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

Optimal. Leaf size=407 \[ \frac{4 b^2 B (c+d x)^3 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{9 g^4 (a+b x)^3 (b c-a d)^3}-\frac{4 B d^3 \log \left (\frac{c+d x}{a+b x}\right ) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{3 b g^4 (b c-a d)^3}+\frac{4 B d^2 (c+d x) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{g^4 (a+b x) (b c-a d)^3}-\frac{2 b B d (c+d x)^2 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{g^4 (a+b x)^2 (b c-a d)^3}-\frac{\left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{3 b g^4 (a+b x)^3}-\frac{8 b^2 B^2 (c+d x)^3}{27 g^4 (a+b x)^3 (b c-a d)^3}-\frac{8 B^2 d^2 (c+d x)}{g^4 (a+b x) (b c-a d)^3}+\frac{4 B^2 d^3 \log ^2\left (\frac{c+d x}{a+b x}\right )}{3 b g^4 (b c-a d)^3}+\frac{2 b B^2 d (c+d x)^2}{g^4 (a+b x)^2 (b c-a d)^3} \]

[Out]

(-8*B^2*d^2*(c + d*x))/((b*c - a*d)^3*g^4*(a + b*x)) + (2*b*B^2*d*(c + d*x)^2)/((b*c - a*d)^3*g^4*(a + b*x)^2)
- (8*b^2*B^2*(c + d*x)^3)/(27*(b*c - a*d)^3*g^4*(a + b*x)^3) + (4*B^2*d^3*Log[(c + d*x)/(a + b*x)]^2)/(3*b*(b
*c - a*d)^3*g^4) + (4*B*d^2*(c + d*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/((b*c - a*d)^3*g^4*(a + b*x))
- (2*b*B*d*(c + d*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/((b*c - a*d)^3*g^4*(a + b*x)^2) + (4*b^2*B*(c
+ d*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(9*(b*c - a*d)^3*g^4*(a + b*x)^3) - (4*B*d^3*Log[(c + d*x)
/(a + b*x)]*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(3*b*(b*c - a*d)^3*g^4) - (A + B*Log[(e*(c + d*x)^2)/(a
+ b*x)^2])^2/(3*b*g^4*(a + b*x)^3)

