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

Optimal. Leaf size=365 \[ -\frac{b^3 (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (a+b x) (b c-a d)^4}-\frac{3 b^2 d \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (b c-a d)^4}+\frac{3 b d^2 (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (c+d x) (b c-a d)^4}-\frac{d^3 (a+b x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^2 i^3 (c+d x)^2 (b c-a d)^4}-\frac{b^3 B (c+d x)}{g^2 i^3 (a+b x) (b c-a d)^4}+\frac{3 b^2 B d \log ^2\left (\frac{a+b x}{c+d x}\right )}{2 g^2 i^3 (b c-a d)^4}-\frac{3 b B d^2 (a+b x)}{g^2 i^3 (c+d x) (b c-a d)^4}+\frac{B d^3 (a+b x)^2}{4 g^2 i^3 (c+d x)^2 (b c-a d)^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 1.11099, antiderivative size = 631, normalized size of antiderivative = 1.73, number of steps used = 32, number of rules used = 11, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.275, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac{3 b^2 B d \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g^2 i^3 (b c-a d)^4}-\frac{3 b^2 B d \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g^2 i^3 (b c-a d)^4}-\frac{3 b^2 d \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (b c-a d)^4}-\frac{b^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (a+b x) (b c-a d)^3}+\frac{3 b^2 d \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (b c-a d)^4}-\frac{2 b d \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^2 i^3 (c+d x) (b c-a d)^3}-\frac{d \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^2 i^3 (c+d x)^2 (b c-a d)^2}-\frac{b^2 B}{g^2 i^3 (a+b x) (b c-a d)^3}+\frac{3 b^2 B d \log ^2(a+b x)}{2 g^2 i^3 (b c-a d)^4}+\frac{3 b^2 B d \log ^2(c+d x)}{2 g^2 i^3 (b c-a d)^4}+\frac{3 b^2 B d \log (a+b x)}{2 g^2 i^3 (b c-a d)^4}-\frac{3 b^2 B d \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g^2 i^3 (b c-a d)^4}-\frac{3 b^2 B d \log (c+d x)}{2 g^2 i^3 (b c-a d)^4}-\frac{3 b^2 B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g^2 i^3 (b c-a d)^4}+\frac{5 b B d}{2 g^2 i^3 (c+d x) (b c-a d)^3}+\frac{B d}{4 g^2 i^3 (c+d x)^2 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2*B)/((b*c - a*d)^3*g^2*i^3*(a + b*x))) + (B*d)/(4*(b*c - a*d)^2*g^2*i^3*(c + d*x)^2) + (5*b*B*d)/(2*(b*c
- a*d)^3*g^2*i^3*(c + d*x)) + (3*b^2*B*d*Log[a + b*x])/(2*(b*c - a*d)^4*g^2*i^3) + (3*b^2*B*d*Log[a + b*x]^2)
/(2*(b*c - a*d)^4*g^2*i^3) - (b^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^3*g^2*i^3*(a + b*x)) - (d
*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^2*g^2*i^3*(c + d*x)^2) - (2*b*d*(A + B*Log[(e*(a + b*x))
/(c + d*x)]))/((b*c - a*d)^3*g^2*i^3*(c + d*x)) - (3*b^2*d*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/
((b*c - a*d)^4*g^2*i^3) - (3*b^2*B*d*Log[c + d*x])/(2*(b*c - a*d)^4*g^2*i^3) - (3*b^2*B*d*Log[-((d*(a + b*x))/
(b*c - a*d))]*Log[c + d*x])/((b*c - a*d)^4*g^2*i^3) + (3*b^2*d*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*
x])/((b*c - a*d)^4*g^2*i^3) + (3*b^2*B*d*Log[c + d*x]^2)/(2*(b*c - a*d)^4*g^2*i^3) - (3*b^2*B*d*Log[a + b*x]*L
og[(b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^4*g^2*i^3) - (3*b^2*B*d*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/
((b*c - a*d)^4*g^2*i^3) - (3*b^2*B*d*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^4*g^2*i^3)

