3.18 \(\int \frac {(A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^5} \, dx\)

Optimal. Leaf size=615 \[ -\frac {b^3 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 g^5 (a+b x)^4 (b c-a d)^4}-\frac {b^3 B n (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{8 g^5 (a+b x)^4 (b c-a d)^4}+\frac {b^2 d (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^5 (a+b x)^3 (b c-a d)^4}+\frac {2 b^2 B d n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 g^5 (a+b x)^3 (b c-a d)^4}+\frac {d^3 (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{g^5 (a+b x) (b c-a d)^4}+\frac {2 B d^3 n (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^5 (a+b x) (b c-a d)^4}-\frac {3 b d^2 (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{2 g^5 (a+b x)^2 (b c-a d)^4}-\frac {3 b B d^2 n (c+d x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 g^5 (a+b x)^2 (b c-a d)^4}-\frac {b^3 B^2 n^2 (c+d x)^4}{32 g^5 (a+b x)^4 (b c-a d)^4}+\frac {2 b^2 B^2 d n^2 (c+d x)^3}{9 g^5 (a+b x)^3 (b c-a d)^4}+\frac {2 B^2 d^3 n^2 (c+d x)}{g^5 (a+b x) (b c-a d)^4}-\frac {3 b B^2 d^2 n^2 (c+d x)^2}{4 g^5 (a+b x)^2 (b c-a d)^4} \]

[Out]

2*B^2*d^3*n^2*(d*x+c)/(-a*d+b*c)^4/g^5/(b*x+a)-3/4*b*B^2*d^2*n^2*(d*x+c)^2/(-a*d+b*c)^4/g^5/(b*x+a)^2+2/9*b^2*
B^2*d*n^2*(d*x+c)^3/(-a*d+b*c)^4/g^5/(b*x+a)^3-1/32*b^3*B^2*n^2*(d*x+c)^4/(-a*d+b*c)^4/g^5/(b*x+a)^4+2*B*d^3*n
*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4/g^5/(b*x+a)-3/2*b*B*d^2*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(
d*x+c))^n))/(-a*d+b*c)^4/g^5/(b*x+a)^2+2/3*b^2*B*d*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4/g^
5/(b*x+a)^3-1/8*b^3*B*n*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^4/g^5/(b*x+a)^4+d^3*(d*x+c)*(A+B*
ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^4/g^5/(b*x+a)-3/2*b*d^2*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-
a*d+b*c)^4/g^5/(b*x+a)^2+b^2*d*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^4/g^5/(b*x+a)^3-1/4*b^3*
(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^4/g^5/(b*x+a)^4

________________________________________________________________________________________

Rubi [C]  time = 1.31, antiderivative size = 826, normalized size of antiderivative = 1.34, number of steps used = 36, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {2525, 12, 2528, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac {B^2 n^2 \log ^2(a+b x) d^4}{4 b (b c-a d)^4 g^5}-\frac {B^2 n^2 \log ^2(c+d x) d^4}{4 b (b c-a d)^4 g^5}+\frac {25 B^2 n^2 \log (a+b x) d^4}{24 b (b c-a d)^4 g^5}+\frac {B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) d^4}{2 b (b c-a d)^4 g^5}-\frac {25 B^2 n^2 \log (c+d x) d^4}{24 b (b c-a d)^4 g^5}+\frac {B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) d^4}{2 b (b c-a d)^4 g^5}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) d^4}{2 b (b c-a d)^4 g^5}+\frac {B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) d^4}{2 b (b c-a d)^4 g^5}+\frac {B^2 n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) d^4}{2 b (b c-a d)^4 g^5}+\frac {B^2 n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) d^4}{2 b (b c-a d)^4 g^5}+\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) d^3}{2 b (b c-a d)^3 g^5 (a+b x)}+\frac {25 B^2 n^2 d^3}{24 b (b c-a d)^3 g^5 (a+b x)}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) d^2}{4 b (b c-a d)^2 g^5 (a+b x)^2}-\frac {13 B^2 n^2 d^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) d}{6 b (b c-a d) g^5 (a+b x)^3}+\frac {7 B^2 n^2 d}{72 b (b c-a d) g^5 (a+b x)^3}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 b g^5 (a+b x)^4}-\frac {B^2 n^2}{32 b g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^5,x]

