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

Optimal. Leaf size=483 \[ -\frac {b^4 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^3 (a+b x)^2 (b c-a d)^5}+\frac {4 b^3 d (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (a+b x) (b c-a d)^5}+\frac {6 b^2 d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (b c-a d)^5}+\frac {d^4 (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^3 (c+d x)^2 (b c-a d)^5}-\frac {4 b d^3 (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (c+d x) (b c-a d)^5}-\frac {b^4 B n (c+d x)^2}{4 g^3 i^3 (a+b x)^2 (b c-a d)^5}+\frac {4 b^3 B d n (c+d x)}{g^3 i^3 (a+b x) (b c-a d)^5}-\frac {3 b^2 B d^2 n \log ^2\left (\frac {a+b x}{c+d x}\right )}{g^3 i^3 (b c-a d)^5}-\frac {B d^4 n (a+b x)^2}{4 g^3 i^3 (c+d x)^2 (b c-a d)^5}+\frac {4 b B d^3 n (a+b x)}{g^3 i^3 (c+d x) (b c-a d)^5} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 1.39, antiderivative size = 701, normalized size of antiderivative = 1.45, number of steps used = 34, number of rules used = 11, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.256, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ \frac {6 b^2 B d^2 n \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{g^3 i^3 (b c-a d)^5}+\frac {6 b^2 B d^2 n \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{g^3 i^3 (b c-a d)^5}+\frac {6 b^2 d^2 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (b c-a d)^5}-\frac {6 b^2 d^2 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (b c-a d)^5}+\frac {3 b^2 d \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (a+b x) (b c-a d)^4}-\frac {b^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^3 (a+b x)^2 (b c-a d)^3}+\frac {3 b d^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^3 i^3 (c+d x) (b c-a d)^4}+\frac {d^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^3 i^3 (c+d x)^2 (b c-a d)^3}-\frac {3 b^2 B d^2 n \log ^2(a+b x)}{g^3 i^3 (b c-a d)^5}-\frac {3 b^2 B d^2 n \log ^2(c+d x)}{g^3 i^3 (b c-a d)^5}+\frac {6 b^2 B d^2 n \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{g^3 i^3 (b c-a d)^5}+\frac {6 b^2 B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{g^3 i^3 (b c-a d)^5}+\frac {7 b^2 B d n}{2 g^3 i^3 (a+b x) (b c-a d)^4}-\frac {b^2 B n}{4 g^3 i^3 (a+b x)^2 (b c-a d)^3}-\frac {7 b B d^2 n}{2 g^3 i^3 (c+d x) (b c-a d)^4}-\frac {B d^2 n}{4 g^3 i^3 (c+d x)^2 (b c-a d)^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(b^2*B*n)/(4*(b*c - a*d)^3*g^3*i^3*(a + b*x)^2) + (7*b^2*B*d*n)/(2*(b*c - a*d)^4*g^3*i^3*(a + b*x)) - (B*d^2*
n)/(4*(b*c - a*d)^3*g^3*i^3*(c + d*x)^2) - (7*b*B*d^2*n)/(2*(b*c - a*d)^4*g^3*i^3*(c + d*x)) - (3*b^2*B*d^2*n*
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))^n]))/(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))^n]))/((b*c - a*d)^4*g^3*i^3*(a + b*x)) + (d^2*(A
+ B*Log[e*((a + b*x)/(c + d*x))^n]))/(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))^n]))/((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))
^n]))/((b*c - a*d)^5*g^3*i^3) + (6*b^2*B*d^2*n*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))^n])*Log[c + d*x])/((b*c - a*d)^5*g^3*i^3) - (3*b^2*B*
d^2*n*Log[c + d*x]^2)/((b*c - a*d)^5*g^3*i^3) + (6*b^2*B*d^2*n*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*n*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*n*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/((b*c - a*d)^5*g^3*i^3)

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 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 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 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]

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 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 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 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 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 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]

