3.175 \(\int \frac {(c i+d i x)^2 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2}{(a g+b g x)^4} \, dx\)

Optimal. Leaf size=157 \[ -\frac {i^2 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 g^4 (a+b x)^3 (b c-a d)}-\frac {2 B i^2 n (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 g^4 (a+b x)^3 (b c-a d)}-\frac {2 B^2 i^2 n^2 (c+d x)^3}{27 g^4 (a+b x)^3 (b c-a d)} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 3.17, antiderivative size = 889, normalized size of antiderivative = 5.66, number of steps used = 86, 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} \[ \frac {B^2 i^2 n^2 \log ^2(a+b x) d^3}{3 b^3 (b c-a d) g^4}+\frac {B^2 i^2 n^2 \log ^2(c+d x) d^3}{3 b^3 (b c-a d) g^4}-\frac {2 B^2 i^2 n^2 \log (a+b x) d^3}{9 b^3 (b c-a d) g^4}-\frac {2 B i^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) d^3}{3 b^3 (b c-a d) g^4}+\frac {2 B^2 i^2 n^2 \log (c+d x) d^3}{9 b^3 (b c-a d) g^4}-\frac {2 B^2 i^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x) d^3}{3 b^3 (b c-a d) g^4}+\frac {2 B i^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x) d^3}{3 b^3 (b c-a d) g^4}-\frac {2 B^2 i^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) d^3}{3 b^3 (b c-a d) g^4}-\frac {2 B^2 i^2 n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) d^3}{3 b^3 (b c-a d) g^4}-\frac {2 B^2 i^2 n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right ) d^3}{3 b^3 (b c-a d) g^4}-\frac {i^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 d^2}{b^3 g^4 (a+b x)}-\frac {2 B i^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) d^2}{3 b^3 g^4 (a+b x)}-\frac {2 B^2 i^2 n^2 d^2}{9 b^3 g^4 (a+b x)}-\frac {(b c-a d) i^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 d}{b^3 g^4 (a+b x)^2}-\frac {2 B (b c-a d) i^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) d}{3 b^3 g^4 (a+b x)^2}-\frac {2 B^2 (b c-a d) i^2 n^2 d}{9 b^3 g^4 (a+b x)^2}-\frac {(b c-a d)^2 i^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {2 B (b c-a d)^2 i^2 n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 b^3 g^4 (a+b x)^3}-\frac {2 B^2 (b c-a d)^2 i^2 n^2}{27 b^3 g^4 (a+b x)^3} \]

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)^4,x]

[Out]

(-2*B^2*(b*c - a*d)^2*i^2*n^2)/(27*b^3*g^4*(a + b*x)^3) - (2*B^2*d*(b*c - a*d)*i^2*n^2)/(9*b^3*g^4*(a + b*x)^2
) - (2*B^2*d^2*i^2*n^2)/(9*b^3*g^4*(a + b*x)) - (2*B^2*d^3*i^2*n^2*Log[a + b*x])/(9*b^3*(b*c - a*d)*g^4) + (B^
2*d^3*i^2*n^2*Log[a + b*x]^2)/(3*b^3*(b*c - a*d)*g^4) - (2*B*(b*c - a*d)^2*i^2*n*(A + B*Log[e*((a + b*x)/(c +
d*x))^n]))/(9*b^3*g^4*(a + b*x)^3) - (2*B*d*(b*c - a*d)*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^3*g
^4*(a + b*x)^2) - (2*B*d^2*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^3*g^4*(a + b*x)) - (2*B*d^3*i^2*
n*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*b^3*(b*c - a*d)*g^4) - ((b*c - a*d)^2*i^2*(A + B*Log
[e*((a + b*x)/(c + d*x))^n])^2)/(3*b^3*g^4*(a + b*x)^3) - (d*(b*c - a*d)*i^2*(A + B*Log[e*((a + b*x)/(c + d*x)
)^n])^2)/(b^3*g^4*(a + b*x)^2) - (d^2*i^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(b^3*g^4*(a + b*x)) + (2*B
^2*d^3*i^2*n^2*Log[c + d*x])/(9*b^3*(b*c - a*d)*g^4) - (2*B^2*d^3*i^2*n^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Lo
g[c + d*x])/(3*b^3*(b*c - a*d)*g^4) + (2*B*d^3*i^2*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x])/(3*b
^3*(b*c - a*d)*g^4) + (B^2*d^3*i^2*n^2*Log[c + d*x]^2)/(3*b^3*(b*c - a*d)*g^4) - (2*B^2*d^3*i^2*n^2*Log[a + b*
x]*Log[(b*(c + d*x))/(b*c - a*d)])/(3*b^3*(b*c - a*d)*g^4) - (2*B^2*d^3*i^2*n^2*PolyLog[2, -((d*(a + b*x))/(b*
c - a*d))])/(3*b^3*(b*c - a*d)*g^4) - (2*B^2*d^3*i^2*n^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(3*b^3*(b*c -
a*d)*g^4)

