3.46 \(\int \frac{A+B \log (\frac{e (a+b x)}{c+d x})}{(a g+b g x)^4 (c i+d i x)^2} \, dx\)

Optimal. Leaf size=457 \[ -\frac{6 b^2 d^2 (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (a+b x) (b c-a d)^5}-\frac{b^4 (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 g^4 i^2 (a+b x)^3 (b c-a d)^5}+\frac{2 b^3 d (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (a+b x)^2 (b c-a d)^5}-\frac{4 b d^3 \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (b c-a d)^5}+\frac{d^4 (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (c+d x) (b c-a d)^5}-\frac{6 b^2 B d^2 (c+d x)}{g^4 i^2 (a+b x) (b c-a d)^5}-\frac{b^4 B (c+d x)^3}{9 g^4 i^2 (a+b x)^3 (b c-a d)^5}+\frac{b^3 B d (c+d x)^2}{g^4 i^2 (a+b x)^2 (b c-a d)^5}-\frac{B d^4 (a+b x)}{g^4 i^2 (c+d x) (b c-a d)^5}+\frac{2 b B d^3 \log ^2\left (\frac{a+b x}{c+d x}\right )}{g^4 i^2 (b c-a d)^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 1.36281, antiderivative size = 705, normalized size of antiderivative = 1.54, number of steps used = 36, 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} \[ -\frac{4 b B d^3 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g^4 i^2 (b c-a d)^5}-\frac{4 b B d^3 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g^4 i^2 (b c-a d)^5}-\frac{4 b d^3 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (b c-a d)^5}-\frac{d^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (c+d x) (b c-a d)^4}+\frac{4 b d^3 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (b c-a d)^5}-\frac{3 b d^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (a+b x) (b c-a d)^4}+\frac{b d \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (a+b x)^2 (b c-a d)^3}-\frac{b \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 g^4 i^2 (a+b x)^3 (b c-a d)^2}+\frac{B d^3}{g^4 i^2 (c+d x) (b c-a d)^4}-\frac{13 b B d^2}{3 g^4 i^2 (a+b x) (b c-a d)^4}+\frac{2 b B d^3 \log ^2(a+b x)}{g^4 i^2 (b c-a d)^5}+\frac{2 b B d^3 \log ^2(c+d x)}{g^4 i^2 (b c-a d)^5}-\frac{10 b B d^3 \log (a+b x)}{3 g^4 i^2 (b c-a d)^5}-\frac{4 b B d^3 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g^4 i^2 (b c-a d)^5}+\frac{10 b B d^3 \log (c+d x)}{3 g^4 i^2 (b c-a d)^5}-\frac{4 b B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g^4 i^2 (b c-a d)^5}+\frac{2 b B d}{3 g^4 i^2 (a+b x)^2 (b c-a d)^3}-\frac{b B}{9 g^4 i^2 (a+b x)^3 (b c-a d)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b*B)/(9*(b*c - a*d)^2*g^4*i^2*(a + b*x)^3) + (2*b*B*d)/(3*(b*c - a*d)^3*g^4*i^2*(a + b*x)^2) - (13*b*B*d^2)/
(3*(b*c - a*d)^4*g^4*i^2*(a + b*x)) + (B*d^3)/((b*c - a*d)^4*g^4*i^2*(c + d*x)) - (10*b*B*d^3*Log[a + b*x])/(3
*(b*c - a*d)^5*g^4*i^2) + (2*b*B*d^3*Log[a + b*x]^2)/((b*c - a*d)^5*g^4*i^2) - (b*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]))/(3*(b*c - a*d)^2*g^4*i^2*(a + b*x)^3) + (b*d*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^3*g^
4*i^2*(a + b*x)^2) - (3*b*d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^4*i^2*(a + b*x)) - (d^3*(
A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^4*i^2*(c + d*x)) - (4*b*d^3*Log[a + b*x]*(A + B*Log[(e*(
a + b*x))/(c + d*x)]))/((b*c - a*d)^5*g^4*i^2) + (10*b*B*d^3*Log[c + d*x])/(3*(b*c - a*d)^5*g^4*i^2) - (4*b*B*
d^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/((b*c - a*d)^5*g^4*i^2) + (4*b*d^3*(A + B*Log[(e*(a + b*x)
)/(c + d*x)])*Log[c + d*x])/((b*c - a*d)^5*g^4*i^2) + (2*b*B*d^3*Log[c + d*x]^2)/((b*c - a*d)^5*g^4*i^2) - (4*
b*B*d^3*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^5*g^4*i^2) - (4*b*B*d^3*PolyLog[2, -((d*(a +
b*x))/(b*c - a*d))])/((b*c - a*d)^5*g^4*i^2) - (4*b*B*d^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)
^5*g^4*i^2)

