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

Optimal. Leaf size=130 \[ -\frac{4 B (c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g^2 (a+b x) (b c-a d)}-\frac{(c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{g^2 (a+b x) (b c-a d)}-\frac{8 B^2 (c+d x)}{g^2 (a+b x) (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 0.888483, antiderivative size = 480, normalized size of antiderivative = 3.69, number of steps used = 26, 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} \[ -\frac{8 B^2 d \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac{8 B^2 d \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac{4 B d \log (a+b x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g^2 (b c-a d)}-\frac{4 B \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g^2 (a+b x)}+\frac{4 B d \log (c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{b g^2 (b c-a d)}-\frac{\left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{b g^2 (a+b x)}+\frac{4 B^2 d \log ^2(a+b x)}{b g^2 (b c-a d)}+\frac{4 B^2 d \log ^2(c+d x)}{b g^2 (b c-a d)}-\frac{8 B^2 d \log (a+b x)}{b g^2 (b c-a d)}-\frac{8 B^2 d \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{b g^2 (b c-a d)}+\frac{8 B^2 d \log (c+d x)}{b g^2 (b c-a d)}-\frac{8 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b g^2 (b c-a d)}-\frac{8 B^2}{b g^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^2} \, dx &=-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{(2 B) \int \frac{2 (b c-a d) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{g (a+b x)^2 (c+d x)} \, dx}{b g}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{(4 B (b c-a d)) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^2 (c+d x)} \, dx}{b g^2}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{(4 B (b c-a d)) \int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d) (a+b x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^2 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b g^2}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{(4 B) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^2} \, dx}{g^2}-\frac{(4 B d) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{a+b x} \, dx}{(b c-a d) g^2}+\frac{\left (4 B d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )}{c+d x} \, dx}{b (b c-a d) g^2}\\ &=-\frac{4 B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac{4 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{4 B d \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{\left (4 B^2\right ) \int \frac{2 (b c-a d)}{(a+b x)^2 (c+d x)} \, dx}{b g^2}+\frac{\left (4 B^2 d\right ) \int \frac{(c+d x)^2 \left (-\frac{2 d e (a+b x)^2}{(c+d x)^3}+\frac{2 b e (a+b x)}{(c+d x)^2}\right ) \log (a+b x)}{e (a+b x)^2} \, dx}{b (b c-a d) g^2}-\frac{\left (4 B^2 d\right ) \int \frac{(c+d x)^2 \left (-\frac{2 d e (a+b x)^2}{(c+d x)^3}+\frac{2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{e (a+b x)^2} \, dx}{b (b c-a d) g^2}\\ &=-\frac{4 B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac{4 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{4 B d \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{\left (8 B^2 (b c-a d)\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{b g^2}+\frac{\left (4 B^2 d\right ) \int \frac{(c+d x)^2 \left (-\frac{2 d e (a+b x)^2}{(c+d x)^3}+\frac{2 b e (a+b x)}{(c+d x)^2}\right ) \log (a+b x)}{(a+b x)^2} \, dx}{b (b c-a d) e g^2}-\frac{\left (4 B^2 d\right ) \int \frac{(c+d x)^2 \left (-\frac{2 d e (a+b x)^2}{(c+d x)^3}+\frac{2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{(a+b x)^2} \, dx}{b (b c-a d) e g^2}\\ &=-\frac{4 B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac{4 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{4 B d \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{\left (8 B^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}{b g^2}+\frac{\left (4 B^2 d\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}{b (b c-a d) e g^2}-\frac{\left (4 B^2 d\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}{b (b c-a d) e g^2}\\ &=-\frac{8 B^2}{b g^2 (a+b x)}-\frac{8 B^2 d \log (a+b x)}{b (b c-a d) g^2}-\frac{4 B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac{4 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{8 B^2 d \log (c+d x)}{b (b c-a d) g^2}+\frac{4 B d \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{\left (8 B^2 d\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{(b c-a d) g^2}-\frac{\left (8 B^2 d\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{(b c-a d) g^2}-\frac{\left (8 B^2 d^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{b (b c-a d) g^2}+\frac{\left (8 B^2 d^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b (b c-a d) g^2}\\ &=-\frac{8 B^2}{b g^2 (a+b x)}-\frac{8 B^2 d \log (a+b x)}{b (b c-a d) g^2}-\frac{4 B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac{4 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{8 B^2 d \log (c+d x)}{b (b c-a d) g^2}-\frac{8 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{4 B d \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}-\frac{8 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d) g^2}+\frac{\left (8 B^2 d\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{(b c-a d) g^2}+\frac{\left (8 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{b (b c-a d) g^2}+\frac{\left (8 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b (b c-a d) g^2}+\frac{\left (8 B^2 d^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b (b c-a d) g^2}\\ &=-\frac{8 B^2}{b g^2 (a+b x)}-\frac{8 B^2 d \log (a+b x)}{b (b c-a d) g^2}+\frac{4 B^2 d \log ^2(a+b x)}{b (b c-a d) g^2}-\frac{4 B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac{4 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{8 B^2 d \log (c+d x)}{b (b c-a d) g^2}-\frac{8 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{4 B d \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{4 B^2 d \log ^2(c+d x)}{b (b c-a d) g^2}-\frac{8 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d) g^2}+\frac{\left (8 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b (b c-a d) g^2}+\frac{\left (8 B^2 d\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b (b c-a d) g^2}\\ &=-\frac{8 B^2}{b g^2 (a+b x)}-\frac{8 B^2 d \log (a+b x)}{b (b c-a d) g^2}+\frac{4 B^2 d \log ^2(a+b x)}{b (b c-a d) g^2}-\frac{4 B \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b g^2 (a+b x)}-\frac{4 B d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d) g^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{b g^2 (a+b x)}+\frac{8 B^2 d \log (c+d x)}{b (b c-a d) g^2}-\frac{8 B^2 d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{4 B d \left (A+B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d) g^2}+\frac{4 B^2 d \log ^2(c+d x)}{b (b c-a d) g^2}-\frac{8 B^2 d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d) g^2}-\frac{8 B^2 d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{b (b c-a d) g^2}-\frac{8 B^2 d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{b (b c-a d) g^2}\\ \end{align*}

