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

Optimal. Leaf size=463 \[ \frac{6 b^2 d^2 \log \left (\frac{a+b x}{c+d x}\right ) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^3 (b c-a d)^5}-\frac{b^4 (c+d x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 i^3 (a+b x)^2 (b c-a d)^5}+\frac{4 b^3 d (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^3 (a+b x) (b c-a d)^5}-\frac{4 b d^3 (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^3 (c+d x) (b c-a d)^5}+\frac{d^4 (a+b x)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 i^3 (c+d x)^2 (b c-a d)^5}-\frac{3 b^2 B d^2 \log ^2\left (\frac{a+b x}{c+d x}\right )}{g^3 i^3 (b c-a d)^5}-\frac{b^4 B (c+d x)^2}{4 g^3 i^3 (a+b x)^2 (b c-a d)^5}+\frac{4 b^3 B d (c+d x)}{g^3 i^3 (a+b x) (b c-a d)^5}+\frac{4 b B d^3 (a+b x)}{g^3 i^3 (c+d x) (b c-a d)^5}-\frac{B d^4 (a+b x)^2}{4 g^3 i^3 (c+d x)^2 (b c-a d)^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 1.40576, antiderivative size = 673, normalized size of antiderivative = 1.45, 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} \[ \frac{6 b^2 B d^2 \text{PolyLog}\left (2,-\frac{d (a+b x)}{b c-a d}\right )}{g^3 i^3 (b c-a d)^5}+\frac{6 b^2 B d^2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{g^3 i^3 (b c-a d)^5}+\frac{6 b^2 d^2 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^3 (b c-a d)^5}-\frac{6 b^2 d^2 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^3 (b c-a d)^5}+\frac{3 b^2 d \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^3 (a+b x) (b c-a d)^4}-\frac{b^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 i^3 (a+b x)^2 (b c-a d)^3}+\frac{3 b d^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^3 (c+d x) (b c-a d)^4}+\frac{d^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 i^3 (c+d x)^2 (b c-a d)^3}-\frac{3 b^2 B d^2 \log ^2(a+b x)}{g^3 i^3 (b c-a d)^5}-\frac{3 b^2 B d^2 \log ^2(c+d x)}{g^3 i^3 (b c-a d)^5}+\frac{6 b^2 B d^2 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{g^3 i^3 (b c-a d)^5}+\frac{6 b^2 B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{g^3 i^3 (b c-a d)^5}+\frac{7 b^2 B d}{2 g^3 i^3 (a+b x) (b c-a d)^4}-\frac{b^2 B}{4 g^3 i^3 (a+b x)^2 (b c-a d)^3}-\frac{7 b B d^2}{2 g^3 i^3 (c+d x) (b c-a d)^4}-\frac{B d^2}{4 g^3 i^3 (c+d x)^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*B)/(4*(b*c - a*d)^3*g^3*i^3*(a + b*x)^2) + (7*b^2*B*d)/(2*(b*c - a*d)^4*g^3*i^3*(a + b*x)) - (B*d^2)/(4*
(b*c - a*d)^3*g^3*i^3*(c + d*x)^2) - (7*b*B*d^2)/(2*(b*c - a*d)^4*g^3*i^3*(c + d*x)) - (3*b^2*B*d^2*Log[a + b*
x]^2)/((b*c - a*d)^5*g^3*i^3) - (b^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^3*g^3*i^3*(a + b*x)^
2) + (3*b^2*d*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^4*g^3*i^3*(a + b*x)) + (d^2*(A + B*Log[(e*(a
+ b*x))/(c + d*x)]))/(2*(b*c - a*d)^3*g^3*i^3*(c + d*x)^2) + (3*b*d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((
b*c - a*d)^4*g^3*i^3*(c + d*x)) + (6*b^2*d^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5
*g^3*i^3) + (6*b^2*B*d^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/((b*c - a*d)^5*g^3*i^3) - (6*b^2*d^2*
(A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[c + d*x])/((b*c - a*d)^5*g^3*i^3) - (3*b^2*B*d^2*Log[c + d*x]^2)/((b*
c - a*d)^5*g^3*i^3) + (6*b^2*B*d^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^5*g^3*i^3) + (6*b
^2*B*d^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/((b*c - a*d)^5*g^3*i^3) + (6*b^2*B*d^2*PolyLog[2, (b*(c + d