________________________________________________________________________________________

Rubi [C]  time = 1.22808, antiderivative size = 692, normalized size of antiderivative = 1.7, number of steps used = 34, number of rules used = 11, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.324, Rules used = {2525, 12, 2528, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{8 B^2 d^3 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{3 b g^4 (b c-a d)^3}-\frac{8 B^2 d^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{3 b g^4 (b c-a d)^3}+\frac{4 B d^3 \log (a+b x) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{3 b g^4 (b c-a d)^3}-\frac{4 B d^3 \log (c+d x) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{3 b g^4 (b c-a d)^3}+\frac{4 B d^2 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{3 b g^4 (a+b x) (b c-a d)^2}-\frac{2 B d \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{3 b g^4 (a+b x)^2 (b c-a d)}-\frac{\left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{3 b g^4 (a+b x)^3}+\frac{4 B \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{9 b g^4 (a+b x)^3}-\frac{44 B^2 d^2}{9 b g^4 (a+b x) (b c-a d)^2}+\frac{4 B^2 d^3 \log ^2(a+b x)}{3 b g^4 (b c-a d)^3}+\frac{4 B^2 d^3 \log ^2(c+d x)}{3 b g^4 (b c-a d)^3}-\frac{44 B^2 d^3 \log (a+b x)}{9 b g^4 (b c-a d)^3}-\frac{8 B^2 d^3 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{3 b g^4 (b c-a d)^3}+\frac{44 B^2 d^3 \log (c+d x)}{9 b g^4 (b c-a d)^3}-\frac{8 B^2 d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{3 b g^4 (b c-a d)^3}+\frac{10 B^2 d}{9 b g^4 (a+b x)^2 (b c-a d)}-\frac{8 B^2}{27 b g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*B^2)/(27*b*g^4*(a + b*x)^3) + (10*B^2*d)/(9*b*(b*c - a*d)*g^4*(a + b*x)^2) - (44*B^2*d^2)/(9*b*(b*c - a*d)
^2*g^4*(a + b*x)) - (44*B^2*d^3*Log[a + b*x])/(9*b*(b*c - a*d)^3*g^4) + (4*B^2*d^3*Log[a + b*x]^2)/(3*b*(b*c -
a*d)^3*g^4) + (44*B^2*d^3*Log[c + d*x])/(9*b*(b*c - a*d)^3*g^4) - (8*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) + (4*B^2*d^3*Log[c + d*x]^2)/(3*b*(b*c - a*d)^3*g^4) - (8*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) + (4*B*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2
]))/(9*b*g^4*(a + b*x)^3) - (2*B*d*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(3*b*(b*c - a*d)*g^4*(a + b*x)^2)
+ (4*B*d^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(3*b*(b*c - a*d)^2*g^4*(a + b*x)) + (4*B*d^3*Log[a + b*x
]*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(3*b*(b*c - a*d)^3*g^4) - (4*B*d^3*Log[c + d*x]*(A + B*Log[(e*(c +
d*x)^2)/(a + b*x)^2]))/(3*b*(b*c - a*d)^3*g^4) - (A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2/(3*b*g^4*(a + b*x
)^3) - (8*B^2*d^3*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(3*b*(b*c - a*d)^3*g^4) - (8*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 (c+d x)^2}{(a+b x)^2}\right )\right )^2}{(a g+b g x)^4} \, dx &=-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{(2 B) \int \frac{2 (b c-a d) \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{g^3 (a+b x)^4 (c+d x)} \, dx}{3 b g}\\ &=-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{(4 B (b c-a d)) \int \frac{-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b g^4}\\ &=-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{(4 B (b c-a d)) \int \left (\frac{b \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d) (a+b x)^4}-\frac{b d \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d)^4 (a+b x)}+\frac{d^4 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b g^4}\\ &=-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{(4 B) \int \frac{-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{(a+b x)^4} \, dx}{3 g^4}-\frac{\left (4 B d^3\right ) \int \frac{-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{a+b x} \, dx}{3 (b c-a d)^3 g^4}+\frac{\left (4 B d^4\right ) \int \frac{-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}+\frac{\left (4 B d^2\right ) \int \frac{-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{(a+b x)^2} \, dx}{3 (b c-a d)^2 g^4}-\frac{(4 B d) \int \frac{-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{(a+b x)^3} \, dx}{3 (b c-a d) g^4}\\ &=\frac{4 B \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{9 b g^4 (a+b x)^3}-\frac{2 B d \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}+\frac{4 B d^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}+\frac{4 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{4 B d^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}-\frac{\left (4 B^2\right ) \int \frac{2 (-b c+a d)}{(a+b x)^4 (c+d x)} \, dx}{9 b g^4}-\frac{\left (4 B^2 d^3\right ) \int \frac{(a+b x)^2 \left (\frac{2 d e (c+d x)}{(a+b x)^2}-\frac{2 b e (c+d x)^2}{(a+b x)^3}\right ) \log (a+b x)}{e (c+d x)^2} \, dx}{3 b (b c-a d)^3 g^4}+\frac{\left (4 B^2 d^3\right ) \int \frac{(a+b x)^2 \left (\frac{2 d e (c+d x)}{(a+b x)^2}-\frac{2 b e (c+d x)^2}{(a+b x)^3}\right ) \log (c+d x)}{e (c+d x)^2} \, dx}{3 b (b c-a d)^3 g^4}-\frac{\left (4 B^2 d^2\right ) \int \frac{2 (-b c+a d)}{(a+b x)^2 (c+d x)} \, dx}{3 b (b c-a d)^2 g^4}+\frac{\left (2 B^2 d\right ) \int \frac{-2 b c+2 a d}{(a+b x)^3 (c+d x)} \, dx}{3 b (b c-a d) g^4}\\ &=\frac{4 B \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{9 b g^4 (a+b x)^3}-\frac{2 B d \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}+\frac{4 B d^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}+\frac{4 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{4 B d^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}-\frac{\left (4 B^2 d\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{3 b g^4}+\frac{\left (8 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 (8 B^2 (b c-a d)\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{9 b g^4}-\frac{\left (4 B^2 d^3\right ) \int \frac{(a+b x)^2 \left (\frac{2 d e (c+d x)}{(a+b x)^2}-\frac{2 b e (c+d x)^2}{(a+b x)^3}\right ) \log (a+b x)}{(c+d x)^2} \, dx}{3 b (b c-a d)^3 e g^4}+\frac{\left (4 B^2 d^3\right ) \int \frac{(a+b x)^2 \left (\frac{2 d e (c+d x)}{(a+b x)^2}-\frac{2 b e (c+d x)^2}{(a+b x)^3}\right ) \log (c+d x)}{(c+d x)^2} \, dx}{3 b (b c-a d)^3 e g^4}\\ &=\frac{4 B \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{9 