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 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 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{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(52 c+52 d x)^3 (a g+b g x)^2} \, dx &=\int \left (\frac{b^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^3 g^2 (a+b x)^2}-\frac{3 b^3 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^2 g^2 (c+d x)^3}+\frac{b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{70304 (b c-a d)^3 g^2 (c+d x)^2}+\frac{3 b^2 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2 (c+d x)}\right ) \, dx\\ &=-\frac{\left (3 b^3 d\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{140608 (b c-a d)^4 g^2}+\frac{\left (3 b^2 d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{140608 (b c-a d)^4 g^2}+\frac{b^3 \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{140608 (b c-a d)^3 g^2}+\frac{\left (b d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{70304 (b c-a d)^3 g^2}+\frac{d^2 \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^3} \, dx}{140608 (b c-a d)^2 g^2}\\ &=-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^3 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{281216 (b c-a d)^2 g^2 (c+d x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{70304 (b c-a d)^3 g^2 (c+d x)}-\frac{3 b^2 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2}+\frac{3 b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac{\left (3 b^2 B d\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}{140608 (b c-a d)^4 g^2}-\frac{\left (3 b^2 B d\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}{140608 (b c-a d)^4 g^2}+\frac{\left (b^2 B\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{140608 (b c-a d)^3 g^2}+\frac{(b B d) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{70304 (b c-a d)^3 g^2}+\frac{(B d) \int \frac{b c-a d}{(a+b x) (c+d x)^3} \, dx}{281216 (b c-a d)^2 g^2}\\ &=-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^3 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{281216 (b c-a d)^2 g^2 (c+d x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{70304 (b c-a d)^3 g^2 (c+d x)}-\frac{3 b^2 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2}+\frac{3 b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac{\left (b^2 B\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{140608 (b c-a d)^2 g^2}+\frac{(b B d) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{70304 (b c-a d)^2 g^2}+\frac{(B d) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{281216 (b c-a d) g^2}+\frac{\left (3 b^2 B d\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}{140608 (b c-a d)^4 e g^2}-\frac{\left (3 b^2 B d\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}{140608 (b c-a d)^4 e g^2}\\ &=-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^3 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{281216 (b c-a d)^2 g^2 (c+d x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{70304 (b c-a d)^3 g^2 (c+d x)}-\frac{3 b^2 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2}+\frac{3 b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac{\left (b^2 B\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}{140608 (b c-a d)^2 g^2}+\frac{(b B d) \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}{70304 (b c-a d)^2 g^2}+\frac{(B d) \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}{281216 (b c-a d) g^2}+\frac{\left (3 b^2 B d\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}{140608 (b c-a d)^4 e g^2}-\frac{\left (3 b^2 B d\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}{140608 (b c-a d)^4 e g^2}\\ &=-\frac{b^2 B}{140608 (b c-a d)^3 g^2 (a+b x)}+\frac{B d}{562432 (b c-a d)^2 g^2 (c+d x)^2}+\frac{5 b B d}{281216 (b c-a d)^3 g^2 (c+d x)}+\frac{3 b^2 B d \log (a+b x)}{281216 (b c-a d)^4 g^2}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^3 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{281216 (b c-a d)^2 g^2 (c+d x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{70304 (b c-a d)^3 g^2 (c+d x)}-\frac{3 b^2 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2}-\frac{3 b^2 B d \log (c+d x)}{281216 (b c-a d)^4 g^2}+\frac{3 b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac{\left (3 b^3 B d\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{140608 (b c-a d)^4 g^2}-\frac{\left (3 b^3 B d\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{140608 (b c-a d)^4 g^2}-\frac{\left (3 b^2 B d^2\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{140608 (b c-a d)^4 g^2}+\frac{\left (3 b^2 B d^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{140608 (b c-a d)^4 g^2}\\ &=-\frac{b^2 B}{140608 (b c-a d)^3 g^2 (a+b x)}+\frac{B d}{562432 (b c-a d)^2 g^2 (c+d x)^2}+\frac{5 b B d}{281216 (b c-a d)^3 g^2 (c+d x)}+\frac{3 b^2 B d \log (a+b x)}{281216 (b c-a d)^4 g^2}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^3 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{281216 (b c-a d)^2 g^2 (c+d x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{70304 (b c-a d)^3 g^2 (c+d x)}-\frac{3 b^2 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2}-\frac{3 b^2 B d \log (c+d x)}{281216 (b c-a d)^4 g^2}-\frac{3 b^2 B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac{3 b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}-\frac{3 b^2 B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{140608 (b c-a d)^4 g^2}+\frac{\left (3 b^2 B d\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{140608 (b c-a d)^4 g^2}+\frac{\left (3 b^2 B d\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{140608 (b c-a d)^4 g^2}+\frac{\left (3 b^3 B d\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{140608 (b c-a d)^4 g^2}+\frac{\left (3 b^2 B d^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{140608 (b c-a d)^4 g^2}\\ &=-\frac{b^2 B}{140608 (b c-a d)^3 g^2 (a+b x)}+\frac{B d}{562432 (b c-a d)^2 g^2 (c+d x)^2}+\frac{5 b B d}{281216 (b c-a d)^3 g^2 (c+d x)}+\frac{3 b^2 B d \log (a+b x)}{281216 (b c-a d)^4 g^2}+\frac{3 b^2 B d \log ^2(a+b x)}{281216 (b c-a d)^4 g^2}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^3 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{281216 (b c-a d)^2 g^2 (c+d x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{70304 (b c-a d)^3 g^2 (c+d x)}-\frac{3 b^2 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2}-\frac{3 b^2 B d \log (c+d x)}{281216 (b c-a d)^4 g^2}-\frac{3 b^2 B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac{3 b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac{3 b^2 B d \log ^2(c+d x)}{281216 (b c-a d)^4 g^2}-\frac{3 b^2 B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{140608 (b c-a d)^4 g^2}+\frac{\left (3 b^2 B 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 )}{140608 (b c-a d)^4 g^2}+\frac{\left (3 b^2 B 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 )}{140608 (b c-a d)^4 g^2}\\ &=-\frac{b^2 B}{140608 (b c-a d)^3 g^2 (a+b x)}+\frac{B d}{562432 (b c-a d)^2 g^2 (c+d x)^2}+\frac{5 b B d}{281216 (b c-a d)^3 g^2 (c+d x)}+\frac{3 b^2 B d \log (a+b x)}{281216 (b c-a d)^4 g^2}+\frac{3 b^2 B d \log ^2(a+b x)}{281216 (b c-a d)^4 g^2}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^3 g^2 (a+b x)}-\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{281216 (b c-a d)^2 g^2 (c+d x)^2}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{70304 (b c-a d)^3 g^2 (c+d x)}-\frac{3 b^2 d \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{140608 (b c-a d)^4 g^2}-\frac{3 b^2 B d \log (c+d x)}{281216 (b c-a d)^4 g^2}-\frac{3 b^2 B d \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac{3 b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{140608 (b c-a d)^4 g^2}+\frac{3 b^2 B d \log ^2(c+d x)}{281216 (b c-a d)^4 g^2}-\frac{3 b^2 B d \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{140608 (b c-a d)^4 g^2}-\frac{3 b^2 B d \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{140608 (b c-a d)^4 g^2}-\frac{3 b^2 B d \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{140608 (b c-a d)^4 g^2}\\ \end{align*}

