3.103 \(\int \frac{(A+B \log (\frac{e (a+b x)}{c+d x}))^2}{(c i+d i x)^3} \, dx\)

Optimal. Leaf size=296 \[ \frac{B d (a+b x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 i^3 (c+d x)^2 (b c-a d)^2}+\frac{b (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{i^3 (c+d x) (b c-a d)^2}-\frac{d (a+b x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{2 i^3 (c+d x)^2 (b c-a d)^2}-\frac{2 A b B (a+b x)}{i^3 (c+d x) (b c-a d)^2}-\frac{2 b B^2 (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )}{i^3 (c+d x) (b c-a d)^2}+\frac{2 b B^2 (a+b x)}{i^3 (c+d x) (b c-a d)^2}-\frac{B^2 d (a+b x)^2}{4 i^3 (c+d x)^2 (b c-a d)^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 0.913585, antiderivative size = 577, normalized size of antiderivative = 1.95, number of steps used = 30, 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} \[ \frac{b^2 B^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{d i^3 (b c-a d)^2}+\frac{b^2 B^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{d i^3 (b c-a d)^2}+\frac{b^2 B \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d i^3 (b c-a d)^2}-\frac{b^2 B \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d i^3 (b c-a d)^2}+\frac{b B \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{d i^3 (c+d x) (b c-a d)}-\frac{\left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{2 d i^3 (c+d x)^2}+\frac{B \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 d i^3 (c+d x)^2}-\frac{b^2 B^2 \log ^2(a+b x)}{2 d i^3 (b c-a d)^2}-\frac{b^2 B^2 \log ^2(c+d x)}{2 d i^3 (b c-a d)^2}-\frac{3 b^2 B^2 \log (a+b x)}{2 d i^3 (b c-a d)^2}+\frac{b^2 B^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{d i^3 (b c-a d)^2}+\frac{3 b^2 B^2 \log (c+d x)}{2 d i^3 (b c-a d)^2}+\frac{b^2 B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{d i^3 (b c-a d)^2}-\frac{3 b B^2}{2 d i^3 (c+d x) (b c-a d)}-\frac{B^2}{4 d i^3 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(c*i + d*i*x)^3,x]

[Out]

-B^2/(4*d*i^3*(c + d*x)^2) - (3*b*B^2)/(2*d*(b*c - a*d)*i^3*(c + d*x)) - (3*b^2*B^2*Log[a + b*x])/(2*d*(b*c -
a*d)^2*i^3) - (b^2*B^2*Log[a + b*x]^2)/(2*d*(b*c - a*d)^2*i^3) + (B*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*d
*i^3*(c + d*x)^2) + (b*B*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d*(b*c - a*d)*i^3*(c + d*x)) + (b^2*B*Log[a +
b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(d*(b*c - a*d)^2*i^3) - (A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(2*d
*i^3*(c + d*x)^2) + (3*b^2*B^2*Log[c + d*x])/(2*d*(b*c - a*d)^2*i^3) + (b^2*B^2*Log[-((d*(a + b*x))/(b*c - a*d
))]*Log[c + d*x])/(d*(b*c - a*d)^2*i^3) - (b^2*B*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/(d*(b*c -
a*d)^2*i^3) - (b^2*B^2*Log[c + d*x]^2)/(2*d*(b*c - a*d)^2*i^3) + (b^2*B^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c
- a*d)])/(d*(b*c - a*d)^2*i^3) + (b^2*B^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(d*(b*c - a*d)^2*i^3) + (b
^2*B^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(d*(b*c - a*d)^2*i^3)

