + 3*a^5*b^3*c*d^2*g^4 - a^6*b^2*d^3*g^4 + (b^8*c^3*g^
4 - 3*a*b^7*c^2*d*g^4 + 3*a^2*b^6*c*d^2*g^4 - a^3*b^5*d^3*g^4)*x^3 + 3*(a*b^7*c^3*g^4 - 3*a^2*b^6*c^2*d*g^4 +
3*a^3*b^5*c*d^2*g^4 - a^4*b^4*d^3*g^4)*x^2 + 3*(a^2*b^6*c^3*g^4 - 3*a^3*b^5*c^2*d*g^4 + 3*a^4*b^4*c*d^2*g^4 -
a^5*b^3*d^3*g^4)*x))*B^2*c*d*i^2 - 1/54*(6*((11*a^2*b^2*c^2 - 7*a^3*b*c*d + 2*a^4*d^2 + 6*(3*b^4*c^2 - 3*a*b^3
*c*d + a^2*b^2*d^2)*x^2 + 3*(9*a*b^3*c^2 - 7*a^2*b^2*c*d + 2*a^3*b*d^2)*x)/((b^8*c^2 - 2*a*b^7*c*d + a^2*b^6*d
^2)*g^4*x^3 + 3*(a*b^7*c^2 - 2*a^2*b^6*c*d + a^3*b^5*d^2)*g^4*x^2 + 3*(a^2*b^6*c^2 - 2*a^3*b^5*c*d + a^4*b^4*d
^2)*g^4*x + (a^3*b^5*c^2 - 2*a^4*b^4*c*d + a^5*b^3*d^2)*g^4) + 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(b*x
+ a)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4) - 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3
)*log(d*x + c)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4))*log(b*e*x/(d*x + c) + a*e/(d*x
+ c)) + (85*a^2*b^3*c^3 - 108*a^3*b^2*c^2*d + 27*a^4*b*c*d^2 - 4*a^5*d^3 + 6*(18*b^5*c^3 - 27*a*b^4*c^2*d + 1
1*a^2*b^3*c*d^2 - 2*a^3*b^2*d^3)*x^2 - 18*(3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2 + a^5*d^3 + (3*b^5*c^2*d - 3*a*b^4*
c*d^2 + a^2*b^3*d^3)*x^3 + 3*(3*a*b^4*c^2*d - 3*a^2*b^3*c*d^2 + a^3*b^2*d^3)*x^2 + 3*(3*a^2*b^3*c^2*d - 3*a^3*
b^2*c*d^2 + a^4*b*d^3)*x)*log(b*x + a)^2 - 18*(3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2 + a^5*d^3 + (3*b^5*c^2*d - 3*a*
b^4*c*d^2 + a^2*b^3*d^3)*x^3 + 3*(3*a*b^4*c^2*d - 3*a^2*b^3*c*d^2 + a^3*b^2*d^3)*x^2 + 3*(3*a^2*b^3*c^2*d - 3*
a^3*b^2*c*d^2 + a^4*b*d^3)*x)*log(d*x + c)^2 + 3*(63*a*b^4*c^3 - 86*a^2*b^3*c^2*d + 27*a^3*b^2*c*d^2 - 4*a^4*b
*d^3)*x + 6*(18*a^3*b^2*c^2*d - 9*a^4*b*c*d^2 + 2*a^5*d^3 + (18*b^5*c^2*d - 9*a*b^4*c*d^2 + 2*a^2*b^3*d^3)*x^3
+ 3*(18*a*b^4*c^2*d - 9*a^2*b^3*c*d^2 + 2*a^3*b^2*d^3)*x^2 + 3*(18*a^2*b^3*c^2*d - 9*a^3*b^2*c*d^2 + 2*a^4*b*
d^3)*x)*log(b*x + a) - 6*(18*a^3*b^2*c^2*d - 9*a^4*b*c*d^2 + 2*a^5*d^3 + (18*b^5*c^2*d - 9*a*b^4*c*d^2 + 2*a^2
*b^3*d^3)*x^3 + 3*(18*a*b^4*c^2*d - 9*a^2*b^3*c*d^2 + 2*a^3*b^2*d^3)*x^2 + 3*(18*a^2*b^3*c^2*d - 9*a^3*b^2*c*d
^2 + 2*a^4*b*d^3)*x - 6*(3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2 + a^5*d^3 + (3*b^5*c^2*d - 3*a*b^4*c*d^2 + a^2*b^3*d^
3)*x^3 + 3*(3*a*b^4*c^2*d - 3*a^2*b^3*c*d^2 + a^3*b^2*d^3)*x^2 + 3*(3*a^2*b^3*c^2*d - 3*a^3*b^2*c*d^2 + a^4*b*
d^3)*x)*log(b*x + a))*log(d*x + c))/(a^3*b^6*c^3*g^4 - 3*a^4*b^5*c^2*d*g^4 + 3*a^5*b^4*c*d^2*g^4 - a^6*b^3*d^3
*g^4 + (b^9*c^3*g^4 - 3*a*b^8*c^2*d*g^4 + 3*a^2*b^7*c*d^2*g^4 - a^3*b^6*d^3*g^4)*x^3 + 3*(a*b^8*c^3*g^4 - 3*a^
2*b^7*c^2*d*g^4 + 3*a^3*b^6*c*d^2*g^4 - a^4*b^5*d^3*g^4)*x^2 + 3*(a^2*b^7*c^3*g^4 - 3*a^3*b^6*c^2*d*g^4 + 3*a^
4*b^5*c*d^2*g^4 - a^5*b^4*d^3*g^4)*x))*B^2*d^2*i^2 - 1/9*A*B*d^2*i^2*(6*(3*b^2*x^2 + 3*a*b*x + a^2)*log(b*e*x/
(d*x + c) + a*e/(d*x + c))/(b^6*g^4*x^3 + 3*a*b^5*g^4*x^2 + 3*a^2*b^4*g^4*x + a^3*b^3*g^4) + (11*a^2*b^2*c^2 -
7*a^3*b*c*d + 2*a^4*d^2 + 6*(3*b^4*c^2 - 3*a*b^3*c*d + a^2*b^2*d^2)*x^2 + 3*(9*a*b^3*c^2 - 7*a^2*b^2*c*d + 2*
a^3*b*d^2)*x)/((b^8*c^2 - 2*a*b^7*c*d + a^2*b^6*d^2)*g^4*x^3 + 3*(a*b^7*c^2 - 2*a^2*b^6*c*d + a^3*b^5*d^2)*g^4
*x^2 + 3*(a^2*b^6*c^2 - 2*a^3*b^5*c*d + a^4*b^4*d^2)*g^4*x + (a^3*b^5*c^2 - 2*a^4*b^4*c*d + a^5*b^3*d^2)*g^4)
+ 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(b*x + a)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d
^3)*g^4) - 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(d*x + c)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 -
a^3*b^3*d^3)*g^4)) - 1/9*A*B*c*d*i^2*(6*(3*b*x + a)*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^5*g^4*x^3 + 3*a*b^
4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) + (5*a*b^2*c^2 - 22*a^2*b*c*d + 5*a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^
2)*x^2 + 3*(3*b^3*c^2 - 16*a*b^2*c*d + 5*a^2*b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b
^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4
*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4) - 6*(3*b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^
2*b^3*c*d^2 - a^3*b^2*d^3)*g^4) + 6*(3*b*c*d^2 - a*d^3)*log(d*x + c)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d
^2 - a^3*b^2*d^3)*g^4)) - 1/9*A*B*c^2*i^2*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d -
5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4
*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) +
6*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) + 6*d^3*l
og(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a
*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4)) - 1/3*B^2*c^2*i^2*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(b^4*
g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) - 1/3*(3*b*x + a)*A^2*c*d*i^2/(b^5*g^4*x^3 + 3*a*b^4*
g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - 1/3*(3*b^2*x^2 + 3*a*b*x + a^2)*A^2*d^2*i^2/(b^6*g^4*x^3 + 3*a*b^5*
g^4*x^2 + 3*a^2*b^4*g^4*x + a^3*b^3*g^4) - 1/3*A^2*c^2*i^2/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x +
a^3*b*g^4)

