3.54 \(\int \frac{A+B \log (\frac{e (a+b x)}{c+d x})}{(a g+b g x)^4 (c i+d i x)^3} \, dx\)

Optimal. Leaf size=563 \[ -\frac{10 b^3 d^2 (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^3 (a+b x) (b c-a d)^6}-\frac{10 b^2 d^3 \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^3 (b c-a d)^6}-\frac{b^5 (c+d x)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{3 g^4 i^3 (a+b x)^3 (b c-a d)^6}+\frac{5 b^4 d (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^4 i^3 (a+b x)^2 (b c-a d)^6}+\frac{5 b d^4 (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^3 (c+d x) (b c-a d)^6}-\frac{d^5 (a+b x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^4 i^3 (c+d x)^2 (b c-a d)^6}-\frac{10 b^3 B d^2 (c+d x)}{g^4 i^3 (a+b x) (b c-a d)^6}+\frac{5 b^2 B d^3 \log ^2\left (\frac{a+b x}{c+d x}\right )}{g^4 i^3 (b c-a d)^6}-\frac{b^5 B (c+d x)^3}{9 g^4 i^3 (a+b x)^3 (b c-a d)^6}+\frac{5 b^4 B d (c+d x)^2}{4 g^4 i^3 (a+b x)^2 (b c-a d)^6}-\frac{5 b B d^4 (a+b x)}{g^4 i^3 (c+d x) (b c-a d)^6}+\frac{B d^5 (a+b x)^2}{4 g^4 i^3 (c+d x)^2 (b c-a d)^6} \]

[Out]

(B*d^5*(a + b*x)^2)/(4*(b*c - a*d)^6*g^4*i^3*(c + d*x)^2) - (5*b*B*d^4*(a + b*x))/((b*c - a*d)^6*g^4*i^3*(c +
d*x)) - (10*b^3*B*d^2*(c + d*x))/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4*B*d*(c + d*x)^2)/(4*(b*c - a*d)^6*
g^4*i^3*(a + b*x)^2) - (b^5*B*(c + d*x)^3)/(9*(b*c - a*d)^6*g^4*i^3*(a + b*x)^3) + (5*b^2*B*d^3*Log[(a + b*x)/
(c + d*x)]^2)/((b*c - a*d)^6*g^4*i^3) - (d^5*(a + b*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^
6*g^4*i^3*(c + d*x)^2) + (5*b*d^4*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^6*g^4*i^3*(c +
d*x)) - (10*b^3*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^6*g^4*i^3*(a + b*x)) + (5*b^4
*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^6*g^4*i^3*(a + b*x)^2) - (b^5*(c + d*x)^3*
(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^6*g^4*i^3*(a + b*x)^3) - (10*b^2*d^3*Log[(a + b*x)/(c + d
*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^6*g^4*i^3)

________________________________________________________________________________________

Rubi [C]  time = 1.69534, antiderivative size = 825, normalized size of antiderivative = 1.47, number of steps used = 40, 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} \[ \frac{5 b^2 B \log ^2(a+b x) d^3}{(b c-a d)^6 g^4 i^3}+\frac{5 b^2 B \log ^2(c+d x) d^3}{(b c-a d)^6 g^4 i^3}-\frac{10 b^2 B \log (a+b x) d^3}{3 (b c-a d)^6 g^4 i^3}-\frac{10 b^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d^3}{(b c-a d)^6 g^4 i^3}-\frac{4 b \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d^3}{(b c-a d)^5 g^4 i^3 (c+d x)}-\frac{\left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d^3}{2 (b c-a d)^4 g^4 i^3 (c+d x)^2}-\frac{10 b^2 B \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x) d^3}{(b c-a d)^6 g^4 i^3}+\frac{10 b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x) d^3}{(b c-a d)^6 g^4 i^3}+\frac{10 b^2 B \log (c+d x) d^3}{3 (b c-a d)^6 g^4 i^3}-\frac{10 b^2 B \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right ) d^3}{(b c-a d)^6 g^4 i^3}-\frac{10 b^2 B \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right ) d^3}{(b c-a d)^6 g^4 i^3}-\frac{10 b^2 B \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right ) d^3}{(b c-a d)^6 g^4 i^3}+\frac{9 b B d^3}{2 (b c-a d)^5 g^4 i^3 (c+d x)}+\frac{B d^3}{4 (b c-a d)^4 g^4 i^3 (c+d x)^2}-\frac{6 b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d^2}{(b c-a d)^5 g^4 i^3 (a+b x)}-\frac{47 b^2 B d^2}{6 (b c-a d)^5 g^4 i^3 (a+b x)}+\frac{3 b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) d}{2 (b c-a d)^4 g^4 i^3 (a+b x)^2}+\frac{11 b^2 B d}{12 (b c-a d)^4 g^4 i^3 (a+b x)^2}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^3 g^4 i^3 (a+b x)^3}-\frac{b^2 B}{9 (b c-a d)^3 g^4 i^3 (a+b x)^3} \]

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)^3),x]

[Out]