Mathematica [C]  time = 0.450889, size = 321, normalized size = 2.47 \[ -\frac{\frac{4 B \left (-B d (a+b x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+B d (a+b x) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+(b c-a d) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+d (a+b x) \log (a+b x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )-d (a+b x) \log (c+d x) \left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )+2 B (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)\right )}{b c-a d}+\left (B \log \left (\frac{e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{b g^2 (a+b x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.09, size = 357, normalized size = 2.8 \begin{align*}{\frac{d{A}^{2}}{{g}^{2} \left ( ad-bc \right ) } \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-1}}-8\,{\frac{d{B}^{2}}{b{g}^{2} \left ( dx+c \right ) } \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-1}}+4\,{\frac{d{B}^{2}}{{g}^{2} \left ( ad-bc \right ) }\ln \left ({\frac{e}{{d}^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{2}} \right ) \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-1}}+{\frac{d{B}^{2}}{{g}^{2} \left ( ad-bc \right ) } \left ( \ln \left ({\frac{e}{{d}^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{2}} \right ) \right ) ^{2} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-1}}-4\,{\frac{dAB}{b{g}^{2} \left ( dx+c \right ) } \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-1}}+2\,{\frac{dAB}{{g}^{2} \left ( ad-bc \right ) }\ln \left ({\frac{e}{{d}^{2}} \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{2}} \right ) \left ({\frac{ad}{dx+c}}-{\frac{bc}{dx+c}}+b \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d/g^2*A^2/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)/(a*d-b*c)-8*d/g^2/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*B^2/b/(d*x+c)+4*d/g^2/
(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*B^2/(a*d-b*c)*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)+d/g^2/(1/(d*x+c)*a*d-b*c
/(d*x+c)+b)*B^2/(a*d-b*c)*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)^2-4*d/g^2/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*A*
B/b/(d*x+c)+2*d/g^2/(1/(d*x+c)*a*d-b*c/(d*x+c)+b)*A*B/(a*d-b*c)*ln(e*(1/(d*x+c)*a*d-b*c/(d*x+c)+b)^2/d^2)