[Out]

-(B^2*n^2)/(32*b*g^5*(a + b*x)^4) + (7*B^2*d*n^2)/(72*b*(b*c - a*d)*g^5*(a + b*x)^3) - (13*B^2*d^2*n^2)/(48*b*
(b*c - a*d)^2*g^5*(a + b*x)^2) + (25*B^2*d^3*n^2)/(24*b*(b*c - a*d)^3*g^5*(a + b*x)) + (25*B^2*d^4*n^2*Log[a +
b*x])/(24*b*(b*c - a*d)^4*g^5) - (B^2*d^4*n^2*Log[a + b*x]^2)/(4*b*(b*c - a*d)^4*g^5) - (B*n*(A + B*Log[e*((a
+ b*x)/(c + d*x))^n]))/(8*b*g^5*(a + b*x)^4) + (B*d*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*b*(b*c - a*d
)*g^5*(a + b*x)^3) - (B*d^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*b*(b*c - a*d)^2*g^5*(a + b*x)^2) + (B
*d^3*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b*(b*c - a*d)^3*g^5*(a + b*x)) + (B*d^4*n*Log[a + b*x]*(A +
B*Log[e*((a + b*x)/(c + d*x))^n]))/(2*b*(b*c - a*d)^4*g^5) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(4*b*g^5
*(a + b*x)^4) - (25*B^2*d^4*n^2*Log[c + d*x])/(24*b*(b*c - a*d)^4*g^5) + (B^2*d^4*n^2*Log[-((d*(a + b*x))/(b*c
- a*d))]*Log[c + d*x])/(2*b*(b*c - a*d)^4*g^5) - (B*d^4*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x]
)/(2*b*(b*c - a*d)^4*g^5) - (B^2*d^4*n^2*Log[c + d*x]^2)/(4*b*(b*c - a*d)^4*g^5) + (B^2*d^4*n^2*Log[a + b*x]*L
og[(b*(c + d*x))/(b*c - a*d)])/(2*b*(b*c - a*d)^4*g^5) + (B^2*d^4*n^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))]
)/(2*b*(b*c - a*d)^4*g^5) + (B^2*d^4*n^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(2*b*(b*c - a*d)^4*g^5)