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Fricas [B]  time = 0.562854, size = 892, normalized size = 6.07 \begin{align*} -\frac{3 \,{\left ({\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c d^{2} -{\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a b^{2} d^{3}\right )} i^{2} x^{2} + 3 \,{\left ({\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c^{2} d -{\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a^{2} b d^{3}\right )} i^{2} x +{\left ({\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c^{3} -{\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a^{3} d^{3}\right )} i^{2} + 9 \,{\left (B^{2} b^{3} d^{3} i^{2} x^{3} + 3 \, B^{2} b^{3} c d^{2} i^{2} x^{2} + 3 \, B^{2} b^{3} c^{2} d i^{2} x + B^{2} b^{3} c^{3} i^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 6 \,{\left ({\left (3 \, A B + B^{2}\right )} b^{3} d^{3} i^{2} x^{3} + 3 \,{\left (3 \, A B + B^{2}\right )} b^{3} c d^{2} i^{2} x^{2} + 3 \,{\left (3 \, A B + B^{2}\right )} b^{3} c^{2} d i^{2} x +{\left (3 \, A B + B^{2}\right )} b^{3} c^{3} i^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{27 \,{\left ({\left (b^{7} c - a b^{6} d\right )} g^{4} x^{3} + 3 \,{\left (a b^{6} c - a^{2} b^{5} d\right )} g^{4} x^{2} + 3 \,{\left (a^{2} b^{5} c - a^{3} b^{4} d\right )} g^{4} x +{\left (a^{3} b^{4} c - a^{4} b^{3} d\right )} g^{4}\right )}} \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)^4,x, algorithm="fricas")