Mathematica [C]  time = 0.787865, size = 452, normalized size = 1.24 \[ \frac{6 b^2 B d \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-6 b^2 B d \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-12 b^2 d \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-\frac{4 b^2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+12 b^2 d \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-\frac{8 b d (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-\frac{2 d (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{(c+d x)^2}-\frac{4 b^3 B c}{a+b x}+\frac{4 a b^2 B d}{a+b x}+6 b^2 B d \log (a+b x)-\frac{8 a b B d^2}{c+d x}+\frac{2 b B d (b c-a d)}{c+d x}+\frac{B d (b c-a d)^2}{(c+d x)^2}+\frac{8 b^2 B c d}{c+d x}-6 b^2 B d \log (c+d x)}{4 g^2 i^3 (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-4*b^3*B*c)/(a + b*x) + (4*a*b^2*B*d)/(a + b*x) + (B*d*(b*c - a*d)^2)/(c + d*x)^2 + (8*b^2*B*c*d)/(c + d*x)
- (8*a*b*B*d^2)/(c + d*x) + (2*b*B*d*(b*c - a*d))/(c + d*x) + 6*b^2*B*d*Log[a + b*x] - (4*b^2*(b*c - a*d)*(A +
B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x) - (2*d*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*
x)^2 - (8*b*d*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x) - 12*b^2*d*Log[a + b*x]*(A + B*Log[(
e*(a + b*x))/(c + d*x)]) - 6*b^2*B*d*Log[c + d*x] + 12*b^2*d*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x]
+ 6*b^2*B*d*(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)]) - 6*b^2*B*d*((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)]))/(4*(b*c - a*d)^4*g^2*i^3)