-(b^2*B)/(9*(b*c - a*d)^3*g^4*i^3*(a + b*x)^3) + (11*b^2*B*d)/(12*(b*c - a*d)^4*g^4*i^3*(a + b*x)^2) - (47*b^2
*B*d^2)/(6*(b*c - a*d)^5*g^4*i^3*(a + b*x)) + (B*d^3)/(4*(b*c - a*d)^4*g^4*i^3*(c + d*x)^2) + (9*b*B*d^3)/(2*(
b*c - a*d)^5*g^4*i^3*(c + d*x)) - (10*b^2*B*d^3*Log[a + b*x])/(3*(b*c - a*d)^6*g^4*i^3) + (5*b^2*B*d^3*Log[a +
b*x]^2)/((b*c - a*d)^6*g^4*i^3) - (b^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^3*g^4*i^3*(a + b*
x)^3) + (3*b^2*d*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^4*g^4*i^3*(a + b*x)^2) - (6*b^2*d^2*(A +
B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5*g^4*i^3*(a + b*x)) - (d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]
))/(2*(b*c - a*d)^4*g^4*i^3*(c + d*x)^2) - (4*b*d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5*g^4*i
^3*(c + d*x)) - (10*b^2*d^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^6*g^4*i^3) + (10*b
^2*B*d^3*Log[c + d*x])/(3*(b*c - a*d)^6*g^4*i^3) - (10*b^2*B*d^3*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x
])/((b*c - a*d)^6*g^4*i^3) + (10*b^2*d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/((b*c - a*d)^6*g^4
*i^3) + (5*b^2*B*d^3*Log[c + d*x]^2)/((b*c - a*d)^6*g^4*i^3) - (10*b^2*B*d^3*Log[a + b*x]*Log[(b*(c + d*x))/(b
*c - a*d)])/((b*c - a*d)^6*g^4*i^3) - (10*b^2*B*d^3*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^6*g
^4*i^3) - (10*b^2*B*d^3*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^6*g^4*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 )}{(54 c+54 d x)^3 (a g+b g x)^4} \, dx &=\int \left (\frac{b^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{157464 (b c-a d)^3 g^4 (a+b x)^4}-\frac{b^3 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{52488 (b c-a d)^4 g^4 (a+b x)^3}+\frac{b^3 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{26244 (b c-a d)^5 g^4 (a+b x)^2}-\frac{5 b^3 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{78732 (b c-a d)^6 g^4 (a+b x)}+\frac{d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{157464 (b c-a d)^4 g^4 (c+d x)^3}+\frac{b d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{39366 (b c-a d)^5 g^4 (c+d x)^2}+\frac{5 b^2 d^4 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{78732 (b c-a d)^6 g^4 (c+d x)}\right ) \, dx\\ &=-\frac{\left (5 b^3 d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{78732 (b c-a d)^6 g^4}+\frac{\left (5 b^2 d^4\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{78732 (b c-a d)^6 g^4}+\frac{\left (b^3 d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{26244 (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)^2} \, dx}{39366 (b c-a d)^5 g^4}-\frac{\left (b^3 d\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{52488 (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)^3} \, dx}{157464 (b c-a d)^4 g^4}+\frac{b^3 \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{157464 (b c-a d)^3 g^4}\\ &=-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{472392 (b c-a d)^3 g^4 (a+b x)^3}+\frac{b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{104976 (b c-a d)^4 g^4 (a+b x)^2}-\frac{b^2 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{26244 (b c-a d)^5 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{314928 (b c-a d)^4 g^4 (c+d x)^2}-\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{39366 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{78732 (b c-a d)^6 g^4}+\frac{5 b^2 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{78732 (b c-a d)^6 g^4}+\frac{\left (5 b^2 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}{78732 (b c-a d)^6 g^4}-\frac{\left (5 b^2 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}{78732 (b c-a d)^6 g^4}+\frac{\left (b^2 B d^2\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{26244 (b c-a d)^5 g^4}+\frac{\left (b B d^3\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{39366 (b c-a d)^5 g^4}-\frac{\left (b^2 B d\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{104976 (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)^3} \, dx}{314928 (b c-a d)^4 g^4}+\frac{\left (b^2 B\right ) \int \frac{b c-a d}{(a+b x)^4 (c+d x)} \, dx}{472392 (b c-a d)^3 g^4}\\ &=-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{472392 (b c-a d)^3 g^4 (a+b x)^3}+\frac{b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{104976 (b c-a d)^4 g^4 (a+b x)^2}-\frac{b^2 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{26244 (b c-a d)^5 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{314928 (b c-a d)^4 g^4 (c+d x)^2}-\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{39366 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{78732 (b c-a d)^6 g^4}+\frac{5 b^2 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{78732 (b c-a d)^6 g^4}+\frac{\left (b^2 B d^2\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{26244 (b c-a d)^4 g^4}+\frac{\left (b B d^3\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{39366 (b c-a d)^4 g^4}-\frac{\left (b^2 B d\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{104976 (b c-a d)^3 g^4}+\frac{\left (B d^3\right ) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{314928 (b c-a d)^3 g^4}+\frac{\left (b^2 B\right ) \int \frac{1}{(a+b x)^4 (c+d x)} \, dx}{472392 (b c-a d)^2 g^4}+\frac{\left (5 b^2 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}{78732 (b c-a d)^6 e g^4}-\frac{\left (5 b^2 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}{78732 (b c-a d)^6 e g^4}\\ &=-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{472392 (b c-a d)^3 g^4 (a+b x)^3}+\frac{b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{104976 (b c-a d)^4 g^4 (a+b x)^2}-\frac{b^2 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{26244 (b c-a d)^5 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{314928 (b c-a d)^4 g^4 (c+d x)^2}-\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{39366 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{78732 (b c-a d)^6 g^4}+\frac{5 b^2 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{78732 (b c-a d)^6 g^4}+\frac{\left (b^2 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}{26244 (b c-a d)^4 g^4}+\frac{\left (b 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}{39366 (b c-a d)^4 g^4}-\frac{\left (b^2 B d\right ) \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}{104976 (b c-a d)^3 g^4}+\frac{\left (B d^3\right ) \int \left (\frac{b^3}{(b c-a d)^3 