*x))/(b*c - a*d)])/((b*c - a*d)^5*g^3*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 )}{(53 c+53 d x)^3 (a g+b g x)^3} \, dx &=\int \left (\frac{b^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^3 g^3 (a+b x)^3}-\frac{3 b^3 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (a+b x)^2}+\frac{6 b^3 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3 (a+b x)}-\frac{d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^3 g^3 (c+d x)^3}-\frac{3 b d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (c+d x)^2}-\frac{6 b^2 d^3 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3 (c+d x)}\right ) \, dx\\ &=\frac{\left (6 b^3 d^2\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{148877 (b c-a d)^5 g^3}-\frac{\left (6 b^2 d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{148877 (b c-a d)^5 g^3}-\frac{\left (3 b^3 d\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{148877 (b c-a d)^4 g^3}-\frac{\left (3 b d^3\right ) \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{148877 (b c-a d)^4 g^3}+\frac{b^3 \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{148877 (b c-a d)^3 g^3}-\frac{d^3 \int \frac{A+B \log \left (\frac{e (a+b x)}{c+d x}\right )}{(c+d x)^3} \, dx}{148877 (b c-a d)^3 g^3}\\ &=-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (c+d x)}+\frac{6 b^2 d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3}-\frac{6 b^2 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac{\left (6 b^2 B d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{148877 (b c-a d)^5 g^3}+\frac{\left (6 b^2 B d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{148877 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B d\right ) \int \frac{b c-a d}{(a+b x)^2 (c+d x)} \, dx}{148877 (b c-a d)^4 g^3}-\frac{\left (3 b B d^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^2} \, dx}{148877 (b c-a d)^4 g^3}+\frac{\left (b^2 B\right ) \int \frac{b c-a d}{(a+b x)^3 (c+d x)} \, dx}{297754 (b c-a d)^3 g^3}-\frac{\left (B d^2\right ) \int \frac{b c-a d}{(a+b x) (c+d x)^3} \, dx}{297754 (b c-a d)^3 g^3}\\ &=-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (c+d x)}+\frac{6 b^2 d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3}-\frac{6 b^2 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B d\right ) \int \frac{1}{(a+b x)^2 (c+d x)} \, dx}{148877 (b c-a d)^3 g^3}-\frac{\left (3 b B d^2\right ) \int \frac{1}{(a+b x) (c+d x)^2} \, dx}{148877 (b c-a d)^3 g^3}+\frac{\left (b^2 B\right ) \int \frac{1}{(a+b x)^3 (c+d x)} \, dx}{297754 (b c-a d)^2 g^3}-\frac{\left (B d^2\right ) \int \frac{1}{(a+b x) (c+d x)^3} \, dx}{297754 (b c-a d)^2 g^3}-\frac{\left (6 b^2 B d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{148877 (b c-a d)^5 e g^3}+\frac{\left (6 b^2 B d^2\right ) \int \frac{(c+d x) \left (-\frac{d e (a+b x)}{(c+d x)^2}+\frac{b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{148877 (b c-a d)^5 e g^3}\\ &=-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (c+d x)}+\frac{6 b^2 d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3}-\frac{6 b^2 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac{\left (3 b^2 B d\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}{148877 (b c-a d)^3 g^3}-\frac{\left (3 b B d^2\right ) \int \left (\frac{b^2}{(b c-a d)^2 (a+b x)}-\frac{d}{(b c-a d) (c+d x)^2}-\frac{b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{148877 (b c-a d)^3 g^3}+\frac{\left (b^2 B\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}{297754 (b c-a d)^2 g^3}-\frac{\left (B d^2\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}{297754 (b c-a d)^2 g^3}-\frac{\left (6 b^2 B d^2\right ) \int \left (\frac{b e \log (a+b x)}{a+b x}-\frac{d e \log (a+b x)}{c+d x}\right ) \, dx}{148877 (b c-a d)^5 e g^3}+\frac{\left (6 b^2 B d^2\right ) \int \left (\frac{b e \log (c+d x)}{a+b x}-\frac{d e \log (c+d x)}{c+d x}\right ) \, dx}{148877 (b c-a d)^5 e g^3}\\ &=-\frac{b^2 B}{595508 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d}{297754 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2}{595508 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2}{297754 (b c-a d)^4 g^3 (c+d x)}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (c+d x)}+\frac{6 b^2 d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3}-\frac{6 b^2 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac{\left (6 b^3 B d^2\right ) \int \frac{\log (a+b x)}{a+b x} \, dx}{148877 (b c-a d)^5 g^3}+\frac{\left (6 b^3 B d^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{148877 (b c-a d)^5 g^3}+\frac{\left (6 b^2 B d^3\right ) \int \frac{\log (a+b x)}{c+d x} \, dx}{148877 (b c-a d)^5 g^3}-\frac{\left (6 b^2 B d^3\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{148877 (b c-a d)^5 g^3}\\ &=-\frac{b^2 B}{595508 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d}{297754 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2}{595508 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2}{297754 (b c-a d)^4 g^3 (c+d x)}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (c+d x)}+\frac{6 b^2 d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3}+\frac{6 b^2 B d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac{6 b^2 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}+\frac{6 b^2 B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{148877 (b c-a d)^5 g^3}-\frac{\left (6 b^2 B d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,a+b x\right )}{148877 (b c-a d)^5 g^3}-\frac{\left (6 b^2 B d^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{148877 (b c-a d)^5 g^3}-\frac{\left (6 b^3 B d^2\right ) \int \frac{\log \left (\frac{b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{148877 (b c-a d)^5 g^3}-\frac{\left (6 b^2 B d^3\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{148877 (b c-a d)^5 g^3}\\ &=-\frac{b^2 B}{595508 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d}{297754 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2}{595508 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2}{297754 (b c-a d)^4 g^3 (c+d x)}-\frac{3 b^2 B d^2 \log ^2(a+b x)}{148877 (b c-a d)^5 g^3}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (c+d x)}+\frac{6 b^2 d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3}+\frac{6 b^2 B d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac{6 b^2 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac{3 b^2 B d^2 \log ^2(c+d x)}{148877 (b c-a d)^5 g^3}+\frac{6 b^2 B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{148877 (b c-a d)^5 g^3}-\frac{\left (6 b^2 B d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{148877 (b c-a d)^5 g^3}-\frac{\left (6 b^2 B d^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{148877 (b c-a d)^5 g^3}\\ &=-\frac{b^2 B}{595508 (b c-a d)^3 g^3 (a+b x)^2}+\frac{7 b^2 B d}{297754 (b c-a d)^4 g^3 (a+b x)}-\frac{B d^2}{595508 (b c-a d)^3 g^3 (c+d x)^2}-\frac{7 b B d^2}{297754 (b c-a d)^4 g^3 (c+d x)}-\frac{3 b^2 B d^2 \log ^2(a+b x)}{148877 (b c-a d)^5 g^3}-\frac{b^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (a+b x)^2}+\frac{3 b^2 d \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (a+b x)}+\frac{d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (c+d x)^2}+\frac{3 b d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (c+d x)}+\frac{6 b^2 d^2 \log (a+b x) \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3}+\frac{6 b^2 B d^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac{6 b^2 d^2 \left (A+B \log \left (\frac{e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac{3 b^2 B d^2 \log ^2(c+d x)}{148877 (b c-a d)^5 g^3}+\frac{6 b^2 B d^2 \log (a+b x) \log \left (\frac{b (c+d x)}{b c-a d}\right )}{148877 (b c-a d)^5 g^3}+\frac{6 b^2 B d^2 \text{Li}_2\left (-\frac{d (a+b x)}{b c-a d}\right )}{148877 (b c-a d)^5 g^3}+\frac{6 b^2 B d^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{148877 (b c-a d)^5 g^3}\\ \end{align*}

