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

Optimal. Leaf size=163 \[ \frac{(a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{i^2 (c+d x) (b c-a d)}-\frac{2 A B n (a+b x)}{i^2 (c+d x) (b c-a d)}-\frac{2 B^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{i^2 (c+d x) (b c-a d)}+\frac{2 B^2 n^2 (a+b x)}{i^2 (c+d x) (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 0.755538, antiderivative size = 514, normalized size of antiderivative = 3.15, number of steps used = 24, number of rules used = 11, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.314, Rules used = {2525, 12, 2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 44} \[ \frac{2 b B^2 n^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{d i^2 (b c-a d)}+\frac{2 b B^2 n^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d i^2 (b c-a d)}+\frac{2 b B n \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d i^2 (b c-a d)}+\frac{2 B n \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d i^2 (c+d x)}-\frac{2 b B n \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{d i^2 (b c-a d)}-\frac{\left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{d i^2 (c+d x)}-\frac{b B^2 n^2 \log ^2(a+b x)}{d i^2 (b c-a d)}-\frac{b B^2 n^2 \log ^2(c+d x)}{d i^2 (b c-a d)}-\frac{2 b B^2 n^2 \log (a+b x)}{d i^2 (b c-a d)}+\frac{2 b B^2 n^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d i^2 (b c-a d)}+\frac{2 b B^2 n^2 \log (c+d x)}{d i^2 (b c-a d)}+\frac{2 b B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d i^2 (b c-a d)}-\frac{2 B^2 n^2}{d i^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*B^2*n^2)/(d*i^2*(c + d*x)) - (2*b*B^2*n^2*Log[a + b*x])/(d*(b*c - a*d)*i^2) - (b*B^2*n^2*Log[a + b*x]^2)/(
d*(b*c - a*d)*i^2) + (2*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(d*i^2*(c + d*x)) + (2*b*B*n*Log[a + b*x]*
(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(d*(b*c - a*d)*i^2) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(d*i^2*
(c + d*x)) + (2*b*B^2*n^2*Log[c + d*x])/(d*(b*c - a*d)*i^2) + (2*b*B^2*n^2*Log[-((d*(a + b*x))/(b*c - a*d))]*L
og[c + d*x])/(d*(b*c - a*d)*i^2) - (2*b*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x])/(d*(b*c - a*d
)*i^2) - (b*B^2*n^2*Log[c + d*x]^2)/(d*(b*c - a*d)*i^2) + (2*b*B^2*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a
*d)])/(d*(b*c - a*d)*i^2) + (2*b*B^2*n^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(d*(b*c - a*d)*i^2) + (2*b*
B^2*n^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(d*(b*c - a*d)*i^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 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]

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])