Rubi steps

\begin {align*} \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(157 c+157 d x)^3 (a g+b g x)^3} \, dx &=\int \left (\frac {b^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^3 g^3 (a+b x)^3}-\frac {3 b^3 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)^2}+\frac {6 b^3 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^3 g^3 (c+d x)^3}-\frac {3 b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (c+d x)^2}-\frac {6 b^2 d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 d^3\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (3 b^3 d\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{3869893 (b c-a d)^4 g^3}-\frac {\left (3 b d^3\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{3869893 (b c-a d)^4 g^3}+\frac {b^3 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{3869893 (b c-a d)^3 g^3}-\frac {d^3 \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{3869893 (b c-a d)^3 g^3}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}+\frac {\left (6 b^2 B d^2 n\right ) \int \frac {(c+d x) \left (-\frac {d (a+b x)}{(c+d x)^2}+\frac {b}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (3 b^2 B d n\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3869893 (b c-a d)^4 g^3}-\frac {\left (3 b B d^2 n\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{3869893 (b c-a d)^4 g^3}+\frac {\left (b^2 B n\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{7739786 (b c-a d)^3 g^3}-\frac {\left (B d^2 n\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{7739786 (b c-a d)^3 g^3}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 n\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{3869893 (b c-a d)^5 g^3}+\frac {\left (6 b^2 B d^2 n\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (3 b^2 B d n\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3869893 (b c-a d)^3 g^3}-\frac {\left (3 b B d^2 n\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{3869893 (b c-a d)^3 g^3}+\frac {\left (b^2 B n\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{7739786 (b c-a d)^2 g^3}-\frac {\left (B d^2 n\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{7739786 (b c-a d)^2 g^3}\\ &=-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^3 B d^2 n\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}+\frac {\left (6 b^3 B d^2 n\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}+\frac {\left (6 b^2 B d^3 n\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^3 n\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (3 b^2 B d n\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}{3869893 (b c-a d)^3 g^3}-\frac {\left (3 b B d^2 n\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}{3869893 (b c-a d)^3 g^3}+\frac {\left (b^2 B n\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}{7739786 (b c-a d)^2 g^3}-\frac {\left (B d^2 n\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}{7739786 (b c-a d)^2 g^3}\\ &=-\frac {b^2 B n}{15479572 (b c-a d)^3 g^3 (a+b x)^2}+\frac {7 b^2 B d n}{7739786 (b c-a d)^4 g^3 (a+b x)}-\frac {B d^2 n}{15479572 (b c-a d)^3 g^3 (c+d x)^2}-\frac {7 b B d^2 n}{7739786 (b c-a d)^4 g^3 (c+d x)}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^3 B d^2 n\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^3 n\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3869893 (b c-a d)^5 g^3}\\ &=-\frac {b^2 B n}{15479572 (b c-a d)^3 g^3 (a+b x)^2}+\frac {7 b^2 B d n}{7739786 (b c-a d)^4 g^3 (a+b x)}-\frac {B d^2 n}{15479572 (b c-a d)^3 g^3 (c+d x)^2}-\frac {7 b B d^2 n}{7739786 (b c-a d)^4 g^3 (c+d x)}-\frac {3 b^2 B d^2 n \log ^2(a+b x)}{3869893 (b c-a d)^5 g^3}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {3 b^2 B d^2 n \log ^2(c+d x)}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3869893 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2 n\right ) \operatorname {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3869893 (b c-a d)^5 g^3}\\ &=-\frac {b^2 B n}{15479572 (b c-a d)^3 g^3 (a+b x)^2}+\frac {7 b^2 B d n}{7739786 (b c-a d)^4 g^3 (a+b x)}-\frac {B d^2 n}{15479572 (b c-a d)^3 g^3 (c+d x)^2}-\frac {7 b B d^2 n}{7739786 (b c-a d)^4 g^3 (c+d x)}-\frac {3 b^2 B d^2 n \log ^2(a+b x)}{3869893 (b c-a d)^5 g^3}-\frac {b^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{7739786 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (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 (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3869893 (b c-a d)^5 g^3}-\frac {3 b^2 B d^2 n \log ^2(c+d x)}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 n \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3869893 (b c-a d)^5 g^3}\\ \end {align*}