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Maple [B]  time = 0.053, size = 1729, normalized size = 4.7 \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)))/(b*g*x+a*g)^2/(d*i*x+c*i)^3,x)

[Out]

-5/2*d/i^3/(a*d-b*c)^5/g^2*A*b^3*c+11/4*d/i^3/(a*d-b*c)^5/g^2*B*b^3*c+5/2*d^2/i^3/(a*d-b*c)^5/g^2*A*b^2*a-11/4
*d^2/i^3/(a*d-b*c)^5/g^2*B*b^2*a-4*d^2/i^3/(a*d-b*c)^5/g^2*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^2/(d*x+c)*a*c-e
*d/i^3/(a*d-b*c)^5/g^2*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a+3/2*d^3/i^3
/(a*d-b*c)^5/g^2*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^2/i^3/(a*d-b*c)^5/g^2*B*ln(b*e/d+(a
*d-b*c)*e/d/(d*x+c))/(d*x+c)^2*a*b^2*c^2-3/2*d^2/i^3/(a*d-b*c)^5/g^2*A/(d*x+c)^2*b^2*c^2*a-3/4*d^3/i^3/(a*d-b*
c)^5/g^2*B/(d*x+c)^2*a^2*b*c+3/2*d^3/i^3/(a*d-b*c)^5/g^2*A/(d*x+c)^2*a^2*b*c+3/4*d^2/i^3/(a*d-b*c)^5/g^2*B/(d*
x+c)^2*a*b^2*c^2+5*d^2/i^3/(a*d-b*c)^5/g^2*B*b^2/(d*x+c)*a*c+e/i^3/(a*d-b*c)^5/g^2*B*b^4/(b*e/d+e/(d*x+c)*a-e/
d/(d*x+c)*b*c)*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c-e*d/i^3/(a*d-b*c)^5/g^2*A*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*
b*c)*a-e*d/i^3/(a*d-b*c)^5/g^2*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*a+2*d^3/i^3/(a*d-b*c)^5/g^2*B*ln(b*e/
d+(a*d-b*c)*e/d/(d*x+c))*b/(d*x+c)*a^2+2*d/i^3/(a*d-b*c)^5/g^2*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^3/(d*x+c)*c
^2+1/2*d/i^3/(a*d-b*c)^5/g^2*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)^2*b^3*c^3+2*d^3/i^3/(a*d-b*c)^5/g^2*A*b
/(d*x+c)*a^2-1/4*d/i^3/(a*d-b*c)^5/g^2*B/(d*x+c)^2*b^3*c^3+e/i^3/(a*d-b*c)^5/g^2*A*b^4/(b*e/d+e/(d*x+c)*a-e/d/
(d*x+c)*b*c)*c+e/i^3/(a*d-b*c)^5/g^2*B*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*c+2*d/i^3/(a*d-b*c)^5/g^2*A*b^3
/(d*x+c)*c^2+1/2*d/i^3/(a*d-b*c)^5/g^2*A/(d*x+c)^2*b^3*c^3-5/2*d^3/i^3/(a*d-b*c)^5/g^2*B*b/(d*x+c)*a^2-5/2*d/i
^3/(a*d-b*c)^5/g^2*B*b^3/(d*x+c)*c^2+3/2*d/i^3/(a*d-b*c)^5/g^2*B*b^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*c+1/4*d
^4/i^3/(a*d-b*c)^5/g^2*B/(d*x+c)^2*a^3-1/2*d^4/i^3/(a*d-b*c)^5/g^2*A/(d*x+c)^2*a^3-4*d^2/i^3/(a*d-b*c)^5/g^2*A
*b^2/(d*x+c)*a*c-3/2*d^2/i^3/(a*d-b*c)^5/g^2*B*b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*a-3*d^2/i^3/(a*d-b*c)^5/g
^2*A*b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a-1/2*d^4/i^3/(a*d-b*c)^5/g^2*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+
c)^2*a^3+5/2*d^2/i^3/(a*d-b*c)^5/g^2*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^2*a-5/2*d/i^3/(a*d-b*c)^5/g^2*B*ln(b*
e/d+(a*d-b*c)*e/d/(d*x+c))*b^3*c+3*d/i^3/(a*d-b*c)^5/g^2*A*b^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c