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 )}{(46 c+46 d x)^2 (a g+b g x)^4} \, dx &=\int \left (\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^2 g^4 (a+b x)^4}-\frac{b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{1058 (b c-a d)^3 g^4 (a+b x)^3}+\frac{3 b^2 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)^2}-\frac{b^2 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4 (a+b x)}+\frac{d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)^2}+\frac{b d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4 (c+d x)}\right ) \, dx\\ &=-\frac{\left (b^2 d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{529 (b c-a d)^5 g^4}+\frac{\left (b d^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{529 (b c-a d)^5 g^4}+\frac{\left (3 b^2 d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{2116 (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)^2} \, dx}{2116 (b c-a d)^4 g^4}-\frac{\left (b^2 d\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{1058 (b c-a d)^3 g^4}+\frac{b^2 \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{2116 (b c-a d)^2 g^4}\\ &=-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac{3 b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac{b d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac{\left (b 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}{529 (b c-a d)^5 g^4}-\frac{\left (b 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}{529 (b c-a d)^5 g^4}+\frac{\left (3 b B d^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{2116 (b c-a d)^4 g^4}+\frac{\left (B d^3\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{2116 (b c-a d)^4 g^4}-\frac{(b B d) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2116 (b c-a d)^3 g^4}+\frac{(b B) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{6348 (b c-a d)^2 g^4}\\ &=-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac{3 b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac{b d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac{\left (3 b B d^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{2116 (b c-a d)^3 g^4}+\frac{\left (B d^3\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{2116 (b c-a d)^3 g^4}-\frac{(b B d) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{2116 (b c-a d)^2 g^4}+\frac{(b B) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{6348 (b c-a d) g^4}+\frac{\left (b 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}{529 (b c-a d)^5 e g^4}-\frac{\left (b 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}{529 (b c-a d)^5 e g^4}\\ &=-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac{3 b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac{b d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac{\left (3 b 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}{2116 (b c-a d)^3 g^4}+\frac{\left (B d^3\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}{2116 (b c-a d)^3 g^4}-\frac{(b 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}{2116 (b c-a d)^2 g^4}+\frac{(b 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}{6348 (b c-a d) g^4}+\frac{\left (b 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}{529 (b c-a d)^5 e g^4}-\frac{\left (b 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}{529 (b c-a d)^5 e g^4}\\ &=-\frac{b B}{19044 (b c-a d)^2 g^4 (a+b x)^3}+\frac{b B d}{3174 (b c-a d)^3 g^4 (a+b x)^2}-\frac{13 b B d^2}{6348 (b c-a d)^4 g^4 (a+b x)}+\frac{B d^3}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac{5 b B d^3 \log (a+b x)}{3174 (b c-a d)^5 