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}{(103 c+103 d x)^3} \, dx &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2185454 d (c+d x)^2}+\frac{B \int \frac{(b c-a d) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{10609 (a+b x) (c+d x)^3} \, dx}{103 d}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2185454 d (c+d x)^2}+\frac{(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)^3} \, dx}{1092727 d}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2185454 d (c+d x)^2}+\frac{(B (b c-a d)) \int \left (\frac{b^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (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)^3}-\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)^2}-\frac{b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{1092727 d}\\ &=-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2185454 d (c+d x)^2}-\frac{B \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^3} \, dx}{1092727}-\frac{\left (b^2 B\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{1092727 (b c-a d)^2}+\frac{\left (b^3 B\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{1092727 d (b c-a d)^2}-\frac{(b B) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{1092727 (b c-a d)}\\ &=\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2185454 d (c+d x)^2}+\frac{b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1092727 d (b c-a d) (c+d x)}+\frac{b^2 B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1092727 d (b c-a d)^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2185454 d (c+d x)^2}-\frac{b^2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1092727 d (b c-a d)^2}-\frac{B^2 \int \frac{b c-a d}{(a+b x) (c+d x)^3} \, dx}{2185454 d}-\frac{\left (b^2 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}{1092727 d (b c-a d)^2}+\frac{\left (b^2 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}{1092727 d (b c-a d)^2}-\frac{\left (b B^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{1092727 d (b c-a d)}\\ &=\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2185454 d (c+d x)^2}+\frac{b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1092727 d (b c-a d) (c+d x)}+\frac{b^2 B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1092727 d (b c-a d)^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2185454 d (c+d x)^2}-\frac{b^2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1092727 d (b c-a d)^2}-\frac{\left (b B^2\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{1092727 d}-\frac{\left (B^2 (b c-a d)\right ) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{2185454 d}-\frac{\left (b^2 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}{1092727 d (b c-a d)^2 e}+\frac{\left (b^2 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}{1092727 d (b c-a d)^2 e}\\ &=\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2185454 d (c+d x)^2}+\frac{b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1092727 d (b c-a d) (c+d x)}+\frac{b^2 B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1092727 d (b c-a d)^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2185454 d (c+d x)^2}-\frac{b^2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1092727 d (b c-a d)^2}-\frac{\left (b B^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}{1092727 d}-\frac{\left (B^2 (b c-a d)\right ) \int \left (\frac{b^3}{(b c-a d)^3 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^3}-\frac{b d}{(b c-a d)^2 (c+d x)^2}-\frac{b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{2185454 d}-\frac{\left (b^2 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}{1092727 d (b c-a d)^2 e}+\frac{\left (b^2 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}{1092727 d (b c-a d)^2 e}\\ &=-\frac{B^2}{4370908 d (c+d x)^2}-\frac{3 b B^2}{2185454 d (b c-a d) (c+d x)}-\frac{3 b^2 B^2 \log (a+b x)}{2185454 d (b c-a d)^2}+\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2185454 d (c+d x)^2}+\frac{b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1092727 d (b c-a d) (c+d