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

Optimal. Leaf size=418 \[ -\frac{b^2 (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{3 g^4 (a+b x)^3 (b c-a d)^3}-\frac{2 b^2 B (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{9 g^4 (a+b x)^3 (b c-a d)^3}-\frac{d^2 (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{g^4 (a+b x) (b c-a d)^3}-\frac{2 B d^2 (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 (a+b x) (b c-a d)^3}+\frac{b d (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{g^4 (a+b x)^2 (b c-a d)^3}+\frac{b B d (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 (a+b x)^2 (b c-a d)^3}-\frac{2 b^2 B^2 (c+d x)^3}{27 g^4 (a+b x)^3 (b c-a d)^3}-\frac{2 B^2 d^2 (c+d x)}{g^4 (a+b x) (b c-a d)^3}+\frac{b B^2 d (c+d x)^2}{2 g^4 (a+b x)^2 (b c-a d)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 1.05899, antiderivative size = 680, normalized size of antiderivative = 1.63, number of steps used = 34, number of rules used = 11, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.344, Rules used = {2525, 12, 2528, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{2 B^2 d^3 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{3 b g^4 (b c-a d)^3}-\frac{2 B^2 d^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{3 b g^4 (b c-a d)^3}-\frac{2 B d^3 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b g^4 (b c-a d)^3}+\frac{2 B d^3 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b g^4 (b c-a d)^3}-\frac{2 B d^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b g^4 (a+b x) (b c-a d)^2}+\frac{B d \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 b g^4 (a+b x)^2 (b c-a d)}-\frac{\left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{3 b g^4 (a+b x)^3}-\frac{2 B \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{9 b g^4 (a+b x)^3}-\frac{11 B^2 d^2}{9 b g^4 (a+b x) (b c-a d)^2}+\frac{B^2 d^3 \log ^2(a+b x)}{3 b g^4 (b c-a d)^3}+\frac{B^2 d^3 \log ^2(c+d x)}{3 b g^4 (b c-a d)^3}-\frac{11 B^2 d^3 \log (a+b x)}{9 b g^4 (b c-a d)^3}-\frac{2 B^2 d^3 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{3 b g^4 (b c-a d)^3}+\frac{11 B^2 d^3 \log (c+d x)}{9 b g^4 (b c-a d)^3}-\frac{2 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b g^4 (b c-a d)^3}+\frac{5 B^2 d}{18 b g^4 (a+b x)^2 (b c-a d)}-\frac{2 B^2}{27 b g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(a*g + b*g*x)^4,x]

[Out]

(-2*B^2)/(27*b*g^4*(a + b*x)^3) + (5*B^2*d)/(18*b*(b*c - a*d)*g^4*(a + b*x)^2) - (11*B^2*d^2)/(9*b*(b*c - a*d)
^2*g^4*(a + b*x)) - (11*B^2*d^3*Log[a + b*x])/(9*b*(b*c - a*d)^3*g^4) + (B^2*d^3*Log[a + b*x]^2)/(3*b*(b*c - a
*d)^3*g^4) - (2*B*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(9*b*g^4*(a + b*x)^3) + (B*d*(A + B*Log[(e*(a + b*x))/
(c + d*x)]))/(3*b*(b*c - a*d)*g^4*(a + b*x)^2) - (2*B*d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b*(b*c - a*
d)^2*g^4*(a + b*x)) - (2*B*d^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*b*(b*c - a*d)^3*g^4) - (A
+ B*Log[(e*(a + b*x))/(c + d*x)])^2/(3*b*g^4*(a + b*x)^3) + (11*B^2*d^3*Log[c + d*x])/(9*b*(b*c - a*d)^3*g^4)
- (2*B^2*d^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(3*b*(b*c - a*d)^3*g^4) + (2*B*d^3*(A + B*Log[(e
*(a + b*x))/(c + d*x)])*Log[c + d*x])/(3*b*(b*c - a*d)^3*g^4) + (B^2*d^3*Log[c + d*x]^2)/(3*b*(b*c - a*d)^3*g^
4) - (2*B^2*d^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(3*b*(b*c - a*d)^3*g^4) - (2*B^2*d^3*PolyLog[2, -
((d*(a + b*x))/(b*c - a*d))])/(3*b*(b*c - a*d)^3*g^4) - (2*B^2*d^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(3*b
*(b*c - a*d)^3*g^4)