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 {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a g+b g x)^5} \, dx &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac {(B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{g^4 (a+b x)^5 (c+d x)} \, dx}{2 b g}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac {(B (b c-a d) n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^5 (c+d x)} \, dx}{2 b g^5}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac {(B (b c-a d) n) \int \left (\frac {b \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (a+b x)^5}-\frac {b d \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 (a+b x)}-\frac {d^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{2 b g^5}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac {(B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^5} \, dx}{2 g^5}+\frac {\left (B d^4 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{2 (b c-a d)^4 g^5}-\frac {\left (B d^5 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}-\frac {\left (B d^3 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^2} \, dx}{2 (b c-a d)^3 g^5}+\frac {\left (B d^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3} \, dx}{2 (b c-a d)^2 g^5}-\frac {(B d n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^4} \, dx}{2 (b c-a d) g^5}\\ &=-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 b g^5 (a+b x)^4}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}+\frac {B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}+\frac {\left (B^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{8 b g^5}-\frac {\left (B^2 d^4 n^2\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}{2 b (b c-a d)^4 g^5}+\frac {\left (B^2 d^4 n^2\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}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^3 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{2 b (b c-a d)^3 g^5}+\frac {\left (B^2 d^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{4 b (b c-a d)^2 g^5}-\frac {\left (B^2 d n^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{6 b (b c-a d) g^5}\\ &=-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 b g^5 (a+b x)^4}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}+\frac {B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d n^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{6 b g^5}-\frac {\left (B^2 d^4 n^2\right ) \int \left (\frac {b \log (a+b x)}{a+b x}-\frac {d \log (a+b x)}{c+d x}\right ) \, dx}{2 b (b c-a d)^4 g^5}+\frac {\left (B^2 d^4 n^2\right ) \int \left (\frac {b \log (c+d x)}{a+b x}-\frac {d \log (c+d x)}{c+d x}\right ) \, dx}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^3 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{2 b (b c-a d)^2 g^5}+\frac {\left (B^2 d^2 n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{4 b (b c-a d) g^5}+\frac {\left (B^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{8 b g^5}\\ &=-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 b g^5 (a+b x)^4}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}+\frac {B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d n^2\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}{6 b g^5}-\frac {\left (B^2 d^4 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{2 (b c-a d)^4 g^5}+\frac {\left (B^2 d^4 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{2 (b c-a d)^4 g^5}+\frac {\left (B^2 d^5 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^5 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^3 n^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}{2 b (b c-a d)^2 g^5}+\frac {\left (B^2 d^2 n^2\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}{4 b (b c-a d) g^5}+\frac {\left (B^2 (b c-a d) n^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{8 b g^5}\\ &=-\frac {B^2 n^2}{32 b g^5 (a+b x)^4}+\frac {7 B^2 d n^2}{72 b (b c-a d) g^5 (a+b x)^3}-\frac {13 B^2 d^2 n^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {25 B^2 d^3 n^2}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac {25 B^2 d^4 n^2 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 b g^5 (a+b x)^4}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}+\frac {B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {25 B^2 d^4 n^2 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac {B^2 d^4 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}+\frac {B^2 d^4 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^4 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{2 (b c-a d)^4 g^5}-\frac {\left (B^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^5 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}\\ &=-\frac {B^2 n^2}{32 b g^5 (a+b x)^4}+\frac {7 B^2 d n^2}{72 b (b c-a d) g^5 (a+b x)^3}-\frac {13 B^2 d^2 n^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {25 B^2 d^3 n^2}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac {25 B^2 d^4 n^2 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac {B^2 d^4 n^2 \log ^2(a+b x)}{4 b (b c-a d)^4 g^5}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 b g^5 (a+b x)^4}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}+\frac {B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {25 B^2 d^4 n^2 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac {B^2 d^4 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {B^2 d^4 n^2 \log ^2(c+d x)}{4 b (b c-a d)^4 g^5}+\frac {B^2 d^4 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^4 n^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 )}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^4 n^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 )}{2 b (b c-a d)^4 g^5}\\ &=-\frac {B^2 n^2}{32 b g^5 (a+b x)^4}+\frac {7 B^2 d n^2}{72 b (b c-a d) g^5 (a+b x)^3}-\frac {13 B^2 d^2 n^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {25 B^2 d^3 n^2}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac {25 B^2 d^4 n^2 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac {B^2 d^4 n^2 \log ^2(a+b x)}{4 b (b c-a d)^4 g^5}-\frac {B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 b g^5 (a+b x)^4}+\frac {B d n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b (b c-a d) g^5 (a+b x)^3}-\frac {B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^3 g^5 (a+b x)}+\frac {B d^4 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {25 B^2 d^4 n^2 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac {B^2 d^4 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {B d^4 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {B^2 d^4 n^2 \log ^2(c+d x)}{4 b (b c-a d)^4 g^5}+\frac {B^2 d^4 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}+\frac {B^2 d^4 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}+\frac {B^2 d^4 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}\\ \end {align*}

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Mathematica [C]  time = 1.02, size = 776, normalized size = 1.26 \[ -\frac {\frac {B n \left (-144 d^4 (a+b x)^4 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+144 d^4 (a+b x)^4 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+144 d^3 (a+b x)^3 (a d-b c) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+72 d^2 (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+36 (b c-a d)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+48 d (a+b x) (a d-b c)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+72 B d^4 n (a+b x)^4 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )-72 B d^4 n (a+b x)^4 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-144 B d^3 n (a+b x)^3 (-d (a+b x) \log (c+d x)+d (a+b x) \log (a+b x)-a d+b c)+36 B d^2 n (a+b x)^2 \left (2 d^2 (a+b x)^2 \log (c+d x)+2 d (a+b x) (a d-b c)+(b c-a d)^2-2 d^2 (a+b x)^2 \log (a+b x)\right )-8 B d n (a+b x) \left (-6 d^3 (a+b x)^3 \log (c+d x)+6 d^2 (a+b x)^2 (b c-a d)-3 d (a+b x) (b c-a d)^2+2 (b c-a d)^3+6 d^3 (a+b x)^3 \log (a+b x)\right )+3 B n \left (12 d^4 (a+b x)^4 \log (c+d x)+12 d^3 (a+b x)^3 (a d-b c)+6 d^2 (a+b x)^2 (b c-a d)^2+4 d (a+b x) (a d-b c)^3+3 (b c-a d)^4-12 d^4 (a+b x)^4 \log (a+b x)\right )\right )}{(b c-a d)^4}+72 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{288 b g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(a*g + b*g*x)^5,x]