x)}+\frac{b^2 B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1092727 d (b c-a d)^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2185454 d (c+d x)^2}+\frac{3 b^2 B^2 \log (c+d x)}{2185454 d (b c-a d)^2}-\frac{b^2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1092727 d (b c-a d)^2}+\frac{\left (b^2 B^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{1092727 (b c-a d)^2}-\frac{\left (b^2 B^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{1092727 (b c-a d)^2}-\frac{\left (b^3 B^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{1092727 d (b c-a d)^2}+\frac{\left (b^3 B^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{1092727 d (b c-a d)^2}\\ &=-\frac{B^2}{4370908 d (c+d x)^2}-\frac{3 b B^2}{2185454 d (b c-a d) (c+d x)}-\frac{3 b^2 B^2 \log (a+b x)}{2185454 d (b c-a d)^2}+\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2185454 d (c+d x)^2}+\frac{b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1092727 d (b c-a d) (c+d x)}+\frac{b^2 B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1092727 d (b c-a d)^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2185454 d (c+d x)^2}+\frac{3 b^2 B^2 \log (c+d x)}{2185454 d (b c-a d)^2}+\frac{b^2 B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1092727 d (b c-a d)^2}-\frac{b^2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1092727 d (b c-a d)^2}+\frac{b^2 B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{1092727 d (b c-a d)^2}-\frac{\left (b^2 B^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{1092727 (b c-a d)^2}-\frac{\left (b^2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{1092727 d (b c-a d)^2}-\frac{\left (b^2 B^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{1092727 d (b c-a d)^2}-\frac{\left (b^3 B^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{1092727 d (b c-a d)^2}\\ &=-\frac{B^2}{4370908 d (c+d x)^2}-\frac{3 b B^2}{2185454 d (b c-a d) (c+d x)}-\frac{3 b^2 B^2 \log (a+b x)}{2185454 d (b c-a d)^2}-\frac{b^2 B^2 \log ^2(a+b x)}{2185454 d (b c-a d)^2}+\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2185454 d (c+d x)^2}+\frac{b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1092727 d (b c-a d) (c+d x)}+\frac{b^2 B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1092727 d (b c-a d)^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2185454 d (c+d x)^2}+\frac{3 b^2 B^2 \log (c+d x)}{2185454 d (b c-a d)^2}+\frac{b^2 B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1092727 d (b c-a d)^2}-\frac{b^2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1092727 d (b c-a d)^2}-\frac{b^2 B^2 \log ^2(c+d x)}{2185454 d (b c-a d)^2}+\frac{b^2 B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{1092727 d (b c-a d)^2}-\frac{\left (b^2 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 )}{1092727 d (b c-a d)^2}-\frac{\left (b^2 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 )}{1092727 d (b c-a d)^2}\\ &=-\frac{B^2}{4370908 d (c+d x)^2}-\frac{3 b B^2}{2185454 d (b c-a d) (c+d x)}-\frac{3 b^2 B^2 \log (a+b x)}{2185454 d (b c-a d)^2}-\frac{b^2 B^2 \log ^2(a+b x)}{2185454 d (b c-a d)^2}+\frac{B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2185454 d (c+d x)^2}+\frac{b B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1092727 d (b c-a d) (c+d x)}+\frac{b^2 B \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1092727 d (b c-a d)^2}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )^2}{2185454 d (c+d x)^2}+\frac{3 b^2 B^2 \log (c+d x)}{2185454 d (b c-a d)^2}+\frac{b^2 B^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{1092727 d (b c-a d)^2}-\frac{b^2 B \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{1092727 d (b c-a d)^2}-\frac{b^2 B^2 \log ^2(c+d x)}{2185454 d (b c-a d)^2}+\frac{b^2 B^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{1092727 d (b c-a d)^2}+\frac{b^2 B^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{1092727 d (b c-a d)^2}+\frac{b^2 B^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{1092727 d (b c-a d)^2}\\ \end{align*}