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 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 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{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^4} \, dx &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{(2 B) \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{g^3 (a+b x)^4 (c+d x)} \, dx}{3 b g}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{(2 B (b c-a d)) \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 g^4}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{(2 B (b c-a d)) \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 g^4}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{(2 B) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{3 g^4}-\frac{\left (2 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 c-a d)^3 g^4}+\frac{\left (2 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 (b c-a d)^3 g^4}+\frac{\left (2 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 c-a d)^2 g^4}-\frac{(2 B d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{3 (b c-a d) g^4}\\ &=-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b g^4 (a+b x)^3}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac{2 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac{2 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{2 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac{\left (2 B^2\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{9 b g^4}+\frac{\left (2 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 (b c-a d)^3 g^4}-\frac{\left (2 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 (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3 b (b c-a d)^2 g^4}-\frac{\left (B^2 d\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{3 b (b c-a d) g^4}\\ &=-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b g^4 (a+b x)^3}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac{2 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac{2 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{2 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}-\frac{\left (B^2 d\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{3 b g^4}+\frac{\left (2 B^2 d^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{3 b (b c-a d) g^4}+\frac{\left (2 B^2 (b c-a d)\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{9 b g^4}+\frac{\left (2 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 (b c-a d)^3 e g^4}-\frac{\left (2 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 (b c-a d)^3 e g^4}\\ &=-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b g^4 (a+b x)^3}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac{2 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac{2 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{2 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}-\frac{\left (B^2 d\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 g^4}+\frac{\left (2 B^2 d^2\right ) \int \left (\frac{b}{(b c-a d) (a+b x)^2}-\frac{b d}{(b c-a d)^2 (a+b x)}+\frac{d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3 b (b c-a d) g^4}+\frac{\left (2 B^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 g^4}+\frac{\left (2 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 (b c-a d)^3 e g^4}-\frac{\left (2 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 (b c-a d)^3 e g^4}\\ &=-\frac{2 B^2}{27 b g^4 (a+b x)^3}+\frac{5 B^2 d}{18 b (b c-a d) g^4 (a+b x)^2}-\frac{11 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac{11 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b g^4 (a+b x)^3}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac{2 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac{2 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{11 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}+\frac{2 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^3\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{3 (b c-a d)^3 g^4}-\frac{\left (2 B^2 d^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{3 (b c-a d)^3 g^4}-\frac{\left (2 B^2 d^4\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^4\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}\\ &=-\frac{2 B^2}{27 b g^4 (a+b x)^3}+\frac{5 B^2 d}{18 b (b c-a d) g^4 (a+b x)^2}-\frac{11 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac{11 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b g^4 (a+b x)^3}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac{2 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac{2 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{11 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac{2 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac{2 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}-\frac{2 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}+\frac{\left (2 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 c-a d)^3 g^4}+\frac{\left (2 B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{3 b (b c-a d)^3 g^4}+\frac{\left (2 B^2 d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{3 b (b c-a d)^3 g^4}+\frac{\left (2 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 (b c-a d)^3 g^4}\\ &=-\frac{2 B^2}{27 b g^4 (a+b x)^3}+\frac{5 B^2 d}{18 b (b c-a d) g^4 (a+b x)^2}-\frac{11 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac{11 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}+\frac{B^2 d^3 \log ^2(a+b x)}{3 b (b c-a d)^3 g^4}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b g^4 (a+b x)^3}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac{2 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac{2 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{11 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac{2 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac{2 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac{B^2 d^3 \log ^2(c+d x)}{3 b (b c-a d)^3 g^4}-\frac{2 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}+\frac{\left (2 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 (b c-a d)^3 g^4}+\frac{\left (2 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 (b c-a d)^3 g^4}\\ &=-\frac{2 B^2}{27 b g^4 (a+b x)^3}+\frac{5 B^2 d}{18 b (b c-a d) g^4 (a+b x)^2}-\frac{11 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac{11 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}+\frac{B^2 d^3 \log ^2(a+b x)}{3 b (b c-a d)^3 g^4}-\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9 b g^4 (a+b x)^3}+\frac{B d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}-\frac{2 B d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}-\frac{2 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{11 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac{2 B^2 d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac{2 B d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac{B^2 d^3 \log ^2(c+d x)}{3 b (b c-a d)^3 g^4}-\frac{2 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}-\frac{2 B^2 d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}-\frac{2 B^2 d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}\\ \end{align*}