Rubi steps

\begin{align*} \int \frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{(197 c+197 d x)^2} \, dx &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}+\frac{(2 B n) \int \frac{(b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{197 (a+b x) (c+d x)^2} \, dx}{197 d}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}+\frac{(2 B (b c-a d) n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^2} \, dx}{38809 d}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}+\frac{(2 B (b c-a d) n) \int \left (\frac{b^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)}-\frac{d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d x)^2}-\frac{b d \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{38809 d}\\ &=-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}-\frac{(2 B n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{38809}-\frac{(2 b B n) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{38809 (b c-a d)}+\frac{\left (2 b^2 B n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{38809 d (b c-a d)}\\ &=\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{38809 d (c+d x)}+\frac{2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{38809 d (b c-a d)}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}-\frac{2 b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{38809 d (b c-a d)}-\frac{\left (2 B^2 n^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{38809 d}-\frac{\left (2 b B^2 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{38809 d (b c-a d)}+\frac{\left (2 b B^2 n^2\right ) \int \frac{(c+d x) \left (-\frac{d (a+b x)}{(c+d x)^2}+\frac{b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{38809 d (b c-a d)}\\ &=\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{38809 d (c+d x)}+\frac{2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{38809 d (b c-a d)}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}-\frac{2 b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{38809 d (b c-a d)}-\frac{\left (2 b B^2 n^2\right ) \int \left (\frac{b \log (a+b x)}{a+b x}-\frac{d \log (a+b x)}{c+d x}\right ) \, dx}{38809 d (b c-a d)}+\frac{\left (2 b B^2 n^2\right ) \int \left (\frac{b \log (c+d x)}{a+b x}-\frac{d \log (c+d x)}{c+d x}\right ) \, dx}{38809 d (b c-a d)}-\frac{\left (2 B^2 (b c-a d) n^2\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{38809 d}\\ &=\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{38809 d (c+d x)}+\frac{2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{38809 d (b c-a d)}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}-\frac{2 b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{38809 d (b c-a d)}+\frac{\left (2 b B^2 n^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{38809 (b c-a d)}-\frac{\left (2 b B^2 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{38809 (b c-a d)}-\frac{\left (2 b^2 B^2 n^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{38809 d (b c-a d)}+\frac{\left (2 b^2 B^2 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{38809 d (b c-a d)}-\frac{\left (2 B^2 (b c-a d) n^2\right ) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{38809 d}\\ &=-\frac{2 B^2 n^2}{38809 d (c+d x)}-\frac{2 b B^2 n^2 \log (a+b x)}{38809 d (b c-a d)}+\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{38809 d (c+d x)}+\frac{2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{38809 d (b c-a d)}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}+\frac{2 b B^2 n^2 \log (c+d x)}{38809 d (b c-a d)}+\frac{2 b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{38809 d (b c-a d)}-\frac{2 b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{38809 d (b c-a d)}+\frac{2 b B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{38809 d (b c-a d)}-\frac{\left (2 b B^2 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{38809 (b c-a d)}-\frac{\left (2 b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{38809 d (b c-a d)}-\frac{\left (2 b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{38809 d (b c-a d)}-\frac{\left (2 b^2 B^2 n^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{38809 d (b c-a d)}\\ &=-\frac{2 B^2 n^2}{38809 d (c+d x)}-\frac{2 b B^2 n^2 \log (a+b x)}{38809 d (b c-a d)}-\frac{b B^2 n^2 \log ^2(a+b x)}{38809 d (b c-a d)}+\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{38809 d (c+d x)}+\frac{2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{38809 d (b c-a d)}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}+\frac{2 b B^2 n^2 \log (c+d x)}{38809 d (b c-a d)}+\frac{2 b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{38809 d (b c-a d)}-\frac{2 b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{38809 d (b c-a d)}-\frac{b B^2 n^2 \log ^2(c+d x)}{38809 d (b c-a d)}+\frac{2 b B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{38809 d (b c-a d)}-\frac{\left (2 b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{38809 d (b c-a d)}-\frac{\left (2 b B^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{38809 d (b c-a d)}\\ &=-\frac{2 B^2 n^2}{38809 d (c+d x)}-\frac{2 b B^2 n^2 \log (a+b x)}{38809 d (b c-a d)}-\frac{b B^2 n^2 \log ^2(a+b x)}{38809 d (b c-a d)}+\frac{2 B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{38809 d (c+d x)}+\frac{2 b B n \log (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{38809 d (b c-a d)}-\frac{\left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{38809 d (c+d x)}+\frac{2 b B^2 n^2 \log (c+d x)}{38809 d (b c-a d)}+\frac{2 b B^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{38809 d (b c-a d)}-\frac{2 b B n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{38809 d (b c-a d)}-\frac{b B^2 n^2 \log ^2(c+d x)}{38809 d (b c-a d)}+\frac{2 b B^2 n^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{38809 d (b c-a d)}+\frac{2 b B^2 n^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{38809 d (b c-a d)}+\frac{2 b B^2 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{38809 d (b c-a d)}\\ \end{align*}