g^4}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac{3 b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac{b d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac{5 b B d^3 \log (c+d x)}{3174 (b c-a d)^5 g^4}+\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac{\left (b^2 B d^3\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{529 (b c-a d)^5 g^4}-\frac{\left (b^2 B d^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{529 (b c-a d)^5 g^4}-\frac{\left (b B d^4\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{529 (b c-a d)^5 g^4}+\frac{\left (b B d^4\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{529 (b c-a d)^5 g^4}\\ &=-\frac{b B}{19044 (b c-a d)^2 g^4 (a+b x)^3}+\frac{b B d}{3174 (b c-a d)^3 g^4 (a+b x)^2}-\frac{13 b B d^2}{6348 (b c-a d)^4 g^4 (a+b x)}+\frac{B d^3}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac{5 b B d^3 \log (a+b x)}{3174 (b c-a d)^5 g^4}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac{3 b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac{b d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac{5 b B d^3 \log (c+d x)}{3174 (b c-a d)^5 g^4}-\frac{b B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}-\frac{b B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{529 (b c-a d)^5 g^4}+\frac{\left (b B d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{529 (b c-a d)^5 g^4}+\frac{\left (b B d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{529 (b c-a d)^5 g^4}+\frac{\left (b^2 B d^3\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{529 (b c-a d)^5 g^4}+\frac{\left (b B d^4\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{529 (b c-a d)^5 g^4}\\ &=-\frac{b B}{19044 (b c-a d)^2 g^4 (a+b x)^3}+\frac{b B d}{3174 (b c-a d)^3 g^4 (a+b x)^2}-\frac{13 b B d^2}{6348 (b c-a d)^4 g^4 (a+b x)}+\frac{B d^3}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac{5 b B d^3 \log (a+b x)}{3174 (b c-a d)^5 g^4}+\frac{b B d^3 \log ^2(a+b x)}{1058 (b c-a d)^5 g^4}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac{3 b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac{b d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac{5 b B d^3 \log (c+d x)}{3174 (b c-a d)^5 g^4}-\frac{b B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac{b B d^3 \log ^2(c+d x)}{1058 (b c-a d)^5 g^4}-\frac{b B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{529 (b c-a d)^5 g^4}+\frac{\left (b 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 )}{529 (b c-a d)^5 g^4}+\frac{\left (b 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 )}{529 (b c-a d)^5 g^4}\\ &=-\frac{b B}{19044 (b c-a d)^2 g^4 (a+b x)^3}+\frac{b B d}{3174 (b c-a d)^3 g^4 (a+b x)^2}-\frac{13 b B d^2}{6348 (b c-a d)^4 g^4 (a+b x)}+\frac{B d^3}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac{5 b B d^3 \log (a+b x)}{3174 (b c-a d)^5 g^4}+\frac{b B d^3 \log ^2(a+b x)}{1058 (b c-a d)^5 g^4}-\frac{b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac{b d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac{3 b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac{b d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac{5 b B d^3 \log (c+d x)}{3174 (b c-a d)^5 g^4}-\frac{b B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac{b B d^3 \log ^2(c+d x)}{1058 (b c-a d)^5 g^4}-\frac{b B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{529 (b c-a d)^5 g^4}-\frac{b B d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{529 (b c-a d)^5 g^4}-\frac{b B d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{529 (b c-a d)^5 g^4}\\ \end{align*}