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Mathematica [C]  time = 1.28, size = 561, normalized size = 1.16 \[ -\frac {-24 b^2 d^2 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+24 b^2 d^2 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-\frac {12 b^2 d (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{a+b x}+\frac {2 b^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(a+b x)^2}-\frac {12 b d^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{c+d x}-\frac {2 d^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{(c+d x)^2}-\frac {12 b^3 B c d n}{a+b x}+12 b^2 B d^2 n \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )-12 b^2 B d^2 n \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-\frac {2 b^2 B d n (b c-a d)}{a+b x}+\frac {b^2 B n (b c-a d)^2}{(a+b x)^2}+\frac {12 a b^2 B d^2 n}{a+b x}-\frac {12 a b B d^3 n}{c+d x}+\frac {2 b B d^2 n (b c-a d)}{c+d x}+\frac {B d^2 n (b c-a d)^2}{(c+d x)^2}+\frac {12 b^2 B c d^2 n}{c+d x}}{4 g^3 i^3 (b c-a d)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((b^2*B*(b*c - a*d)^2*n)/(a + b*x)^2 - (12*b^3*B*c*d*n)/(a + b*x) + (12*a*b^2*B*d^2*n)/(a + b*x) - (2*b^2
*B*d*(b*c - a*d)*n)/(a + b*x) + (B*d^2*(b*c - a*d)^2*n)/(c + d*x)^2 + (12*b^2*B*c*d^2*n)/(c + d*x) - (12*a*b*B
*d^3*n)/(c + d*x) + (2*b*B*d^2*(b*c - a*d)*n)/(c + d*x) + (2*b^2*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*
x))^n]))/(a + b*x)^2 - (12*b^2*d*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(a + b*x) - (2*d^2*(b*c -
a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(c + d*x)^2 - (12*b*d^2*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c
+ d*x))^n]))/(c + d*x) - 24*b^2*d^2*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 24*b^2*d^2*(A + B*L
og[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 12*b^2*B*d^2*n*(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*n*((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)]))/((b*c - a*d)^5*g^3*i^3)

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fricas [B]  time = 1.05, size = 1416, normalized size = 2.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^3/(d*i*x+c*i)^3,x, algorithm="fricas")

[Out]

