3.38 \(\int \frac{A+B \log (\frac{e (a+b x)}{c+d x})}{(a g+b g x)^4 (c i+d i x)} \, dx\)

Optimal. Leaf size=373 \[ \frac{3 b^2 d (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^4 i (a+b x)^2 (b c-a d)^4}-\frac{b^3 (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 g^4 i (a+b x)^3 (b c-a d)^4}-\frac{d^3 \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i (b c-a d)^4}-\frac{3 b d^2 (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i (a+b x) (b c-a d)^4}+\frac{3 b^2 B d (c+d x)^2}{4 g^4 i (a+b x)^2 (b c-a d)^4}-\frac{b^3 B (c+d x)^3}{9 g^4 i (a+b x)^3 (b c-a d)^4}-\frac{3 b B d^2 (c+d x)}{g^4 i (a+b x) (b c-a d)^4}+\frac{B d^3 \log ^2\left (\frac{a+b x}{c+d x}\right )}{2 g^4 i (b c-a d)^4} \]

[Out]

(-3*b*B*d^2*(c + d*x))/((b*c - a*d)^4*g^4*i*(a + b*x)) + (3*b^2*B*d*(c + d*x)^2)/(4*(b*c - a*d)^4*g^4*i*(a + b
*x)^2) - (b^3*B*(c + d*x)^3)/(9*(b*c - a*d)^4*g^4*i*(a + b*x)^3) + (B*d^3*Log[(a + b*x)/(c + d*x)]^2)/(2*(b*c
- a*d)^4*g^4*i) - (3*b*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^4*i*(a + b*x)) + (
3*b^2*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^4*g^4*i*(a + b*x)^2) - (b^3*(c + d*x)
^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^4*g^4*i*(a + b*x)^3) - (d^3*Log[(a + b*x)/(c + d*x)]*(
A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^4*i)

________________________________________________________________________________________

Rubi [C]  time = 1.07811, antiderivative size = 620, normalized size of antiderivative = 1.66, 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} \[ -\frac{B d^3 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g^4 i (b c-a d)^4}-\frac{B d^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g^4 i (b c-a d)^4}-\frac{d^3 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i (b c-a d)^4}+\frac{d^3 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i (b c-a d)^4}-\frac{d^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i (a+b x) (b c-a d)^3}+\frac{d \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^4 i (a+b x)^2 (b c-a d)^2}-\frac{B \log \left (\frac{e (a+b x)}{c+d x}\right )+A}{3 g^4 i (a+b x)^3 (b c-a d)}-\frac{11 B d^2}{6 g^4 i (a+b x) (b c-a d)^3}+\frac{B d^3 \log ^2(a+b x)}{2 g^4 i (b c-a d)^4}+\frac{B d^3 \log ^2(c+d x)}{2 g^4 i (b c-a d)^4}-\frac{11 B d^3 \log (a+b x)}{6 g^4 i (b c-a d)^4}-\frac{B d^3 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g^4 i (b c-a d)^4}+\frac{11 B d^3 \log (c+d x)}{6 g^4 i (b c-a d)^4}-\frac{B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g^4 i (b c-a d)^4}+\frac{5 B d}{12 g^4 i (a+b x)^2 (b c-a d)^2}-\frac{B}{9 g^4 i (a+b x)^3 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-B/(9*(b*c - a*d)*g^4*i*(a + b*x)^3) + (5*B*d)/(12*(b*c - a*d)^2*g^4*i*(a + b*x)^2) - (11*B*d^2)/(6*(b*c - a*d
)^3*g^4*i*(a + b*x)) - (11*B*d^3*Log[a + b*x])/(6*(b*c - a*d)^4*g^4*i) + (B*d^3*Log[a + b*x]^2)/(2*(b*c - a*d)
^4*g^4*i) - (A + B*Log[(e*(a + b*x))/(c + d*x)])/(3*(b*c - a*d)*g^4*i*(a + b*x)^3) + (d*(A + B*Log[(e*(a + b*x
))/(c + d*x)]))/(2*(b*c - a*d)^2*g^4*i*(a + b*x)^2) - (d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^
3*g^4*i*(a + b*x)) - (d^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^4*i) + (11*B*d^3
*Log[c + d*x])/(6*(b*c - a*d)^4*g^4*i) - (B*d^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/((b*c - a*d)^4
*g^4*i) + (d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/((b*c - a*d)^4*g^4*i) + (B*d^3*Log[c + d*x]^
2)/(2*(b*c - a*d)^4*g^4*i) - (B*d^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^4*g^4*i) - (B*d^
3*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^4*g^4*i) - (B*d^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d
)])/((b*c - a*d)^4*g^4*i)

