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

Optimal. Leaf size=319 \[ -\frac {b i^2 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 g^5 (a+b x)^4 (b c-a d)^2}-\frac {b B i^2 n (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{8 g^5 (a+b x)^4 (b c-a d)^2}+\frac {d i^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 g^5 (a+b x)^3 (b c-a d)^2}+\frac {2 B d i^2 n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 g^5 (a+b x)^3 (b c-a d)^2}-\frac {b B^2 i^2 n^2 (c+d x)^4}{32 g^5 (a+b x)^4 (b c-a d)^2}+\frac {2 B^2 d i^2 n^2 (c+d x)^3}{27 g^5 (a+b x)^3 (b c-a d)^2} \]

[Out]

2/27*B^2*d*i^2*n^2*(d*x+c)^3/(-a*d+b*c)^2/g^5/(b*x+a)^3-1/32*b*B^2*i^2*n^2*(d*x+c)^4/(-a*d+b*c)^2/g^5/(b*x+a)^
4+2/9*B*d*i^2*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^2/g^5/(b*x+a)^3-1/8*b*B*i^2*n*(d*x+c)^4*(
A+B*ln(e*((b*x+a)/(d*x+c))^n))/(-a*d+b*c)^2/g^5/(b*x+a)^4+1/3*d*i^2*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^
2/(-a*d+b*c)^2/g^5/(b*x+a)^3-1/4*b*i^2*(d*x+c)^4*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/(-a*d+b*c)^2/g^5/(b*x+a)^4

________________________________________________________________________________________

Rubi [C]  time = 3.79, antiderivative size = 989, normalized size of antiderivative = 3.10, number of steps used = 98, number of rules used = 11, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.244, Rules used = {2528, 2525, 12, 44, 2524, 2418, 2390, 2301, 2394, 2393, 2391} \[ -\frac {B^2 i^2 n^2 \log ^2(a+b x) d^4}{12 b^3 (b c-a d)^2 g^5}-\frac {B^2 i^2 n^2 \log ^2(c+d x) d^4}{12 b^3 (b c-a d)^2 g^5}+\frac {7 B^2 i^2 n^2 \log (a+b x) d^4}{72 b^3 (b c-a d)^2 g^5}+\frac {B i^2 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}{6 b^3 (b c-a d)^2 g^5}-\frac {7 B^2 i^2 n^2 \log (c+d x) d^4}{72 b^3 (b c-a d)^2 g^5}+\frac {B^2 i^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) d^4}{6 b^3 (b c-a d)^2 g^5}-\frac {B i^2 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}{6 b^3 (b c-a d)^2 g^5}+\frac {B^2 i^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) d^4}{6 b^3 (b c-a d)^2 g^5}+\frac {B^2 i^2 n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) d^4}{6 b^3 (b c-a d)^2 g^5}+\frac {B^2 i^2 n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) d^4}{6 b^3 (b c-a d)^2 g^5}+\frac {B i^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) d^3}{6 b^3 (b c-a d) g^5 (a+b x)}+\frac {7 B^2 i^2 n^2 d^3}{72 b^3 (b c-a d) g^5 (a+b x)}-\frac {i^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 d^2}{2 b^3 g^5 (a+b x)^2}-\frac {B i^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) d^2}{12 b^3 g^5 (a+b x)^2}+\frac {5 B^2 i^2 n^2 d^2}{144 b^3 g^5 (a+b x)^2}-\frac {2 (b c-a d) i^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 d}{3 b^3 g^5 (a+b x)^3}-\frac {5 B (b c-a d) i^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) d}{18 b^3 g^5 (a+b x)^3}-\frac {11 B^2 (b c-a d) i^2 n^2 d}{216 b^3 g^5 (a+b x)^3}-\frac {(b c-a d)^2 i^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 b^3 g^5 (a+b x)^4}-\frac {B (b c-a d)^2 i^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{8 b^3 g^5 (a+b x)^4}-\frac {B^2 (b c-a d)^2 i^2 n^2}{32 b^3 g^5 (a+b x)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(B^2*(b*c - a*d)^2*i^2*n^2)/(32*b^3*g^5*(a + b*x)^4) - (11*B^2*d*(b*c - a*d)*i^2*n^2)/(216*b^3*g^5*(a + b*x)^
3) + (5*B^2*d^2*i^2*n^2)/(144*b^3*g^5*(a + b*x)^2) + (7*B^2*d^3*i^2*n^2)/(72*b^3*(b*c - a*d)*g^5*(a + b*x)) +
(7*B^2*d^4*i^2*n^2*Log[a + b*x])/(72*b^3*(b*c - a*d)^2*g^5) - (B^2*d^4*i^2*n^2*Log[a + b*x]^2)/(12*b^3*(b*c -
a*d)^2*g^5) - (B*(b*c - a*d)^2*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(8*b^3*g^5*(a + b*x)^4) - (5*B*d*
(b*c - a*d)*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(18*b^3*g^5*(a + b*x)^3) - (B*d^2*i^2*n*(A + B*Log[e
*((a + b*x)/(c + d*x))^n]))/(12*b^3*g^5*(a + b*x)^2) + (B*d^3*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6
*b^3*(b*c - a*d)*g^5*(a + b*x)) + (B*d^4*i^2*n*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*b^3*(b*
c - a*d)^2*g^5) - ((b*c - a*d)^2*i^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*b^3*g^5*(a + b*x)^4) - (2*d*
(b*c - a*d)*i^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(3*b^3*g^5*(a + b*x)^3) - (d^2*i^2*(A + B*Log[e*((a
+ b*x)/(c + d*x))^n])^2)/(2*b^3*g^5*(a + b*x)^2) - (7*B^2*d^4*i^2*n^2*Log[c + d*x])/(72*b^3*(b*c - a*d)^2*g^5)
+ (B^2*d^4*i^2*n^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(6*b^3*(b*c - a*d)^2*g^5) - (B*d^4*i^2*n*(
A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x])/(6*b^3*(b*c - a*d)^2*g^5) - (B^2*d^4*i^2*n^2*Log[c + d*x]^
2)/(12*b^3*(b*c - a*d)^2*g^5) + (B^2*d^4*i^2*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)])/(6*b^3*(b*c - a*
d)^2*g^5) + (B^2*d^4*i^2*n^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(6*b^3*(b*c - a*d)^2*g^5) + (B^2*d^4*i^