b g^4 (a+b x)^3}-\frac{2 B d \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}+\frac{4 B d^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}+\frac{4 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{4 B d^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}-\frac{\left (4 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 (8 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 (8 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 (4 B^2 d^3\right ) \int \left (-\frac{2 b e \log (a+b x)}{a+b x}+\frac{2 d e \log (a+b x)}{c+d x}\right ) \, dx}{3 b (b c-a d)^3 e g^4}+\frac{\left (4 B^2 d^3\right ) \int \left (-\frac{2 b e \log (c+d x)}{a+b x}+\frac{2 d e \log (c+d x)}{c+d x}\right ) \, dx}{3 b (b c-a d)^3 e g^4}\\ &=-\frac{8 B^2}{27 b g^4 (a+b x)^3}+\frac{10 B^2 d}{9 b (b c-a d) g^4 (a+b x)^2}-\frac{44 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac{44 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}+\frac{44 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}+\frac{4 B \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{9 b g^4 (a+b x)^3}-\frac{2 B d \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}+\frac{4 B d^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}+\frac{4 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{4 B d^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{\left (8 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 (8 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 (8 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 (8 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{8 B^2}{27 b g^4 (a+b x)^3}+\frac{10 B^2 d}{9 b (b c-a d) g^4 (a+b x)^2}-\frac{44 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac{44 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}+\frac{44 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac{8 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{8 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{4 B \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{9 b g^4 (a+b x)^3}-\frac{2 B d \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}+\frac{4 B d^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}+\frac{4 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{4 B d^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{\left (8 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 (8 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 (8 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 (8 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{8 B^2}{27 b g^4 (a+b x)^3}+\frac{10 B^2 d}{9 b (b c-a d) g^4 (a+b x)^2}-\frac{44 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac{44 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}+\frac{4 B^2 d^3 \log ^2(a+b x)}{3 b (b c-a d)^3 g^4}+\frac{44 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac{8 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{4 B^2 d^3 \log ^2(c+d x)}{3 b (b c-a d)^3 g^4}-\frac{8 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{4 B \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{9 b g^4 (a+b x)^3}-\frac{2 B d \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}+\frac{4 B d^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}+\frac{4 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{4 B d^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac{\left (8 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 (8 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{8 B^2}{27 b g^4 (a+b x)^3}+\frac{10 B^2 d}{9 b (b c-a d) g^4 (a+b x)^2}-\frac{44 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac{44 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}+\frac{4 B^2 d^3 \log ^2(a+b x)}{3 b (b c-a d)^3 g^4}+\frac{44 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac{8 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{4 B^2 d^3 \log ^2(c+d x)}{3 b (b c-a d)^3 g^4}-\frac{8 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{4 B \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{9 b g^4 (a+b x)^3}-\frac{2 B d \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}+\frac{4 B d^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}+\frac{4 B d^3 \log (a+b x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{4 B d^3 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac{\left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{3 b g^4 (a+b x)^3}-\frac{8 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{8 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 = 0.720682, size = 598, normalized size = 1.47 \[ -\frac{\frac{2 B \left (-18 B d^3 (a+b x)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+18 B d^3 (a+b x)^3 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+18 d^2 (a+b x)^2 (a d-b c) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )-18 d^3 (a+b x)^3 \log (a+b x) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )+18 d^3 (a+b x)^3 \log (c+d x) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )-6 (b c-a d)^3 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )+9 d (a+b x) (b c-a d)^2 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )+36 B d^2 (a+b x)^2 (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)-9 B d (a+b x) \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )+2 B \left (6 d^2 (a+b x)^2 (b c-a d)-6 d^3 (a+b x)^3 \log (c+d x)-3 d (a+b x) (b c-a d)^2+2 (b c-a d)^3+6 d^3 (a+b x)^3 \log (a+b x)\right )\right )}{(b c-a d)^3}+9 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{27 b g^4 (a+b x)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(9*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2 + (2*B*(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]) - 6*(b*c - a*d)
^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]) + 9*d*(b*c - a*d)^2*(a + b*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)
^2]) + 18*d^2*(-(b*c) + a*d)*(a + b*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]) - 18*d^3*(a + b*x)^3*Log[a +
b*x]*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]) + 18*d^3*(a + b*x)^3*Log[c + d*x]*(A + B*Log[(e*(c + d*x)^2)/(a
+ b*x)^2]) - 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)/(27*b*g^4*(a + b*x)^3)