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 )}{(38 c+38 d x) (a g+b g x)^4} \, dx &=\int \left (\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d) g^4 (a+b x)^4}-\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^2 g^4 (a+b x)^3}+\frac{b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^3 g^4 (a+b x)^2}-\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4 (a+b x)}+\frac{d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4 (c+d x)}\right ) \, dx\\ &=-\frac{\left (b d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{38 (b c-a d)^4 g^4}+\frac{d^4 \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{38 (b c-a d)^4 g^4}+\frac{\left (b d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{38 (b c-a d)^3 g^4}-\frac{(b d) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{38 (b c-a d)^2 g^4}+\frac{b \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{38 (b c-a d) g^4}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{114 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{76 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac{\left (B d^3\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}{38 (b c-a d)^4 g^4}-\frac{\left (B d^3\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}{38 (b c-a d)^4 g^4}+\frac{\left (B d^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{38 (b c-a d)^3 g^4}-\frac{(B d) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{76 (b c-a d)^2 g^4}+\frac{B \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{114 (b c-a d) g^4}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{114 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{76 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac{B \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{114 g^4}+\frac{\left (B d^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{38 (b c-a d)^2 g^4}-\frac{(B d) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{76 (b c-a d) g^4}+\frac{\left (B d^3\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}{38 (b c-a d)^4 e g^4}-\frac{\left (B d^3\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}{38 (b c-a d)^4 e g^4}\\ &=-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{114 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{76 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac{B \int \left (\frac{b}{(b c-a d) (a+b x)^4}-\frac{b d}{(b c-a d)^2 (a+b x)^3}+\frac{b d^2}{(b c-a d)^3 (a+b x)^2}-\frac{b d^3}{(b c-a d)^4 (a+b x)}+\frac{d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{114 g^4}+\frac{\left (B d^2\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}{38 (b c-a d)^2 g^4}-\frac{(B d) \int \left (\frac{b}{(b c-a d) (a+b x)^3}-\frac{b d}{(b c-a d)^2 (a+b x)^2}+\frac{b d^2}{(b c-a d)^3 (a+b x)}-\frac{d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{76 (b c-a d) g^4}+\frac{\left (B d^3\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}{38 (b c-a d)^4 e g^4}-\frac{\left (B d^3\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}{38 (b c-a d)^4 e g^4}\\ &=-\frac{B}{342 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d}{456 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2}{228 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 \log (a+b x)}{228 (b c-a d)^4 g^4}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{114 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{76 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4}+\frac{11 B d^3 \log (c+d x)}{228 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac{\left (b B d^3\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{38 (b c-a d)^4 g^4}-\frac{\left (b B d^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{38 (b c-a d)^4 g^4}-\frac{\left (B d^4\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{38 (b c-a d)^4 g^4}+\frac{\left (B d^4\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{38 (b c-a d)^4 g^4}\\ &=-\frac{B}{342 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d}{456 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2}{228 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 \log (a+b x)}{228 (b c-a d)^4 g^4}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{114 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{76 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4}+\frac{11 B d^3 \log (c+d x)}{228 (b c-a d)^4 g^4}-\frac{B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}-\frac{B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{38 (b c-a d)^4 g^4}+\frac{\left (B d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{38 (b c-a d)^4 g^4}+\frac{\left (B d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{38 (b c-a d)^4 g^4}+\frac{\left (b B d^3\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{38 (b c-a d)^4 g^4}+\frac{\left (B d^4\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{38 (b c-a d)^4 g^4}\\ &=-\frac{B}{342 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d}{456 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2}{228 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 \log (a+b x)}{228 (b c-a d)^4 g^4}+\frac{B d^3 \log ^2(a+b x)}{76 (b c-a d)^4 g^4}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{114 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{76 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4}+\frac{11 B d^3 \log (c+d x)}{228 (b c-a d)^4 g^4}-\frac{B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac{B d^3 \log ^2(c+d x)}{76 (b c-a d)^4 g^4}-\frac{B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{38 (b c-a d)^4 g^4}+\frac{\left (B d^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{38 (b c-a d)^4 g^4}+\frac{\left (B d^3\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{38 (b c-a d)^4 g^4}\\ &=-\frac{B}{342 (b c-a d) g^4 (a+b x)^3}+\frac{5 B d}{456 (b c-a d)^2 g^4 (a+b x)^2}-\frac{11 B d^2}{228 (b c-a d)^3 g^4 (a+b x)}-\frac{11 B d^3 \log (a+b x)}{228 (b c-a d)^4 g^4}+\frac{B d^3 \log ^2(a+b x)}{76 (b c-a d)^4 g^4}-\frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{114 (b c-a d) g^4 (a+b x)^3}+\frac{d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{76 (b c-a d)^2 g^4 (a+b x)^2}-\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^3 g^4 (a+b x)}-\frac{d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4}+\frac{11 B d^3 \log (c+d x)}{228 (b c-a d)^4 g^4}-\frac{B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac{B d^3 \log ^2(c+d x)}{76 (b c-a d)^4 g^4}-\frac{B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{38 (b c-a d)^4 g^4}-\frac{B d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{38 (b c-a d)^4 g^4}-\frac{B d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{38 (b c-a d)^4 g^4}\\ \end{align*}

