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

Optimal. Leaf size=152 \[ \frac{(a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{i^2 (c+d x) (b c-a d)}-\frac{2 A B (a+b x)}{i^2 (c+d x) (b c-a d)}-\frac{2 B^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{i^2 (c+d x) (b c-a d)}+\frac{2 B^2 (a+b x)}{i^2 (c+d x) (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 0.782059, antiderivative size = 472, normalized size of antiderivative = 3.11, number of steps used = 26, number of rules used = 11, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.344, Rules used = {2525, 12, 2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 44} \[ \frac{2 b B^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{d i^2 (b c-a d)}+\frac{2 b B^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d i^2 (b c-a d)}+\frac{2 b B \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d i^2 (b c-a d)}+\frac{2 B \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d i^2 (c+d x)}-\frac{2 b B \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d i^2 (b c-a d)}-\frac{\left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{d i^2 (c+d x)}-\frac{b B^2 \log ^2(a+b x)}{d i^2 (b c-a d)}-\frac{b B^2 \log ^2(c+d x)}{d i^2 (b c-a d)}-\frac{2 b B^2 \log (a+b x)}{d i^2 (b c-a d)}+\frac{2 b B^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d i^2 (b c-a d)}+\frac{2 b B^2 \log (c+d x)}{d i^2 (b c-a d)}+\frac{2 b B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d i^2 (b c-a d)}-\frac{2 B^2}{d i^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*B^2)/(d*i^2*(c + d*x)) - (2*b*B^2*Log[a + b*x])/(d*(b*c - a*d)*i^2) - (b*B^2*Log[a + b*x]^2)/(d*(b*c - a*d
)*i^2) + (2*B*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d*i^2*(c + d*x)) + (2*b*B*Log[a + b*x]*(A + B*Log[(e*(a +
b*x))/(c + d*x)]))/(d*(b*c - a*d)*i^2) - (A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(d*i^2*(c + d*x)) + (2*b*B^2*
Log[c + d*x])/(d*(b*c - a*d)*i^2) + (2*b*B^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(d*(b*c - a*d)*i^
2) - (2*b*B*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(d*(b*c - a*d)*i^2) - (b*B^2*Log[c + d*x]^2)/(d
*(b*c - a*d)*i^2) + (2*b*B^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*PolyL
og[2, -((d*(a + b*x))/(b*c - a*d))])/(d*(b*c - a*d)*i^2) + (2*b*B^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 (\frac{e (a+b x)}{c+d x}\right )\right )^2}{(95 c+95 d x)^2} \, dx &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac{(2 B) \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{95 (a+b x) (c+d x)^2} \, dx}{95 d}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\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) (c+d x)^2} \, dx}{9025 d}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac{(2 B (b c-a d)) \int \left (\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d) (c+d x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{9025 d}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}-\frac{(2 B) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{9025}-\frac{(2 b B) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{9025 (b c-a d)}+\frac{\left (2 b^2 B\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{9025 d (b c-a d)}\\ &=\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac{2 b B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac{\left (2 B^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{9025 d}-\frac{\left (2 b B^2\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}{9025 d (b c-a d)}+\frac{\left (2 b B^2\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}{9025 d (b c-a d)}\\ &=\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac{2 b B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac{\left (2 B^2 (b c-a d)\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{9025 d}-\frac{\left (2 b B^2\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}{9025 d (b c-a d) e}+\frac{\left (2 b B^2\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}{9025 d (b c-a d) e}\\ &=\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac{2 b B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac{\left (2 B^2 (b c-a d)\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}{9025 d}-\frac{\left (2 b B^2\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}{9025 d (b c-a d) e}+\frac{\left (2 b B^2\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}{9025 d (b c-a d) e}\\ &=-\frac{2 B^2}{9025 d (c+d x)}-\frac{2 b B^2 \log (a+b x)}{9025 d (b c-a d)}+\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac{2 b B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac{2 b B^2 \log (c+d x)}{9025 d (b c-a d)}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}+\frac{\left (2 b B^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{9025 (b c-a d)}-\frac{\left (2 b B^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{9025 (b c-a d)}-\frac{\left (2 b^2 B^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{9025 d (b c-a d)}+\frac{\left (2 b^2 B^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{9025 d (b c-a d)}\\ &=-\frac{2 B^2}{9025 d (c+d x)}-\frac{2 b B^2 \log (a+b x)}{9025 d (b c-a d)}+\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac{2 b B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac{2 b B^2 \log (c+d x)}{9025 d (b c-a d)}+\frac{2 b B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}+\frac{2 b B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{9025 d (b c-a d)}-\frac{\left (2 b B^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{9025 (b c-a d)}-\frac{\left (2 b B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{9025 d (b c-a d)}-\frac{\left (2 b B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{9025 d (b c-a d)}-\frac{\left (2 b^2 B^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{9025 d (b c-a d)}\\ &=-\frac{2 B^2}{9025 d (c+d x)}-\frac{2 b B^2 \log (a+b x)}{9025 d (b c-a d)}-\frac{b B^2 \log ^2(a+b x)}{9025 d (b c-a d)}+\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac{2 b B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac{2 b B^2 \log (c+d x)}{9025 d (b c-a d)}+\frac{2 b B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac{b B^2 \log ^2(c+d x)}{9025 d (b c-a d)}+\frac{2 b B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{9025 d (b c-a d)}-\frac{\left (2 b B^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 )}{9025 d (b c-a d)}-\frac{\left (2 b B^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 )}{9025 d (b c-a d)}\\ &=-\frac{2 B^2}{9025 d (c+d x)}-\frac{2 b B^2 \log (a+b x)}{9025 d (b c-a d)}-\frac{b B^2 \log ^2(a+b x)}{9025 d (b c-a d)}+\frac{2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9025 d (c+d x)}+\frac{2 b B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{9025 d (b c-a d)}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{9025 d (c+d x)}+\frac{2 b B^2 \log (c+d x)}{9025 d (b c-a d)}+\frac{2 b B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac{2 b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{9025 d (b c-a d)}-\frac{b B^2 \log ^2(c+d x)}{9025 d (b c-a d)}+\frac{2 b B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{9025 d (b c-a d)}+\frac{2 b B^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{9025 d (b c-a d)}+\frac{2 b B^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{9025 d (b c-a d)}\\ \end{align*}