Mathematica [C]  time = 0.44917, size = 444, normalized size = 1.5 \[ \frac{\frac{B \left (-2 b^2 B (c+d x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+2 b^2 B (c+d x)^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+4 b^2 (c+d x)^2 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-4 b^2 (c+d x)^2 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+4 b (c+d x) (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-B \left (2 b^2 (c+d x)^2 \log (a+b x)+2 b (c+d x) (b c-a d)+(b c-a d)^2-2 b^2 (c+d x)^2 \log (c+d x)\right )-4 b B (c+d x) (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)\right )}{(b c-a d)^2}-2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )^2}{4 d i^3 (c+d x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])^2/(c*i + d*i*x)^3,x]

[Out]

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

________________________________________________________________________________________

Maple [B]  time = 0.055, size = 1917, normalized size = 6.5 \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)^3,x)

[Out]

-1/2*d^2/(a*d-b*c)^3/i^3*B^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/(d*x+c)^2*a^3-3/(a*d-b*c)^3/i^3*B^2*b^2/(d*x+c)
*c*a-1/2/d/(a*d-b*c)^3/i^3*B^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*b^3*c+1/(a*d-b*c)^3/i^3*A*B*ln(b*e/d+(a*d-b*c
)*e/d/(d*x+c))*b^2*a+1/4/d/(a*d-b*c)^3/i^3*B^2/(d*x+c)^2*b^3*c^3+1/2*d^2/(a*d-b*c)^3/i^3*A*B/(d*x+c)^2*a^3-3/(
a*d-b*c)^3/i^3*A*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)^2*a*b^2*c^2+1/d/(a*d-b*c)^3/i^3*A*B*ln(b*e/d+(a*d-b
*c)*e/d/(d*x+c))/(d*x+c)^2*b^3*c^3+1/2/(a*d-b*c)^3/i^3*B^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*b^2*a-3/2/(a*d-b*
c)^3/i^3*B^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^2*a-1/4*d^2/(a*d-b*c)^3/i^3*B^2/(d*x+c)^2*a^3-1/2*d^2/(a*d-b*c)
^3/i^3*A^2/(d*x+c)^2*a^3+3/2/d/(a*d-b*c)^3/i^3*B^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^3*c+1/2*d^2/(a*d-b*c)^3/i
^3*B^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)^2*a^3-1/2/d/(a*d-b*c)^3/i^3*A^2*b^3*c+7/4/(a*d-b*c)^3/i^3*B^2*b
^2*a-7/4/d/(a*d-b*c)^3/i^3*B^2*b^3*c+1/2/(a*d-b*c)^3/i^3*A^2*b^2*a+3/2*d/(a*d-b*c)^3/i^3*B^2*ln(b*e/d+(a*d-b*c
)*e/d/(d*x+c))^2/(d*x+c)^2*a^2*b*c-3/2*d/(a*d-b*c)^3/i^3*B^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)^2*a^2*b*c
-3/2*d/(a*d-b*c)^3/i^3*A*B/(d*x+c)^2*a^2*b*c+3*d/(a*d-b*c)^3/i^3*A*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)^2
*a^2*b*c+3/2*d/(a*d-b*c)^3/i^3*A^2/(d*x+c)^2*a^2*b*c+2/(a*d-b*c)^3/i^3*A*B*b^2/(d*x+c)*a*c+3/2/(a*d-b*c)^3/i^3
*A*B/(d*x+c)^2*b^2*c^2*a-1/d/(a*d-b*c)^3/i^3*B^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^3/(d*x+c)*c^2-1/d/(a*d-b*c)
^3/i^3*A*B*b^3/(d*x+c)*c^2+1/2/d/(a*d-b*c)^3/i^3*A^2/(d*x+c)^2*b^3*c^3+3/2*d/(a*d-b*c)^3/i^3*B^2*b/(d*x+c)*a^2
+3/2/d/(a*d-b*c)^3/i^3*B^2*b^3/(d*x+c)*c^2-3/2/(a*d-b*c)^3/i^3*A^2/(d*x+c)^2*b^2*c^2*a-3/4/(a*d-b*c)^3/i^3*B^2
/(d*x+c)^2*b^2*c^2*a+2/(a*d-b*c)^3/i^3*B^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^2/(d*x+c)*a*c-d^2/(a*d-b*c)^3/i^3
*A*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)^2*a^3-1/d/(a*d-b*c)^3/i^3*A*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^3
*c-d/(a*d-b*c)^3/i^3*B^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b/(d*x+c)*a^2-3/2/(a*d-b*c)^3/i^3*A*B*b^2*a+3/2/d/(a*
d-b*c)^3/i^3*A*B*b^3*c+1/2/d/(a*d-b*c)^3/i^3*B^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2/(d*x+c)^2*b^3*c^3+3/2/(a*d-
b*c)^3/i^3*B^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)^2*b^2*c^2*a-3/2/(a*d-b*c)^3/i^3*B^2*ln(b*e/d+(a*d-b*c)*
e/d/(d*x+c))^2/(d*x+c)^2*b^2*c^2*a-1/2/d/(a*d-b*c)^3/i^3*A*B/(d*x+c)^2*b^3*c^3-d/(a*d-b*c)^3/i^3*A*B*b/(d*x+c)
*a^2+3/4*d/(a*d-b*c)^3/i^3*B^2/(d*x+c)^2*a^2*b*c-1/2/d/(a*d-b*c)^3/i^3*B^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*
x+c)^2*b^3*c^3

________________________________________________________________________________________

Maxima [B]  time = 1.55489, size = 1145, normalized size = 3.87 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(d*i*x+c*i)^3,x, algorithm="maxima")

[Out]