Mathematica [C]  time = 0.724539, size = 492, normalized size = 1.32 \[ \frac{-36 B d^3 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )-36 B d^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\frac{36 A d^2 (a d-b c)}{a+b x}+\frac{18 A d (b c-a d)^2}{(a+b x)^2}-\frac{12 A (b c-a d)^3}{(a+b x)^3}-36 A d^3 \log (a+b x)-36 B d^3 \log (a+b x) \log \left (\frac{e (a+b x)}{c+d x}\right )+36 B d^3 \log (c+d x) \log \left (\frac{e (a+b x)}{c+d x}\right )+\frac{36 B d^2 (a d-b c) \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x}+\frac{66 B d^2 (a d-b c)}{a+b x}-36 B d^3 \log (c+d x) \log \left (\frac{d (a+b x)}{a d-b c}\right )-36 B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )+\frac{18 B d (b c-a d)^2 \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2}-\frac{12 B (b c-a d)^3 \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3}+\frac{15 B d (b c-a d)^2}{(a+b x)^2}-\frac{4 B (b c-a d)^3}{(a+b x)^3}+18 B d^3 \log ^2(a+b x)-66 B d^3 \log (a+b x)+36 A d^3 \log (c+d x)+18 B d^3 \log ^2(c+d x)+66 B d^3 \log (c+d x)}{36 g^4 i (b c-a d)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-12*A*(b*c - a*d)^3)/(a + b*x)^3 - (4*B*(b*c - a*d)^3)/(a + b*x)^3 + (18*A*d*(b*c - a*d)^2)/(a + b*x)^2 + (1
5*B*d*(b*c - a*d)^2)/(a + b*x)^2 + (36*A*d^2*(-(b*c) + a*d))/(a + b*x) + (66*B*d^2*(-(b*c) + a*d))/(a + b*x) -
36*A*d^3*Log[a + b*x] - 66*B*d^3*Log[a + b*x] + 18*B*d^3*Log[a + b*x]^2 - (12*B*(b*c - a*d)^3*Log[(e*(a + b*x
))/(c + d*x)])/(a + b*x)^3 + (18*B*d*(b*c - a*d)^2*Log[(e*(a + b*x))/(c + d*x)])/(a + b*x)^2 + (36*B*d^2*(-(b*
c) + a*d)*Log[(e*(a + b*x))/(c + d*x)])/(a + b*x) - 36*B*d^3*Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)] + 36*A*
d^3*Log[c + d*x] + 66*B*d^3*Log[c + d*x] - 36*B*d^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] + 36*B*d^3*
Log[(e*(a + b*x))/(c + d*x)]*Log[c + d*x] + 18*B*d^3*Log[c + d*x]^2 - 36*B*d^3*Log[a + b*x]*Log[(b*(c + d*x))/
(b*c - a*d)] - 36*B*d^3*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] - 36*B*d^3*PolyLog[2, (b*(c + d*x))/(b*c - a*
d)])/(36*(b*c - a*d)^4*g^4*i)