[Out]

-1/288*(72*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + (B*n*(36*(b*c - a*d)^4*(A + B*Log[e*((a + b*x)/(c + d*x)
)^n]) + 48*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 72*d^2*(b*c - a*d)^2*(a + b*x
)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 144*d^3*(-(b*c) + a*d)*(a + b*x)^3*(A + B*Log[e*((a + b*x)/(c + d
*x))^n]) - 144*d^4*(a + b*x)^4*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 144*d^4*(a + b*x)^4*(A +
B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - 144*B*d^3*n*(a + b*x)^3*(b*c - a*d + d*(a + b*x)*Log[a + b*x]
- d*(a + b*x)*Log[c + d*x]) + 36*B*d^2*n*(a + b*x)^2*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x) - 2*d^2*(a
+ b*x)^2*Log[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d*x]) - 8*B*d*n*(a + b*x)*(2*(b*c - a*d)^3 - 3*d*(b*c - a*d
)^2*(a + b*x) + 6*d^2*(b*c - a*d)*(a + b*x)^2 + 6*d^3*(a + b*x)^3*Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[c + d*x
]) + 3*B*n*(3*(b*c - a*d)^4 + 4*d*(-(b*c) + a*d)^3*(a + b*x) + 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 12*d^3*(-(b*c
) + a*d)*(a + b*x)^3 - 12*d^4*(a + b*x)^4*Log[a + b*x] + 12*d^4*(a + b*x)^4*Log[c + d*x]) + 72*B*d^4*n*(a + b*
x)^4*(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)]) - 72*B*d^4*n*(a + b*x)^4*((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)^4)/(b*g^5*(a + b*x)^4)

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fricas [B]  time = 1.08, size = 1762, normalized size = 2.87 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^5,x, algorithm="fricas")

[Out]