-1/4*(2*A*b^4*c^4 - 16*A*a*b^3*c^3*d + 16*A*a^3*b*c*d^3 - 2*A*a^4*d^4 - 24*(A*b^4*c*d^3 - A*a*b^3*d^4)*x^3 - 1
2*(3*A*b^4*c^2*d^2 - 3*A*a^2*b^2*d^4 + (B*b^4*c^2*d^2 - 2*B*a*b^3*c*d^3 + B*a^2*b^2*d^4)*n)*x^2 - 12*(B*b^4*d^
4*n*x^4 + B*a^2*b^2*c^2*d^2*n + 2*(B*b^4*c*d^3 + B*a*b^3*d^4)*n*x^3 + (B*b^4*c^2*d^2 + 4*B*a*b^3*c*d^3 + B*a^2
*b^2*d^4)*n*x^2 + 2*(B*a*b^3*c^2*d^2 + B*a^2*b^2*c*d^3)*n*x)*log((b*x + a)/(d*x + c))^2 + (B*b^4*c^4 - 16*B*a*
b^3*c^3*d + 30*B*a^2*b^2*c^2*d^2 - 16*B*a^3*b*c*d^3 + B*a^4*d^4)*n - 4*(2*A*b^4*c^3*d + 12*A*a*b^3*c^2*d^2 - 1
2*A*a^2*b^2*c*d^3 - 2*A*a^3*b*d^4 + 3*(B*b^4*c^3*d - B*a*b^3*c^2*d^2 - B*a^2*b^2*c*d^3 + B*a^3*b*d^4)*n)*x + 2
*(B*b^4*c^4 - 8*B*a*b^3*c^3*d + 8*B*a^3*b*c*d^3 - B*a^4*d^4 - 12*(B*b^4*c*d^3 - B*a*b^3*d^4)*x^3 - 18*(B*b^4*c
^2*d^2 - B*a^2*b^2*d^4)*x^2 - 4*(B*b^4*c^3*d + 6*B*a*b^3*c^2*d^2 - 6*B*a^2*b^2*c*d^3 - B*a^3*b*d^4)*x - 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*x + a)/(d*x + c)))*log(e) - 2*(12*A*b^4*d^4*x^
4 + 12*A*a^2*b^2*c^2*d^2 + 12*(2*A*b^4*c*d^3 + 2*A*a*b^3*d^4 + (B*b^4*c*d^3 - B*a*b^3*d^4)*n)*x^3 + 6*(2*A*b^4
*c^2*d^2 + 8*A*a*b^3*c*d^3 + 2*A*a^2*b^2*d^4 + 3*(B*b^4*c^2*d^2 - B*a^2*b^2*d^4)*n)*x^2 - (B*b^4*c^4 - 8*B*a*b
^3*c^3*d + 8*B*a^3*b*c*d^3 - B*a^4*d^4)*n + 4*(6*A*a*b^3*c^2*d^2 + 6*A*a^2*b^2*c*d^3 + (B*b^4*c^3*d + 6*B*a*b^
3*c^2*d^2 - 6*B*a^2*b^2*c*d^3 - B*a^3*b*d^4)*n)*x)*log((b*x + a)/(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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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(b*g*x+a*g)^3/(d*i*x+c*i)^3,x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.47, size = 0, normalized size = 0.00 \[ \int \frac {B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A}{\left (b g x +a g \right )^{3} \left (d i x +c i \right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*ln(e*((b*x+a)/(d*x+c))^n)+A)/(b*g*x+a*g)^3/(d*i*x+c*i)^3,x)

[Out]

int((B*ln(e*((b*x+a)/(d*x+c))^n)+A)/(b*g*x+a*g)^3/(d*i*x+c*i)^3,x)

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maxima [B]  time = 2.52, size = 2383, normalized size = 4.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))/(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(e*(b*x/(d*x + c) + a/(d*x + c))^n) - 1/4*(b^4*c^4 - 1
6*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*n/(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) + 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))