[Out]

-1/27*(3*((9*A^2 + 6*A*B + 2*B^2)*b^3*c*d^2 - (9*A^2 + 6*A*B + 2*B^2)*a*b^2*d^3)*i^2*x^2 + 3*((9*A^2 + 6*A*B +
2*B^2)*b^3*c^2*d - (9*A^2 + 6*A*B + 2*B^2)*a^2*b*d^3)*i^2*x + ((9*A^2 + 6*A*B + 2*B^2)*b^3*c^3 - (9*A^2 + 6*A
*B + 2*B^2)*a^3*d^3)*i^2 + 9*(B^2*b^3*d^3*i^2*x^3 + 3*B^2*b^3*c*d^2*i^2*x^2 + 3*B^2*b^3*c^2*d*i^2*x + B^2*b^3*
c^3*i^2)*log((b*e*x + a*e)/(d*x + c))^2 + 6*((3*A*B + B^2)*b^3*d^3*i^2*x^3 + 3*(3*A*B + B^2)*b^3*c*d^2*i^2*x^2
+ 3*(3*A*B + B^2)*b^3*c^2*d*i^2*x + (3*A*B + B^2)*b^3*c^3*i^2)*log((b*e*x + a*e)/(d*x + c)))/((b^7*c - a*b^6*
d)*g^4*x^3 + 3*(a*b^6*c - a^2*b^5*d)*g^4*x^2 + 3*(a^2*b^5*c - a^3*b^4*d)*g^4*x + (a^3*b^4*c - a^4*b^3*d)*g^4)

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Sympy [B]  time = 69.1096, size = 1178, normalized size = 8.01 \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*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**4,x)

[Out]