-1/288*(72*A^2*b^4*c^4 - 288*A^2*a*b^3*c^3*d + 432*A^2*a^2*b^2*c^2*d^2 - 288*A^2*a^3*b*c*d^3 + 72*A^2*a^4*d^4
- 12*(25*(B^2*b^4*c*d^3 - B^2*a*b^3*d^4)*n^2 + 12*(A*B*b^4*c*d^3 - A*B*a*b^3*d^4)*n)*x^3 + (9*B^2*b^4*c^4 - 64
*B^2*a*b^3*c^3*d + 216*B^2*a^2*b^2*c^2*d^2 - 576*B^2*a^3*b*c*d^3 + 415*B^2*a^4*d^4)*n^2 + 6*((13*B^2*b^4*c^2*d
^2 - 176*B^2*a*b^3*c*d^3 + 163*B^2*a^2*b^2*d^4)*n^2 + 12*(A*B*b^4*c^2*d^2 - 8*A*B*a*b^3*c*d^3 + 7*A*B*a^2*b^2*
d^4)*n)*x^2 + 72*(B^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2*d^2 - 4*B^2*a^3*b*c*d^3 + B^2*a^4*d^4)*l
og(e)^2 - 72*(B^2*b^4*d^4*n^2*x^4 + 4*B^2*a*b^3*d^4*n^2*x^3 + 6*B^2*a^2*b^2*d^4*n^2*x^2 + 4*B^2*a^3*b*d^4*n^2*
x - (B^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2*d^2 - 4*B^2*a^3*b*c*d^3)*n^2)*log((b*x + a)/(d*x + c)
)^2 + 12*(3*A*B*b^4*c^4 - 16*A*B*a*b^3*c^3*d + 36*A*B*a^2*b^2*c^2*d^2 - 48*A*B*a^3*b*c*d^3 + 25*A*B*a^4*d^4)*n
- 4*((7*B^2*b^4*c^3*d - 60*B^2*a*b^3*c^2*d^2 + 324*B^2*a^2*b^2*c*d^3 - 271*B^2*a^3*b*d^4)*n^2 + 12*(A*B*b^4*c
^3*d - 6*A*B*a*b^3*c^2*d^2 + 18*A*B*a^2*b^2*c*d^3 - 13*A*B*a^3*b*d^4)*n)*x + 12*(12*A*B*b^4*c^4 - 48*A*B*a*b^3
*c^3*d + 72*A*B*a^2*b^2*c^2*d^2 - 48*A*B*a^3*b*c*d^3 + 12*A*B*a^4*d^4 - 12*(B^2*b^4*c*d^3 - B^2*a*b^3*d^4)*n*x
^3 + 6*(B^2*b^4*c^2*d^2 - 8*B^2*a*b^3*c*d^3 + 7*B^2*a^2*b^2*d^4)*n*x^2 - 4*(B^2*b^4*c^3*d - 6*B^2*a*b^3*c^2*d^
2 + 18*B^2*a^2*b^2*c*d^3 - 13*B^2*a^3*b*d^4)*n*x + (3*B^2*b^4*c^4 - 16*B^2*a*b^3*c^3*d + 36*B^2*a^2*b^2*c^2*d^
2 - 48*B^2*a^3*b*c*d^3 + 25*B^2*a^4*d^4)*n - 12*(B^2*b^4*d^4*n*x^4 + 4*B^2*a*b^3*d^4*n*x^3 + 6*B^2*a^2*b^2*d^4
*n*x^2 + 4*B^2*a^3*b*d^4*n*x - (B^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d + 6*B^2*a^2*b^2*c^2*d^2 - 4*B^2*a^3*b*c*d^3)*n
)*log((b*x + a)/(d*x + c)))*log(e) - 12*((25*B^2*b^4*d^4*n^2 + 12*A*B*b^4*d^4*n)*x^4 + 4*(12*A*B*a*b^3*d^4*n +
(3*B^2*b^4*c*d^3 + 22*B^2*a*b^3*d^4)*n^2)*x^3 - (3*B^2*b^4*c^4 - 16*B^2*a*b^3*c^3*d + 36*B^2*a^2*b^2*c^2*d^2
- 48*B^2*a^3*b*c*d^3)*n^2 + 6*(12*A*B*a^2*b^2*d^4*n - (B^2*b^4*c^2*d^2 - 8*B^2*a*b^3*c*d^3 - 18*B^2*a^2*b^2*d^
4)*n^2)*x^2 - 12*(A*B*b^4*c^4 - 4*A*B*a*b^3*c^3*d + 6*A*B*a^2*b^2*c^2*d^2 - 4*A*B*a^3*b*c*d^3)*n + 4*(12*A*B*a
^3*b*d^4*n + (B^2*b^4*c^3*d - 6*B^2*a*b^3*c^2*d^2 + 18*B^2*a^2*b^2*c*d^3 + 12*B^2*a^3*b*d^4)*n^2)*x)*log((b*x
+ a)/(d*x + c)))/((b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*g^5*x^4 + 4*(a
*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*g^5*x^3 + 6*(a^2*b^7*c^4 - 4*a
^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a^5*b^4*c*d^3 + a^6*b^3*d^4)*g^5*x^2 + 4*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d +
6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7*b^2*d^4)*g^5*x + (a^4*b^5*c^4 - 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2
- 4*a^7*b^2*c*d^3 + a^8*b*d^4)*g^5)

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giac [A]  time = 17.90, size = 1166, normalized size = 1.90 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^5,x, algorithm="giac")

[Out]