2*n^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(6*b^3*(b*c - a*d)^2*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 {(176 c+176 d x)^2 \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 &=\int \left (\frac {30976 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^5 (a+b x)^5}+\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^5 (a+b x)^4}+\frac {30976 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^5 (a+b x)^3}\right ) \, dx\\ &=\frac {\left (30976 d^2\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^3} \, dx}{b^2 g^5}+\frac {(61952 d (b c-a d)) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^4} \, dx}{b^2 g^5}+\frac {\left (30976 (b c-a d)^2\right ) \int \frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(a+b x)^5} \, dx}{b^2 g^5}\\ &=-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}+\frac {\left (30976 B d^2 n\right ) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac {(123904 B d (b c-a d) 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 )}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^5}+\frac {\left (15488 B (b c-a d)^2 n\right ) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}+\frac {\left (30976 B d^2 (b c-a d) n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (123904 B d (b c-a 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)^4 (c+d x)} \, dx}{3 b^3 g^5}+\frac {\left (15488 B (b c-a 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)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}+\frac {\left (30976 B d^2 (b c-a d) n\right ) \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)^3}-\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)^2}+\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)}-\frac {d^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b^3 g^5}+\frac {\left (123904 B d (b c-a d)^2 n\right ) \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)^4}-\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)^3}+\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)^2}-\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)}+\frac {d^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b^3 g^5}+\frac {\left (15488 B (b c-a d)^3 n\right ) \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}{b^3 g^5}\\ &=-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}+\frac {\left (15488 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}{b^2 g^5}+\frac {\left (30976 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}{b^2 g^5}-\frac {\left (123904 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}{3 b^2 g^5}+\frac {\left (15488 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}{b^2 (b c-a d)^2 g^5}+\frac {\left (30976 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}{b^2 (b c-a d)^2 g^5}-\frac {\left (123904 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}{3 b^2 (b c-a d)^2 g^5}-\frac {\left (15488 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}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 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}{b^3 (b c-a d)^2 g^5}+\frac {\left (123904 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}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (15488 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}{b^2 (b c-a d) g^5}-\frac {\left (30976 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}{b^2 (b c-a d) g^5}+\frac {\left (123904 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}{3 b^2 (b c-a d) g^5}-\frac {(15488 B d (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)^4} \, dx}{b^2 g^5}+\frac {(123904 B d (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)^4} \, dx}{3 b^2 g^5}+\frac {\left (15488 B (b c-a 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)^5} \, dx}{b^2 g^5}\\ &=-\frac {3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac {7744 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac {15488 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac {15488 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 )}{3 b^3 (b c-a d)^2 g^5}-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {15488 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)}{3 b^3 (b c-a d)^2 g^5}+\frac {\left (7744 B^2 d^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (15488 B^2 d^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}-\frac {\left (61952 B^2 d^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{3 b^3 g^5}-\frac {\left (15488 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}{b^3 (b c-a d)^2 g^5}+\frac {\left (15488 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}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 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}{b^3 (b c-a d)^2 g^5}+\frac {\left (30976 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}{b^3 (b c-a d)^2 g^5}+\frac {\left (123904 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}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (123904 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}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (15488 B^2 d^3 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^5}-\frac {\left (30976 B^2 d^3 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{b^3 (b c-a d) g^5}+\frac {\left (123904 B^2 d^3 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3 b^3 (b c-a d) g^5}-\frac {\left (15488 B^2 d (b c-a d) n^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^5}+\frac {\left (123904 B^2 d (b c-a d) n^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{9 b^3 g^5}+\frac {\left (3872 B^2 (b c-a d)^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac {3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac {7744 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac {15488 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac {15488 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 )}{3 b^3 (b c-a d)^2 g^5}-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {15488 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)}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (15488 B^2 d^3 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^5}-\frac {\left (30976 B^2 d^3 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (123904 B^2 d^3 n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3 b^3 g^5}-\frac {\left (15488 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}{b^3 (b c-a d)^2 g^5}+\frac {\left (15488 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}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 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}{b^3 (b c-a d)^2 g^5}+\frac {\left (30976 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}{b^3 (b c-a d)^2 g^5}+\frac {\left (123904 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}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (123904 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}{3 b^3 (b c-a d)^2 g^5}+\frac {\left (7744 B^2 d^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}+\frac {\left (15488 B^2 d^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^5}-\frac {\left (61952 B^2 d^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{3 b^3 g^5}-\frac {\left (15488 B^2 d (b c-a d)^2 n^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b^3 g^5}+\frac {\left (123904 B^2 d (b c-a d)^2 n^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{9 b^3 g^5}+\frac {\left (3872 B^2 (b c-a d)^3 n^2\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{b^3 g^5}\\ &=-\frac {3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac {7744 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac {15488 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac {15488 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 )}{3 b^3 (b c-a d)^2 g^5}-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {15488 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)}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (15488 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}{b^3 g^5}-\frac {\left (30976 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}{b^3 g^5}+\frac {\left (123904 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}{3 b^3 g^5}-\frac {\left (15488 B^2 d^4 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac {\left (15488 B^2 d^4 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}-\frac {\left (30976 B^2 d^4 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac {\left (30976 B^2 d^4 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{b^2 (b c-a d)^2 g^5}+\frac {\left (123904 B^2 d^4 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3 b^2 (b c-a d)^2 g^5}-\frac {\left (123904 B^2 d^4 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 b^2 (b c-a d)^2 g^5}+\frac {\left (15488 B^2 d^5 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (15488 B^2 d^5 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}+\frac {\left (30976 B^2 d^5 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 B^2 d^5 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b^3 (b c-a d)^2 g^5}-\frac {\left (123904 B^2 d^5 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 b^3 (b c-a d)^2 g^5}+\frac {\left (123904 B^2 d^5 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b^3 (b c-a d)^2 g^5}+\frac {\left (7744 B^2 d^2 (b c-a d) 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}{b^3 g^5}+\frac {\left (15488 B^2 d^2 (b c-a d) 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}{b^3 g^5}-\frac {\left (61952 B^2 d^2 (b c-a d) 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}{3 b^3 g^5}-\frac {\left (15488 B^2 d (b c-a d)^2 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}{3 b^3 g^5}+\frac {\left (123904 B^2 d (b c-a d)^2 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}{9 b^3 g^5}+\frac {\left (3872 B^2 (b c-a d)^3 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}{b^3 g^5}\\ &=-\frac {968 B^2 (b c-a d)^2 n^2}{b^3 g^5 (a+b x)^4}-\frac {42592 B^2 d (b c-a d) n^2}{27 b^3 g^5 (a+b x)^3}+\frac {9680 B^2 d^2 n^2}{9 b^3 g^5 (a+b x)^2}+\frac {27104 B^2 d^3 n^2}{9 b^3 (b c-a d) g^5 (a+b x)}+\frac {27104 B^2 d^4 n^2 \log (a+b x)}{9 b^3 (b c-a d)^2 g^5}-\frac {3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac {7744 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac {15488 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac {15488 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 )}{3 b^3 (b c-a d)^2 g^5}-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {27104 B^2 d^4 n^2 \log (c+d x)}{9 b^3 (b c-a d)^2 g^5}+\frac {15488 B^2 d^4 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac {15488 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)}{3 b^3 (b c-a d)^2 g^5}+\frac {15488 B^2 d^4 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (15488 B^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (15488 B^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 B^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 B^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b^3 (b c-a d)^2 g^5}+\frac {\left (123904 B^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 b^3 (b c-a d)^2 g^5}+\frac {\left (123904 B^2 d^4 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (15488 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}{b^2 (b c-a d)^2 g^5}-\frac {\left (30976 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}{b^2 (b c-a d)^2 g^5}+\frac {\left (123904 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}{3 b^2 (b c-a d)^2 g^5}-\frac {\left (15488 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}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 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}{b^3 (b c-a d)^2 g^5}+\frac {\left (123904 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}{3 b^3 (b c-a d)^2 g^5}\\ &=-\frac {968 B^2 (b c-a d)^2 n^2}{b^3 g^5 (a+b x)^4}-\frac {42592 B^2 d (b c-a d) n^2}{27 b^3 g^5 (a+b x)^3}+\frac {9680 B^2 d^2 n^2}{9 b^3 g^5 (a+b x)^2}+\frac {27104 B^2 d^3 n^2}{9 b^3 (b c-a d) g^5 (a+b x)}+\frac {27104 B^2 d^4 n^2 \log (a+b x)}{9 b^3 (b c-a d)^2 g^5}-\frac {7744 B^2 d^4 n^2 \log ^2(a+b x)}{3 b^3 (b c-a d)^2 g^5}-\frac {3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac {7744 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac {15488 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac {15488 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 )}{3 b^3 (b c-a d)^2 g^5}-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {27104 B^2 d^4 n^2 \log (c+d x)}{9 b^3 (b c-a d)^2 g^5}+\frac {15488 B^2 d^4 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac {15488 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)}{3 b^3 (b c-a d)^2 g^5}-\frac {7744 B^2 d^4 n^2 \log ^2(c+d x)}{3 b^3 (b c-a d)^2 g^5}+\frac {15488 B^2 d^4 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d)^2 g^5}-\frac {\left (15488 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 )}{b^3 (b c-a d)^2 g^5}-\frac {\left (15488 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 )}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 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 )}{b^3 (b c-a d)^2 g^5}-\frac {\left (30976 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 )}{b^3 (b c-a d)^2 g^5}+\frac {\left (123904 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 )}{3 b^3 (b c-a d)^2 g^5}+\frac {\left (123904 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 )}{3 b^3 (b c-a d)^2 g^5}\\ &=-\frac {968 B^2 (b c-a d)^2 n^2}{b^3 g^5 (a+b x)^4}-\frac {42592 B^2 d (b c-a d) n^2}{27 b^3 g^5 (a+b x)^3}+\frac {9680 B^2 d^2 n^2}{9 b^3 g^5 (a+b x)^2}+\frac {27104 B^2 d^3 n^2}{9 b^3 (b c-a d) g^5 (a+b x)}+\frac {27104 B^2 d^4 n^2 \log (a+b x)}{9 b^3 (b c-a d)^2 g^5}-\frac {7744 B^2 d^4 n^2 \log ^2(a+b x)}{3 b^3 (b c-a d)^2 g^5}-\frac {3872 B (b c-a d)^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 g^5 (a+b x)^4}-\frac {77440 B d (b c-a d) n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^5 (a+b x)^3}-\frac {7744 B d^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 g^5 (a+b x)^2}+\frac {15488 B d^3 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 b^3 (b c-a d) g^5 (a+b x)}+\frac {15488 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 )}{3 b^3 (b c-a d)^2 g^5}-\frac {7744 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^4}-\frac {61952 d (b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^5 (a+b x)^3}-\frac {15488 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^5 (a+b x)^2}-\frac {27104 B^2 d^4 n^2 \log (c+d x)}{9 b^3 (b c-a d)^2 g^5}+\frac {15488 B^2 d^4 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^3 (b c-a d)^2 g^5}-\frac {15488 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)}{3 b^3 (b c-a d)^2 g^5}-\frac {7744 B^2 d^4 n^2 \log ^2(c+d x)}{3 b^3 (b c-a d)^2 g^5}+\frac {15488 B^2 d^4 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d)^2 g^5}+\frac {15488 B^2 d^4 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3 b^3 (b c-a d)^2 g^5}+\frac {15488 B^2 d^4 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d)^2 g^5}\\ \end {align*}

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Mathematica [C]  time = 3.28, size = 1860, normalized size = 5.83 \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-1/864*(i^2*(216*(b*c - a*d)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - 576*d*(-(b*c) + a*d)^3*(a + b*x)*(A