Mathematica [C]  time = 1.27403, size = 533, normalized size = 1.15 \[ -\frac{12 b^2 B d^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac{b (c+d x)}{b c-a d}\right )\right )-2 \text{PolyLog}\left (2,\frac{d (a+b x)}{a d-b c}\right )\right )-12 b^2 B d^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-24 b^2 d^2 \log (a+b x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )+24 b^2 d^2 \log (c+d x) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )-\frac{12 b^2 d (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{a+b x}+\frac{2 b^2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{(a+b x)^2}-\frac{12 b d^2 (b c-a d) \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{c+d x}-\frac{2 d^2 (b c-a d)^2 \left (B \log \left (\frac{e (a+b x)}{c+d x}\right )+A\right )}{(c+d x)^2}-\frac{12 b^3 B c d}{a+b x}-\frac{2 b^2 B d (b c-a d)}{a+b x}+\frac{b^2 B (b c-a d)^2}{(a+b x)^2}+\frac{12 a b^2 B d^2}{a+b x}-\frac{12 a b B d^3}{c+d x}+\frac{2 b B d^2 (b c-a d)}{c+d x}+\frac{B d^2 (b c-a d)^2}{(c+d x)^2}+\frac{12 b^2 B c d^2}{c+d x}}{4 g^3 i^3 (b c-a d)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2*B*(b*c - a*d)^2)/(a + b*x)^2 - (12*b^3*B*c*d)/(a + b*x) + (12*a*b^2*B*d^2)/(a + b*x) - (2*b^2*B*d*(b*c
- a*d))/(a + b*x) + (B*d^2*(b*c - a*d)^2)/(c + d*x)^2 + (12*b^2*B*c*d^2)/(c + d*x) - (12*a*b*B*d^3)/(c + d*x)
+ (2*b*B*d^2*(b*c - a*d))/(c + d*x) + (2*b^2*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x)^2 -
(12*b^2*d*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x) - (2*d^2*(b*c - a*d)^2*(A + B*Log[(e*(a
+ b*x))/(c + d*x)]))/(c + d*x)^2 - (12*b*d^2*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x) - 24
*b^2*d^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 24*b^2*d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)])*L
og[c + d*x] + 12*b^2*B*d^2*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(
a + b*x))/(-(b*c) + a*d)]) - 12*b^2*B*d^2*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] +
2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)]))/(4*(b*c - a*d)^5*g^3*i^3)