-1/288*(72*(B^2*b^3*n^2 - 4*(b*x + a)*B^2*b^2*d*n^2/(d*x + c) + 6*(b*x + a)^2*B^2*b*d^2*n^2/(d*x + c)^2 - 4*(b
*x + a)^3*B^2*d^3*n^2/(d*x + c)^3)*log((b*x + a)/(d*x + c))^2/((b*x + a)^4*b^3*c^3*g^5/(d*x + c)^4 - 3*(b*x +
a)^4*a*b^2*c^2*d*g^5/(d*x + c)^4 + 3*(b*x + a)^4*a^2*b*c*d^2*g^5/(d*x + c)^4 - (b*x + a)^4*a^3*d^3*g^5/(d*x +
c)^4) + 12*(3*B^2*b^3*n^2 - 16*(b*x + a)*B^2*b^2*d*n^2/(d*x + c) + 36*(b*x + a)^2*B^2*b*d^2*n^2/(d*x + c)^2 -
48*(b*x + a)^3*B^2*d^3*n^2/(d*x + c)^3 + 12*A*B*b^3*n + 12*B^2*b^3*n - 48*(b*x + a)*A*B*b^2*d*n/(d*x + c) - 48
*(b*x + a)*B^2*b^2*d*n/(d*x + c) + 72*(b*x + a)^2*A*B*b*d^2*n/(d*x + c)^2 + 72*(b*x + a)^2*B^2*b*d^2*n/(d*x +
c)^2 - 48*(b*x + a)^3*A*B*d^3*n/(d*x + c)^3 - 48*(b*x + a)^3*B^2*d^3*n/(d*x + c)^3)*log((b*x + a)/(d*x + c))/(
(b*x + a)^4*b^3*c^3*g^5/(d*x + c)^4 - 3*(b*x + a)^4*a*b^2*c^2*d*g^5/(d*x + c)^4 + 3*(b*x + a)^4*a^2*b*c*d^2*g^
5/(d*x + c)^4 - (b*x + a)^4*a^3*d^3*g^5/(d*x + c)^4) + (9*B^2*b^3*n^2 - 64*(b*x + a)*B^2*b^2*d*n^2/(d*x + c) +
216*(b*x + a)^2*B^2*b*d^2*n^2/(d*x + c)^2 - 576*(b*x + a)^3*B^2*d^3*n^2/(d*x + c)^3 + 36*A*B*b^3*n + 36*B^2*b
^3*n - 192*(b*x + a)*A*B*b^2*d*n/(d*x + c) - 192*(b*x + a)*B^2*b^2*d*n/(d*x + c) + 432*(b*x + a)^2*A*B*b*d^2*n
/(d*x + c)^2 + 432*(b*x + a)^2*B^2*b*d^2*n/(d*x + c)^2 - 576*(b*x + a)^3*A*B*d^3*n/(d*x + c)^3 - 576*(b*x + a)
^3*B^2*d^3*n/(d*x + c)^3 + 72*A^2*b^3 + 144*A*B*b^3 + 72*B^2*b^3 - 288*(b*x + a)*A^2*b^2*d/(d*x + c) - 576*(b*
x + a)*A*B*b^2*d/(d*x + c) - 288*(b*x + a)*B^2*b^2*d/(d*x + c) + 432*(b*x + a)^2*A^2*b*d^2/(d*x + c)^2 + 864*(
b*x + a)^2*A*B*b*d^2/(d*x + c)^2 + 432*(b*x + a)^2*B^2*b*d^2/(d*x + c)^2 - 288*(b*x + a)^3*A^2*d^3/(d*x + c)^3
- 576*(b*x + a)^3*A*B*d^3/(d*x + c)^3 - 288*(b*x + a)^3*B^2*d^3/(d*x + c)^3)/((b*x + a)^4*b^3*c^3*g^5/(d*x +
c)^4 - 3*(b*x + a)^4*a*b^2*c^2*d*g^5/(d*x + c)^4 + 3*(b*x + a)^4*a^2*b*c*d^2*g^5/(d*x + c)^4 - (b*x + a)^4*a^3
*d^3*g^5/(d*x + c)^4))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

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maple [F]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (b g x +a g \right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*ln(e*((b*x+a)/(d*x+c))^n)+A)^2/(b*g*x+a*g)^5,x)

[Out]

int((B*ln(e*((b*x+a)/(d*x+c))^n)+A)^2/(b*g*x+a*g)^5,x)

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maxima [B]  time = 2.98, size = 2136, normalized size = 3.47 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(b*g*x+a*g)^5,x, algorithm="maxima")

[Out]