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^3}-\frac{b d}{(b c-a d)^2 (c+d x)^2}-\frac{b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{314928 (b c-a d)^3 g^4}+\frac{\left (b^2 B\right ) \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}{472392 (b c-a d)^2 g^4}+\frac{\left (5 b^2 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}{78732 (b c-a d)^6 e g^4}-\frac{\left (5 b^2 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}{78732 (b c-a d)^6 e g^4}\\ &=-\frac{b^2 B}{1417176 (b c-a d)^3 g^4 (a+b x)^3}+\frac{11 b^2 B d}{1889568 (b c-a d)^4 g^4 (a+b x)^2}-\frac{47 b^2 B d^2}{944784 (b c-a d)^5 g^4 (a+b x)}+\frac{B d^3}{629856 (b c-a d)^4 g^4 (c+d x)^2}+\frac{b B d^3}{34992 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 B d^3 \log (a+b x)}{236196 (b c-a d)^6 g^4}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{472392 (b c-a d)^3 g^4 (a+b x)^3}+\frac{b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{104976 (b c-a d)^4 g^4 (a+b x)^2}-\frac{b^2 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{26244 (b c-a d)^5 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{314928 (b c-a d)^4 g^4 (c+d x)^2}-\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{39366 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{78732 (b c-a d)^6 g^4}+\frac{5 b^2 B d^3 \log (c+d x)}{236196 (b c-a d)^6 g^4}+\frac{5 b^2 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{78732 (b c-a d)^6 g^4}+\frac{\left (5 b^3 B d^3\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{78732 (b c-a d)^6 g^4}-\frac{\left (5 b^3 B d^3\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{78732 (b c-a d)^6 g^4}-\frac{\left (5 b^2 B d^4\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{78732 (b c-a d)^6 g^4}+\frac{\left (5 b^2 B d^4\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{78732 (b c-a d)^6 g^4}\\ &=-\frac{b^2 B}{1417176 (b c-a d)^3 g^4 (a+b x)^3}+\frac{11 b^2 B d}{1889568 (b c-a d)^4 g^4 (a+b x)^2}-\frac{47 b^2 B d^2}{944784 (b c-a d)^5 g^4 (a+b x)}+\frac{B d^3}{629856 (b c-a d)^4 g^4 (c+d x)^2}+\frac{b B d^3}{34992 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 B d^3 \log (a+b x)}{236196 (b c-a d)^6 g^4}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{472392 (b c-a d)^3 g^4 (a+b x)^3}+\frac{b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{104976 (b c-a d)^4 g^4 (a+b x)^2}-\frac{b^2 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{26244 (b c-a d)^5 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{314928 (b c-a d)^4 g^4 (c+d x)^2}-\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{39366 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{78732 (b c-a d)^6 g^4}+\frac{5 b^2 B d^3 \log (c+d x)}{236196 (b c-a d)^6 g^4}-\frac{5 b^2 B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{78732 (b c-a d)^6 g^4}+\frac{5 b^2 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{78732 (b c-a d)^6 g^4}-\frac{5 b^2 B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{78732 (b c-a d)^6 g^4}+\frac{\left (5 b^2 B d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{78732 (b c-a d)^6 g^4}+\frac{\left (5 b^2 B d^3\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{78732 (b c-a d)^6 g^4}+\frac{\left (5 b^3 B d^3\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{78732 (b c-a d)^6 g^4}+\frac{\left (5 b^2 B d^4\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{78732 (b c-a d)^6 g^4}\\ &=-\frac{b^2 B}{1417176 (b c-a d)^3 g^4 (a+b x)^3}+\frac{11 b^2 B d}{1889568 (b c-a d)^4 g^4 (a+b x)^2}-\frac{47 b^2 B d^2}{944784 (b c-a d)^5 g^4 (a+b x)}+\frac{B d^3}{629856 (b c-a d)^4 g^4 (c+d x)^2}+\frac{b B d^3}{34992 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 B d^3 \log (a+b x)}{236196 (b c-a d)^6 g^4}+\frac{5 b^2 B d^3 \log ^2(a+b x)}{157464 (b c-a d)^6 g^4}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{472392 (b c-a d)^3 g^4 (a+b x)^3}+\frac{b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{104976 (b c-a d)^4 g^4 (a+b x)^2}-\frac{b^2 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{26244 (b c-a d)^5 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{314928 (b c-a d)^4 g^4 (c+d x)^2}-\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{39366 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{78732 (b c-a d)^6 g^4}+\frac{5 b^2 B d^3 \log (c+d x)}{236196 (b c-a d)^6 g^4}-\frac{5 b^2 B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{78732 (b c-a d)^6 g^4}+\frac{5 b^2 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{78732 (b c-a d)^6 g^4}+\frac{5 b^2 B d^3 \log ^2(c+d x)}{157464 (b c-a d)^6 g^4}-\frac{5 b^2 B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{78732 (b c-a d)^6 g^4}+\frac{\left (5 b^2 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 )}{78732 (b c-a d)^6 g^4}+\frac{\left (5 b^2 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 )}{78732 (b c-a d)^6 g^4}\\ &=-\frac{b^2 B}{1417176 (b c-a d)^3 g^4 (a+b x)^3}+\frac{11 b^2 B d}{1889568 (b c-a d)^4 g^4 (a+b x)^2}-\frac{47 b^2 B d^2}{944784 (b c-a d)^5 g^4 (a+b x)}+\frac{B d^3}{629856 (b c-a d)^4 g^4 (c+d x)^2}+\frac{b B d^3}{34992 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 B d^3 \log (a+b x)}{236196 (b c-a d)^6 g^4}+\frac{5 b^2 B d^3 \log ^2(a+b x)}{157464 (b c-a d)^6 g^4}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{472392 (b c-a d)^3 g^4 (a+b x)^3}+\frac{b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{104976 (b c-a d)^4 g^4 (a+b x)^2}-\frac{b^2 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{26244 (b c-a d)^5 g^4 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{314928 (b c-a d)^4 g^4 (c+d x)^2}-\frac{b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{39366 (b c-a d)^5 g^4 (c+d x)}-\frac{5 b^2 d^3 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{78732 (b c-a d)^6 g^4}+\frac{5 b^2 B d^3 \log (c+d x)}{236196 (b c-a d)^6 g^4}-\frac{5 b^2 B d^3 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{78732 (b c-a d)^6 g^4}+\frac{5 b^2 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{78732 (b c-a d)^6 g^4}+\frac{5 b^2 B d^3 \log ^2(c+d x)}{157464 (b c-a d)^6 g^4}-\frac{5 b^2 B d^3 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{78732 (b c-a d)^6 g^4}-\frac{5 b^2 B d^3 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{78732 (b c-a d)^6 g^4}-\frac{5 b^2 B d^3 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{78732 (b c-a d)^6 g^4}\\ \end{align*}