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Maxima [B]  time = 1.90353, size = 2323, normalized size = 6.36 \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)))/(b*g*x+a*g)^2/(d*i*x+c*i)^3,x, algorithm="maxima")

[Out]

-1/2*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 + 5*a*b*c*d - a^2*d^2 + 3*(3*b^2*c*d + a*b*d^2)*x)/((b^4*c^3*d^2 - 3*a*b^3*
c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*g^2*i^3*x^3 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3*
b*c*d^4 - a^4*d^5)*g^2*i^3*x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*g
^2*i^3*x + (a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3)*g^2*i^3) + 6*b^2*d*log(b*x + a)/((b^4
*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3) - 6*b^2*d*log(d*x + c)/((b^4*c^4
- 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3))*log(b*e*x/(d*x + c) + a*e/(d*x + c))
- 1/2*A*((6*b^2*d^2*x^2 + 2*b^2*c^2 + 5*a*b*c*d - a^2*d^2 + 3*(3*b^2*c*d + a*b*d^2)*x)/((b^4*c^3*d^2 - 3*a*b^3
*c^2*d^3 + 3*a^2*b^2*c*d^4 - a^3*b*d^5)*g^2*i^3*x^3 + (2*b^4*c^4*d - 5*a*b^3*c^3*d^2 + 3*a^2*b^2*c^2*d^3 + a^3
*b*c*d^4 - a^4*d^5)*g^2*i^3*x^2 + (b^4*c^5 - a*b^3*c^4*d - 3*a^2*b^2*c^3*d^2 + 5*a^3*b*c^2*d^3 - 2*a^4*c*d^4)*
g^2*i^3*x + (a*b^3*c^5 - 3*a^2*b^2*c^4*d + 3*a^3*b*c^3*d^2 - a^4*c^2*d^3)*g^2*i^3) + 6*b^2*d*log(b*x + a)/((b^
4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3) - 6*b^2*d*log(d*x + c)/((b^4*c^4
- 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*g^2*i^3)) - 1/4*(4*b^3*c^3 - 15*a*b^2*c^2*d +
12*a^2*b*c*d^2 - a^3*d^3 - 6*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2
*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x + a)^2 - 6*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^
2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(d*x + c)^2 - 3*(b^3*c^2*d + 2*a*b^2*c*d^2 - 3*a^2*b*d^3)*x - 6
*(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x + a) + 6*
(b^3*d^3*x^3 + a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x + 2*(b^3*d^3*x^3 +
a*b^2*c^2*d + (2*b^3*c*d^2 + a*b^2*d^3)*x^2 + (b^3*c^2*d + 2*a*b^2*c*d^2)*x)*log(b*x + a))*log(d*x + c))*B/(a*
b^4*c^6*g^2*i^3 - 4*a^2*b^3*c^5*d*g^2*i^3 + 6*a^3*b^2*c^4*d^2*g^2*i^3 - 4*a^4*b*c^3*d^3*g^2*i^3 + a^5*c^2*d^4*
g^2*i^3 + (b^5*c^4*d^2*g^2*i^3 - 4*a*b^4*c^3*d^3*g^2*i^3 + 6*a^2*b^3*c^2*d^4*g^2*i^3 - 4*a^3*b^2*c*d^5*g^2*i^3
+ a^4*b*d^6*g^2*i^3)*x^3 + (2*b^5*c^5*d*g^2*i^3 - 7*a*b^4*c^4*d^2*g^2*i^3 + 8*a^2*b^3*c^3*d^3*g^2*i^3 - 2*a^3
*b^2*c^2*d^4*g^2*i^3 - 2*a^4*b*c*d^5*g^2*i^3 + a^5*d^6*g^2*i^3)*x^2 + (b^5*c^6*g^2*i^3 - 2*a*b^4*c^5*d*g^2*i^3
- 2*a^2*b^3*c^4*d^2*g^2*i^3 + 8*a^3*b^2*c^3*d^3*g^2*i^3 - 7*a^4*b*c^2*d^4*g^2*i^3 + 2*a^5*c*d^5*g^2*i^3)*x)