Mathematica [C]  time = 0.466976, size = 331, normalized size = 2.03 \[ \frac{\frac{B n \left (-b B n (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+b B n (c+d x) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+2 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 b (c+d x) \log (a+b x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-2 b (c+d x) \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-2 B n (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)\right )}{b c-a d}-\left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{d i^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.532, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dix+ci \right ) ^{2}} \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 1.30244, size = 578, normalized size = 3.55 \begin{align*} 2 \, A B n{\left (\frac{1}{d^{2} i^{2} x + c d i^{2}} + \frac{b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} i^{2}} - \frac{b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} i^{2}}\right )} +{\left (2 \, n{\left (\frac{1}{d^{2} i^{2} x + c d i^{2}} + \frac{b \log \left (b x + a\right )}{{\left (b c d - a d^{2}\right )} i^{2}} - \frac{b \log \left (d x + c\right )}{{\left (b c d - a d^{2}\right )} i^{2}}\right )} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right ) - \frac{{\left ({\left (b d x + b c\right )} \log \left (b x + a\right )^{2} +{\left (b d x + b c\right )} \log \left (d x + c\right )^{2} + 2 \, b c - 2 \, a d + 2 \,{\left (b d x + b c\right )} \log \left (b x + a\right ) - 2 \,{\left (b d x + b c +{\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} n^{2}}{b c^{2} d i^{2} - a c d^{2} i^{2} +{\left (b c d^{2} i^{2} - a d^{3} i^{2}\right )} x}\right )} B^{2} - \frac{B^{2} \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right )^{2}}{d^{2} i^{2} x + c d i^{2}} - \frac{2 \, A B \log \left (e{\left (\frac{b x}{d x + c} + \frac{a}{d x + c}\right )}^{n}\right )}{d^{2} i^{2} x + c d i^{2}} - \frac{A^{2}}{d^{2} i^{2} x + c d i^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*A*B*n*(1/(d^2*i^2*x + c*d*i^2) + b*log(b*x + a)/((b*c*d - a*d^2)*i^2) - b*log(d*x + c)/((b*c*d - a*d^2)*i^2)
) + (2*n*(1/(d^2*i^2*x + c*d*i^2) + b*log(b*x + a)/((b*c*d - a*d^2)*i^2) - b*log(d*x + c)/((b*c*d - a*d^2)*i^2
))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) - ((b*d*x + b*c)*log(b*x + a)^2 + (b*d*x + b*c)*log(d*x + c)^2 + 2*b
*c - 2*a*d + 2*(b*d*x + b*c)*log(b*x + a) - 2*(b*d*x + b*c + (b*d*x + b*c)*log(b*x + a))*log(d*x + c))*n^2/(b*
c^2*d*i^2 - a*c*d^2*i^2 + (b*c*d^2*i^2 - a*d^3*i^2)*x))*B^2 - B^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)^2/(d^
2*i^2*x + c*d*i^2) - 2*A*B*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(d^2*i^2*x + c*d*i^2) - A^2/(d^2*i^2*x + c*d
*i^2)

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Fricas [A]  time = 0.529081, size = 555, normalized size = 3.4 \begin{align*} -\frac{A^{2} b c - A^{2} a d + 2 \,{\left (B^{2} b c - B^{2} a d\right )} n^{2} +{\left (B^{2} b c - B^{2} a d\right )} \log \left (e\right )^{2} -{\left (B^{2} b d n^{2} x + B^{2} a d n^{2}\right )} \log \left (\frac{b x + a}{d x + c}\right )^{2} - 2 \,{\left (A B b c - A B a d\right )} n + 2 \,{\left (A B b c - A B a d -{\left (B^{2} b c - B^{2} a d\right )} n -{\left (B^{2} b d n x + B^{2} a d n\right )} \log \left (\frac{b x + a}{d x + c}\right )\right )} \log \left (e\right ) + 2 \,{\left (B^{2} a d n^{2} - A B a d n +{\left (B^{2} b d n^{2} - A B b d n\right )} x\right )} \log \left (\frac{b x + a}{d x + c}\right )}{{\left (b c d^{2} - a d^{3}\right )} i^{2} x +{\left (b c^{2} d - a c d^{2}\right )} i^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(A^2*b*c - A^2*a*d + 2*(B^2*b*c - B^2*a*d)*n^2 + (B^2*b*c - B^2*a*d)*log(e)^2 - (B^2*b*d*n^2*x + B^2*a*d*n^2)
*log((b*x + a)/(d*x + c))^2 - 2*(A*B*b*c - A*B*a*d)*n + 2*(A*B*b*c - A*B*a*d - (B^2*b*c - B^2*a*d)*n - (B^2*b*
d*n*x + B^2*a*d*n)*log((b*x + a)/(d*x + c)))*log(e) + 2*(B^2*a*d*n^2 - A*B*a*d*n + (B^2*b*d*n^2 - A*B*b*d*n)*x
)*log((b*x + a)/(d*x + c)))/((b*c*d^2 - a*d^3)*i^2*x + (b*c^2*d - a*c*d^2)*i^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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