Mathematica [C]  time = 1.99388, size = 637, normalized size = 1.13 \[ -\frac{-180 b^2 B d^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )+180 b^2 B d^3 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+360 b^2 d^3 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-360 b^2 d^3 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+\frac{216 b^2 d^2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{a+b x}-\frac{54 b^2 d (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{(a+b x)^2}+\frac{12 b^2 (b c-a d)^3 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{(a+b x)^3}+\frac{144 b d^3 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{c+d x}+\frac{18 d^3 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{(c+d x)^2}+\frac{216 b^3 B c d^2}{a+b x}+\frac{66 b^2 B d^2 (b c-a d)}{a+b x}-\frac{33 b^2 B d (b c-a d)^2}{(a+b x)^2}+\frac{4 b^2 B (b c-a d)^3}{(a+b x)^3}-\frac{216 a b^2 B d^3}{a+b x}+120 b^2 B d^3 \log (a+b x)+\frac{144 a b B d^4}{c+d x}-\frac{18 b B d^3 (b c-a d)}{c+d x}-\frac{9 B d^3 (b c-a d)^2}{(c+d x)^2}-\frac{144 b^2 B c d^3}{c+d x}-120 b^2 B d^3 \log (c+d x)}{36 g^4 i^3 (b c-a d)^6} \]