1/4*(2*((2*b*d*x + 3*b*c - a*d)/((b*c*d^3 - a*d^4)*i^3*x^2 + 2*(b*c^2*d^2 - a*c*d^3)*i^3*x + (b*c^3*d - a*c^2*
d^2)*i^3) + 2*b^2*log(b*x + a)/((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*i^3) - 2*b^2*log(d*x + c)/((b^2*c^2*d - 2*
a*b*c*d^2 + a^2*d^3)*i^3))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - (7*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(b^2*d^
2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b*x + a)^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(d*x + c)^2 + 6*(b^
2*c*d - a*b*d^2)*x + 6*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b*x + a) - 2*(3*b^2*d^2*x^2 + 6*b^2*c*d*x + 3
*b^2*c^2 + 2*(b^2*d^2*x^2 + 2*b^2*c*d*x + b^2*c^2)*log(b*x + a))*log(d*x + c))/(b^2*c^4*d*i^3 - 2*a*b*c^3*d^2*
i^3 + a^2*c^2*d^3*i^3 + (b^2*c^2*d^3*i^3 - 2*a*b*c*d^4*i^3 + a^2*d^5*i^3)*x^2 + 2*(b^2*c^3*d^2*i^3 - 2*a*b*c^2
*d^3*i^3 + a^2*c*d^4*i^3)*x))*B^2 + 1/2*A*B*((2*b*d*x + 3*b*c - a*d)/((b*c*d^3 - a*d^4)*i^3*x^2 + 2*(b*c^2*d^2
- a*c*d^3)*i^3*x + (b*c^3*d - a*c^2*d^2)*i^3) - 2*log(b*e*x/(d*x + c) + a*e/(d*x + c))/(d^3*i^3*x^2 + 2*c*d^2
*i^3*x + c^2*d*i^3) + 2*b^2*log(b*x + a)/((b^2*c^2*d - 2*a*b*c*d^2 + a^2*d^3)*i^3) - 2*b^2*log(d*x + c)/((b^2*
c^2*d - 2*a*b*c*d^2 + a^2*d^3)*i^3)) - 1/2*B^2*log(b*e*x/(d*x + c) + a*e/(d*x + c))^2/(d^3*i^3*x^2 + 2*c*d^2*i
^3*x + c^2*d*i^3) - 1/2*A^2/(d^3*i^3*x^2 + 2*c*d^2*i^3*x + c^2*d*i^3)

________________________________________________________________________________________

Fricas [A]  time = 0.516859, size = 767, normalized size = 2.59 \begin{align*} -\frac{{\left (2 \, A^{2} - 6 \, A B + 7 \, B^{2}\right )} b^{2} c^{2} - 4 \,{\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a b c d +{\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} a^{2} d^{2} - 2 \,{\left (B^{2} b^{2} d^{2} x^{2} + 2 \, B^{2} b^{2} c d x + 2 \, B^{2} a b c d - B^{2} a^{2} d^{2}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} - 2 \,{\left ({\left (2 \, A B - 3 \, B^{2}\right )} b^{2} c d -{\left (2 \, A B - 3 \, B^{2}\right )} a b d^{2}\right )} x - 2 \,{\left ({\left (2 \, A B - 3 \, B^{2}\right )} b^{2} d^{2} x^{2} + 4 \,{\left (A B - B^{2}\right )} a b c d -{\left (2 \, A B - B^{2}\right )} a^{2} d^{2} - 2 \,{\left (B^{2} a b d^{2} - 2 \,{\left (A B - B^{2}\right )} b^{2} c d\right )} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{4 \,{\left ({\left (b^{2} c^{2} d^{3} - 2 \, a b c d^{4} + a^{2} d^{5}\right )} i^{3} x^{2} + 2 \,{\left (b^{2} c^{3} d^{2} - 2 \, a b c^{2} d^{3} + a^{2} c d^{4}\right )} i^{3} x +{\left (b^{2} c^{4} d - 2 \, a b c^{3} d^{2} + a^{2} c^{2} d^{3}\right )} i^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))^2/(d*i*x+c*i)^3,x, algorithm="fricas")

[Out]