1/24*A*B*n*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^
2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3
*b^5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 -
3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2
- a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) + 12*d^4*log(b*x +
a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b
^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5)) + 1/288*(12*n*((12*b^3*d^3*x^3
- 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d -
5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*g^5*x^4 + 4*(a*b^
7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^
4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*g^5*x + (a^
4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*
d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*
a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) - (9*b^4*c^4 - 64*
a*b^3*c^3*d + 216*a^2*b^2*c^2*d^2 - 576*a^3*b*c*d^3 + 415*a^4*d^4 - 300*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 6*(13*b^
4*c^2*d^2 - 176*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4
*a^3*b*d^4*x + a^4*d^4)*log(b*x + a)^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x
+ a^4*d^4)*log(d*x + c)^2 - 4*(7*b^4*c^3*d - 60*a*b^3*c^2*d^2 + 324*a^2*b^2*c*d^3 - 271*a^3*b*d^4)*x - 300*(b
^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a) + 12*(25*b^4*d^4*x^4
+ 100*a*b^3*d^4*x^3 + 150*a^2*b^2*d^4*x^2 + 100*a^3*b*d^4*x + 25*a^4*d^4 - 12*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 +
6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a))*log(d*x + c))*n^2/(a^4*b^5*c^4*g^5 - 4*a^5*b^4*c^3
*d*g^5 + 6*a^6*b^3*c^2*d^2*g^5 - 4*a^7*b^2*c*d^3*g^5 + a^8*b*d^4*g^5 + (b^9*c^4*g^5 - 4*a*b^8*c^3*d*g^5 + 6*a^
2*b^7*c^2*d^2*g^5 - 4*a^3*b^6*c*d^3*g^5 + a^4*b^5*d^4*g^5)*x^4 + 4*(a*b^8*c^4*g^5 - 4*a^2*b^7*c^3*d*g^5 + 6*a^
3*b^6*c^2*d^2*g^5 - 4*a^4*b^5*c*d^3*g^5 + a^5*b^4*d^4*g^5)*x^3 + 6*(a^2*b^7*c^4*g^5 - 4*a^3*b^6*c^3*d*g^5 + 6*
a^4*b^5*c^2*d^2*g^5 - 4*a^5*b^4*c*d^3*g^5 + a^6*b^3*d^4*g^5)*x^2 + 4*(a^3*b^6*c^4*g^5 - 4*a^4*b^5*c^3*d*g^5 +
6*a^5*b^4*c^2*d^2*g^5 - 4*a^6*b^3*c*d^3*g^5 + a^7*b^2*d^4*g^5)*x))*B^2 - 1/4*B^2*log(e*(b*x/(d*x + c) + a/(d*x
+ c))^n)^2/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) - 1/2*A*B*log(e*
(b*x/(d*x + c) + a/(d*x + c))^n)/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*
g^5) - 1/4*A^2/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5)

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mupad [B]  time = 9.22, size = 1769, normalized size = 2.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log(e*((a + b*x)/(c + d*x))^n))^2/(a*g + b*g*x)^5,x)

[Out]