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)^3),x]

[Out]

-((4*b^2*B*(b*c - a*d)^3)/(a + b*x)^3 - (33*b^2*B*d*(b*c - a*d)^2)/(a + b*x)^2 + (216*b^3*B*c*d^2)/(a + b*x) -
(216*a*b^2*B*d^3)/(a + b*x) + (66*b^2*B*d^2*(b*c - a*d))/(a + b*x) - (9*B*d^3*(b*c - a*d)^2)/(c + d*x)^2 - (1
44*b^2*B*c*d^3)/(c + d*x) + (144*a*b*B*d^4)/(c + d*x) - (18*b*B*d^3*(b*c - a*d))/(c + d*x) + 120*b^2*B*d^3*Log
[a + b*x] + (12*b^2*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x)^3 - (54*b^2*d*(b*c - a*d)^2*
(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x)^2 + (216*b^2*d^2*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x
)]))/(a + b*x) + (18*d^3*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x)^2 + (144*b*d^3*(b*c - a
*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x) + 360*b^2*d^3*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d
*x)]) - 120*b^2*B*d^3*Log[c + d*x] - 360*b^2*d^3*(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x] - 180*b^2*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)]) + 180*b^2*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)]))/(36*(b*c - a*d)^6*g^4*i^3)

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Maple [B]  time = 0.054, size = 2616, 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)^4/(d*i*x+c*i)^3,x)

[Out]