Mathematica [C]  time = 1.42348, size = 520, normalized size = 1.14 \[ -\frac{-18 b B d^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+18 b B d^3 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+36 b d^3 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-\frac{9 d^3 (a d-b c) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-36 b d^3 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+\frac{27 b d^2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac{9 b d (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{(a+b x)^2}+\frac{3 b (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{(a+b x)^3}+\frac{27 b^2 B c d^2}{a+b x}+\frac{12 b B d^2 (b c-a d)}{a+b x}-\frac{6 b B d (b c-a d)^2}{(a+b x)^2}+\frac{b B (b c-a d)^3}{(a+b x)^3}-\frac{27 a b B d^3}{a+b x}+30 b B d^3 \log (a+b x)+\frac{9 a B d^4}{c+d x}-\frac{9 b B c d^3}{c+d x}-30 b B d^3 \log (c+d x)}{9 g^4 i^2 (b c-a d)^5} \]

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)^2),x]

[Out]

-((b*B*(b*c - a*d)^3)/(a + b*x)^3 - (6*b*B*d*(b*c - a*d)^2)/(a + b*x)^2 + (27*b^2*B*c*d^2)/(a + b*x) - (27*a*b
*B*d^3)/(a + b*x) + (12*b*B*d^2*(b*c - a*d))/(a + b*x) - (9*b*B*c*d^3)/(c + d*x) + (9*a*B*d^4)/(c + d*x) + 30*
b*B*d^3*Log[a + b*x] + (3*b*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x)^3 - (9*b*d*(b*c - a*
d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x)^2 + (27*b*d^2*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d
*x)]))/(a + b*x) - (9*d^3*(-(b*c) + a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x) + 36*b*d^3*Log[a + b*
x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 30*b*B*d^3*Log[c + d*x] - 36*b*d^3*(A + B*Log[(e*(a + b*x))/(c + d*x
)])*Log[c + d*x] - 18*b*B*d^3*(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)]) + 18*b*B*d^3*((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)]))/(9*(b*c - a*d)^5*g^4*i^2)

________________________________________________________________________________________

Maple [B]  time = 0.055, size = 2068, normalized size = 4.5 \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)))/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x)

[Out]

-2*e^2*d^2/i^2/(a*d-b*c)^6/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))*a+2
*e^2*d/i^2/(a*d-b*c)^6/g^4*B*b^4/(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-6*e*d
^2/i^2/(a*d-b*c)^6/g^4*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))*c+d^4/i^2/(a*
d-b*c)^6/g^4*B*b*a-d^3/i^2/(a*d-b*c)^6/g^4*B*b^2*c-d^4/i^2/(a*d-b*c)^6/g^4*A*b*a+d^3/i^2/(a*d-b*c)^6/g^4*A*b^2
*c-d^3/i^2/(a*d-b*c)^6/g^4*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)*b^2*c^2-1/3*e^3/i^2/(a*d-b*c)^6/g^4*B*b^5
/(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-2*e^2*d^2/i^2/(a*d-b*c)^6/g^4*A*b^3/(
b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*a+2*d^4/i^2/(a*d-b*c)^6/g^4*A/(d*x+c)*a*b*c-d^5/i^2/(a*d-b*c)^6/g^4*A/(d*
x+c)*a^2+d^5/i^2/(a*d-b*c)^6/g^4*B/(d*x+c)*a^2+6*e*d^3/i^2/(a*d-b*c)^6/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))*a+1/3*e^3*d/i^2/(a*d-b*c)^6/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))*a+1/9*e^3*d/i^2/(a*d-b*c)^6/g^4*B*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)
^3*a-4*d^3/i^2/(a*d-b*c)^6/g^4*A*b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c+2*d^4/i^2/(a*d-b*c)^6/g^4*B*b*ln(b*e/d+
(a*d-b*c)*e/d/(d*x+c))^2*a-2*d^3/i^2/(a*d-b*c)^6/g^4*B*b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*c-d^5/i^2/(a*d-b*
c)^6/g^4*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)*a^2-d^4/i^2/(a*d-b*c)^6/g^4*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c
))*b*a+d^3/i^2/(a*d-b*c)^6/g^4*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^2*c+4*d^4/i^2/(a*d-b*c)^6/g^4*A*b*ln(b*e/d+
(a*d-b*c)*e/d/(d*x+c))*a+d^3/i^2/(a*d-b*c)^6/g^4*B/(d*x+c)*b^2*c^2-1/9*e^3/i^2/(a*d-b*c)^6/g^4*B*b^5/(b*e/d+e/
(d*x+c)*a-e/d/(d*x+c)*b*c)^3*c-1/3*e^3/i^2/(a*d-b*c)^6/g^4*A*b^5/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*c-d^3/i
^2/(a*d-b*c)^6/g^4*A/(d*x+c)*b^2*c^2+6*e*d^3/i^2/(a*d-b*c)^6/g^4*A*b^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*a-2
*d^4/i^2/(a*d-b*c)^6/g^4*B/(d*x+c)*a*b*c-6*e*d^2/i^2/(a*d-b*c)^6/g^4*A*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)
*c-e^2*d^2/i^2/(a*d-b*c)^6/g^4*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*a+e^2*d/i^2/(a*d-b*c)^6/g^4*B*b^4/(
b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*c+2*e^2*d/i^2/(a*d-b*c)^6/g^4*A*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2
*c+1/3*e^3*d/i^2/(a*d-b*c)^6/g^4*A*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*a+6*e*d^3/i^2/(a*d-b*c)^6/g^4*B*b
^2/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*a-6*e*d^2/i^2/(a*d-b*c)^6/g^4*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)
*c+2*d^4/i^2/(a*d-b*c)^6/g^4*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)*a*b*c