+ B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 432*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n
])^2 + 216*B*d^2*n*(a + b*x)^2*(2*(b*c - a*d)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 4*d*(-(b*c) + a*d)*(a
+ b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 4*d^2*(a + b*x)^2*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d
*x))^n]) + 4*d^2*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - 4*B*d*n*(a + b*x)*(b*c - a*
d + d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) + B*n*((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]) + 2*B*d^2*n*(a + b*x)^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)]) - 2*B*d^2*n*(a + b*x
)^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
)])) + 32*B*d*n*(a + b*x)*(12*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 18*d*(b*c - a*d)^2*(a + b
*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 36*d^2*(b*c - a*d)*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))
^n]) + 36*d^3*(a + b*x)^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 36*d^3*(a + b*x)^3*(A + B*Log[
e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 36*B*d^2*n*(a + b*x)^2*(b*c - a*d + d*(a + b*x)*Log[a + b*x] - d*(a
+ b*x)*Log[c + d*x]) - 9*B*d*n*(a + b*x)*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x) - 2*d^2*(a + b*x)^2*Lo
g[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d*x]) + 2*B*n*(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]) - 18*B*d^3*n*(a + b*x
)^3*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d
)]) + 18*B*d^3*n*(a + b*x)^3*((2*Log[(d*(a + b*x))/(-(b*c) + a*d)] - Log[c + d*x])*Log[c + d*x] + 2*PolyLog[2,
(b*(c + d*x))/(b*c - a*d)])) + 3*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]*(L
og[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^3*(b*c - a*d)^2*g^5*(a + b*x)^4)

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fricas [B]  time = 0.94, size = 1729, normalized size = 5.42 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/864*((27*B^2*b^4*c^4 - 64*B^2*a*b^3*c^3*d + 37*B^2*a^4*d^4)*i^2*n^2 + 12*(9*A*B*b^4*c^4 - 16*A*B*a*b^3*c^3*
d + 7*A*B*a^4*d^4)*i^2*n - 12*(7*(B^2*b^4*c*d^3 - B^2*a*b^3*d^4)*i^2*n^2 + 12*(A*B*b^4*c*d^3 - A*B*a*b^3*d^4)*
i^2*n)*x^3 + 72*(3*A^2*b^4*c^4 - 4*A^2*a*b^3*c^3*d + A^2*a^4*d^4)*i^2 - 6*((5*B^2*b^4*c^2*d^2 + 32*B^2*a*b^3*c
*d^3 - 37*B^2*a^2*b^2*d^4)*i^2*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)*i^2*n - 72*(
A^2*b^4*c^2*d^2 - 2*A^2*a*b^3*c*d^3 + A^2*a^2*b^2*d^4)*i^2)*x^2 + 72*(6*(B^2*b^4*c^2*d^2 - 2*B^2*a*b^3*c*d^3 +
B^2*a^2*b^2*d^4)*i^2*x^2 + 4*(2*B^2*b^4*c^3*d - 3*B^2*a*b^3*c^2*d^2 + B^2*a^3*b*d^4)*i^2*x + (3*B^2*b^4*c^4 -
4*B^2*a*b^3*c^3*d + B^2*a^4*d^4)*i^2)*log(e)^2 - 72*(B^2*b^4*d^4*i^2*n^2*x^4 + 4*B^2*a*b^3*d^4*i^2*n^2*x^3 -
6*(B^2*b^4*c^2*d^2 - 2*B^2*a*b^3*c*d^3)*i^2*n^2*x^2 - 4*(2*B^2*b^4*c^3*d - 3*B^2*a*b^3*c^2*d^2)*i^2*n^2*x - (3
*B^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d)*i^2*n^2)*log((b*x + a)/(d*x + c))^2 + 4*((11*B^2*b^4*c^3*d - 48*B^2*a*b^3*c^
2*d^2 + 37*B^2*a^3*b*d^4)*i^2*n^2 + 12*(5*A*B*b^4*c^3*d - 12*A*B*a*b^3*c^2*d^2 + 7*A*B*a^3*b*d^4)*i^2*n + 72*(
2*A^2*b^4*c^3*d - 3*A^2*a*b^3*c^2*d^2 + A^2*a^3*b*d^4)*i^2)*x - 12*(12*(B^2*b^4*c*d^3 - B^2*a*b^3*d^4)*i^2*n*x
^3 - (9*B^2*b^4*c^4 - 16*B^2*a*b^3*c^3*d + 7*B^2*a^4*d^4)*i^2*n - 12*(3*A*B*b^4*c^4 - 4*A*B*a*b^3*c^3*d + A*B*
a^4*d^4)*i^2 - 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)*i^2*n + 12*(A*B*b^4*c^2*d^2 - 2*A*
B*a*b^3*c*d^3 + A*B*a^2*b^2*d^4)*i^2)*x^2 - 4*((5*B^2*b^4*c^3*d - 12*B^2*a*b^3*c^2*d^2 + 7*B^2*a^3*b*d^4)*i^2*
n + 12*(2*A*B*b^4*c^3*d - 3*A*B*a*b^3*c^2*d^2 + A*B*a^3*b*d^4)*i^2)*x + 12*(B^2*b^4*d^4*i^2*n*x^4 + 4*B^2*a*b^
3*d^4*i^2*n*x^3 - 6*(B^2*b^4*c^2*d^2 - 2*B^2*a*b^3*c*d^3)*i^2*n*x^2 - 4*(2*B^2*b^4*c^3*d - 3*B^2*a*b^3*c^2*d^2
)*i^2*n*x - (3*B^2*b^4*c^4 - 4*B^2*a*b^3*c^3*d)*i^2*n)*log((b*x + a)/(d*x + c)))*log(e) + 12*((9*B^2*b^4*c^4 -
16*B^2*a*b^3*c^3*d)*i^2*n^2 - (7*B^2*b^4*d^4*i^2*n^2 + 12*A*B*b^4*d^4*i^2*n)*x^4 + 12*(3*A*B*b^4*c^4 - 4*A*B*
a*b^3*c^3*d)*i^2*n - 4*(12*A*B*a*b^3*d^4*i^2*n + (3*B^2*b^4*c*d^3 + 4*B^2*a*b^3*d^4)*i^2*n^2)*x^3 + 6*((B^2*b^
4*c^2*d^2 - 8*B^2*a*b^3*c*d^3)*i^2*n^2 + 12*(A*B*b^4*c^2*d^2 - 2*A*B*a*b^3*c*d^3)*i^2*n)*x^2 + 4*((5*B^2*b^4*c
^3*d - 12*B^2*a*b^3*c^2*d^2)*i^2*n^2 + 12*(2*A*B*b^4*c^3*d - 3*A*B*a*b^3*c^2*d^2)*i^2*n)*x)*log((b*x + a)/(d*x
+ c)))/((b^9*c^2 - 2*a*b^8*c*d + a^2*b^7*d^2)*g^5*x^4 + 4*(a*b^8*c^2 - 2*a^2*b^7*c*d + a^3*b^6*d^2)*g^5*x^3 +
6*(a^2*b^7*c^2 - 2*a^3*b^6*c*d + a^4*b^5*d^2)*g^5*x^2 + 4*(a^3*b^6*c^2 - 2*a^4*b^5*c*d + a^5*b^4*d^2)*g^5*x +
(a^4*b^5*c^2 - 2*a^5*b^4*c*d + a^6*b^3*d^2)*g^5)