________________________________________________________________________________________

Maple [B]  time = 0.058, size = 2182, 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)^3/(d*i*x+c*i)^3,x)

[Out]

7/2*d^3/i^3/(a*d-b*c)^6/g^3*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^2*a-7/2*d^2/i^3/(a*d-b*c)^6/g^3*B*ln(b*e/d+(a*
d-b*c)*e/d/(d*x+c))*b^3*c+6*d^2/i^3/(a*d-b*c)^6/g^3*A*b^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*c-3*d^3/i^3/(a*d-b*c
)^6/g^3*B*b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*a+3*d^2/i^3/(a*d-b*c)^6/g^3*B*b^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+
c))^2*c-1/2*d^5/i^3/(a*d-b*c)^6/g^3*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)^2*a^3-6*d^3/i^3/(a*d-b*c)^6/g^3*
A*b^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*a+1/2*d^2/i^3/(a*d-b*c)^6/g^3*A/(d*x+c)^2*b^3*c^3-1/4*d^2/i^3/(a*d-b*c)^
6/g^3*B/(d*x+c)^2*b^3*c^3-1/2*e^2/i^3/(a*d-b*c)^6/g^3*A*b^5/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*c-1/4*e^2/i^
3/(a*d-b*c)^6/g^3*B*b^5/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*c+3*d^4/i^3/(a*d-b*c)^6/g^3*A*b/(d*x+c)*a^2-7/2*
d^2/i^3/(a*d-b*c)^6/g^3*B*b^3/(d*x+c)*c^2-7/2*d^4/i^3/(a*d-b*c)^6/g^3*B*b/(d*x+c)*a^2+3*d^2/i^3/(a*d-b*c)^6/g^
3*A*b^3/(d*x+c)*c^2-15/4*d^3/i^3/(a*d-b*c)^6/g^3*B*b^2*a+15/4*d^2/i^3/(a*d-b*c)^6/g^3*B*b^3*c+7/2*d^3/i^3/(a*d
-b*c)^6/g^3*A*b^2*a-7/2*d^2/i^3/(a*d-b*c)^6/g^3*A*b^3*c-6*d^3/i^3/(a*d-b*c)^6/g^3*A*b^2/(d*x+c)*a*c+1/2*d^2/i^
3/(a*d-b*c)^6/g^3*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))/(d*x+c)^2*b^3*c^3-6*d^3/i^3/(a*d-b*c)^6/g^3*B*ln(b*e/d+(a*
d-b*c)*e/d/(d*x+c))*b^2/(d*x+c)*c*a-4*e*d^2/i^3/(a*d-b*c)^6/g^3*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+4*e*d/i^3/(a*d-b*c)^6/g^3*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+1/2*e^2*d/i^3/(a*d-b*c)^6/g^3*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+3/2*d^4/i^3/(a*d-b*c)^6/g^3*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^3/i^
3/(a*d-b*c)^6/g^3*A/(d*x+c)^2*a*b^2*c^2-3/4*d^4/i^3/(a*d-b*c)^6/g^3*B/(d*x+c)^2*a^2*b*c+1/4*d^5/i^3/(a*d-b*c)^
6/g^3*B/(d*x+c)^2*a^3-1/2*d^5/i^3/(a*d-b*c)^6/g^3*A/(d*x+c)^2*a^3+1/2*e^2*d/i^3/(a*d-b*c)^6/g^3*A*b^4/(b*e/d+e
/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*a+3*d^2/i^3/(a*d-b*c)^6/g^3*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b^3/(d*x+c)*c^2+3*
d^4/i^3/(a*d-b*c)^6/g^3*B*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))*b/(d*x+c)*a^2+1/4*e^2*d/i^3/(a*d-b*c)^6/g^3*B*b^4/(b
*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)^2*a+7*d^3/i^3/(a*d-b*c)^6/g^3*B*b^2/(d*x+c)*c*a+3/4*d^3/i^3/(a*d-b*c)^6/g^3*
B/(d*x+c)^2*b^2*c^2*a-4*e*d^2/i^3/(a*d-b*c)^6/g^3*A*b^3/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*a+4*e*d/i^3/(a*d-b
*c)^6/g^3*A*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*c-4*e*d^2/i^3/(a*d-b*c)^6/g^3*B*b^3/(b*e/d+e/(d*x+c)*a-e/d
/(d*x+c)*b*c)*a+4*e*d/i^3/(a*d-b*c)^6/g^3*B*b^4/(b*e/d+e/(d*x+c)*a-e/d/(d*x+c)*b*c)*c+3/2*d^4/i^3/(a*d-b*c)^6/
g^3*A/(d*x+c)^2*a^2*b*c-1/2*e^2/i^3/(a*d-b*c)^6/g^3*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^3/i^3/(a*d-b*c)^6/g^3*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 = 2.0952, size = 3213, normalized size = 6.94 \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)^3/(d*i*x+c*i)^3,x, algorithm="maxima")