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Maxima [B]  time = 2.42841, size = 3456, normalized size = 7.56 \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)^2,x, algorithm="maxima")

[Out]

-1/3*B*((12*b^3*d^3*x^3 + b^3*c^3 - 5*a*b^2*c^2*d + 13*a^2*b*c*d^2 + 3*a^3*d^3 + 6*(b^3*c*d^2 + 5*a*b^2*d^3)*x
^2 - 2*(b^3*c^2*d - 8*a*b^2*c*d^2 - 11*a^2*b*d^3)*x)/((b^7*c^4*d - 4*a*b^6*c^3*d^2 + 6*a^2*b^5*c^2*d^3 - 4*a^3
*b^4*c*d^4 + a^4*b^3*d^5)*g^4*i^2*x^4 + (b^7*c^5 - a*b^6*c^4*d - 6*a^2*b^5*c^3*d^2 + 14*a^3*b^4*c^2*d^3 - 11*a
^4*b^3*c*d^4 + 3*a^5*b^2*d^5)*g^4*i^2*x^3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^4*d + 2*a^3*b^4*c^3*d^2 + 2*a^4*b^3*c^2
*d^3 - 3*a^5*b^2*c*d^4 + a^6*b*d^5)*g^4*i^2*x^2 + (3*a^2*b^5*c^5 - 11*a^3*b^4*c^4*d + 14*a^4*b^3*c^3*d^2 - 6*a
^5*b^2*c^2*d^3 - a^6*b*c*d^4 + a^7*d^5)*g^4*i^2*x + (a^3*b^4*c^5 - 4*a^4*b^3*c^4*d + 6*a^5*b^2*c^3*d^2 - 4*a^6
*b*c^2*d^3 + a^7*c*d^4)*g^4*i^2) + 12*b*d^3*log(b*x + a)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a
^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4*i^2) - 12*b*d^3*log(d*x + c)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*
b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4*i^2))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) - 1
/3*A*((12*b^3*d^3*x^3 + b^3*c^3 - 5*a*b^2*c^2*d + 13*a^2*b*c*d^2 + 3*a^3*d^3 + 6*(b^3*c*d^2 + 5*a*b^2*d^3)*x^2
- 2*(b^3*c^2*d - 8*a*b^2*c*d^2 - 11*a^2*b*d^3)*x)/((b^7*c^4*d - 4*a*b^6*c^3*d^2 + 6*a^2*b^5*c^2*d^3 - 4*a^3*b
^4*c*d^4 + a^4*b^3*d^5)*g^4*i^2*x^4 + (b^7*c^5 - a*b^6*c^4*d - 6*a^2*b^5*c^3*d^2 + 14*a^3*b^4*c^2*d^3 - 11*a^4
*b^3*c*d^4 + 3*a^5*b^2*d^5)*g^4*i^2*x^3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^4*d + 2*a^3*b^4*c^3*d^2 + 2*a^4*b^3*c^2*d
^3 - 3*a^5*b^2*c*d^4 + a^6*b*d^5)*g^4*i^2*x^2 + (3*a^2*b^5*c^5 - 11*a^3*b^4*c^4*d + 14*a^4*b^3*c^3*d^2 - 6*a^5
*b^2*c^2*d^3 - a^6*b*c*d^4 + a^7*d^5)*g^4*i^2*x + (a^3*b^4*c^5 - 4*a^4*b^3*c^4*d + 6*a^5*b^2*c^3*d^2 - 4*a^6*b
*c^2*d^3 + a^7*c*d^4)*g^4*i^2) + 12*b*d^3*log(b*x + a)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3