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Maple [B]  time = 0.075, size = 947, normalized size = 2.3 \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*(d*x+c)^2/(b*x+a)^2))^2/(b*g*x+a*g)^4,x)

[Out]

-1/3/b/(b*x+a)^3/g^4*A^2-8/27/b/g^4*B^2/(b*x+a)^3+4/9/b/g^4*B^2/(b*x+a)^3*ln(e*(1/(b*x+a)*a*d-b*c/(b*x+a)-d)^2
/b^2)-1/3/b/g^4*B^2/(b*x+a)^3*ln(e*(1/(b*x+a)*a*d-b*c/(b*x+a)-d)^2/b^2)^2-10/9/b/g^4*B^2*d/(a*d-b*c)/(b*x+a)^2
-44/9/b/g^4*B^2*d^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(b*x+a)-22/9/b/g^4*d^3*B^2/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*
d-b^3*c^3)*ln(e*(1/(b*x+a)*a*d-b*c/(b*x+a)-d)^2/b^2)+1/3/b/g^4*d^3*B^2/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^
3*c^3)*ln(e*(1/(b*x+a)*a*d-b*c/(b*x+a)-d)^2/b^2)^2+2/3/b/g^4*B^2*d/(a*d-b*c)/(b*x+a)^2*ln(e*(1/(b*x+a)*a*d-b*c
/(b*x+a)-d)^2/b^2)+4/3/b/g^4*B^2*d^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(b*x+a)*ln(e*(1/(b*x+a)*a*d-b*c/(b*x+a)-d)^2/
b^2)-2/3/b/g^4*A*B/(b*x+a)^3*ln(e*(1/(b*x+a)*a*d-b*c/(b*x+a)-d)^2/b^2)+4/9/b/g^4*A*B*a^3*d^3/(a*d-b*c)^3/(b*x+
a)^3-4/3/g^4*A*B*a^2*d^2/(a*d-b*c)^3/(b*x+a)^3*c+4/3*b/g^4*A*B*a*d/(a*d-b*c)^3/(b*x+a)^3*c^2+2/3/b/g^4*A*B*a^2
*d^3/(a*d-b*c)^3/(b*x+a)^2-4/3/g^4*A*B*a*d^2/(a*d-b*c)^3/(b*x+a)^2*c+4/3/b/g^4*A*B*a*d^3/(a*d-b*c)^3/(b*x+a)+4
/3/b/g^4*A*B*a*d^4/(a*d-b*c)^4*ln(1/(b*x+a)*a*d-b*c/(b*x+a)-d)-4/9*b^2/g^4*A*B*c^3/(a*d-b*c)^3/(b*x+a)^3+2/3*b
/g^4*A*B*c^2/(a*d-b*c)^3/(b*x+a)^2*d-4/3/g^4*A*B*c/(a*d-b*c)^3/(b*x+a)*d^2-4/3/g^4*A*B*c*d^3/(a*d-b*c)^4*ln(1/
(b*x+a)*a*d-b*c/(b*x+a)-d)