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Fricas [A]  time = 0.567175, size = 1368, normalized size = 3.75 \begin{align*} -\frac{4 \,{\left (A + B\right )} b^{3} c^{3} + 3 \,{\left (2 \, A - 5 \, B\right )} a b^{2} c^{2} d - 12 \,{\left (A - B\right )} a^{2} b c d^{2} +{\left (2 \, A - B\right )} a^{3} d^{3} + 6 \,{\left ({\left (2 \, A - B\right )} b^{3} c d^{2} -{\left (2 \, A - B\right )} a b^{2} d^{3}\right )} x^{2} + 6 \,{\left (B b^{3} d^{3} x^{3} + B a b^{2} c^{2} d +{\left (2 \, B b^{3} c d^{2} + B a b^{2} d^{3}\right )} x^{2} +{\left (B b^{3} c^{2} d + 2 \, B a b^{2} c d^{2}\right )} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} + 3 \,{\left ({\left (6 \, A - B\right )} b^{3} c^{2} d - 2 \,{\left (2 \, A + B\right )} a b^{2} c d^{2} -{\left (2 \, A - 3 \, B\right )} a^{2} b d^{3}\right )} x + 2 \,{\left (3 \,{\left (2 \, A - B\right )} b^{3} d^{3} x^{3} + 2 \, B b^{3} c^{3} + 6 \, A a b^{2} c^{2} d - 6 \, B a^{2} b c d^{2} + B a^{3} d^{3} + 3 \,{\left (4 \, A b^{3} c d^{2} +{\left (2 \, A - 3 \, B\right )} a b^{2} d^{3}\right )} x^{2} + 3 \,{\left (2 \,{\left (A + B\right )} b^{3} c^{2} d + 4 \,{\left (A - B\right )} a b^{2} c d^{2} - B a^{2} b d^{3}\right )} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{4 \,{\left ({\left (b^{5} c^{4} d^{2} - 4 \, a b^{4} c^{3} d^{3} + 6 \, a^{2} b^{3} c^{2} d^{4} - 4 \, a^{3} b^{2} c d^{5} + a^{4} b d^{6}\right )} g^{2} i^{3} x^{3} +{\left (2 \, b^{5} c^{5} d - 7 \, a b^{4} c^{4} d^{2} + 8 \, a^{2} b^{3} c^{3} d^{3} - 2 \, a^{3} b^{2} c^{2} d^{4} - 2 \, a^{4} b c d^{5} + a^{5} d^{6}\right )} g^{2} i^{3} x^{2} +{\left (b^{5} c^{6} - 2 \, a b^{4} c^{5} d - 2 \, a^{2} b^{3} c^{4} d^{2} + 8 \, a^{3} b^{2} c^{3} d^{3} - 7 \, a^{4} b c^{2} d^{4} + 2 \, a^{5} c d^{5}\right )} g^{2} i^{3} x +{\left (a b^{4} c^{6} - 4 \, a^{2} b^{3} c^{5} d + 6 \, a^{3} b^{2} c^{4} d^{2} - 4 \, a^{4} b c^{3} d^{3} + a^{5} c^{2} d^{4}\right )} g^{2} 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)))/(b*g*x+a*g)^2/(d*i*x+c*i)^3,x, algorithm="fricas")