________________________________________________________________________________________

Maple [B]  time = 0.053, size = 1474, normalized size = 4. \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)^4/(d*i*x+c*i),x)

[Out]

-d^4/i/(a*d-b*c)^5/g^4*A*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a+d^3/i/(a*d-b*c)^5/g^4*A*ln(b*e/d+(a*d-b*c)*e/d/(d*x
+c))*b*c-3*e*d^3/i/(a*d-b*c)^5/g^4*A*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*a+3*e*d^2/i/(a*d-b*c)^5/g^4*A*b^2/(
b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*c+3/2*e^2*d^2/i/(a*d-b*c)^5/g^4*A*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2
*a-3/2*e^2*d/i/(a*d-b*c)^5/g^4*A*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*c-1/3*e^3*d/i/(a*d-b*c)^5/g^4*A*b^3
/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*a+1/3*e^3/i/(a*d-b*c)^5/g^4*A*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3
*c-1/2*d^4/i/(a*d-b*c)^5/g^4*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*a+1/2*d^3/i/(a*d-b*c)^5/g^4*B*ln(b*e/d+(a*d-b
*c)*e/d/(d*x+c))^2*b*c-3*e*d^3/i/(a*d-b*c)^5/g^4*B*b/(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*e*d^2/i/(a*d-b*c)^5/g^4*B*b^2/(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-3*e*d^3/i/(a*d-b*c)^5/g^4*B*b/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*a+3*e*d^2/i/(a*d-b*c)^5/g^4*B*b^2/(b*e/
d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*c+3/2*e^2*d^2/i/(a*d-b*c)^5/g^4*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*ln(
b*e/d+(a*d-b*c)*e/d/(d*x+c))*a-3/2*e^2*d/i/(a*d-b*c)^5/g^4*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*ln(b*e/
d+(a*d-b*c)*e/d/(d*x+c))*c+3/4*e^2*d^2/i/(a*d-b*c)^5/g^4*B*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*a-3/4*e^2
*d/i/(a*d-b*c)^5/g^4*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*c-1/3*e^3*d/i/(a*d-b*c)^5/g^4*B*b^3/(b*e/d+e/
(d*x+c)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a+1/3*e^3/i/(a*d-b*c)^5/g^4*B*b^4/(b*e/d+e/(d*x+c
)*a-e/d/(d*x+c)*b*c)^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c-1/9*e^3*d/i/(a*d-b*c)^5/g^4*B*b^3/(b*e/d+e/(d*x+c)*a-
e/d/(d*x+c)*b*c)^3*a+1/9*e^3/i/(a*d-b*c)^5/g^4*B*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*c

________________________________________________________________________________________

Maxima [B]  time = 1.97217, size = 1983, normalized size = 5.32 \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)^4/(d*i*x+c*i),x, algorithm="maxima")

[Out]