Mathematica [C]  time = 0.766238, size = 585, normalized size = 1.4 \[ -\frac{\frac{B \left (-18 B d^3 (a+b x)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+18 B d^3 (a+b x)^3 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+36 d^2 (a+b x)^2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+36 d^3 (a+b x)^3 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-36 d^3 (a+b x)^3 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+12 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-18 d (a+b x) (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+36 B d^2 (a+b x)^2 (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)-9 B d (a+b x) \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )+2 B \left (6 d^2 (a+b x)^2 (b c-a d)-6 d^3 (a+b x)^3 \log (c+d x)-3 d (a+b x) (b c-a d)^2+2 (b c-a d)^3+6 d^3 (a+b x)^3 \log (a+b x)\right )\right )}{(b c-a d)^3}+18 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{54 b g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(a*g + b*g*x)^4,x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.05, size = 2624, normalized size = 6.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x)

[Out]

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

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Maxima [B]  time = 1.85789, size = 1916, normalized size = 4.58 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x, algorithm="maxima")

[Out]

-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 - 8
5*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 - 1/9*A*B*((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*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)) - 1/3*B^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*A^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 [A]  time = 1.10476, size = 1388, normalized size = 3.32 \begin{align*} -\frac{2 \,{\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} b^{3} c^{3} - 27 \,{\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a b^{2} c^{2} d + 54 \,{\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a^{2} b c d^{2} -{\left (18 \, A^{2} + 66 \, A B + 85 \, B^{2}\right )} a^{3} d^{3} + 6 \,{\left ({\left (6 \, A B + 11 \, B^{2}\right )} b^{3} c d^{2} -{\left (6 \, A B + 11 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} + 18 \,{\left (B^{2} b^{3} d^{3} x^{3} + 3 \, B^{2} a b^{2} d^{3} x^{2} + 3 \, B^{2} a^{2} b d^{3} x + B^{2} b^{3} c^{3} - 3 \, B^{2} a b^{2} c^{2} d + 3 \, B^{2} a^{2} b c d^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} - 3 \,{\left ({\left (6 \, A B + 5 \, B^{2}\right )} b^{3} c^{2} d - 18 \,{\left (2 \, A B + 3 \, B^{2}\right )} a b^{2} c d^{2} +{\left (30 \, A B + 49 \, B^{2}\right )} a^{2} b d^{3}\right )} x + 6 \,{\left ({\left (6 \, A B + 11 \, B^{2}\right )} b^{3} d^{3} x^{3} + 2 \,{\left (3 \, A B + B^{2}\right )} b^{3} c^{3} - 9 \,{\left (2 \, A B + B^{2}\right )} a b^{2} c^{2} d + 18 \,{\left (A B + B^{2}\right )} a^{2} b c d^{2} + 3 \,{\left (2 \, B^{2} b^{3} c d^{2} + 3 \,{\left (2 \, A B + 3 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} - 3 \,{\left (B^{2} b^{3} c^{2} d - 6 \, B^{2} a b^{2} c d^{2} - 6 \,{\left (A B + B^{2}\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{54 \,{\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} g^{4} x^{3} + 3 \,{\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} g^{4} x^{2} + 3 \,{\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} g^{4} x +{\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} g^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x, algorithm="fricas")