*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4*i^2) - 12*b*d^3*log(d*x + c)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^
3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4*i^2)) - 1/9*(b^4*c^4 - 9*a*b^3*c^3*d + 54*a^2*b^
2*c^2*d^2 - 55*a^3*b*c*d^3 + 9*a^4*d^4 + 30*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 3*(11*b^4*c^2*d^2 + 8*a*b^3*c*d^3 -
19*a^2*b^2*d^4)*x^2 - 18*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2
*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a)^2 - 18*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a
*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(d*x + c)^2 - (5*b^4*c
^3*d - 81*a*b^3*c^2*d^2 + 57*a^2*b^2*c*d^3 + 19*a^3*b*d^4)*x + 30*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*
a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a) - 6*(5*b^4*
d^4*x^4 + 5*a^3*b*c*d^3 + 5*(b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 15*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + 5*(3*a^2*b^2*
c*d^3 + a^3*b*d^4)*x - 6*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2
*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a))*log(d*x + c))*B/(a^3*b^5*c^6*g^4*i^2 - 5*a^4*b^4*c^
5*d*g^4*i^2 + 10*a^5*b^3*c^4*d^2*g^4*i^2 - 10*a^6*b^2*c^3*d^3*g^4*i^2 + 5*a^7*b*c^2*d^4*g^4*i^2 - a^8*c*d^5*g^
4*i^2 + (b^8*c^5*d*g^4*i^2 - 5*a*b^7*c^4*d^2*g^4*i^2 + 10*a^2*b^6*c^3*d^3*g^4*i^2 - 10*a^3*b^5*c^2*d^4*g^4*i^2
+ 5*a^4*b^4*c*d^5*g^4*i^2 - a^5*b^3*d^6*g^4*i^2)*x^4 + (b^8*c^6*g^4*i^2 - 2*a*b^7*c^5*d*g^4*i^2 - 5*a^2*b^6*c
^4*d^2*g^4*i^2 + 20*a^3*b^5*c^3*d^3*g^4*i^2 - 25*a^4*b^4*c^2*d^4*g^4*i^2 + 14*a^5*b^3*c*d^5*g^4*i^2 - 3*a^6*b^
2*d^6*g^4*i^2)*x^3 + 3*(a*b^7*c^6*g^4*i^2 - 4*a^2*b^6*c^5*d*g^4*i^2 + 5*a^3*b^5*c^4*d^2*g^4*i^2 - 5*a^5*b^3*c^
2*d^4*g^4*i^2 + 4*a^6*b^2*c*d^5*g^4*i^2 - a^7*b*d^6*g^4*i^2)*x^2 + (3*a^2*b^6*c^6*g^4*i^2 - 14*a^3*b^5*c^5*d*g
^4*i^2 + 25*a^4*b^4*c^4*d^2*g^4*i^2 - 20*a^5*b^3*c^3*d^3*g^4*i^2 + 5*a^6*b^2*c^2*d^4*g^4*i^2 + 2*a^7*b*c*d^5*g
^4*i^2 - a^8*d^6*g^4*i^2)*x)