-1/6*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^3 - 3*a*b^5*c
^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4*i*x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d
^3)*g^4*i*x^2 + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*g^4*i*x + (a^3*b^3*c^3 - 3*a^4
*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*g^4*i) + 6*d^3*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^4*i) - 6*d^3*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^4*i))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 1/6*A*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c
*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4*i*
x^3 + 3*(a*b^5*c^3 - 3*a^2*b^4*c^2*d + 3*a^3*b^3*c*d^2 - a^4*b^2*d^3)*g^4*i*x^2 + 3*(a^2*b^4*c^3 - 3*a^3*b^3*c
^2*d + 3*a^4*b^2*c*d^2 - a^5*b*d^3)*g^4*i*x + (a^3*b^3*c^3 - 3*a^4*b^2*c^2*d + 3*a^5*b*c*d^2 - a^6*d^3)*g^4*i)
+ 6*d^3*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^4*i) - 6*d^3*
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^4*i)) - 1/36*(4*b^3*c^
3 - 27*a*b^2*c^2*d + 108*a^2*b*c*d^2 - 85*a^3*d^3 + 66*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 18*(b^3*d^3*x^3 + 3*a*b^2
*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a)^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d
^3)*log(d*x + c)^2 - 3*(5*b^3*c^2*d - 54*a*b^2*c*d^2 + 49*a^2*b*d^3)*x + 66*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3
*a^2*b*d^3*x + a^3*d^3)*log(b*x + a) - 6*(11*b^3*d^3*x^3 + 33*a*b^2*d^3*x^2 + 33*a^2*b*d^3*x + 11*a^3*d^3 - 6*
(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a))*log(d*x + c))*B/(a^3*b^4*c^4*g^4*i - 4
*a^4*b^3*c^3*d*g^4*i + 6*a^5*b^2*c^2*d^2*g^4*i - 4*a^6*b*c*d^3*g^4*i + a^7*d^4*g^4*i + (b^7*c^4*g^4*i - 4*a*b^
6*c^3*d*g^4*i + 6*a^2*b^5*c^2*d^2*g^4*i - 4*a^3*b^4*c*d^3*g^4*i + a^4*b^3*d^4*g^4*i)*x^3 + 3*(a*b^6*c^4*g^4*i
- 4*a^2*b^5*c^3*d*g^4*i + 6*a^3*b^4*c^2*d^2*g^4*i - 4*a^4*b^3*c*d^3*g^4*i + a^5*b^2*d^4*g^4*i)*x^2 + 3*(a^2*b^
5*c^4*g^4*i - 4*a^3*b^4*c^3*d*g^4*i + 6*a^4*b^3*c^2*d^2*g^4*i - 4*a^5*b^2*c*d^3*g^4*i + a^6*b*d^4*g^4*i)*x)

________________________________________________________________________________________

Fricas [A]  time = 0.558678, size = 1285, normalized size = 3.45 \begin{align*} -\frac{4 \,{\left (3 \, A + B\right )} b^{3} c^{3} - 27 \,{\left (2 \, A + B\right )} a b^{2} c^{2} d + 108 \,{\left (A + B\right )} a^{2} b c d^{2} -{\left (66 \, A + 85 \, B\right )} a^{3} d^{3} + 6 \,{\left ({\left (6 \, A + 11 \, B\right )} b^{3} c d^{2} -{\left (6 \, A + 11 \, B\right )} a b^{2} d^{3}\right )} x^{2} + 18 \,{\left (B b^{3} d^{3} x^{3} + 3 \, B a b^{2} d^{3} x^{2} + 3 \, B a^{2} b d^{3} x + B a^{3} d^{3}\right )} \log \left (\frac{b e x + a e}{d x + c}\right )^{2} - 3 \,{\left ({\left (6 \, A + 5 \, B\right )} b^{3} c^{2} d - 18 \,{\left (2 \, A + 3 \, B\right )} a b^{2} c d^{2} +{\left (30 \, A + 49 \, B\right )} a^{2} b d^{3}\right )} x + 6 \,{\left ({\left (6 \, A + 11 \, B\right )} b^{3} d^{3} x^{3} + 2 \, B b^{3} c^{3} - 9 \, B a b^{2} c^{2} d + 18 \, B a^{2} b c d^{2} + 6 \, A a^{3} d^{3} + 3 \,{\left (2 \, B b^{3} c d^{2} + 3 \,{\left (2 \, A + 3 \, B\right )} a b^{2} d^{3}\right )} x^{2} - 3 \,{\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} - 6 \,{\left (A + B\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac{b e x + a e}{d x + c}\right )}{36 \,{\left ({\left (b^{7} c^{4} - 4 \, a b^{6} c^{3} d + 6 \, a^{2} b^{5} c^{2} d^{2} - 4 \, a^{3} b^{4} c d^{3} + a^{4} b^{3} d^{4}\right )} g^{4} i x^{3} + 3 \,{\left (a b^{6} c^{4} - 4 \, a^{2} b^{5} c^{3} d + 6 \, a^{3} b^{4} c^{2} d^{2} - 4 \, a^{4} b^{3} c d^{3} + a^{5} b^{2} d^{4}\right )} g^{4} i x^{2} + 3 \,{\left (a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}\right )} g^{4} i x +{\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )} g^{4} i\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)^4/(d*i*x+c*i),x, algorithm="fricas")