-1/2*d^6/i^3/(a*d-b*c)^7/g^4*A/(d*x+c)^2*a^3-10*e*d^3/i^3/(a*d-b*c)^7/g^4*B*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)
*b*c)*a-1/3*e^3*d/i^3/(a*d-b*c)^7/g^4*A*b^5/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*a+5/4*e^2*d^2/i^3/(a*d-b*c)^
7/g^4*B*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*a+1/2*d^3/i^3/(a*d-b*c)^7/g^4*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+
c))/(d*x+c)^2*b^3*c^3+4*d^5/i^3/(a*d-b*c)^7/g^4*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b/(d*x+c)*a^2+1/3*e^3/i^3/(a
*d-b*c)^7/g^4*B*b^6/(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+5/2*e^2*d^2/i^3/(a
*d-b*c)^7/g^4*A*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*a-5/4*e^2*d/i^3/(a*d-b*c)^7/g^4*B*b^5/(b*e/d+e/(d*x+
c)*a-e/d/(d*x+c)*b*c)^2*c+9/2*d^4/i^3/(a*d-b*c)^7/g^4*A*b^2*a-9/2*d^3/i^3/(a*d-b*c)^7/g^4*A*b^3*c-19/4*d^4/i^3
/(a*d-b*c)^7/g^4*B*b^2*a+19/4*d^3/i^3/(a*d-b*c)^7/g^4*B*b^3*c-1/2*d^6/i^3/(a*d-b*c)^7/g^4*B*ln(b*e/d+(a*d-b*c)
*e/d/(d*x+c))/(d*x+c)^2*a^3-5*d^4/i^3/(a*d-b*c)^7/g^4*B*b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*a+9/2*d^4/i^3/(a
*d-b*c)^7/g^4*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^2*a+5*d^3/i^3/(a*d-b*c)^7/g^4*B*b^3*ln(b*e/d+(a*d-b*c)*e/d/(
d*x+c))^2*c+10*d^3/i^3/(a*d-b*c)^7/g^4*A*b^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c-9/2*d^3/i^3/(a*d-b*c)^7/g^4*B*l
n(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^3*c-10*d^4/i^3/(a*d-b*c)^7/g^4*A*b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a+4*d^5/
i^3/(a*d-b*c)^7/g^4*A*b/(d*x+c)*a^2+4*d^3/i^3/(a*d-b*c)^7/g^4*A*b^3/(d*x+c)*c^2-1/4*d^3/i^3/(a*d-b*c)^7/g^4*B/
(d*x+c)^2*b^3*c^3+1/3*e^3/i^3/(a*d-b*c)^7/g^4*A*b^6/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^3*c-9/2*d^3/i^3/(a*d-b
*c)^7/g^4*B*b^3/(d*x+c)*c^2-1/3*e^3*d/i^3/(a*d-b*c)^7/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))*a+5/2*e^2*d^2/i^3/(a*d-b*c)^7/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))*a-10*e*d^3/i^3/(a*d-b*c)^7/g^4*A*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*a+10*e*d^2/i^
3/(a*d-b*c)^7/g^4*A*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*c-10*e*d^3/i^3/(a*d-b*c)^7/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))*a+10*e*d^2/i^3/(a*d-b*c)^7/g^4*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+1/9*e^3/i^3/(a*d-b*c)^7/g^4*B*b^6/(b*e/d+e/(d*x+c)*a-e/d/(
d*x+c)*b*c)^3*c+1/2*d^3/i^3/(a*d-b*c)^7/g^4*A/(d*x+c)^2*b^3*c^3-9/2*d^5/i^3/(a*d-b*c)^7/g^4*B*b/(d*x+c)*a^2+1/
4*d^6/i^3/(a*d-b*c)^7/g^4*B/(d*x+c)^2*a^3-1/9*e^3*d/i^3/(a*d-b*c)^7/g^4*B*b^5/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b
*c)^3*a-8*d^4/i^3/(a*d-b*c)^7/g^4*A*b^2/(d*x+c)*c*a-3/2*d^4/i^3/(a*d-b*c)^7/g^4*A/(d*x+c)^2*b^2*c^2*a+3/4*d^4/
i^3/(a*d-b*c)^7/g^4*B/(d*x+c)^2*b^2*c^2*a-3/4*d^5/i^3/(a*d-b*c)^7/g^4*B/(d*x+c)^2*a^2*b*c+9*d^4/i^3/(a*d-b*c)^
7/g^4*B*b^2/(d*x+c)*a*c+3/2*d^5/i^3/(a*d-b*c)^7/g^4*A/(d*x+c)^2*a^2*b*c+4*d^3/i^3/(a*d-b*c)^7/g^4*B*ln(b*e/d+(
a*d-b*c)*e/d/(d*x+c))*b^3/(d*x+c)*c^2-5/2*e^2*d/i^3/(a*d-b*c)^7/g^4*A*b^5/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^
2*c+10*e*d^2/i^3/(a*d-b*c)^7/g^4*B*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*c-8*d^4/i^3/(a*d-b*c)^7/g^4*B*ln(b*
e/d+(a*d-b*c)*e/d/(d*x+c))*b^2/(d*x+c)*a*c-5/2*e^2*d/i^3/(a*d-b*c)^7/g^4*B*b^5/(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+3/2*d^5/i^3/(a*d-b*c)^7/g^4*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^4/i^3/(a*d-b*c)^7/g^4*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)^2*a*b^2*c^2

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Maxima [B]  time = 3.54157, size = 5152, normalized size = 9.15 \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)^3,x, algorithm="maxima")

[Out]