[Out]

1/2*B*((12*b^3*d^3*x^3 - b^3*c^3 + 7*a*b^2*c^2*d + 7*a^2*b*c*d^2 - a^3*d^3 + 18*(b^3*c*d^2 + a*b^2*d^3)*x^2 +
4*(b^3*c^2*d + 7*a*b^2*c*d^2 + a^2*b*d^3)*x)/((b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c
*d^5 + a^4*b^2*d^6)*g^3*i^3*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a
^4*b^2*c*d^5 + a^5*b*d^6)*g^3*i^3*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2*c^2*d^4
+ a^6*d^6)*g^3*i^3*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 + 2*a^4*b^2*c^3*d^3 - 3*a^5*b*c^2*
d^4 + a^6*c*d^5)*g^3*i^3*x + (a^2*b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^
4)*g^3*i^3) + 12*b^2*d^2*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^3*i^3) - 12*b^2*d^2*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^3*i^3))*log(b*e*x/(d*x + c) + a*e/(d*x + c)) + 1/2*A*((12*b^3*d^3
*x^3 - b^3*c^3 + 7*a*b^2*c^2*d + 7*a^2*b*c*d^2 - a^3*d^3 + 18*(b^3*c*d^2 + a*b^2*d^3)*x^2 + 4*(b^3*c^2*d + 7*a
*b^2*c*d^2 + a^2*b*d^3)*x)/((b^6*c^4*d^2 - 4*a*b^5*c^3*d^3 + 6*a^2*b^4*c^2*d^4 - 4*a^3*b^3*c*d^5 + a^4*b^2*d^6
)*g^3*i^3*x^4 + 2*(b^6*c^5*d - 3*a*b^5*c^4*d^2 + 2*a^2*b^4*c^3*d^3 + 2*a^3*b^3*c^2*d^4 - 3*a^4*b^2*c*d^5 + a^5
*b*d^6)*g^3*i^3*x^3 + (b^6*c^6 - 9*a^2*b^4*c^4*d^2 + 16*a^3*b^3*c^3*d^3 - 9*a^4*b^2*c^2*d^4 + a^6*d^6)*g^3*i^3
*x^2 + 2*(a*b^5*c^6 - 3*a^2*b^4*c^5*d + 2*a^3*b^3*c^4*d^2 + 2*a^4*b^2*c^3*d^3 - 3*a^5*b*c^2*d^4 + a^6*c*d^5)*g
^3*i^3*x + (a^2*b^4*c^6 - 4*a^3*b^3*c^5*d + 6*a^4*b^2*c^4*d^2 - 4*a^5*b*c^3*d^3 + a^6*c^2*d^4)*g^3*i^3) + 12*b
^2*d^2*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^3*i^3) - 12*b^2*d^2*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^3*i^3)) - 1/4*(b^4*c^4 - 16*a*b^3*c^3*d + 30*a^2*b^2*c^2*d^2 - 16*a^3*b*c*d^3 + a^4
*d^4 - 12*(b^4*c^2*d^2 - 2*a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + 12*(b^4*d^4*x^4 + a^2*b^2*c^2*d^2 + 2*(b^4*c*d^3 +
a*b^3*d^4)*x^3 + (b^4*c^2*d^2 + 4*a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + 2*(a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*x)*log(b
*x + a)^2 - 24*(b^4*d^4*x^4 + a^2*b^2*c^2*d^2 + 2*(b^4*c*d^3 + a*b^3*d^4)*x^3 + (b^4*c^2*d^2 + 4*a*b^3*c*d^3 +
a^2*b^2*d^4)*x^2 + 2*(a*b^3*c^2*d^2 + a^2*b^2*c*d^3)*x)*log(b*x + a)*log(d*x + c) + 12*(b^4*d^4*x^4 + a^2*b^2
*c^2*d^2 + 2*(b^4*c*d^3 + a*b^3*d^4)*x^3 + (b^4*c^2*d^2 + 4*a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + 2*(a*b^3*c^2*d^2
+ a^2*b^2*c*d^3)*x)*log(d*x + c)^2 - 12*(b^4*c^3*d - a*b^3*c^2*d^2 - a^2*b^2*c*d^3 + a^3*b*d^4)*x)*B/(a^2*b^5*
c^7*g^3*i^3 - 5*a^3*b^4*c^6*d*g^3*i^3 + 10*a^4*b^3*c^5*d^2*g^3*i^3 - 10*a^5*b^2*c^4*d^3*g^3*i^3 + 5*a^6*b*c^3*
d^4*g^3*i^3 - a^7*c^2*d^5*g^3*i^3 + (b^7*c^5*d^2*g^3*i^3 - 5*a*b^6*c^4*d^3*g^3*i^3 + 10*a^2*b^5*c^3*d^4*g^3*i^
3 - 10*a^3*b^4*c^2*d^5*g^3*i^3 + 5*a^4*b^3*c*d^6*g^3*i^3 - a^5*b^2*d^7*g^3*i^3)*x^4 + 2*(b^7*c^6*d*g^3*i^3 - 4
*a*b^6*c^5*d^2*g^3*i^3 + 5*a^2*b^5*c^4*d^3*g^3*i^3 - 5*a^4*b^3*c^2*d^5*g^3*i^3 + 4*a^5*b^2*c*d^6*g^3*i^3 - a^6
*b*d^7*g^3*i^3)*x^3 + (b^7*c^7*g^3*i^3 - a*b^6*c^6*d*g^3*i^3 - 9*a^2*b^5*c^5*d^2*g^3*i^3 + 25*a^3*b^4*c^4*d^3*
g^3*i^3 - 25*a^4*b^3*c^3*d^4*g^3*i^3 + 9*a^5*b^2*c^2*d^5*g^3*i^3 + a^6*b*c*d^6*g^3*i^3 - a^7*d^7*g^3*i^3)*x^2
+ 2*(a*b^6*c^7*g^3*i^3 - 4*a^2*b^5*c^6*d*g^3*i^3 + 5*a^3*b^4*c^5*d^2*g^3*i^3 - 5*a^5*b^2*c^3*d^4*g^3*i^3 + 4*a
^6*b*c^2*d^5*g^3*i^3 - a^7*c*d^6*g^3*i^3)*x)

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Fricas [B]  time = 0.6042, size = 2037, normalized size = 4.4 \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)^3/(d*i*x+c*i)^3,x, algorithm="fricas")

[Out]