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giac [A]  time = 163.53, size = 461, normalized size = 1.45 \[ \frac {1}{864} \, {\left (\frac {72 \, {\left (3 \, B^{2} b n^{2} - \frac {4 \, {\left (b x + a\right )} B^{2} d n^{2}}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{\frac {{\left (b x + a\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x + a\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {12 \, {\left (9 \, B^{2} b n^{2} - \frac {16 \, {\left (b x + a\right )} B^{2} d n^{2}}{d x + c} + 36 \, A B b n + 36 \, B^{2} b n - \frac {48 \, {\left (b x + a\right )} A B d n}{d x + c} - \frac {48 \, {\left (b x + a\right )} B^{2} d n}{d x + c}\right )} \log \left (\frac {b x + a}{d x + c}\right )}{\frac {{\left (b x + a\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x + a\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {27 \, B^{2} b n^{2} - \frac {64 \, {\left (b x + a\right )} B^{2} d n^{2}}{d x + c} + 108 \, A B b n + 108 \, B^{2} b n - \frac {192 \, {\left (b x + a\right )} A B d n}{d x + c} - \frac {192 \, {\left (b x + a\right )} B^{2} d n}{d x + c} + 216 \, A^{2} b + 432 \, A B b + 216 \, B^{2} b - \frac {288 \, {\left (b x + a\right )} A^{2} d}{d x + c} - \frac {576 \, {\left (b x + a\right )} A B d}{d x + c} - \frac {288 \, {\left (b x + a\right )} B^{2} d}{d x + c}}{\frac {{\left (b x + a\right )}^{4} b c g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x + a\right )}^{4} a d g^{5}}{{\left (d x + c\right )}^{4}}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/864*(72*(3*B^2*b*n^2 - 4*(b*x + a)*B^2*d*n^2/(d*x + c))*log((b*x + a)/(d*x + c))^2/((b*x + a)^4*b*c*g^5/(d*x
+ c)^4 - (b*x + a)^4*a*d*g^5/(d*x + c)^4) + 12*(9*B^2*b*n^2 - 16*(b*x + a)*B^2*d*n^2/(d*x + c) + 36*A*B*b*n +
36*B^2*b*n - 48*(b*x + a)*A*B*d*n/(d*x + c) - 48*(b*x + a)*B^2*d*n/(d*x + c))*log((b*x + a)/(d*x + c))/((b*x
+ a)^4*b*c*g^5/(d*x + c)^4 - (b*x + a)^4*a*d*g^5/(d*x + c)^4) + (27*B^2*b*n^2 - 64*(b*x + a)*B^2*d*n^2/(d*x +
c) + 108*A*B*b*n + 108*B^2*b*n - 192*(b*x + a)*A*B*d*n/(d*x + c) - 192*(b*x + a)*B^2*d*n/(d*x + c) + 216*A^2*b
+ 432*A*B*b + 216*B^2*b - 288*(b*x + a)*A^2*d/(d*x + c) - 576*(b*x + a)*A*B*d/(d*x + c) - 288*(b*x + a)*B^2*d
/(d*x + c))/((b*x + a)^4*b*c*g^5/(d*x + c)^4 - (b*x + a)^4*a*d*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 = 0.46, size = 0, normalized size = 0.00 \[ \int \frac {\left (d i x +c i \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (b g x +a g \right )^{5}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((d*i*x+c*i)^2*(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 = 7.61, size = 8087, normalized size = 25.35 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*i*x+c*i)^2*(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*c^2*i^2*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*lo
g(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)) - 1/72*A*B*d^2*i^2*n*(
(13*a^2*b^3*c^3 - 75*a^3*b^2*c^2*d + 33*a^4*b*c*d^2 - 7*a^5*d^3 - 12*(6*b^5*c^2*d - 4*a*b^4*c*d^2 + a^2*b^3*d^
3)*x^3 + 6*(6*b^5*c^3 - 46*a*b^4*c^2*d + 29*a^2*b^3*c*d^2 - 7*a^3*b^2*d^3)*x^2 + 4*(10*a*b^4*c^3 - 63*a^2*b^3*
c^2*d + 33*a^3*b^2*c*d^2 - 7*a^4*b*d^3)*x)/((b^10*c^3 - 3*a*b^9*c^2*d + 3*a^2*b^8*c*d^2 - a^3*b^7*d^3)*g^5*x^4
+ 4*(a*b^9*c^3 - 3*a^2*b^8*c^2*d + 3*a^3*b^7*c*d^2 - a^4*b^6*d^3)*g^5*x^3 + 6*(a^2*b^8*c^3 - 3*a^3*b^7*c^2*d
+ 3*a^4*b^6*c*d^2 - a^5*b^5*d^3)*g^5*x^2 + 4*(a^3*b^7*c^3 - 3*a^4*b^6*c^2*d + 3*a^5*b^5*c*d^2 - a^6*b^4*d^3)*g
^5*x + (a^4*b^6*c^3 - 3*a^5*b^5*c^2*d + 3*a^6*b^4*c*d^2 - a^7*b^3*d^3)*g^5) - 12*(6*b^2*c^2*d^2 - 4*a*b*c*d^3
+ a^2*d^4)*log(b*x + a)/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^5) +
12*(6*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(d*x + c)/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*
b^4*c*d^3 + a^4*b^3*d^4)*g^5)) - 1/36*A*B*c*d*i^2*n*((7*a*b^3*c^3 - 33*a^2*b^2*c^2*d + 75*a^3*b*c*d^2 - 13*a^4
*d^3 + 12*(4*b^4*c*d^2 - a*b^3*d^3)*x^3 - 6*(4*b^4*c^2*d - 29*a*b^3*c*d^2 + 7*a^2*b^2*d^3)*x^2 + 4*(4*b^4*c^3
- 21*a*b^3*c^2*d + 57*a^2*b^2*c*d^2 - 13*a^3*b*d^3)*x)/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d
^3)*g^5*x^4 + 4*(a*b^8*c^3 - 3*a^2*b^7*c^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3
*b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5*b^4*d^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6
*b^3*d^3)*g^5*x + (a^4*b^5*c^3 - 3*a^5*b^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*g^5) + 12*(4*b*c*d^3 - a*d^4
)*log(b*x + a)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5) - 12*(4*b*c
*d^3 - a*d^4)*log(d*x + c)/((b^6*c^4 - 4*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5)
) - 1/6*(4*b*x + a)*B^2*c*d*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)^2/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^
2*b^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) - 1/12*(6*b^2*x^2 + 4*a*b*x + a^2)*B^2*d^2*i^2*log(e*(b*x/(d*x
+ c) + a/(d*x + c))^n)^2/(b^7*g^5*x^4 + 4*a*b^6*g^5*x^3 + 6*a^2*b^5*g^5*x^2 + 4*a^3*b^4*g^5*x + a^4*b^3*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*c^2*i^2 -
1/432*(12*n*((7*a*b^3*c^3 - 33*a^2*b^2*c^2*d + 75*a^3*b*c*d^2 - 13*a^4*d^3 + 12*(4*b^4*c*d^2 - a*b^3*d^3)*x^3
- 6*(4*b^4*c^2*d - 29*a*b^3*c*d^2 + 7*a^2*b^2*d^3)*x^2 + 4*(4*b^4*c^3 - 21*a*b^3*c^2*d + 57*a^2*b^2*c*d^2 - 13
*a^3*b*d^3)*x)/((b^9*c^3 - 3*a*b^8*c^2*d + 3*a^2*b^7*c*d^2 - a^3*b^6*d^3)*g^5*x^4 + 4*(a*b^8*c^3 - 3*a^2*b^7*c