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Fricas [B]  time = 0.612324, size = 2101, normalized size = 4.6 \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)^2,x, algorithm="fricas")

[Out]

-1/9*((3*A + B)*b^4*c^4 - 9*(2*A + B)*a*b^3*c^3*d + 54*(A + B)*a^2*b^2*c^2*d^2 - 5*(6*A + 11*B)*a^3*b*c*d^3 -
9*(A - B)*a^4*d^4 + 6*((6*A + 5*B)*b^4*c*d^3 - (6*A + 5*B)*a*b^3*d^4)*x^3 + 3*((6*A + 11*B)*b^4*c^2*d^2 + 8*(3
*A + B)*a*b^3*c*d^3 - (30*A + 19*B)*a^2*b^2*d^4)*x^2 + 18*(B*b^4*d^4*x^4 + B*a^3*b*c*d^3 + (B*b^4*c*d^3 + 3*B*
a*b^3*d^4)*x^3 + 3*(B*a*b^3*c*d^3 + B*a^2*b^2*d^4)*x^2 + (3*B*a^2*b^2*c*d^3 + B*a^3*b*d^4)*x)*log((b*e*x + a*e
)/(d*x + c))^2 - ((6*A + 5*B)*b^4*c^3*d - 27*(2*A + 3*B)*a*b^3*c^2*d^2 - 3*(6*A - 19*B)*a^2*b^2*c*d^3 + (66*A
+ 19*B)*a^3*b*d^4)*x + 3*(2*(6*A + 5*B)*b^4*d^4*x^4 + B*b^4*c^4 - 6*B*a*b^3*c^3*d + 18*B*a^2*b^2*c^2*d^2 + 12*
A*a^3*b*c*d^3 - 3*B*a^4*d^4 + 2*((6*A + 11*B)*b^4*c*d^3 + 9*(2*A + B)*a*b^3*d^4)*x^3 + 6*(B*b^4*c^2*d^2 + 3*(2
*A + 3*B)*a*b^3*c*d^3 + 6*A*a^2*b^2*d^4)*x^2 - 2*(B*b^4*c^3*d - 9*B*a*b^3*c^2*d^2 - 18*(A + B)*a^2*b^2*c*d^3 -
6*(A - B)*a^3*b*d^4)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^8*c^5*d - 5*a*b^7*c^4*d^2 + 10*a^2*b^6*c^3*d^3 - 10
*a^3*b^5*c^2*d^4 + 5*a^4*b^4*c*d^5 - a^5*b^3*d^6)*g^4*i^2*x^4 + (b^8*c^6 - 2*a*b^7*c^5*d - 5*a^2*b^6*c^4*d^2 +
20*a^3*b^5*c^3*d^3 - 25*a^4*b^4*c^2*d^4 + 14*a^5*b^3*c*d^5 - 3*a^6*b^2*d^6)*g^4*i^2*x^3 + 3*(a*b^7*c^6 - 4*a^
2*b^6*c^5*d + 5*a^3*b^5*c^4*d^2 - 5*a^5*b^3*c^2*d^4 + 4*a^6*b^2*c*d^5 - a^7*b*d^6)*g^4*i^2*x^2 + (3*a^2*b^6*c^
6 - 14*a^3*b^5*c^5*d + 25*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 5*a^6*b^2*c^2*d^4 + 2*a^7*b*c*d^5 - a^8*d^6)*
g^4*i^2*x + (a^3*b^5*c^6 - 5*a^4*b^4*c^5*d + 10*a^5*b^3*c^4*d^2 - 10*a^6*b^2*c^3*d^3 + 5*a^7*b*c^2*d^4 - a^8*c
*d^5)*g^4*i^2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**4/(d*i*x+c*i)**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{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 )}^{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)))/(b*g*x+a*g)^4/(d*i*x+c*i)^2,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)^2), x)