[Out]

-1/4*(4*(A + B)*b^3*c^3 + 3*(2*A - 5*B)*a*b^2*c^2*d - 12*(A - B)*a^2*b*c*d^2 + (2*A - B)*a^3*d^3 + 6*((2*A - B
)*b^3*c*d^2 - (2*A - B)*a*b^2*d^3)*x^2 + 6*(B*b^3*d^3*x^3 + B*a*b^2*c^2*d + (2*B*b^3*c*d^2 + B*a*b^2*d^3)*x^2
+ (B*b^3*c^2*d + 2*B*a*b^2*c*d^2)*x)*log((b*e*x + a*e)/(d*x + c))^2 + 3*((6*A - B)*b^3*c^2*d - 2*(2*A + B)*a*b
^2*c*d^2 - (2*A - 3*B)*a^2*b*d^3)*x + 2*(3*(2*A - B)*b^3*d^3*x^3 + 2*B*b^3*c^3 + 6*A*a*b^2*c^2*d - 6*B*a^2*b*c
*d^2 + B*a^3*d^3 + 3*(4*A*b^3*c*d^2 + (2*A - 3*B)*a*b^2*d^3)*x^2 + 3*(2*(A + B)*b^3*c^2*d + 4*(A - B)*a*b^2*c*
d^2 - B*a^2*b*d^3)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^5*c^4*d^2 - 4*a*b^4*c^3*d^3 + 6*a^2*b^3*c^2*d^4 - 4*a^
3*b^2*c*d^5 + a^4*b*d^6)*g^2*i^3*x^3 + (2*b^5*c^5*d - 7*a*b^4*c^4*d^2 + 8*a^2*b^3*c^3*d^3 - 2*a^3*b^2*c^2*d^4
- 2*a^4*b*c*d^5 + a^5*d^6)*g^2*i^3*x^2 + (b^5*c^6 - 2*a*b^4*c^5*d - 2*a^2*b^3*c^4*d^2 + 8*a^3*b^2*c^3*d^3 - 7*
a^4*b*c^2*d^4 + 2*a^5*c*d^5)*g^2*i^3*x + (a*b^4*c^6 - 4*a^2*b^3*c^5*d + 6*a^3*b^2*c^4*d^2 - 4*a^4*b*c^3*d^3 +
a^5*c^2*d^4)*g^2*i^3)

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Sympy [B]  time = 33.0362, size = 1562, normalized size = 4.28 \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)))/(b*g*x+a*g)**2/(d*i*x+c*i)**3,x)

[Out]