-1/4*((2*A^2 - 6*A*B + 7*B^2)*b^2*c^2 - 4*(A^2 - 2*A*B + 2*B^2)*a*b*c*d + (2*A^2 - 2*A*B + B^2)*a^2*d^2 - 2*(B
^2*b^2*d^2*x^2 + 2*B^2*b^2*c*d*x + 2*B^2*a*b*c*d - B^2*a^2*d^2)*log((b*e*x + a*e)/(d*x + c))^2 - 2*((2*A*B - 3
*B^2)*b^2*c*d - (2*A*B - 3*B^2)*a*b*d^2)*x - 2*((2*A*B - 3*B^2)*b^2*d^2*x^2 + 4*(A*B - B^2)*a*b*c*d - (2*A*B -
B^2)*a^2*d^2 - 2*(B^2*a*b*d^2 - 2*(A*B - B^2)*b^2*c*d)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^2*c^2*d^3 - 2*a*b
*c*d^4 + a^2*d^5)*i^3*x^2 + 2*(b^2*c^3*d^2 - 2*a*b*c^2*d^3 + a^2*c*d^4)*i^3*x + (b^2*c^4*d - 2*a*b*c^3*d^2 + a
^2*c^2*d^3)*i^3)

________________________________________________________________________________________

Sympy [B]  time = 6.61356, size = 892, normalized size = 3.01 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))**2/(d*i*x+c*i)**3,x)

[Out]

-B*b**2*(2*A - 3*B)*log(x + (2*A*B*a*b**2*d + 2*A*B*b**3*c - 3*B**2*a*b**2*d - 3*B**2*b**3*c - B*a**3*b**2*d**
3*(2*A - 3*B)/(a*d - b*c)**2 + 3*B*a**2*b**3*c*d**2*(2*A - 3*B)/(a*d - b*c)**2 - 3*B*a*b**4*c**2*d*(2*A - 3*B)
/(a*d - b*c)**2 + B*b**5*c**3*(2*A - 3*B)/(a*d - b*c)**2)/(4*A*B*b**3*d - 6*B**2*b**3*d))/(2*d*i**3*(a*d - b*c
)**2) + B*b**2*(2*A - 3*B)*log(x + (2*A*B*a*b**2*d + 2*A*B*b**3*c - 3*B**2*a*b**2*d - 3*B**2*b**3*c + B*a**3*b
**2*d**3*(2*A - 3*B)/(a*d - b*c)**2 - 3*B*a**2*b**3*c*d**2*(2*A - 3*B)/(a*d - b*c)**2 + 3*B*a*b**4*c**2*d*(2*A
- 3*B)/(a*d - b*c)**2 - B*b**5*c**3*(2*A - 3*B)/(a*d - b*c)**2)/(4*A*B*b**3*d - 6*B**2*b**3*d))/(2*d*i**3*(a*
d - b*c)**2) + (-B**2*a**2*d + 2*B**2*a*b*c + 2*B**2*b**2*c*x + B**2*b**2*d*x**2)*log(e*(a + b*x)/(c + d*x))**
2/(2*a**2*c**2*d**2*i**3 + 4*a**2*c*d**3*i**3*x + 2*a**2*d**4*i**3*x**2 - 4*a*b*c**3*d*i**3 - 8*a*b*c**2*d**2*
i**3*x - 4*a*b*c*d**3*i**3*x**2 + 2*b**2*c**4*i**3 + 4*b**2*c**3*d*i**3*x + 2*b**2*c**2*d**2*i**3*x**2) + (-2*
A*B*a*d + 2*A*B*b*c + B**2*a*d - 3*B**2*b*c - 2*B**2*b*d*x)*log(e*(a + b*x)/(c + d*x))/(2*a*c**2*d**2*i**3 + 4
*a*c*d**3*i**3*x + 2*a*d**4*i**3*x**2 - 2*b*c**3*d*i**3 - 4*b*c**2*d**2*i**3*x - 2*b*c*d**3*i**3*x**2) - (2*A*
*2*a*d - 2*A**2*b*c - 2*A*B*a*d + 6*A*B*b*c + B**2*a*d - 7*B**2*b*c + x*(4*A*B*b*d - 6*B**2*b*d))/(4*a*c**2*d*
*2*i**3 - 4*b*c**3*d*i**3 + x**2*(4*a*d**4*i**3 - 4*b*c*d**3*i**3) + x*(8*a*c*d**3*i**3 - 8*b*c**2*d**2*i**3))

________________________________________________________________________________________

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 )}^{3}}\,{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)^3,x, algorithm="giac")

[Out]

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