^2*d + 3*a^3*b^6*c*d^2 - a^4*b^5*d^3)*g^5*x^3 + 6*(a^2*b^7*c^3 - 3*a^3*b^6*c^2*d + 3*a^4*b^5*c*d^2 - a^5*b^4*d
^3)*g^5*x^2 + 4*(a^3*b^6*c^3 - 3*a^4*b^5*c^2*d + 3*a^5*b^4*c*d^2 - a^6*b^3*d^3)*g^5*x + (a^4*b^5*c^3 - 3*a^5*b
^4*c^2*d + 3*a^6*b^3*c*d^2 - a^7*b^2*d^3)*g^5) + 12*(4*b*c*d^3 - a*d^4)*log(b*x + a)/((b^6*c^4 - 4*a*b^5*c^3*d
+ 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5) - 12*(4*b*c*d^3 - a*d^4)*log(d*x + c)/((b^6*c^4 - 4
*a*b^5*c^3*d + 6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*g^5))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)
+ (37*a*b^4*c^4 - 304*a^2*b^3*c^3*d + 1512*a^3*b^2*c^2*d^2 - 1360*a^4*b*c*d^3 + 115*a^5*d^4 + 12*(88*b^5*c^2*
d^2 - 101*a*b^4*c*d^3 + 13*a^2*b^3*d^4)*x^3 - 6*(40*b^5*c^3*d - 609*a*b^4*c^2*d^2 + 648*a^2*b^3*c*d^3 - 79*a^3
*b^2*d^4)*x^2 - 72*(4*a^4*b*c*d^3 - a^5*d^4 + (4*b^5*c*d^3 - a*b^4*d^4)*x^4 + 4*(4*a*b^4*c*d^3 - a^2*b^3*d^4)*
x^3 + 6*(4*a^2*b^3*c*d^3 - a^3*b^2*d^4)*x^2 + 4*(4*a^3*b^2*c*d^3 - a^4*b*d^4)*x)*log(b*x + a)^2 - 72*(4*a^4*b*
c*d^3 - a^5*d^4 + (4*b^5*c*d^3 - a*b^4*d^4)*x^4 + 4*(4*a*b^4*c*d^3 - a^2*b^3*d^4)*x^3 + 6*(4*a^2*b^3*c*d^3 - a
^3*b^2*d^4)*x^2 + 4*(4*a^3*b^2*c*d^3 - a^4*b*d^4)*x)*log(d*x + c)^2 + 4*(16*b^5*c^4 - 163*a*b^4*c^3*d + 1068*a
^2*b^3*c^2*d^2 - 1036*a^3*b^2*c*d^3 + 115*a^4*b*d^4)*x + 12*(88*a^4*b*c*d^3 - 13*a^5*d^4 + (88*b^5*c*d^3 - 13*
a*b^4*d^4)*x^4 + 4*(88*a*b^4*c*d^3 - 13*a^2*b^3*d^4)*x^3 + 6*(88*a^2*b^3*c*d^3 - 13*a^3*b^2*d^4)*x^2 + 4*(88*a
^3*b^2*c*d^3 - 13*a^4*b*d^4)*x)*log(b*x + a) - 12*(88*a^4*b*c*d^3 - 13*a^5*d^4 + (88*b^5*c*d^3 - 13*a*b^4*d^4)
*x^4 + 4*(88*a*b^4*c*d^3 - 13*a^2*b^3*d^4)*x^3 + 6*(88*a^2*b^3*c*d^3 - 13*a^3*b^2*d^4)*x^2 + 4*(88*a^3*b^2*c*d
^3 - 13*a^4*b*d^4)*x - 12*(4*a^4*b*c*d^3 - a^5*d^4 + (4*b^5*c*d^3 - a*b^4*d^4)*x^4 + 4*(4*a*b^4*c*d^3 - a^2*b^
3*d^4)*x^3 + 6*(4*a^2*b^3*c*d^3 - a^3*b^2*d^4)*x^2 + 4*(4*a^3*b^2*c*d^3 - a^4*b*d^4)*x)*log(b*x + a))*log(d*x
+ c))*n^2/(a^4*b^6*c^4*g^5 - 4*a^5*b^5*c^3*d*g^5 + 6*a^6*b^4*c^2*d^2*g^5 - 4*a^7*b^3*c*d^3*g^5 + a^8*b^2*d^4*g
^5 + (b^10*c^4*g^5 - 4*a*b^9*c^3*d*g^5 + 6*a^2*b^8*c^2*d^2*g^5 - 4*a^3*b^7*c*d^3*g^5 + a^4*b^6*d^4*g^5)*x^4 +
4*(a*b^9*c^4*g^5 - 4*a^2*b^8*c^3*d*g^5 + 6*a^3*b^7*c^2*d^2*g^5 - 4*a^4*b^6*c*d^3*g^5 + a^5*b^5*d^4*g^5)*x^3 +
6*(a^2*b^8*c^4*g^5 - 4*a^3*b^7*c^3*d*g^5 + 6*a^4*b^6*c^2*d^2*g^5 - 4*a^5*b^5*c*d^3*g^5 + a^6*b^4*d^4*g^5)*x^2
+ 4*(a^3*b^7*c^4*g^5 - 4*a^4*b^6*c^3*d*g^5 + 6*a^5*b^5*c^2*d^2*g^5 - 4*a^6*b^4*c*d^3*g^5 + a^7*b^3*d^4*g^5)*x)
)*B^2*c*d*i^2 - 1/864*(12*n*((13*a^2*b^3*c^3 - 75*a^3*b^2*c^2*d + 33*a^4*b*c*d^2 - 7*a^5*d^3 - 12*(6*b^5*c^2*d
- 4*a*b^4*c*d^2 + a^2*b^3*d^3)*x^3 + 6*(6*b^5*c^3 - 46*a*b^4*c^2*d + 29*a^2*b^3*c*d^2 - 7*a^3*b^2*d^3)*x^2 +
4*(10*a*b^4*c^3 - 63*a^2*b^3*c^2*d + 33*a^3*b^2*c*d^2 - 7*a^4*b*d^3)*x)/((b^10*c^3 - 3*a*b^9*c^2*d + 3*a^2*b^8
*c*d^2 - a^3*b^7*d^3)*g^5*x^4 + 4*(a*b^9*c^3 - 3*a^2*b^8*c^2*d + 3*a^3*b^7*c*d^2 - a^4*b^6*d^3)*g^5*x^3 + 6*(a
^2*b^8*c^3 - 3*a^3*b^7*c^2*d + 3*a^4*b^6*c*d^2 - a^5*b^5*d^3)*g^5*x^2 + 4*(a^3*b^7*c^3 - 3*a^4*b^6*c^2*d + 3*a
^5*b^5*c*d^2 - a^6*b^4*d^3)*g^5*x + (a^4*b^6*c^3 - 3*a^5*b^5*c^2*d + 3*a^6*b^4*c*d^2 - a^7*b^3*d^3)*g^5) - 12*
(6*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(b*x + a)/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4
*c*d^3 + a^4*b^3*d^4)*g^5) + 12*(6*b^2*c^2*d^2 - 4*a*b*c*d^3 + a^2*d^4)*log(d*x + c)/((b^7*c^4 - 4*a*b^6*c^3*d
+ 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^5))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + (115*a^2*
b^4*c^4 - 1360*a^3*b^3*c^3*d + 1512*a^4*b^2*c^2*d^2 - 304*a^5*b*c*d^3 + 37*a^6*d^4 - 12*(108*b^6*c^3*d - 148*a
*b^5*c^2*d^2 + 47*a^2*b^4*c*d^3 - 7*a^3*b^3*d^4)*x^3 + 6*(36*b^6*c^4 - 712*a*b^5*c^3*d + 903*a^2*b^4*c^2*d^2 -
264*a^3*b^3*c*d^3 + 37*a^4*b^2*d^4)*x^2 + 72*(6*a^4*b^2*c^2*d^2 - 4*a^5*b*c*d^3 + a^6*d^4 + (6*b^6*c^2*d^2 -
4*a*b^5*c*d^3 + a^2*b^4*d^4)*x^4 + 4*(6*a*b^5*c^2*d^2 - 4*a^2*b^4*c*d^3 + a^3*b^3*d^4)*x^3 + 6*(6*a^2*b^4*c^2*
d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*x^2 + 4*(6*a^3*b^3*c^2*d^2 - 4*a^4*b^2*c*d^3 + a^5*b*d^4)*x)*log(b*x + a)
^2 + 72*(6*a^4*b^2*c^2*d^2 - 4*a^5*b*c*d^3 + a^6*d^4 + (6*b^6*c^2*d^2 - 4*a*b^5*c*d^3 + a^2*b^4*d^4)*x^4 + 4*(
6*a*b^5*c^2*d^2 - 4*a^2*b^4*c*d^3 + a^3*b^3*d^4)*x^3 + 6*(6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^3 + a^4*b^2*d^4)*x
^2 + 4*(6*a^3*b^3*c^2*d^2 - 4*a^4*b^2*c*d^3 + a^5*b*d^4)*x)*log(d*x + c)^2 + 4*(76*a*b^5*c^4 - 1057*a^2*b^4*c^
3*d + 1248*a^3*b^3*c^2*d^2 - 304*a^4*b^2*c*d^3 + 37*a^5*b*d^4)*x - 12*(108*a^4*b^2*c^2*d^2 - 40*a^5*b*c*d^3 +
7*a^6*d^4 + (108*b^6*c^2*d^2 - 40*a*b^5*c*d^3 + 7*a^2*b^4*d^4)*x^4 + 4*(108*a*b^5*c^2*d^2 - 40*a^2*b^4*c*d^3 +
7*a^3*b^3*d^4)*x^3 + 6*(108*a^2*b^4*c^2*d^2 - 40*a^3*b^3*c*d^3 + 7*a^4*b^2*d^4)*x^2 + 4*(108*a^3*b^3*c^2*d^2
- 40*a^4*b^2*c*d^3 + 7*a^5*b*d^4)*x)*log(b*x + a) + 12*(108*a^4*b^2*c^2*d^2 - 40*a^5*b*c*d^3 + 7*a^6*d^4 + (10
8*b^6*c^2*d^2 - 40*a*b^5*c*d^3 + 7*a^2*b^4*d^4)*x^4 + 4*(108*a*b^5*c^2*d^2 - 40*a^2*b^4*c*d^3 + 7*a^3*b^3*d^4)