-3*B*b**2*d*log(e*(a + b*x)/(c + d*x))**2/(2*a**4*d**4*g**2*i**3 - 8*a**3*b*c*d**3*g**2*i**3 + 12*a**2*b**2*c*
*2*d**2*g**2*i**3 - 8*a*b**3*c**3*d*g**2*i**3 + 2*b**4*c**4*g**2*i**3) + 3*b**2*d*(2*A - B)*log(x + (6*A*a*b**
2*d**2 + 6*A*b**3*c*d - 3*B*a*b**2*d**2 - 3*B*b**3*c*d - 3*a**5*b**2*d**6*(2*A - B)/(a*d - b*c)**4 + 15*a**4*b
**3*c*d**5*(2*A - B)/(a*d - b*c)**4 - 30*a**3*b**4*c**2*d**4*(2*A - B)/(a*d - b*c)**4 + 30*a**2*b**5*c**3*d**3
*(2*A - B)/(a*d - b*c)**4 - 15*a*b**6*c**4*d**2*(2*A - B)/(a*d - b*c)**4 + 3*b**7*c**5*d*(2*A - B)/(a*d - b*c)
**4)/(12*A*b**3*d**2 - 6*B*b**3*d**2))/(2*g**2*i**3*(a*d - b*c)**4) - 3*b**2*d*(2*A - B)*log(x + (6*A*a*b**2*d
**2 + 6*A*b**3*c*d - 3*B*a*b**2*d**2 - 3*B*b**3*c*d + 3*a**5*b**2*d**6*(2*A - B)/(a*d - b*c)**4 - 15*a**4*b**3
*c*d**5*(2*A - B)/(a*d - b*c)**4 + 30*a**3*b**4*c**2*d**4*(2*A - B)/(a*d - b*c)**4 - 30*a**2*b**5*c**3*d**3*(2
*A - B)/(a*d - b*c)**4 + 15*a*b**6*c**4*d**2*(2*A - B)/(a*d - b*c)**4 - 3*b**7*c**5*d*(2*A - B)/(a*d - b*c)**4
)/(12*A*b**3*d**2 - 6*B*b**3*d**2))/(2*g**2*i**3*(a*d - b*c)**4) + (-B*a**2*d**2 + 5*B*a*b*c*d + 3*B*a*b*d**2*
x + 2*B*b**2*c**2 + 9*B*b**2*c*d*x + 6*B*b**2*d**2*x**2)*log(e*(a + b*x)/(c + d*x))/(2*a**4*c**2*d**3*g**2*i**
3 + 4*a**4*c*d**4*g**2*i**3*x + 2*a**4*d**5*g**2*i**3*x**2 - 6*a**3*b*c**3*d**2*g**2*i**3 - 10*a**3*b*c**2*d**
3*g**2*i**3*x - 2*a**3*b*c*d**4*g**2*i**3*x**2 + 2*a**3*b*d**5*g**2*i**3*x**3 + 6*a**2*b**2*c**4*d*g**2*i**3 +
6*a**2*b**2*c**3*d**2*g**2*i**3*x - 6*a**2*b**2*c**2*d**3*g**2*i**3*x**2 - 6*a**2*b**2*c*d**4*g**2*i**3*x**3
- 2*a*b**3*c**5*g**2*i**3 + 2*a*b**3*c**4*d*g**2*i**3*x + 10*a*b**3*c**3*d**2*g**2*i**3*x**2 + 6*a*b**3*c**2*d
**3*g**2*i**3*x**3 - 2*b**4*c**5*g**2*i**3*x - 4*b**4*c**4*d*g**2*i**3*x**2 - 2*b**4*c**3*d**2*g**2*i**3*x**3)
+ (-2*A*a**2*d**2 + 10*A*a*b*c*d + 4*A*b**2*c**2 + B*a**2*d**2 - 11*B*a*b*c*d + 4*B*b**2*c**2 + x**2*(12*A*b*
*2*d**2 - 6*B*b**2*d**2) + x*(6*A*a*b*d**2 + 18*A*b**2*c*d - 9*B*a*b*d**2 - 3*B*b**2*c*d))/(4*a**4*c**2*d**3*g
**2*i**3 - 12*a**3*b*c**3*d**2*g**2*i**3 + 12*a**2*b**2*c**4*d*g**2*i**3 - 4*a*b**3*c**5*g**2*i**3 + x**3*(4*a
**3*b*d**5*g**2*i**3 - 12*a**2*b**2*c*d**4*g**2*i**3 + 12*a*b**3*c**2*d**3*g**2*i**3 - 4*b**4*c**3*d**2*g**2*i
**3) + x**2*(4*a**4*d**5*g**2*i**3 - 4*a**3*b*c*d**4*g**2*i**3 - 12*a**2*b**2*c**2*d**3*g**2*i**3 + 20*a*b**3*
c**3*d**2*g**2*i**3 - 8*b**4*c**4*d*g**2*i**3) + x*(8*a**4*c*d**4*g**2*i**3 - 20*a**3*b*c**2*d**3*g**2*i**3 +
12*a**2*b**2*c**3*d**2*g**2*i**3 + 4*a*b**3*c**4*d*g**2*i**3 - 4*b**4*c**5*g**2*i**3))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{B \log \left (\frac{{\left (b x + a\right )} e}{d x + c}\right ) + A}{{\left (b g x + a g\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)))/(b*g*x+a*g)^2/(d*i*x+c*i)^3,x, algorithm="giac")

[Out]

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