________________________________________________________________________________________

Maxima [B]  time = 1.4819, size = 775, normalized size = 5.96 \begin{align*} -4 \,{\left ({\left (\frac{1}{b^{2} g^{2} x + a b g^{2}} + \frac{d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac{d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} \log \left (\frac{b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - \frac{{\left (b d x + a d\right )} \log \left (b x + a\right )^{2} +{\left (b d x + a d\right )} \log \left (d x + c\right )^{2} - 2 \, b c + 2 \, a d - 2 \,{\left (b d x + a d\right )} \log \left (b x + a\right ) + 2 \,{\left (b d x + a d -{\left (b d x + a d\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a b^{2} c g^{2} - a^{2} b d g^{2} +{\left (b^{3} c g^{2} - a b^{2} d g^{2}\right )} x}\right )} B^{2} - 2 \, A B{\left (\frac{\log \left (\frac{b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{b^{2} g^{2} x + a b g^{2}} + \frac{2}{b^{2} g^{2} x + a b g^{2}} + \frac{2 \, d \log \left (b x + a\right )}{{\left (b^{2} c - a b d\right )} g^{2}} - \frac{2 \, d \log \left (d x + c\right )}{{\left (b^{2} c - a b d\right )} g^{2}}\right )} - \frac{B^{2} \log \left (\frac{b^{2} e x^{2}}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{2 \, a b e x}{d^{2} x^{2} + 2 \, c d x + c^{2}} + \frac{a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2}}{b^{2} g^{2} x + a b g^{2}} - \frac{A^{2}}{b^{2} g^{2} x + a b g^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 0.998379, size = 416, normalized size = 3.2 \begin{align*} -\frac{{\left (A^{2} + 4 \, A B + 8 \, B^{2}\right )} b c -{\left (A^{2} + 4 \, A B + 8 \, B^{2}\right )} a d +{\left (B^{2} b d x + B^{2} b c\right )} \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} + 2 \,{\left ({\left (A B + 2 \, B^{2}\right )} b d x +{\left (A B + 2 \, B^{2}\right )} b c\right )} \log \left (\frac{b^{2} e x^{2} + 2 \, a b e x + a^{2} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{{\left (b^{3} c - a b^{2} d\right )} g^{2} x +{\left (a b^{2} c - a^{2} b d\right )} g^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((A^2 + 4*A*B + 8*B^2)*b*c - (A^2 + 4*A*B + 8*B^2)*a*d + (B^2*b*d*x + B^2*b*c)*log((b^2*e*x^2 + 2*a*b*e*x + a
^2*e)/(d^2*x^2 + 2*c*d*x + c^2))^2 + 2*((A*B + 2*B^2)*b*d*x + (A*B + 2*B^2)*b*c)*log((b^2*e*x^2 + 2*a*b*e*x +
a^2*e)/(d^2*x^2 + 2*c*d*x + c^2)))/((b^3*c - a*b^2*d)*g^2*x + (a*b^2*c - a^2*b*d)*g^2)