-1/6*B*((60*b^4*d^4*x^4 + 2*b^4*c^4 - 13*a*b^3*c^3*d + 47*a^2*b^2*c^2*d^2 + 27*a^3*b*c*d^3 - 3*a^4*d^4 + 30*(3
*b^4*c*d^3 + 5*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 + 23*a*b^3*c*d^3 + 11*a^2*b^2*d^4)*x^2 - 5*(b^4*c^3*d - 11*a
*b^3*c^2*d^2 - 35*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x)/((b^8*c^5*d^2 - 5*a*b^7*c^4*d^3 + 10*a^2*b^6*c^3*d^4 - 10*a^
3*b^5*c^2*d^5 + 5*a^4*b^4*c*d^6 - a^5*b^3*d^7)*g^4*i^3*x^5 + (2*b^8*c^6*d - 7*a*b^7*c^5*d^2 + 5*a^2*b^6*c^4*d^
3 + 10*a^3*b^5*c^3*d^4 - 20*a^4*b^4*c^2*d^5 + 13*a^5*b^3*c*d^6 - 3*a^6*b^2*d^7)*g^4*i^3*x^4 + (b^8*c^7 + a*b^7
*c^6*d - 17*a^2*b^6*c^5*d^2 + 35*a^3*b^5*c^4*d^3 - 25*a^4*b^4*c^3*d^4 - a^5*b^3*c^2*d^5 + 9*a^6*b^2*c*d^6 - 3*
a^7*b*d^7)*g^4*i^3*x^3 + (3*a*b^7*c^7 - 9*a^2*b^6*c^6*d + a^3*b^5*c^5*d^2 + 25*a^4*b^4*c^4*d^3 - 35*a^5*b^3*c^
3*d^4 + 17*a^6*b^2*c^2*d^5 - a^7*b*c*d^6 - a^8*d^7)*g^4*i^3*x^2 + (3*a^2*b^6*c^7 - 13*a^3*b^5*c^6*d + 20*a^4*b
^4*c^5*d^2 - 10*a^5*b^3*c^4*d^3 - 5*a^6*b^2*c^3*d^4 + 7*a^7*b*c^2*d^5 - 2*a^8*c*d^6)*g^4*i^3*x + (a^3*b^5*c^7
- 5*a^4*b^4*c^6*d + 10*a^5*b^3*c^5*d^2 - 10*a^6*b^2*c^4*d^3 + 5*a^7*b*c^3*d^4 - a^8*c^2*d^5)*g^4*i^3) + 60*b^2
*d^3*log(b*x + a)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6
*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3) - 60*b^2*d^3*log(d*x + c)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20
*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3))*log(b*e*x/(d*x + c) + a*e/(d*x + c)
) - 1/6*A*((60*b^4*d^4*x^4 + 2*b^4*c^4 - 13*a*b^3*c^3*d + 47*a^2*b^2*c^2*d^2 + 27*a^3*b*c*d^3 - 3*a^4*d^4 + 30
*(3*b^4*c*d^3 + 5*a*b^3*d^4)*x^3 + 10*(2*b^4*c^2*d^2 + 23*a*b^3*c*d^3 + 11*a^2*b^2*d^4)*x^2 - 5*(b^4*c^3*d - 1
1*a*b^3*c^2*d^2 - 35*a^2*b^2*c*d^3 - 3*a^3*b*d^4)*x)/((b^8*c^5*d^2 - 5*a*b^7*c^4*d^3 + 10*a^2*b^6*c^3*d^4 - 10
*a^3*b^5*c^2*d^5 + 5*a^4*b^4*c*d^6 - a^5*b^3*d^7)*g^4*i^3*x^5 + (2*b^8*c^6*d - 7*a*b^7*c^5*d^2 + 5*a^2*b^6*c^4
*d^3 + 10*a^3*b^5*c^3*d^4 - 20*a^4*b^4*c^2*d^5 + 13*a^5*b^3*c*d^6 - 3*a^6*b^2*d^7)*g^4*i^3*x^4 + (b^8*c^7 + a*
b^7*c^6*d - 17*a^2*b^6*c^5*d^2 + 35*a^3*b^5*c^4*d^3 - 25*a^4*b^4*c^3*d^4 - a^5*b^3*c^2*d^5 + 9*a^6*b^2*c*d^6 -
3*a^7*b*d^7)*g^4*i^3*x^3 + (3*a*b^7*c^7 - 9*a^2*b^6*c^6*d + a^3*b^5*c^5*d^2 + 25*a^4*b^4*c^4*d^3 - 35*a^5*b^3
*c^3*d^4 + 17*a^6*b^2*c^2*d^5 - a^7*b*c*d^6 - a^8*d^7)*g^4*i^3*x^2 + (3*a^2*b^6*c^7 - 13*a^3*b^5*c^6*d + 20*a^
4*b^4*c^5*d^2 - 10*a^5*b^3*c^4*d^3 - 5*a^6*b^2*c^3*d^4 + 7*a^7*b*c^2*d^5 - 2*a^8*c*d^6)*g^4*i^3*x + (a^3*b^5*c
^7 - 5*a^4*b^4*c^6*d + 10*a^5*b^3*c^5*d^2 - 10*a^6*b^2*c^4*d^3 + 5*a^7*b*c^3*d^4 - a^8*c^2*d^5)*g^4*i^3) + 60*
b^2*d^3*log(b*x + a)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 - 20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4
- 6*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3) - 60*b^2*d^3*log(d*x + c)/((b^6*c^6 - 6*a*b^5*c^5*d + 15*a^2*b^4*c^4*d^2 -
20*a^3*b^3*c^3*d^3 + 15*a^4*b^2*c^2*d^4 - 6*a^5*b*c*d^5 + a^6*d^6)*g^4*i^3)) - 1/36*(4*b^5*c^5 - 45*a*b^4*c^4
*d + 360*a^2*b^3*c^3*d^2 - 490*a^3*b^2*c^2*d^3 + 180*a^4*b*c*d^4 - 9*a^5*d^5 + 120*(b^5*c*d^4 - a*b^4*d^5)*x^4
+ 120*(3*b^5*c^2*d^3 - 2*a*b^4*c*d^4 - a^2*b^3*d^5)*x^3 + 20*(11*b^5*c^3*d^2 + 21*a*b^4*c^2*d^3 - 39*a^2*b^3*
c*d^4 + 7*a^3*b^2*d^5)*x^2 - 180*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d
^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c
^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(b*x + a)^2 - 180*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^5)
*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)*x
^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(d*x + c)^2 - 5*(5*b^5*c^4*d - 108*a*b^4*c^3*d^2 + 78*a^2*b^3
*c^2*d^3 + 52*a^3*b^2*c*d^4 - 27*a^4*b*d^5)*x + 120*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^4 + 3*a*b^4*d^
5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4 + a^3*b^2*d^5)
*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(b*x + a) - 120*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d
^4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4
+ a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x - 3*(b^5*d^5*x^5 + a^3*b^2*c^2*d^3 + (2*b^5*c*d^
4 + 3*a*b^4*d^5)*x^4 + (b^5*c^2*d^3 + 6*a*b^4*c*d^4 + 3*a^2*b^3*d^5)*x^3 + (3*a*b^4*c^2*d^3 + 6*a^2*b^3*c*d^4
+ a^3*b^2*d^5)*x^2 + (3*a^2*b^3*c^2*d^3 + 2*a^3*b^2*c*d^4)*x)*log(b*x + a))*log(d*x + c))*B/(a^3*b^6*c^8*g^4*i
^3 - 6*a^4*b^5*c^7*d*g^4*i^3 + 15*a^5*b^4*c^6*d^2*g^4*i^3 - 20*a^6*b^3*c^5*d^3*g^4*i^3 + 15*a^7*b^2*c^4*d^4*g^
4*i^3 - 6*a^8*b*c^3*d^5*g^4*i^3 + a^9*c^2*d^6*g^4*i^3 + (b^9*c^6*d^2*g^4*i^3 - 6*a*b^8*c^5*d^3*g^4*i^3 + 15*a^
2*b^7*c^4*d^4*g^4*i^3 - 20*a^3*b^6*c^3*d^5*g^4*i^3 + 15*a^4*b^5*c^2*d^6*g^4*i^3 - 6*a^5*b^4*c*d^7*g^4*i^3 + a^
6*b^3*d^8*g^4*i^3)*x^5 + (2*b^9*c^7*d*g^4*i^3 - 9*a*b^8*c^6*d^2*g^4*i^3 + 12*a^2*b^7*c^5*d^3*g^4*i^3 + 5*a^3*b
^6*c^4*d^4*g^4*i^3 - 30*a^4*b^5*c^3*d^5*g^4*i^3 + 33*a^5*b^4*c^2*d^6*g^4*i^3 - 16*a^6*b^3*c*d^7*g^4*i^3 + 3*a^
7*b^2*d^8*g^4*i^3)*x^4 + (b^9*c^8*g^4*i^3 - 18*a^2*b^7*c^6*d^2*g^4*i^3 + 52*a^3*b^6*c^5*d^3*g^4*i^3 - 60*a^4*b
^5*c^4*d^4*g^4*i^3 + 24*a^5*b^4*c^3*d^5*g^4*i^3 + 10*a^6*b^3*c^2*d^6*g^4*i^3 - 12*a^7*b^2*c*d^7*g^4*i^3 + 3*a^
8*b*d^8*g^4*i^3)*x^3 + (3*a*b^8*c^8*g^4*i^3 - 12*a^2*b^7*c^7*d*g^4*i^3 + 10*a^3*b^6*c^6*d^2*g^4*i^3 + 24*a^4*b
^5*c^5*d^3*g^4*i^3 - 60*a^5*b^4*c^4*d^4*g^4*i^3 + 52*a^6*b^3*c^3*d^5*g^4*i^3 - 18*a^7*b^2*c^2*d^6*g^4*i^3 + a^
9*d^8*g^4*i^3)*x^2 + (3*a^2*b^7*c^8*g^4*i^3 - 16*a^3*b^6*c^7*d*g^4*i^3 + 33*a^4*b^5*c^6*d^2*g^4*i^3 - 30*a^5*b
^4*c^5*d^3*g^4*i^3 + 5*a^6*b^3*c^4*d^4*g^4*i^3 + 12*a^7*b^2*c^3*d^5*g^4*i^3 - 9*a^8*b*c^2*d^6*g^4*i^3 + 2*a^9*
c*d^7*g^4*i^3)*x)