[Out]

-1/54*(2*(9*A^2 + 6*A*B + 2*B^2)*b^3*c^3 - 27*(2*A^2 + 2*A*B + B^2)*a*b^2*c^2*d + 54*(A^2 + 2*A*B + 2*B^2)*a^2
*b*c*d^2 - (18*A^2 + 66*A*B + 85*B^2)*a^3*d^3 + 6*((6*A*B + 11*B^2)*b^3*c*d^2 - (6*A*B + 11*B^2)*a*b^2*d^3)*x^
2 + 18*(B^2*b^3*d^3*x^3 + 3*B^2*a*b^2*d^3*x^2 + 3*B^2*a^2*b*d^3*x + B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2*a^
2*b*c*d^2)*log((b*e*x + a*e)/(d*x + c))^2 - 3*((6*A*B + 5*B^2)*b^3*c^2*d - 18*(2*A*B + 3*B^2)*a*b^2*c*d^2 + (3
0*A*B + 49*B^2)*a^2*b*d^3)*x + 6*((6*A*B + 11*B^2)*b^3*d^3*x^3 + 2*(3*A*B + B^2)*b^3*c^3 - 9*(2*A*B + B^2)*a*b
^2*c^2*d + 18*(A*B + B^2)*a^2*b*c*d^2 + 3*(2*B^2*b^3*c*d^2 + 3*(2*A*B + 3*B^2)*a*b^2*d^3)*x^2 - 3*(B^2*b^3*c^2
*d - 6*B^2*a*b^2*c*d^2 - 6*(A*B + B^2)*a^2*b*d^3)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^7*c^3 - 3*a*b^6*c^2*d +
3*a^2*b^5*c*d^2 - a^3*b^4*d^3)*g^4*x^3 + 3*(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*g^4*
x^2 + 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*g^4*x + (a^3*b^4*c^3 - 3*a^4*b^3*c^2*d
+ 3*a^5*b^2*c*d^2 - a^6*b*d^3)*g^4)

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Sympy [B]  time = 36.9113, size = 1544, normalized size = 3.69 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))**2/(b*g*x+a*g)**4,x)

[Out]