-1/4*((2*A + B)*b^4*c^4 - 16*(A + B)*a*b^3*c^3*d + 30*B*a^2*b^2*c^2*d^2 + 16*(A - B)*a^3*b*c*d^3 - (2*A - B)*a
^4*d^4 - 24*(A*b^4*c*d^3 - A*a*b^3*d^4)*x^3 - 12*((3*A + B)*b^4*c^2*d^2 - 2*B*a*b^3*c*d^3 - (3*A - B)*a^2*b^2*
d^4)*x^2 - 12*(B*b^4*d^4*x^4 + B*a^2*b^2*c^2*d^2 + 2*(B*b^4*c*d^3 + B*a*b^3*d^4)*x^3 + (B*b^4*c^2*d^2 + 4*B*a*
b^3*c*d^3 + B*a^2*b^2*d^4)*x^2 + 2*(B*a*b^3*c^2*d^2 + B*a^2*b^2*c*d^3)*x)*log((b*e*x + a*e)/(d*x + c))^2 - 4*(
(2*A + 3*B)*b^4*c^3*d + 3*(4*A - B)*a*b^3*c^2*d^2 - 3*(4*A + B)*a^2*b^2*c*d^3 - (2*A - 3*B)*a^3*b*d^4)*x - 2*(
12*A*b^4*d^4*x^4 - B*b^4*c^4 + 8*B*a*b^3*c^3*d + 12*A*a^2*b^2*c^2*d^2 - 8*B*a^3*b*c*d^3 + B*a^4*d^4 + 12*((2*A
+ B)*b^4*c*d^3 + (2*A - B)*a*b^3*d^4)*x^3 + 6*((2*A + 3*B)*b^4*c^2*d^2 + 8*A*a*b^3*c*d^3 + (2*A - 3*B)*a^2*b^
2*d^4)*x^2 + 4*(B*b^4*c^3*d + 6*(A + B)*a*b^3*c^2*d^2 + 6*(A - B)*a^2*b^2*c*d^3 - B*a^3*b*d^4)*x)*log((b*e*x +
a*e)/(d*x + c)))/((b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 - 10*a^3*b^4*c^2*d^5 + 5*a^4*b^3*c*d^6
- a^5*b^2*d^7)*g^3*i^3*x^4 + 2*(b^7*c^6*d - 4*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 - 5*a^4*b^3*c^2*d^5 + 4*a^5*b^
2*c*d^6 - a^6*b*d^7)*g^3*i^3*x^3 + (b^7*c^7 - a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 25*a^3*b^4*c^4*d^3 - 25*a^4*b^
3*c^3*d^4 + 9*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 - a^7*d^7)*g^3*i^3*x^2 + 2*(a*b^6*c^7 - 4*a^2*b^5*c^6*d + 5*a^3*b^
4*c^5*d^2 - 5*a^5*b^2*c^3*d^4 + 4*a^6*b*c^2*d^5 - a^7*c*d^6)*g^3*i^3*x + (a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a
^4*b^3*c^5*d^2 - 10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5)*g^3*i^3)

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Sympy [B]  time = 65.1957, size = 2106, normalized size = 4.55 \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)**3/(d*i*x+c*i)**3,x)

[Out]