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Fricas [B]  time = 0.661795, size = 3140, normalized size = 5.58 \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)^3,x, algorithm="fricas")

[Out]

-1/36*(4*(3*A + B)*b^5*c^5 - 45*(2*A + B)*a*b^4*c^4*d + 360*(A + B)*a^2*b^3*c^3*d^2 - 10*(12*A + 49*B)*a^3*b^2
*c^2*d^3 - 180*(A - B)*a^4*b*c*d^4 + 9*(2*A - B)*a^5*d^5 + 120*((3*A + B)*b^5*c*d^4 - (3*A + B)*a*b^4*d^5)*x^4
+ 60*(3*(3*A + 2*B)*b^5*c^2*d^3 + 2*(3*A - 2*B)*a*b^4*c*d^4 - (15*A + 2*B)*a^2*b^3*d^5)*x^3 + 20*((6*A + 11*B
)*b^5*c^3*d^2 + 21*(3*A + B)*a*b^4*c^2*d^3 - 3*(12*A + 13*B)*a^2*b^3*c*d^4 - (33*A - 7*B)*a^3*b^2*d^5)*x^2 + 1
80*(B*b^5*d^5*x^5 + B*a^3*b^2*c^2*d^3 + (2*B*b^5*c*d^4 + 3*B*a*b^4*d^5)*x^4 + (B*b^5*c^2*d^3 + 6*B*a*b^4*c*d^4
+ 3*B*a^2*b^3*d^5)*x^3 + (3*B*a*b^4*c^2*d^3 + 6*B*a^2*b^3*c*d^4 + B*a^3*b^2*d^5)*x^2 + (3*B*a^2*b^3*c^2*d^3 +
2*B*a^3*b^2*c*d^4)*x)*log((b*e*x + a*e)/(d*x + c))^2 - 5*((6*A + 5*B)*b^5*c^4*d - 36*(2*A + 3*B)*a*b^4*c^3*d^
2 - 6*(24*A - 13*B)*a^2*b^3*c^2*d^3 + 4*(48*A + 13*B)*a^3*b^2*c*d^4 + 9*(2*A - 3*B)*a^4*b*d^5)*x + 6*(20*(3*A
+ B)*b^5*d^5*x^5 + 2*B*b^5*c^5 - 15*B*a*b^4*c^4*d + 60*B*a^2*b^3*c^3*d^2 + 60*A*a^3*b^2*c^2*d^3 - 30*B*a^4*b*c
*d^4 + 3*B*a^5*d^5 + 20*((6*A + 5*B)*b^5*c*d^4 + 9*A*a*b^4*d^5)*x^4 + 10*((6*A + 11*B)*b^5*c^2*d^3 + 18*(2*A +
B)*a*b^4*c*d^4 + 9*(2*A - B)*a^2*b^3*d^5)*x^3 + 10*(2*B*b^5*c^3*d^2 + 9*(2*A + 3*B)*a*b^4*c^2*d^3 + 36*A*a^2*
b^3*c*d^4 + 3*(2*A - 3*B)*a^3*b^2*d^5)*x^2 - 5*(B*b^5*c^4*d - 12*B*a*b^4*c^3*d^2 - 36*(A + B)*a^2*b^3*c^2*d^3
- 24*(A - B)*a^3*b^2*c*d^4 + 3*B*a^4*b*d^5)*x)*log((b*e*x + a*e)/(d*x + c)))/((b^9*c^6*d^2 - 6*a*b^8*c^5*d^3 +
15*a^2*b^7*c^4*d^4 - 20*a^3*b^6*c^3*d^5 + 15*a^4*b^5*c^2*d^6 - 6*a^5*b^4*c*d^7 + a^6*b^3*d^8)*g^4*i^3*x^5 + (
2*b^9*c^7*d - 9*a*b^8*c^6*d^2 + 12*a^2*b^7*c^5*d^3 + 5*a^3*b^6*c^4*d^4 - 30*a^4*b^5*c^3*d^5 + 33*a^5*b^4*c^2*d
^6 - 16*a^6*b^3*c*d^7 + 3*a^7*b^2*d^8)*g^4*i^3*x^4 + (b^9*c^8 - 18*a^2*b^7*c^6*d^2 + 52*a^3*b^6*c^5*d^3 - 60*a
^4*b^5*c^4*d^4 + 24*a^5*b^4*c^3*d^5 + 10*a^6*b^3*c^2*d^6 - 12*a^7*b^2*c*d^7 + 3*a^8*b*d^8)*g^4*i^3*x^3 + (3*a*
b^8*c^8 - 12*a^2*b^7*c^7*d + 10*a^3*b^6*c^6*d^2 + 24*a^4*b^5*c^5*d^3 - 60*a^5*b^4*c^4*d^4 + 52*a^6*b^3*c^3*d^5
- 18*a^7*b^2*c^2*d^6 + a^9*d^8)*g^4*i^3*x^2 + (3*a^2*b^7*c^8 - 16*a^3*b^6*c^7*d + 33*a^4*b^5*c^6*d^2 - 30*a^5
*b^4*c^5*d^3 + 5*a^6*b^3*c^4*d^4 + 12*a^7*b^2*c^3*d^5 - 9*a^8*b*c^2*d^6 + 2*a^9*c*d^7)*g^4*i^3*x + (a^3*b^6*c^
8 - 6*a^4*b^5*c^7*d + 15*a^5*b^4*c^6*d^2 - 20*a^6*b^3*c^5*d^3 + 15*a^7*b^2*c^4*d^4 - 6*a^8*b*c^3*d^5 + a^9*c^2
*d^6)*g^4*i^3)

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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)**3,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 )}^{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)^4/(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)^4*(d*i*x + c*i)^3), x)