[Out]

-1/36*(4*(3*A + B)*b^3*c^3 - 27*(2*A + B)*a*b^2*c^2*d + 108*(A + B)*a^2*b*c*d^2 - (66*A + 85*B)*a^3*d^3 + 6*((
6*A + 11*B)*b^3*c*d^2 - (6*A + 11*B)*a*b^2*d^3)*x^2 + 18*(B*b^3*d^3*x^3 + 3*B*a*b^2*d^3*x^2 + 3*B*a^2*b*d^3*x
+ B*a^3*d^3)*log((b*e*x + a*e)/(d*x + c))^2 - 3*((6*A + 5*B)*b^3*c^2*d - 18*(2*A + 3*B)*a*b^2*c*d^2 + (30*A +
49*B)*a^2*b*d^3)*x + 6*((6*A + 11*B)*b^3*d^3*x^3 + 2*B*b^3*c^3 - 9*B*a*b^2*c^2*d + 18*B*a^2*b*c*d^2 + 6*A*a^3*
d^3 + 3*(2*B*b^3*c*d^2 + 3*(2*A + 3*B)*a*b^2*d^3)*x^2 - 3*(B*b^3*c^2*d - 6*B*a*b^2*c*d^2 - 6*(A + B)*a^2*b*d^3
)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d
^4)*g^4*i*x^3 + 3*(a*b^6*c^4 - 4*a^2*b^5*c^3*d + 6*a^3*b^4*c^2*d^2 - 4*a^4*b^3*c*d^3 + a^5*b^2*d^4)*g^4*i*x^2
+ 3*(a^2*b^5*c^4 - 4*a^3*b^4*c^3*d + 6*a^4*b^3*c^2*d^2 - 4*a^5*b^2*c*d^3 + a^6*b*d^4)*g^4*i*x + (a^3*b^4*c^4 -
4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4*a^6*b*c*d^3 + a^7*d^4)*g^4*i)

________________________________________________________________________________________

Sympy [B]  time = 22.1794, size = 1392, normalized size = 3.73 \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)**4/(d*i*x+c*i),x)

[Out]