-B*d**3*(6*A + 11*B)*log(x + (6*A*B*a*d**4 + 6*A*B*b*c*d**3 + 11*B**2*a*d**4 + 11*B**2*b*c*d**3 - B*a**4*d**7*
(6*A + 11*B)/(a*d - b*c)**3 + 4*B*a**3*b*c*d**6*(6*A + 11*B)/(a*d - b*c)**3 - 6*B*a**2*b**2*c**2*d**5*(6*A + 1
1*B)/(a*d - b*c)**3 + 4*B*a*b**3*c**3*d**4*(6*A + 11*B)/(a*d - b*c)**3 - B*b**4*c**4*d**3*(6*A + 11*B)/(a*d -
b*c)**3)/(12*A*B*b*d**4 + 22*B**2*b*d**4))/(9*b*g**4*(a*d - b*c)**3) + B*d**3*(6*A + 11*B)*log(x + (6*A*B*a*d*
*4 + 6*A*B*b*c*d**3 + 11*B**2*a*d**4 + 11*B**2*b*c*d**3 + B*a**4*d**7*(6*A + 11*B)/(a*d - b*c)**3 - 4*B*a**3*b
*c*d**6*(6*A + 11*B)/(a*d - b*c)**3 + 6*B*a**2*b**2*c**2*d**5*(6*A + 11*B)/(a*d - b*c)**3 - 4*B*a*b**3*c**3*d*
*4*(6*A + 11*B)/(a*d - b*c)**3 + B*b**4*c**4*d**3*(6*A + 11*B)/(a*d - b*c)**3)/(12*A*B*b*d**4 + 22*B**2*b*d**4
))/(9*b*g**4*(a*d - b*c)**3) + (3*B**2*a**2*c*d**2 + 3*B**2*a**2*d**3*x - 3*B**2*a*b*c**2*d + 3*B**2*a*b*d**3*
x**2 + B**2*b**2*c**3 + B**2*b**2*d**3*x**3)*log(e*(a + b*x)/(c + d*x))**2/(3*a**6*d**3*g**4 - 9*a**5*b*c*d**2
*g**4 + 9*a**5*b*d**3*g**4*x + 9*a**4*b**2*c**2*d*g**4 - 27*a**4*b**2*c*d**2*g**4*x + 9*a**4*b**2*d**3*g**4*x*
*2 - 3*a**3*b**3*c**3*g**4 + 27*a**3*b**3*c**2*d*g**4*x - 27*a**3*b**3*c*d**2*g**4*x**2 + 3*a**3*b**3*d**3*g**
4*x**3 - 9*a**2*b**4*c**3*g**4*x + 27*a**2*b**4*c**2*d*g**4*x**2 - 9*a**2*b**4*c*d**2*g**4*x**3 - 9*a*b**5*c**
3*g**4*x**2 + 9*a*b**5*c**2*d*g**4*x**3 - 3*b**6*c**3*g**4*x**3) + (-6*A*B*a**2*d**2 + 12*A*B*a*b*c*d - 6*A*B*
b**2*c**2 - 11*B**2*a**2*d**2 + 7*B**2*a*b*c*d - 15*B**2*a*b*d**2*x - 2*B**2*b**2*c**2 + 3*B**2*b**2*c*d*x - 6
*B**2*b**2*d**2*x**2)*log(e*(a + b*x)/(c + d*x))/(9*a**5*b*d**2*g**4 - 18*a**4*b**2*c*d*g**4 + 27*a**4*b**2*d*
*2*g**4*x + 9*a**3*b**3*c**2*g**4 - 54*a**3*b**3*c*d*g**4*x + 27*a**3*b**3*d**2*g**4*x**2 + 27*a**2*b**4*c**2*
g**4*x - 54*a**2*b**4*c*d*g**4*x**2 + 9*a**2*b**4*d**2*g**4*x**3 + 27*a*b**5*c**2*g**4*x**2 - 18*a*b**5*c*d*g*
*4*x**3 + 9*b**6*c**2*g**4*x**3) - (18*A**2*a**2*d**2 - 36*A**2*a*b*c*d + 18*A**2*b**2*c**2 + 66*A*B*a**2*d**2
- 42*A*B*a*b*c*d + 12*A*B*b**2*c**2 + 85*B**2*a**2*d**2 - 23*B**2*a*b*c*d + 4*B**2*b**2*c**2 + x**2*(36*A*B*b
**2*d**2 + 66*B**2*b**2*d**2) + x*(90*A*B*a*b*d**2 - 18*A*B*b**2*c*d + 147*B**2*a*b*d**2 - 15*B**2*b**2*c*d))/
(54*a**5*b*d**2*g**4 - 108*a**4*b**2*c*d*g**4 + 54*a**3*b**3*c**2*g**4 + x**3*(54*a**2*b**4*d**2*g**4 - 108*a*
b**5*c*d*g**4 + 54*b**6*c**2*g**4) + x**2*(162*a**3*b**3*d**2*g**4 - 324*a**2*b**4*c*d*g**4 + 162*a*b**5*c**2*
g**4) + x*(162*a**4*b**2*d**2*g**4 - 324*a**3*b**3*c*d*g**4 + 162*a**2*b**4*c**2*g**4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\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((A+B*log(e*(b*x+a)/(d*x+c)))^2/(b*g*x+a*g)^4,x, algorithm="giac")

[Out]

integrate((B*log((b*x + a)*e/(d*x + c)) + A)^2/(b*g*x + a*g)^4, x)