*x^3 + 6*(108*a^2*b^4*c^2*d^2 - 40*a^3*b^3*c*d^3 + 7*a^4*b^2*d^4)*x^2 + 4*(108*a^3*b^3*c^2*d^2 - 40*a^4*b^2*c*
d^3 + 7*a^5*b*d^4)*x - 12*(6*a^4*b^2*c^2*d^2 - 4*a^5*b*c*d^3 + a^6*d^4 + (6*b^6*c^2*d^2 - 4*a*b^5*c*d^3 + a^2*
b^4*d^4)*x^4 + 4*(6*a*b^5*c^2*d^2 - 4*a^2*b^4*c*d^3 + a^3*b^3*d^4)*x^3 + 6*(6*a^2*b^4*c^2*d^2 - 4*a^3*b^3*c*d^
3 + a^4*b^2*d^4)*x^2 + 4*(6*a^3*b^3*c^2*d^2 - 4*a^4*b^2*c*d^3 + a^5*b*d^4)*x)*log(b*x + a))*log(d*x + c))*n^2/
(a^4*b^7*c^4*g^5 - 4*a^5*b^6*c^3*d*g^5 + 6*a^6*b^5*c^2*d^2*g^5 - 4*a^7*b^4*c*d^3*g^5 + a^8*b^3*d^4*g^5 + (b^11
*c^4*g^5 - 4*a*b^10*c^3*d*g^5 + 6*a^2*b^9*c^2*d^2*g^5 - 4*a^3*b^8*c*d^3*g^5 + a^4*b^7*d^4*g^5)*x^4 + 4*(a*b^10
*c^4*g^5 - 4*a^2*b^9*c^3*d*g^5 + 6*a^3*b^8*c^2*d^2*g^5 - 4*a^4*b^7*c*d^3*g^5 + a^5*b^6*d^4*g^5)*x^3 + 6*(a^2*b
^9*c^4*g^5 - 4*a^3*b^8*c^3*d*g^5 + 6*a^4*b^7*c^2*d^2*g^5 - 4*a^5*b^6*c*d^3*g^5 + a^6*b^5*d^4*g^5)*x^2 + 4*(a^3
*b^8*c^4*g^5 - 4*a^4*b^7*c^3*d*g^5 + 6*a^5*b^6*c^2*d^2*g^5 - 4*a^6*b^5*c*d^3*g^5 + a^7*b^4*d^4*g^5)*x))*B^2*d^
2*i^2 - 1/3*(4*b*x + a)*A*B*c*d*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*
a^2*b^4*g^5*x^2 + 4*a^3*b^3*g^5*x + a^4*b^2*g^5) - 1/6*(6*b^2*x^2 + 4*a*b*x + a^2)*A*B*d^2*i^2*log(e*(b*x/(d*x
+ c) + a/(d*x + c))^n)/(b^7*g^5*x^4 + 4*a*b^6*g^5*x^3 + 6*a^2*b^5*g^5*x^2 + 4*a^3*b^4*g^5*x + a^4*b^3*g^5) -
1/4*B^2*c^2*i^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) - 1/6*(4*b*x + a)*A^2*c*d*i^2/(b^6*g^5*x^4 + 4*a*b^5*g^5*x^3 + 6*a^2*b^4*g^5*x^2
+ 4*a^3*b^3*g^5*x + a^4*b^2*g^5) - 1/12*(6*b^2*x^2 + 4*a*b*x + a^2)*A^2*d^2*i^2/(b^7*g^5*x^4 + 4*a*b^6*g^5*x^3
+ 6*a^2*b^5*g^5*x^2 + 4*a^3*b^4*g^5*x + a^4*b^3*g^5) - 1/2*A*B*c^2*i^2*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*c^2*i^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 = 9.42, size = 1934, normalized size = 6.06 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- log(e*((a + b*x)/(c + d*x))^n)*((a*(A*B*a*d^2*i^2 - (B^2*a*d^2*i^2*n)/2 + (B^2*b*c*d*i^2*n)/2 + 2*A*B*b*c*d*
i^2) + x*(b*(A*B*a*d^2*i^2 - (B^2*a*d^2*i^2*n)/2 + (B^2*b*c*d*i^2*n)/2 + 2*A*B*b*c*d*i^2) + 3*A*B*a*b*d^2*i^2
+ 6*A*B*b^2*c*d*i^2 - (3*B^2*a*b*d^2*i^2*n)/2 + (3*B^2*b^2*c*d*i^2*n)/2) + 3*A*B*b^2*c^2*i^2 - B^2*a^2*d^2*i^2
*n + (B^2*b^2*c^2*i^2*n)/2 + 6*A*B*b^2*d^2*i^2*x^2 + (B^2*a*b*c*d*i^2*n)/2)/(6*a^4*b^3*g^5 + 6*b^7*g^5*x^4 + 2
4*a^3*b^4*g^5*x + 24*a*b^6*g^5*x^3 + 36*a^2*b^5*g^5*x^2) + (B^2*d^4*i^2*(x^2*(b*(b*((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)) + (3*a*b^4*g^5*n*(a*d - b*c))/d + (b^4*g^5*n*(a*d - b
*c)*(4*a*d - b*c))/d^2) + (9*a*b^5*g^5*n*(a*d - b*c))/(2*d) + (3*b^5*g^5*n*(a*d - b*c)*(4*a*d - b*c))/(2*d^2))
+ a*(a*((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)) + (b^3*g^5*n*(a*d
- b*c)*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(2*d^3)) + x*(a*(b*((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)) + (3*a*b^4*g^5*n*(a*d - b*c))/d + (b^4*g^5*n*(a*d - b*c)*(4*a*d - b*c))/d^2
) + b*(a*((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)) + (b^3*g^5*n*(a*d
- b*c)*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*d))/(2*d^3)) + (3*b^4*g^5*n*(a*d - b*c)*(6*a^2*d^2 + b^2*c^2 - 4*a*b*c*
d))/(2*d^3)) + (3*b^3*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) + (6*b^
6*g^5*n*x^3*(a*d - b*c))/d))/(6*b^3*g^5*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)*(6*a^4*b^3*g^5 + 6*b^7*g^5*x^4 + 24*a^
3*b^4*g^5*x + 24*a*b^6*g^5*x^3 + 36*a^2*b^5*g^5*x^2))) - ((72*A^2*a^3*d^3*i^2 - 216*A^2*b^3*c^3*i^2 + 37*B^2*a
^3*d^3*i^2*n^2 - 27*B^2*b^3*c^3*i^2*n^2 + 72*A^2*a*b^2*c^2*d*i^2 + 72*A^2*a^2*b*c*d^2*i^2 + 84*A*B*a^3*d^3*i^2
*n - 108*A*B*b^3*c^3*i^2*n + 37*B^2*a*b^2*c^2*d*i^2*n^2 + 37*B^2*a^2*b*c*d^2*i^2*n^2 + 84*A*B*a*b^2*c^2*d*i^2*
n + 84*A*B*a^2*b*c*d^2*i^2*n)/(12*(a*d - b*c)) + (x^3*(7*B^2*b^3*d^3*i^2*n^2 + 12*A*B*b^3*d^3*i^2*n))/(a*d - b
*c) + (x*(72*A^2*a^2*b*d^3*i^2 - 144*A^2*b^3*c^2*d*i^2 + 72*A^2*a*b^2*c*d^2*i^2 + 37*B^2*a^2*b*d^3*i^2*n^2 - 1
1*B^2*b^3*c^2*d*i^2*n^2 - 60*A*B*b^3*c^2*d*i^2*n + 37*B^2*a*b^2*c*d^2*i^2*n^2 + 84*A*B*a^2*b*d^3*i^2*n + 84*A*
B*a*b^2*c*d^2*i^2*n))/(3*(a*d - b*c)) + (x^2*(72*A^2*a*b^2*d^3*i^2 - 72*A^2*b^3*c*d^2*i^2 + 37*B^2*a*b^2*d^3*i
^2*n^2 + 5*B^2*b^3*c*d^2*i^2*n^2 - 12*A*B*b^3*c*d^2*i^2*n + 84*A*B*a*b^2*d^3*i^2*n))/(2*(a*d - b*c)))/(72*a^4*
b^3*g^5 + 72*b^7*g^5*x^4 + 288*a^3*b^4*g^5*x + 288*a*b^6*g^5*x^3 + 432*a^2*b^5*g^5*x^2) - log(e*((a + b*x)/(c
+ d*x))^n)^2*((a*((B^2*c*d*i^2)/(6*b^2) + (B^2*a*d^2*i^2)/(12*b^3)) + x*(b*((B^2*c*d*i^2)/(6*b^2) + (B^2*a*d^2
*i^2)/(12*b^3)) + (B^2*c*d*i^2)/(2*b) + (B^2*a*d^2*i^2)/(4*b^2)) + (B^2*c^2*i^2)/(4*b) + (B^2*d^2*i^2*x^2)/(2*
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*i^2)/(12*b^3*g^5*
(a^2*d^2 + b^2*c^2 - 2*a*b*c*d))) - (B*d^4*i^2*n*atan(((2*b*d*x - (72*b^5*c^2*g^5 - 72*a^2*b^3*d^2*g^5)/(72*b^
3*g^5*(a*d - b*c)))*1i)/(a*d - b*c))*(12*A + 7*B*n)*1i)/(36*b^3*g^5*(a*d - b*c)^2)

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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((d*i*x+c*i)**2*(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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