-2*B*d**3*i**2*(3*A + B)*log(x + (6*A*B*a*d**4*i**2 + 6*A*B*b*c*d**3*i**2 + 2*B**2*a*d**4*i**2 + 2*B**2*b*c*d*
*3*i**2 - 2*B*a**2*d**5*i**2*(3*A + B)/(a*d - b*c) + 4*B*a*b*c*d**4*i**2*(3*A + B)/(a*d - b*c) - 2*B*b**2*c**2
*d**3*i**2*(3*A + B)/(a*d - b*c))/(12*A*B*b*d**4*i**2 + 4*B**2*b*d**4*i**2))/(9*b**3*g**4*(a*d - b*c)) + 2*B*d
**3*i**2*(3*A + B)*log(x + (6*A*B*a*d**4*i**2 + 6*A*B*b*c*d**3*i**2 + 2*B**2*a*d**4*i**2 + 2*B**2*b*c*d**3*i**
2 + 2*B*a**2*d**5*i**2*(3*A + B)/(a*d - b*c) - 4*B*a*b*c*d**4*i**2*(3*A + B)/(a*d - b*c) + 2*B*b**2*c**2*d**3*
i**2*(3*A + B)/(a*d - b*c))/(12*A*B*b*d**4*i**2 + 4*B**2*b*d**4*i**2))/(9*b**3*g**4*(a*d - b*c)) + (B**2*c**3*
i**2 + 3*B**2*c**2*d*i**2*x + 3*B**2*c*d**2*i**2*x**2 + B**2*d**3*i**2*x**3)*log(e*(a + b*x)/(c + d*x))**2/(3*
a**4*d*g**4 - 3*a**3*b*c*g**4 + 9*a**3*b*d*g**4*x - 9*a**2*b**2*c*g**4*x + 9*a**2*b**2*d*g**4*x**2 - 9*a*b**3*
c*g**4*x**2 + 3*a*b**3*d*g**4*x**3 - 3*b**4*c*g**4*x**3) - (9*A**2*a**2*d**2*i**2 + 9*A**2*a*b*c*d*i**2 + 9*A*
*2*b**2*c**2*i**2 + 6*A*B*a**2*d**2*i**2 + 6*A*B*a*b*c*d*i**2 + 6*A*B*b**2*c**2*i**2 + 2*B**2*a**2*d**2*i**2 +
2*B**2*a*b*c*d*i**2 + 2*B**2*b**2*c**2*i**2 + x**2*(27*A**2*b**2*d**2*i**2 + 18*A*B*b**2*d**2*i**2 + 6*B**2*b
**2*d**2*i**2) + x*(27*A**2*a*b*d**2*i**2 + 27*A**2*b**2*c*d*i**2 + 18*A*B*a*b*d**2*i**2 + 18*A*B*b**2*c*d*i**
2 + 6*B**2*a*b*d**2*i**2 + 6*B**2*b**2*c*d*i**2))/(27*a**3*b**3*g**4 + 81*a**2*b**4*g**4*x + 81*a*b**5*g**4*x*
*2 + 27*b**6*g**4*x**3) + (-6*A*B*a**2*d**2*i**2 - 6*A*B*a*b*c*d*i**2 - 18*A*B*a*b*d**2*i**2*x - 6*A*B*b**2*c*
*2*i**2 - 18*A*B*b**2*c*d*i**2*x - 18*A*B*b**2*d**2*i**2*x**2 - 2*B**2*a**2*d**2*i**2 - 2*B**2*a*b*c*d*i**2 -
6*B**2*a*b*d**2*i**2*x - 2*B**2*b**2*c**2*i**2 - 6*B**2*b**2*c*d*i**2*x - 6*B**2*b**2*d**2*i**2*x**2)*log(e*(a
+ b*x)/(c + d*x))/(9*a**3*b**3*g**4 + 27*a**2*b**4*g**4*x + 27*a*b**5*g**4*x**2 + 9*b**6*g**4*x**3)

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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 )}^{4}}\,{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)^4,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)^4, x)