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Maxima [B]  time = 1.88066, size = 2128, normalized size = 5.23 \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*(d*x+c)^2/(b*x+a)^2))^2/(b*g*x+a*g)^4,x, algorithm="maxima")

[Out]

2/27*(3*((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(d^2*e*x^2/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x + a^2) + c^2*e/(b^
2*x^2 + 2*a*b*x + a^2)) - (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 + 2/9*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) - 3*log(d^2*e*x^2/
(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x + a^2) + c^2*e/(b^2*x^2 + 2*a*b*x + 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) + 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(d^2*e*x^2/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x + a^2) + c^2*e/(b^2*x^2 + 2*a*b
*x + a^2))^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 = 1.12827, size = 1474, normalized size = 3.62 \begin{align*} -\frac{{\left (9 \, A^{2} - 12 \, A B + 8 \, B^{2}\right )} b^{3} c^{3} - 27 \,{\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a b^{2} c^{2} d + 27 \,{\left (A^{2} - 4 \, A B + 8 \, B^{2}\right )} a^{2} b c d^{2} -{\left (9 \, A^{2} - 66 \, A B + 170 \, B^{2}\right )} a^{3} d^{3} - 12 \,{\left ({\left (3 \, A B - 11 \, B^{2}\right )} b^{3} c d^{2} -{\left (3 \, A B - 11 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} + 9 \,{\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{d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )^{2} + 6 \,{\left ({\left (3 \, A B - 5 \, B^{2}\right )} b^{3} c^{2} d - 18 \,{\left (A B - 3 \, B^{2}\right )} a b^{2} c d^{2} +{\left (15 \, A B - 49 \, B^{2}\right )} a^{2} b d^{3}\right )} x + 6 \,{\left ({\left (3 \, A B - 11 \, B^{2}\right )} b^{3} d^{3} x^{3} +{\left (3 \, A B - 2 \, B^{2}\right )} b^{3} c^{3} - 9 \,{\left (A B - B^{2}\right )} a b^{2} c^{2} d + 9 \,{\left (A B - 2 \, B^{2}\right )} a^{2} b c d^{2} - 3 \,{\left (2 \, B^{2} b^{3} c d^{2} - 3 \,{\left (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} + 3 \,{\left (A B - 2 \, B^{2}\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac{d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )}{27 \,{\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*(d*x+c)^2/(b*x+a)^2))^2/(b*g*x+a*g)^4,x, algorithm="fricas")

[Out]

-1/27*((9*A^2 - 12*A*B + 8*B^2)*b^3*c^3 - 27*(A^2 - 2*A*B + 2*B^2)*a*b^2*c^2*d + 27*(A^2 - 4*A*B + 8*B^2)*a^2*
b*c*d^2 - (9*A^2 - 66*A*B + 170*B^2)*a^3*d^3 - 12*((3*A*B - 11*B^2)*b^3*c*d^2 - (3*A*B - 11*B^2)*a*b^2*d^3)*x^
2 + 9*(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((d^2*e*x^2 + 2*c*d*e*x + c^2*e)/(b^2*x^2 + 2*a*b*x + a^2))^2 + 6*((3*A*B - 5*B^2)*b^3*c^2*d - 18
*(A*B - 3*B^2)*a*b^2*c*d^2 + (15*A*B - 49*B^2)*a^2*b*d^3)*x + 6*((3*A*B - 11*B^2)*b^3*d^3*x^3 + (3*A*B - 2*B^2
)*b^3*c^3 - 9*(A*B - B^2)*a*b^2*c^2*d + 9*(A*B - 2*B^2)*a^2*b*c*d^2 - 3*(2*B^2*b^3*c*d^2 - 3*(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 + 3*(A*B - 2*B^2)*a^2*b*d^3)*x)*log((d^2*e*x^2 + 2*c*d*e*x
+ c^2*e)/(b^2*x^2 + 2*a*b*x + a^2)))/((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 = 35.508, size = 1561, normalized size = 3.84 \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*(d*x+c)**2/(b*x+a)**2))**2/(b*g*x+a*g)**4,x)