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mupad [B]  time = 7.93, size = 1341, normalized size = 2.78 \[ \frac {\frac {2\,x\,\left (2\,A\,a^2\,b\,d^3+2\,A\,b^3\,c^2\,d+14\,A\,a\,b^2\,c\,d^2-3\,B\,a^2\,b\,d^3\,n+3\,B\,b^3\,c^2\,d\,n\right )}{a\,d-b\,c}-\frac {2\,A\,a^3\,d^3+2\,A\,b^3\,c^3-B\,a^3\,d^3\,n+B\,b^3\,c^3\,n-14\,A\,a\,b^2\,c^2\,d-14\,A\,a^2\,b\,c\,d^2-15\,B\,a\,b^2\,c^2\,d\,n+15\,B\,a^2\,b\,c\,d^2\,n}{2\,\left (a\,d-b\,c\right )}+\frac {6\,x^2\,\left (3\,A\,a\,b^2\,d^3+3\,A\,b^3\,c\,d^2-B\,a\,b^2\,d^3\,n+B\,b^3\,c\,d^2\,n\right )}{a\,d-b\,c}+\frac {12\,A\,b^3\,d^3\,x^3}{a\,d-b\,c}}{x^4\,\left (2\,a^3\,b^2\,d^5\,g^3\,i^3-6\,a^2\,b^3\,c\,d^4\,g^3\,i^3+6\,a\,b^4\,c^2\,d^3\,g^3\,i^3-2\,b^5\,c^3\,d^2\,g^3\,i^3\right )-x\,\left (-4\,a^5\,c\,d^4\,g^3\,i^3+8\,a^4\,b\,c^2\,d^3\,g^3\,i^3-8\,a^2\,b^3\,c^4\,d\,g^3\,i^3+4\,a\,b^4\,c^5\,g^3\,i^3\right )+x^3\,\left (4\,a^4\,b\,d^5\,g^3\,i^3-8\,a^3\,b^2\,c\,d^4\,g^3\,i^3+8\,a\,b^4\,c^3\,d^2\,g^3\,i^3-4\,b^5\,c^4\,d\,g^3\,i^3\right )+x^2\,\left (2\,a^5\,d^5\,g^3\,i^3+2\,a^4\,b\,c\,d^4\,g^3\,i^3-16\,a^3\,b^2\,c^2\,d^3\,g^3\,i^3+16\,a^2\,b^3\,c^3\,d^2\,g^3\,i^3-2\,a\,b^4\,c^4\,d\,g^3\,i^3-2\,b^5\,c^5\,g^3\,i^3\right )-2\,a^2\,b^3\,c^5\,g^3\,i^3+2\,a^5\,c^2\,d^3\,g^3\,i^3+6\,a^3\,b^2\,c^4\,d\,g^3\,i^3-6\,a^4\,b\,c^3\,d^2\,g^3\,i^3}+\frac {\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (x\,\left (\frac {3\,B\,b\,d\,{\left (a\,d+b\,c\right )}^2}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}-\frac {B\,b\,d}{a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2}+\frac {6\,B\,a\,b^2\,c\,d^2}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}\right )-\frac {B\,\left (a\,d+b\,c\right )}{2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {6\,B\,b^3\,d^3\,x^3}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}+\frac {9\,B\,b^2\,d^2\,x^2\,\left (a\,d+b\,c\right )}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}+\frac {3\,B\,a\,b\,c\,d\,\left (a\,d+b\,c\right )}{{\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}^2}\right )}{x\,\left (2\,d\,a^2\,c\,g^3\,i^3+2\,b\,a\,c^2\,g^3\,i^3\right )+x^3\,\left (2\,c\,b^2\,d\,g^3\,i^3+2\,a\,b\,d^2\,g^3\,i^3\right )+x^2\,\left (a^2\,d^2\,g^3\,i^3+4\,a\,b\,c\,d\,g^3\,i^3+b^2\,c^2\,g^3\,i^3\right )+a^2\,c^2\,g^3\,i^3+b^2\,d^2\,g^3\,i^3\,x^4}-\frac {3\,B\,b^2\,d^2\,{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2}{g^3\,i^3\,n\,{\left (a\,d-b\,c\right )}^5}+\frac {A\,b^2\,d^2\,\mathrm {atan}\left (\frac {\left (a^5\,d^5\,g^3\,i^3-3\,a^4\,b\,c\,d^4\,g^3\,i^3+2\,a^3\,b^2\,c^2\,d^3\,g^3\,i^3+2\,a^2\,b^3\,c^3\,d^2\,g^3\,i^3-3\,a\,b^4\,c^4\,d\,g^3\,i^3+b^5\,c^5\,g^3\,i^3\right )\,1{}\mathrm {i}}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^5}+\frac {b\,d\,x\,\left (a^4\,d^4\,g^3\,i^3-4\,a^3\,b\,c\,d^3\,g^3\,i^3+6\,a^2\,b^2\,c^2\,d^2\,g^3\,i^3-4\,a\,b^3\,c^3\,d\,g^3\,i^3+b^4\,c^4\,g^3\,i^3\right )\,2{}\mathrm {i}}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^5}\right )\,12{}\mathrm {i}}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((2*x*(2*A*a^2*b*d^3 + 2*A*b^3*c^2*d + 14*A*a*b^2*c*d^2 - 3*B*a^2*b*d^3*n + 3*B*b^3*c^2*d*n))/(a*d - b*c) - (2
*A*a^3*d^3 + 2*A*b^3*c^3 - B*a^3*d^3*n + B*b^3*c^3*n - 14*A*a*b^2*c^2*d - 14*A*a^2*b*c*d^2 - 15*B*a*b^2*c^2*d*
n + 15*B*a^2*b*c*d^2*n)/(2*(a*d - b*c)) + (6*x^2*(3*A*a*b^2*d^3 + 3*A*b^3*c*d^2 - B*a*b^2*d^3*n + B*b^3*c*d^2*
n))/(a*d - b*c) + (12*A*b^3*d^3*x^3)/(a*d - b*c))/(x^4*(2*a^3*b^2*d^5*g^3*i^3 - 2*b^5*c^3*d^2*g^3*i^3 + 6*a*b^
4*c^2*d^3*g^3*i^3 - 6*a^2*b^3*c*d^4*g^3*i^3) - x*(4*a*b^4*c^5*g^3*i^3 - 4*a^5*c*d^4*g^3*i^3 - 8*a^2*b^3*c^4*d*
g^3*i^3 + 8*a^4*b*c^2*d^3*g^3*i^3) + x^3*(4*a^4*b*d^5*g^3*i^3 - 4*b^5*c^4*d*g^3*i^3 + 8*a*b^4*c^3*d^2*g^3*i^3
- 8*a^3*b^2*c*d^4*g^3*i^3) + x^2*(2*a^5*d^5*g^3*i^3 - 2*b^5*c^5*g^3*i^3 - 2*a*b^4*c^4*d*g^3*i^3 + 2*a^4*b*c*d^
4*g^3*i^3 + 16*a^2*b^3*c^3*d^2*g^3*i^3 - 16*a^3*b^2*c^2*d^3*g^3*i^3) - 2*a^2*b^3*c^5*g^3*i^3 + 2*a^5*c^2*d^3*g
^3*i^3 + 6*a^3*b^2*c^4*d*g^3*i^3 - 6*a^4*b*c^3*d^2*g^3*i^3) + (log(e*((a + b*x)/(c + d*x))^n)*(x*((3*B*b*d*(a*
d + b*c)^2)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2 - (B*b*d)/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d) + (6*B*a*b^2*c*d^2)/(a
^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2) - (B*(a*d + b*c))/(2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (6*B*b^3*d^3*x^3)/(a^
2*d^2 + b^2*c^2 - 2*a*b*c*d)^2 + (9*B*b^2*d^2*x^2*(a*d + b*c))/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2 + (3*B*a*b*c*
d*(a*d + b*c))/(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)^2))/(x*(2*a*b*c^2*g^3*i^3 + 2*a^2*c*d*g^3*i^3) + x^3*(2*a*b*d^2
*g^3*i^3 + 2*b^2*c*d*g^3*i^3) + x^2*(a^2*d^2*g^3*i^3 + b^2*c^2*g^3*i^3 + 4*a*b*c*d*g^3*i^3) + a^2*c^2*g^3*i^3
+ b^2*d^2*g^3*i^3*x^4) + (A*b^2*d^2*atan(((a^5*d^5*g^3*i^3 + b^5*c^5*g^3*i^3 - 3*a*b^4*c^4*d*g^3*i^3 - 3*a^4*b
*c*d^4*g^3*i^3 + 2*a^2*b^3*c^3*d^2*g^3*i^3 + 2*a^3*b^2*c^2*d^3*g^3*i^3)*1i)/(g^3*i^3*(a*d - b*c)^5) + (b*d*x*(
a^4*d^4*g^3*i^3 + b^4*c^4*g^3*i^3 - 4*a*b^3*c^3*d*g^3*i^3 - 4*a^3*b*c*d^3*g^3*i^3 + 6*a^2*b^2*c^2*d^2*g^3*i^3)
*2i)/(g^3*i^3*(a*d - b*c)^5))*12i)/(g^3*i^3*(a*d - b*c)^5) - (3*B*b^2*d^2*log(e*((a + b*x)/(c + d*x))^n)^2)/(g
^3*i^3*n*(a*d - b*c)^5)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))/(b*g*x+a*g)**3/(d*i*x+c*i)**3,x)

[Out]

Timed out

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