(B*d^4*n*atan((B*d^4*n*(12*A + 25*B*n)*(24*b^5*c^4*g^5 - 24*a^4*b*d^4*g^5 - 48*a*b^4*c^3*d*g^5 + 48*a^3*b^2*c*
d^3*g^5)*1i)/(24*b*g^5*(25*B^2*d^4*n^2 + 12*A*B*d^4*n)*(a*d - b*c)^4) + (B*d^5*n*x*(12*A + 25*B*n)*(b^4*c^3*g^
5 - a^3*b*d^3*g^5 - 3*a*b^3*c^2*d*g^5 + 3*a^2*b^2*c*d^2*g^5)*2i)/(g^5*(25*B^2*d^4*n^2 + 12*A*B*d^4*n)*(a*d - b
*c)^4))*(12*A + 25*B*n)*1i)/(12*b*g^5*(a*d - b*c)^4) - ((72*A^2*a^3*d^3 - 72*A^2*b^3*c^3 + 415*B^2*a^3*d^3*n^2
- 9*B^2*b^3*c^3*n^2 + 216*A^2*a*b^2*c^2*d - 216*A^2*a^2*b*c*d^2 + 300*A*B*a^3*d^3*n - 36*A*B*b^3*c^3*n + 55*B
^2*a*b^2*c^2*d*n^2 - 161*B^2*a^2*b*c*d^2*n^2 + 156*A*B*a*b^2*c^2*d*n - 276*A*B*a^2*b*c*d^2*n)/(12*(a*d - b*c))
+ (x^2*(163*B^2*a*b^2*d^3*n^2 - 13*B^2*b^3*c*d^2*n^2 + 84*A*B*a*b^2*d^3*n - 12*A*B*b^3*c*d^2*n))/(2*(a*d - b*
c)) + (x*(271*B^2*a^2*b*d^3*n^2 + 7*B^2*b^3*c^2*d*n^2 - 53*B^2*a*b^2*c*d^2*n^2 + 156*A*B*a^2*b*d^3*n + 12*A*B*
b^3*c^2*d*n - 60*A*B*a*b^2*c*d^2*n))/(3*(a*d - b*c)) + (d*x^3*(25*B^2*b^3*d^2*n^2 + 12*A*B*b^3*d^2*n))/(a*d -
b*c))/(x*(96*a^3*b^4*c^2*g^5 + 96*a^5*b^2*d^2*g^5 - 192*a^4*b^3*c*d*g^5) + x^3*(96*a*b^6*c^2*g^5 + 96*a^3*b^4*
d^2*g^5 - 192*a^2*b^5*c*d*g^5) + x^4*(24*b^7*c^2*g^5 + 24*a^2*b^5*d^2*g^5 - 48*a*b^6*c*d*g^5) + x^2*(144*a^2*b
^5*c^2*g^5 + 144*a^4*b^3*d^2*g^5 - 288*a^3*b^4*c*d*g^5) + 24*a^6*b*d^2*g^5 + 24*a^4*b^3*c^2*g^5 - 48*a^5*b^2*c
*d*g^5) - log(e*((a + b*x)/(c + d*x))^n)^2*(B^2/(4*b*(a^4*g^5 + b^4*g^5*x^4 + 4*a*b^3*g^5*x^3 + 6*a^2*b^2*g^5*
x^2 + 4*a^3*b*g^5*x)) - (B^2*d^4)/(4*b*g^5*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*
d^3))) - log(e*((a + b*x)/(c + d*x))^n)*((A*B)/(2*a^4*b*g^5 + 2*b^5*g^5*x^4 + 8*a^3*b^2*g^5*x + 8*a*b^4*g^5*x^
3 + 12*a^2*b^3*g^5*x^2) + (B^2*d^4*(x*(b*(a*((b*g^5*n*(a*d - b*c)*(4*a*d - b*c))/(6*d^2) + (a*b*g^5*n*(a*d - b
*c))/(2*d)) + (b*g^5*n*(a*d - b*c)*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(6*d^3)) + a*(b*((b*g^5*n*(a*d - b*c)*(4
*a*d - b*c))/(6*d^2) + (a*b*g^5*n*(a*d - b*c))/(2*d)) + (a*b^2*g^5*n*(a*d - b*c))/d + (b^2*g^5*n*(a*d - b*c)*(
4*a*d - b*c))/(3*d^2)) + (b^2*g^5*n*(a*d - b*c)*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(2*d^3)) + a*(a*((b*g^5*n*(
a*d - b*c)*(4*a*d - b*c))/(6*d^2) + (a*b*g^5*n*(a*d - b*c))/(2*d)) + (b*g^5*n*(a*d - b*c)*(6*a^2*d^2 + b^2*c^2
- 4*a*b*c*d))/(6*d^3)) + x^2*(b*(b*((b*g^5*n*(a*d - b*c)*(4*a*d - b*c))/(6*d^2) + (a*b*g^5*n*(a*d - b*c))/(2*
d)) + (a*b^2*g^5*n*(a*d - b*c))/d + (b^2*g^5*n*(a*d - b*c)*(4*a*d - b*c))/(3*d^2)) + (3*a*b^3*g^5*n*(a*d - b*c
))/(2*d) + (b^3*g^5*n*(a*d - b*c)*(4*a*d - b*c))/(2*d^2)) + (2*b^4*g^5*n*x^3*(a*d - b*c))/d + (b*g^5*n*(a*d -
b*c)*(4*a^3*d^3 - b^3*c^3 + 4*a*b^2*c^2*d - 6*a^2*b*c*d^2))/(2*d^4)))/(2*b*g^5*(2*a^4*b*g^5 + 2*b^5*g^5*x^4 +
8*a^3*b^2*g^5*x + 8*a*b^4*g^5*x^3 + 12*a^2*b^3*g^5*x^2)*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d
- 4*a^3*b*c*d^3)))

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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))**2/(b*g*x+a*g)**5,x)

[Out]

Timed out

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