[Out]

4*B*d**3*(3*A - 11*B)*log(x + (12*A*B*a*d**4 + 12*A*B*b*c*d**3 - 44*B**2*a*d**4 - 44*B**2*b*c*d**3 - 4*B*a**4*
d**7*(3*A - 11*B)/(a*d - b*c)**3 + 16*B*a**3*b*c*d**6*(3*A - 11*B)/(a*d - b*c)**3 - 24*B*a**2*b**2*c**2*d**5*(
3*A - 11*B)/(a*d - b*c)**3 + 16*B*a*b**3*c**3*d**4*(3*A - 11*B)/(a*d - b*c)**3 - 4*B*b**4*c**4*d**3*(3*A - 11*
B)/(a*d - b*c)**3)/(24*A*B*b*d**4 - 88*B**2*b*d**4))/(9*b*g**4*(a*d - b*c)**3) - 4*B*d**3*(3*A - 11*B)*log(x +
(12*A*B*a*d**4 + 12*A*B*b*c*d**3 - 44*B**2*a*d**4 - 44*B**2*b*c*d**3 + 4*B*a**4*d**7*(3*A - 11*B)/(a*d - b*c)
**3 - 16*B*a**3*b*c*d**6*(3*A - 11*B)/(a*d - b*c)**3 + 24*B*a**2*b**2*c**2*d**5*(3*A - 11*B)/(a*d - b*c)**3 -
16*B*a*b**3*c**3*d**4*(3*A - 11*B)/(a*d - b*c)**3 + 4*B*b**4*c**4*d**3*(3*A - 11*B)/(a*d - b*c)**3)/(24*A*B*b*
d**4 - 88*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*(c + d*x)**2/(a + b*x)**2)**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 + 22*B**2*a**2*d**2 - 14*B**2*a*b*c*d + 30*B**2*a*b*d**2*x + 4*B**2*b**2*
c**2 - 6*B**2*b**2*c*d*x + 12*B**2*b**2*d**2*x**2)*log(e*(c + d*x)**2/(a + b*x)**2)/(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) + (-9*A**2*a**2*d**2 + 18*A**2*a*b*c*d - 9
*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 - 170*B**2*a**2*d**2 + 46*B**2*a*b*c*d
- 8*B**2*b**2*c**2 + x**2*(36*A*B*b**2*d**2 - 132*B**2*b**2*d**2) + x*(90*A*B*a*b*d**2 - 18*A*B*b**2*c*d - 294
*B**2*a*b*d**2 + 30*B**2*b**2*c*d))/(27*a**5*b*d**2*g**4 - 54*a**4*b**2*c*d*g**4 + 27*a**3*b**3*c**2*g**4 + x*
*3*(27*a**2*b**4*d**2*g**4 - 54*a*b**5*c*d*g**4 + 27*b**6*c**2*g**4) + x**2*(81*a**3*b**3*d**2*g**4 - 162*a**2
*b**4*c*d*g**4 + 81*a*b**5*c**2*g**4) + x*(81*a**4*b**2*d**2*g**4 - 162*a**3*b**3*c*d*g**4 + 81*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 (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\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*(d*x+c)^2/(b*x+a)^2))^2/(b*g*x+a*g)^4,x, algorithm="giac")

[Out]

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