-B*d**3*log(e*(a + b*x)/(c + d*x))**2/(2*a**4*d**4*g**4*i - 8*a**3*b*c*d**3*g**4*i + 12*a**2*b**2*c**2*d**2*g*
*4*i - 8*a*b**3*c**3*d*g**4*i + 2*b**4*c**4*g**4*i) + d**3*(6*A + 11*B)*log(x + (6*A*a*d**4 + 6*A*b*c*d**3 + 1
1*B*a*d**4 + 11*B*b*c*d**3 - a**5*d**8*(6*A + 11*B)/(a*d - b*c)**4 + 5*a**4*b*c*d**7*(6*A + 11*B)/(a*d - b*c)*
*4 - 10*a**3*b**2*c**2*d**6*(6*A + 11*B)/(a*d - b*c)**4 + 10*a**2*b**3*c**3*d**5*(6*A + 11*B)/(a*d - b*c)**4 -
5*a*b**4*c**4*d**4*(6*A + 11*B)/(a*d - b*c)**4 + b**5*c**5*d**3*(6*A + 11*B)/(a*d - b*c)**4)/(12*A*b*d**4 + 2
2*B*b*d**4))/(6*g**4*i*(a*d - b*c)**4) - d**3*(6*A + 11*B)*log(x + (6*A*a*d**4 + 6*A*b*c*d**3 + 11*B*a*d**4 +
11*B*b*c*d**3 + a**5*d**8*(6*A + 11*B)/(a*d - b*c)**4 - 5*a**4*b*c*d**7*(6*A + 11*B)/(a*d - b*c)**4 + 10*a**3*
b**2*c**2*d**6*(6*A + 11*B)/(a*d - b*c)**4 - 10*a**2*b**3*c**3*d**5*(6*A + 11*B)/(a*d - b*c)**4 + 5*a*b**4*c**
4*d**4*(6*A + 11*B)/(a*d - b*c)**4 - b**5*c**5*d**3*(6*A + 11*B)/(a*d - b*c)**4)/(12*A*b*d**4 + 22*B*b*d**4))/
(6*g**4*i*(a*d - b*c)**4) + (11*B*a**2*d**2 - 7*B*a*b*c*d + 15*B*a*b*d**2*x + 2*B*b**2*c**2 - 3*B*b**2*c*d*x +
6*B*b**2*d**2*x**2)*log(e*(a + b*x)/(c + d*x))/(6*a**6*d**3*g**4*i - 18*a**5*b*c*d**2*g**4*i + 18*a**5*b*d**3
*g**4*i*x + 18*a**4*b**2*c**2*d*g**4*i - 54*a**4*b**2*c*d**2*g**4*i*x + 18*a**4*b**2*d**3*g**4*i*x**2 - 6*a**3
*b**3*c**3*g**4*i + 54*a**3*b**3*c**2*d*g**4*i*x - 54*a**3*b**3*c*d**2*g**4*i*x**2 + 6*a**3*b**3*d**3*g**4*i*x
**3 - 18*a**2*b**4*c**3*g**4*i*x + 54*a**2*b**4*c**2*d*g**4*i*x**2 - 18*a**2*b**4*c*d**2*g**4*i*x**3 - 18*a*b*
*5*c**3*g**4*i*x**2 + 18*a*b**5*c**2*d*g**4*i*x**3 - 6*b**6*c**3*g**4*i*x**3) + (66*A*a**2*d**2 - 42*A*a*b*c*d
+ 12*A*b**2*c**2 + 85*B*a**2*d**2 - 23*B*a*b*c*d + 4*B*b**2*c**2 + x**2*(36*A*b**2*d**2 + 66*B*b**2*d**2) + x
*(90*A*a*b*d**2 - 18*A*b**2*c*d + 147*B*a*b*d**2 - 15*B*b**2*c*d))/(36*a**6*d**3*g**4*i - 108*a**5*b*c*d**2*g*
*4*i + 108*a**4*b**2*c**2*d*g**4*i - 36*a**3*b**3*c**3*g**4*i + x**3*(36*a**3*b**3*d**3*g**4*i - 108*a**2*b**4
*c*d**2*g**4*i + 108*a*b**5*c**2*d*g**4*i - 36*b**6*c**3*g**4*i) + x**2*(108*a**4*b**2*d**3*g**4*i - 324*a**3*
b**3*c*d**2*g**4*i + 324*a**2*b**4*c**2*d*g**4*i - 108*a*b**5*c**3*g**4*i) + x*(108*a**5*b*d**3*g**4*i - 324*a
**4*b**2*c*d**2*g**4*i + 324*a**3*b**3*c**2*d*g**4*i - 108*a**2*b**4*c**3*g**4*i))

________________________________________________________________________________________

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 )}^{4}{\left (d i x + c i\right )}}\,{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)^4/(d*i*x+c*i),x, algorithm="giac")

[Out]

integrate((B*log((b*x + a)*e/(d*x + c)) + A)/((b*g*x + a*g)^4*(d*i*x + c*i)), x)