6*A*b**2*d**2*log(x + (-6*A*a**6*b**2*d**8/(a*d - b*c)**5 + 36*A*a**5*b**3*c*d**7/(a*d - b*c)**5 - 90*A*a**4*b
**4*c**2*d**6/(a*d - b*c)**5 + 120*A*a**3*b**5*c**3*d**5/(a*d - b*c)**5 - 90*A*a**2*b**6*c**4*d**4/(a*d - b*c)
**5 + 36*A*a*b**7*c**5*d**3/(a*d - b*c)**5 + 6*A*a*b**2*d**3 - 6*A*b**8*c**6*d**2/(a*d - b*c)**5 + 6*A*b**3*c*
d**2)/(12*A*b**3*d**3))/(g**3*i**3*(a*d - b*c)**5) - 6*A*b**2*d**2*log(x + (6*A*a**6*b**2*d**8/(a*d - b*c)**5
- 36*A*a**5*b**3*c*d**7/(a*d - b*c)**5 + 90*A*a**4*b**4*c**2*d**6/(a*d - b*c)**5 - 120*A*a**3*b**5*c**3*d**5/(
a*d - b*c)**5 + 90*A*a**2*b**6*c**4*d**4/(a*d - b*c)**5 - 36*A*a*b**7*c**5*d**3/(a*d - b*c)**5 + 6*A*a*b**2*d*
*3 + 6*A*b**8*c**6*d**2/(a*d - b*c)**5 + 6*A*b**3*c*d**2)/(12*A*b**3*d**3))/(g**3*i**3*(a*d - b*c)**5) - 3*B*b
**2*d**2*log(e*(a + b*x)/(c + d*x))**2/(a**5*d**5*g**3*i**3 - 5*a**4*b*c*d**4*g**3*i**3 + 10*a**3*b**2*c**2*d*
*3*g**3*i**3 - 10*a**2*b**3*c**3*d**2*g**3*i**3 + 5*a*b**4*c**4*d*g**3*i**3 - b**5*c**5*g**3*i**3) + (-B*a**3*
d**3 + 7*B*a**2*b*c*d**2 + 4*B*a**2*b*d**3*x + 7*B*a*b**2*c**2*d + 28*B*a*b**2*c*d**2*x + 18*B*a*b**2*d**3*x**
2 - B*b**3*c**3 + 4*B*b**3*c**2*d*x + 18*B*b**3*c*d**2*x**2 + 12*B*b**3*d**3*x**3)*log(e*(a + b*x)/(c + d*x))/
(2*a**6*c**2*d**4*g**3*i**3 + 4*a**6*c*d**5*g**3*i**3*x + 2*a**6*d**6*g**3*i**3*x**2 - 8*a**5*b*c**3*d**3*g**3
*i**3 - 12*a**5*b*c**2*d**4*g**3*i**3*x + 4*a**5*b*d**6*g**3*i**3*x**3 + 12*a**4*b**2*c**4*d**2*g**3*i**3 + 8*
a**4*b**2*c**3*d**3*g**3*i**3*x - 18*a**4*b**2*c**2*d**4*g**3*i**3*x**2 - 12*a**4*b**2*c*d**5*g**3*i**3*x**3 +
2*a**4*b**2*d**6*g**3*i**3*x**4 - 8*a**3*b**3*c**5*d*g**3*i**3 + 8*a**3*b**3*c**4*d**2*g**3*i**3*x + 32*a**3*
b**3*c**3*d**3*g**3*i**3*x**2 + 8*a**3*b**3*c**2*d**4*g**3*i**3*x**3 - 8*a**3*b**3*c*d**5*g**3*i**3*x**4 + 2*a
**2*b**4*c**6*g**3*i**3 - 12*a**2*b**4*c**5*d*g**3*i**3*x - 18*a**2*b**4*c**4*d**2*g**3*i**3*x**2 + 8*a**2*b**
4*c**3*d**3*g**3*i**3*x**3 + 12*a**2*b**4*c**2*d**4*g**3*i**3*x**4 + 4*a*b**5*c**6*g**3*i**3*x - 12*a*b**5*c**
4*d**2*g**3*i**3*x**3 - 8*a*b**5*c**3*d**3*g**3*i**3*x**4 + 2*b**6*c**6*g**3*i**3*x**2 + 4*b**6*c**5*d*g**3*i*
*3*x**3 + 2*b**6*c**4*d**2*g**3*i**3*x**4) + (-2*A*a**3*d**3 + 14*A*a**2*b*c*d**2 + 14*A*a*b**2*c**2*d - 2*A*b
**3*c**3 + 24*A*b**3*d**3*x**3 + B*a**3*d**3 - 15*B*a**2*b*c*d**2 + 15*B*a*b**2*c**2*d - B*b**3*c**3 + x**2*(3
6*A*a*b**2*d**3 + 36*A*b**3*c*d**2 - 12*B*a*b**2*d**3 + 12*B*b**3*c*d**2) + x*(8*A*a**2*b*d**3 + 56*A*a*b**2*c
*d**2 + 8*A*b**3*c**2*d - 12*B*a**2*b*d**3 + 12*B*b**3*c**2*d))/(4*a**6*c**2*d**4*g**3*i**3 - 16*a**5*b*c**3*d
**3*g**3*i**3 + 24*a**4*b**2*c**4*d**2*g**3*i**3 - 16*a**3*b**3*c**5*d*g**3*i**3 + 4*a**2*b**4*c**6*g**3*i**3
+ x**4*(4*a**4*b**2*d**6*g**3*i**3 - 16*a**3*b**3*c*d**5*g**3*i**3 + 24*a**2*b**4*c**2*d**4*g**3*i**3 - 16*a*b
**5*c**3*d**3*g**3*i**3 + 4*b**6*c**4*d**2*g**3*i**3) + x**3*(8*a**5*b*d**6*g**3*i**3 - 24*a**4*b**2*c*d**5*g*
*3*i**3 + 16*a**3*b**3*c**2*d**4*g**3*i**3 + 16*a**2*b**4*c**3*d**3*g**3*i**3 - 24*a*b**5*c**4*d**2*g**3*i**3
+ 8*b**6*c**5*d*g**3*i**3) + x**2*(4*a**6*d**6*g**3*i**3 - 36*a**4*b**2*c**2*d**4*g**3*i**3 + 64*a**3*b**3*c**
3*d**3*g**3*i**3 - 36*a**2*b**4*c**4*d**2*g**3*i**3 + 4*b**6*c**6*g**3*i**3) + x*(8*a**6*c*d**5*g**3*i**3 - 24
*a**5*b*c**2*d**4*g**3*i**3 + 16*a**4*b**2*c**3*d**3*g**3*i**3 + 16*a**3*b**3*c**4*d**2*g**3*i**3 - 24*a**2*b*
*4*c**5*d*g**3*i**3 + 8*a*b**5*c**6*g**3*i**3))

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