________________________________________________________________________________________

Sympy [B]  time = 3.79672, size = 452, normalized size = 3.48 \begin{align*} - \frac{4 B d \left (A + 2 B\right ) \log{\left (x + \frac{4 A B a d^{2} + 4 A B b c d + 8 B^{2} a d^{2} + 8 B^{2} b c d - \frac{4 B a^{2} d^{3} \left (A + 2 B\right )}{a d - b c} + \frac{8 B a b c d^{2} \left (A + 2 B\right )}{a d - b c} - \frac{4 B b^{2} c^{2} d \left (A + 2 B\right )}{a d - b c}}{8 A B b d^{2} + 16 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac{4 B d \left (A + 2 B\right ) \log{\left (x + \frac{4 A B a d^{2} + 4 A B b c d + 8 B^{2} a d^{2} + 8 B^{2} b c d + \frac{4 B a^{2} d^{3} \left (A + 2 B\right )}{a d - b c} - \frac{8 B a b c d^{2} \left (A + 2 B\right )}{a d - b c} + \frac{4 B b^{2} c^{2} d \left (A + 2 B\right )}{a d - b c}}{8 A B b d^{2} + 16 B^{2} b d^{2}} \right )}}{b g^{2} \left (a d - b c\right )} + \frac{\left (- 2 A B - 4 B^{2}\right ) \log{\left (\frac{e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}}{a b g^{2} + b^{2} g^{2} x} + \frac{\left (B^{2} c + B^{2} d x\right ) \log{\left (\frac{e \left (a + b x\right )^{2}}{\left (c + d x\right )^{2}} \right )}^{2}}{a^{2} d g^{2} - a b c g^{2} + a b d g^{2} x - b^{2} c g^{2} x} - \frac{A^{2} + 4 A B + 8 B^{2}}{a b g^{2} + b^{2} g^{2} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*B*d*(A + 2*B)*log(x + (4*A*B*a*d**2 + 4*A*B*b*c*d + 8*B**2*a*d**2 + 8*B**2*b*c*d - 4*B*a**2*d**3*(A + 2*B)/
(a*d - b*c) + 8*B*a*b*c*d**2*(A + 2*B)/(a*d - b*c) - 4*B*b**2*c**2*d*(A + 2*B)/(a*d - b*c))/(8*A*B*b*d**2 + 16
*B**2*b*d**2))/(b*g**2*(a*d - b*c)) + 4*B*d*(A + 2*B)*log(x + (4*A*B*a*d**2 + 4*A*B*b*c*d + 8*B**2*a*d**2 + 8*
B**2*b*c*d + 4*B*a**2*d**3*(A + 2*B)/(a*d - b*c) - 8*B*a*b*c*d**2*(A + 2*B)/(a*d - b*c) + 4*B*b**2*c**2*d*(A +
2*B)/(a*d - b*c))/(8*A*B*b*d**2 + 16*B**2*b*d**2))/(b*g**2*(a*d - b*c)) + (-2*A*B - 4*B**2)*log(e*(a + b*x)**
2/(c + d*x)**2)/(a*b*g**2 + b**2*g**2*x) + (B**2*c + B**2*d*x)*log(e*(a + b*x)**2/(c + d*x)**2)**2/(a**2*d*g**
2 - a*b*c*g**2 + a*b*d*g**2*x - b**2*c*g**2*x) - (A**2 + 4*A*B + 8*B**2)/(a*b*g**2 + b**2*g**2*x)

________________________________________________________________________________________

Giac [B]  time = 1.47634, size = 510, normalized size = 3.92 \begin{align*} -{\left (\frac{B^{2} d}{b^{2} c g^{2} - a b d g^{2}} + \frac{B^{2}}{{\left (b g x + a g\right )} b g}\right )} \log \left (\frac{b^{2}}{\frac{b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac{2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac{a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac{2 \, b c d g}{b g x + a g} - \frac{2 \, a d^{2} g}{b g x + a g} + d^{2}}\right )^{2} + \frac{4 \,{\left (A B d + 3 \, B^{2} d\right )} \log \left (\frac{b c g}{b g x + a g} - \frac{a d g}{b g x + a g} + d\right )}{b^{2} c g^{2} - a b d g^{2}} - \frac{2 \,{\left (A B + 3 \, B^{2}\right )} \log \left (\frac{b^{2}}{\frac{b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac{2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac{a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac{2 \, b c d g}{b g x + a g} - \frac{2 \, a d^{2} g}{b g x + a g} + d^{2}}\right )}{{\left (b g x + a g\right )} b g} - \frac{A^{2} + 6 \, A B + 13 \, B^{2}}{{\left (b g x + a g\right )} b g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(B^2*d/(b^2*c*g^2 - a*b*d*g^2) + B^2/((b*g*x + a*g)*b*g))*log(b^2/(b^2*c^2*g^2/(b*g*x + a*g)^2 - 2*a*b*c*d*g^
2/(b*g*x + a*g)^2 + a^2*d^2*g^2/(b*g*x + a*g)^2 + 2*b*c*d*g/(b*g*x + a*g) - 2*a*d^2*g/(b*g*x + a*g) + d^2))^2
+ 4*(A*B*d + 3*B^2*d)*log(b*c*g/(b*g*x + a*g) - a*d*g/(b*g*x + a*g) + d)/(b^2*c*g^2 - a*b*d*g^2) - 2*(A*B + 3*
B^2)*log(b^2/(b^2*c^2*g^2/(b*g*x + a*g)^2 - 2*a*b*c*d*g^2/(b*g*x + a*g)^2 + a^2*d^2*g^2/(b*g*x + a*g)^2 + 2*b*
c*d*g/(b*g*x + a*g) - 2*a*d^2*g/(b*g*x + a*g) + d^2))/((b*g*x + a*g)*b*g) - (A^2 + 6*A*B + 13*B^2)/((b*g*x + a
*g)*b*g)