Mathematica [C]  time = 0.436611, size = 315, normalized size = 2.07 \[ \frac{\frac{B \left (-b B (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+b B (c+d x) \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2 b (c+d x) \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-2 b (c+d x) \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-2 B (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)\right )}{b c-a d}-\left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{d i^2 (c+d x)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + (B*(2*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 2*b*(c + d
*x)*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 2*b*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log
[c + d*x] - 2*B*(b*c - a*d + b*(c + d*x)*Log[a + b*x] - b*(c + d*x)*Log[c + d*x]) - b*B*(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*(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 [B]  time = 0.052, size = 1236, normalized size = 8.1 \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/(d*i*x+c*i)^2,x)

[Out]

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

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Maxima [B]  time = 1.33913, size = 562, normalized size = 3.7 \begin{align*}{\left (2 \,{\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 (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right ) - \frac{{\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 )}{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} - 2 \, A B{\left (\frac{\log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right )}{d^{2} i^{2} x + c d i^{2}} - \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 )} - \frac{B^{2} \log \left (\frac{b e x}{d x + c} + \frac{a e}{d x + c}\right )^{2}}{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)))^2/(d*i*x+c*i)^2,x, algorithm="maxima")

[Out]

(2*(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
(b*e*x/(d*x + c) + a*e/(d*x + c)) - ((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))/(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 - 2*A*B*(log(b*e*x/(d*x + c) + a*e/(d*x + c))/(d^2*i^2*x + c*d
*i^2) - 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))
- B^2*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(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 = 0.512753, size = 319, normalized size = 2.1 \begin{align*} -\frac{{\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} b c -{\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a d -{\left (B^{2} b d x + B^{2} a d\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} - 2 \,{\left ({\left (A B - B^{2}\right )} b d x +{\left (A B - B^{2}\right )} a d\right )} \log \left (\frac{b e x + a e}{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)))^2/(d*i*x+c*i)^2,x, algorithm="fricas")

[Out]

-((A^2 - 2*A*B + 2*B^2)*b*c - (A^2 - 2*A*B + 2*B^2)*a*d - (B^2*b*d*x + B^2*a*d)*log((b*e*x + a*e)/(d*x + c))^2
- 2*((A*B - B^2)*b*d*x + (A*B - B^2)*a*d)*log((b*e*x + a*e)/(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 [B]  time = 3.85915, size = 432, normalized size = 2.84 \begin{align*} \frac{2 B b \left (A - B\right ) \log{\left (x + \frac{2 A B a b d + 2 A B b^{2} c - 2 B^{2} a b d - 2 B^{2} b^{2} c - \frac{2 B a^{2} b d^{2} \left (A - B\right )}{a d - b c} + \frac{4 B a b^{2} c d \left (A - B\right )}{a d - b c} - \frac{2 B b^{3} c^{2} \left (A - B\right )}{a d - b c}}{4 A B b^{2} d - 4 B^{2} b^{2} d} \right )}}{d i^{2} \left (a d - b c\right )} - \frac{2 B b \left (A - B\right ) \log{\left (x + \frac{2 A B a b d + 2 A B b^{2} c - 2 B^{2} a b d - 2 B^{2} b^{2} c + \frac{2 B a^{2} b d^{2} \left (A - B\right )}{a d - b c} - \frac{4 B a b^{2} c d \left (A - B\right )}{a d - b c} + \frac{2 B b^{3} c^{2} \left (A - B\right )}{a d - b c}}{4 A B b^{2} d - 4 B^{2} b^{2} d} \right )}}{d i^{2} \left (a d - b c\right )} + \frac{\left (- 2 A B + 2 B^{2}\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}}{c d i^{2} + d^{2} i^{2} x} + \frac{\left (- B^{2} a - B^{2} b x\right ) \log{\left (\frac{e \left (a + b x\right )}{c + d x} \right )}^{2}}{a c d i^{2} + a d^{2} i^{2} x - b c^{2} i^{2} - b c d i^{2} x} - \frac{A^{2} - 2 A B + 2 B^{2}}{c d i^{2} + d^{2} i^{2} x} \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/(d*i*x+c*i)**2,x)

[Out]

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

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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 (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)))^2/(d*i*x+c*i)^2,x, algorithm="giac")

[Out]

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