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 {(175 c+175 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)^4} \, dx &=\int \left (\frac {30625 (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^4 (a+b x)^4}+\frac {61250 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^4 (a+b x)^3}+\frac {30625 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^2 g^4 (a+b x)^2}\right ) \, dx\\ &=\frac {\left (30625 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)^2} \, dx}{b^2 g^4}+\frac {(61250 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)^3} \, dx}{b^2 g^4}+\frac {\left (30625 (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)^4} \, dx}{b^2 g^4}\\ &=-\frac {30625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {30625 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^3 g^4 (a+b x)^2}-\frac {30625 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {\left (61250 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)^2 (c+d x)} \, dx}{b^3 g^4}+\frac {(61250 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)^3 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (61250 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)^4 (c+d x)} \, dx}{3 b^3 g^4}\\ &=-\frac {30625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {30625 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^3 g^4 (a+b x)^2}-\frac {30625 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {\left (61250 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)^2 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (61250 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)^3 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (61250 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)^4 (c+d x)} \, dx}{3 b^3 g^4}\\ &=-\frac {30625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {30625 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^3 g^4 (a+b x)^2}-\frac {30625 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {\left (61250 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)^2}-\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)}+\frac {d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b^3 g^4}+\frac {\left (61250 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)^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^4}+\frac {\left (61250 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)^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^4}\\ &=-\frac {30625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {30625 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^3 g^4 (a+b x)^2}-\frac {30625 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {\left (61250 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)^2} \, dx}{3 b^2 g^4}-\frac {\left (61250 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} \, dx}{3 b^2 (b c-a d) g^4}+\frac {\left (61250 B d^4 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) g^4}-\frac {(61250 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)^3} \, dx}{3 b^2 g^4}+\frac {(61250 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)^3} \, dx}{b^2 g^4}+\frac {\left (61250 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)^4} \, dx}{3 b^2 g^4}\\ &=-\frac {61250 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 )}{9 b^3 g^4 (a+b x)^3}-\frac {61250 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 )}{3 b^3 g^4 (a+b x)^2}-\frac {61250 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^4 (a+b x)}-\frac {61250 B d^3 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) g^4}-\frac {30625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {30625 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^3 g^4 (a+b x)^2}-\frac {30625 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {61250 B d^3 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) g^4}+\frac {\left (61250 B^2 d^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{3 b^3 g^4}+\frac {\left (61250 B^2 d^3 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) g^4}-\frac {\left (61250 B^2 d^3 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) g^4}-\frac {\left (30625 B^2 d (b c-a d) n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{3 b^3 g^4}+\frac {\left (30625 B^2 d (b c-a d) n^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (61250 B^2 (b c-a d)^2 n^2\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{9 b^3 g^4}\\ &=-\frac {61250 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 )}{9 b^3 g^4 (a+b x)^3}-\frac {61250 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 )}{3 b^3 g^4 (a+b x)^2}-\frac {61250 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^4 (a+b x)}-\frac {61250 B d^3 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) g^4}-\frac {30625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {30625 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^3 g^4 (a+b x)^2}-\frac {30625 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {61250 B d^3 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) g^4}+\frac {\left (61250 B^2 d^3 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) g^4}-\frac {\left (61250 B^2 d^3 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) g^4}+\frac {\left (61250 B^2 d^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3 b^3 g^4}-\frac {\left (30625 B^2 d (b c-a d)^2 n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{3 b^3 g^4}+\frac {\left (30625 B^2 d (b c-a d)^2 n^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b^3 g^4}+\frac {\left (61250 B^2 (b c-a d)^3 n^2\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{9 b^3 g^4}\\ &=-\frac {61250 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 )}{9 b^3 g^4 (a+b x)^3}-\frac {61250 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 )}{3 b^3 g^4 (a+b x)^2}-\frac {61250 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^4 (a+b x)}-\frac {61250 B d^3 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) g^4}-\frac {30625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {30625 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^3 g^4 (a+b x)^2}-\frac {30625 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {61250 B d^3 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) g^4}+\frac {\left (61250 B^2 d^3 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3 b^2 (b c-a d) g^4}-\frac {\left (61250 B^2 d^3 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 b^2 (b c-a d) g^4}-\frac {\left (61250 B^2 d^4 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 b^3 (b c-a d) g^4}+\frac {\left (61250 B^2 d^4 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b^3 (b c-a d) g^4}+\frac {\left (61250 B^2 d^2 (b c-a d) 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^4}-\frac {\left (30625 B^2 d (b c-a 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}{3 b^3 g^4}+\frac {\left (30625 B^2 d (b c-a 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}{b^3 g^4}+\frac {\left (61250 B^2 (b c-a d)^3 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^4}\\ &=-\frac {61250 B^2 (b c-a d)^2 n^2}{27 b^3 g^4 (a+b x)^3}-\frac {61250 B^2 d (b c-a d) n^2}{9 b^3 g^4 (a+b x)^2}-\frac {61250 B^2 d^2 n^2}{9 b^3 g^4 (a+b x)}-\frac {61250 B^2 d^3 n^2 \log (a+b x)}{9 b^3 (b c-a d) g^4}-\frac {61250 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 )}{9 b^3 g^4 (a+b x)^3}-\frac {61250 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 )}{3 b^3 g^4 (a+b x)^2}-\frac {61250 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^4 (a+b x)}-\frac {61250 B d^3 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) g^4}-\frac {30625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {30625 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^3 g^4 (a+b x)^2}-\frac {30625 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {61250 B^2 d^3 n^2 \log (c+d x)}{9 b^3 (b c-a d) g^4}-\frac {61250 B^2 d^3 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) g^4}+\frac {61250 B d^3 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) g^4}-\frac {61250 B^2 d^3 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) g^4}+\frac {\left (61250 B^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 b^3 (b c-a d) g^4}+\frac {\left (61250 B^2 d^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b^3 (b c-a d) g^4}+\frac {\left (61250 B^2 d^3 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) g^4}+\frac {\left (61250 B^2 d^4 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) g^4}\\ &=-\frac {61250 B^2 (b c-a d)^2 n^2}{27 b^3 g^4 (a+b x)^3}-\frac {61250 B^2 d (b c-a d) n^2}{9 b^3 g^4 (a+b x)^2}-\frac {61250 B^2 d^2 n^2}{9 b^3 g^4 (a+b x)}-\frac {61250 B^2 d^3 n^2 \log (a+b x)}{9 b^3 (b c-a d) g^4}+\frac {30625 B^2 d^3 n^2 \log ^2(a+b x)}{3 b^3 (b c-a d) g^4}-\frac {61250 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 )}{9 b^3 g^4 (a+b x)^3}-\frac {61250 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 )}{3 b^3 g^4 (a+b x)^2}-\frac {61250 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^4 (a+b x)}-\frac {61250 B d^3 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) g^4}-\frac {30625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {30625 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^3 g^4 (a+b x)^2}-\frac {30625 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {61250 B^2 d^3 n^2 \log (c+d x)}{9 b^3 (b c-a d) g^4}-\frac {61250 B^2 d^3 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) g^4}+\frac {61250 B d^3 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) g^4}+\frac {30625 B^2 d^3 n^2 \log ^2(c+d x)}{3 b^3 (b c-a d) g^4}-\frac {61250 B^2 d^3 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) g^4}+\frac {\left (61250 B^2 d^3 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) g^4}+\frac {\left (61250 B^2 d^3 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) g^4}\\ &=-\frac {61250 B^2 (b c-a d)^2 n^2}{27 b^3 g^4 (a+b x)^3}-\frac {61250 B^2 d (b c-a d) n^2}{9 b^3 g^4 (a+b x)^2}-\frac {61250 B^2 d^2 n^2}{9 b^3 g^4 (a+b x)}-\frac {61250 B^2 d^3 n^2 \log (a+b x)}{9 b^3 (b c-a d) g^4}+\frac {30625 B^2 d^3 n^2 \log ^2(a+b x)}{3 b^3 (b c-a d) g^4}-\frac {61250 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 )}{9 b^3 g^4 (a+b x)^3}-\frac {61250 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 )}{3 b^3 g^4 (a+b x)^2}-\frac {61250 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^4 (a+b x)}-\frac {61250 B d^3 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) g^4}-\frac {30625 (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^3 g^4 (a+b x)^3}-\frac {30625 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^3 g^4 (a+b x)^2}-\frac {30625 d^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{b^3 g^4 (a+b x)}+\frac {61250 B^2 d^3 n^2 \log (c+d x)}{9 b^3 (b c-a d) g^4}-\frac {61250 B^2 d^3 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) g^4}+\frac {61250 B d^3 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) g^4}+\frac {30625 B^2 d^3 n^2 \log ^2(c+d x)}{3 b^3 (b c-a d) g^4}-\frac {61250 B^2 d^3 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) g^4}-\frac {61250 B^2 d^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}-\frac {61250 B^2 d^3 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b^3 (b c-a d) g^4}\\ \end {align*}

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Mathematica [C]  time = 2.24, size = 1415, normalized size = 9.01 \[ \text {result too large to display} \]

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)^4,x]

[Out]

-1/54*(i^2*(18*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 54*d*(b*c - a*d)^2*(a + b*x)*(A + B*Lo
g[e*((a + b*x)/(c + d*x))^n])^2 - 54*d^2*(-(b*c) + a*d)*(a + b*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 +
54*B*d^2*n*(a + b*x)^2*(2*(b*c - a*d)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 2*d*(a + b*x)*Log[a + b*x]*(A
+ B*Log[e*((a + b*x)/(c + d*x))^n]) - 2*d*(a + b*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] + 2*B*
n*(b*c - a*d + d*(a + b*x)*Log[a + b*x] - d*(a + b*x)*Log[c + d*x]) - B*d*n*(a + b*x)*(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)]) + B*d*n*(a + b*x)*((2*Lo
g[(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)])) + 27*
B*d*n*(a + b*x)*(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)])) + B*n*(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])*Lo
g[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*Log[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)])))
)/(b^3*(b*c - a*d)*g^4*(a + b*x)^3)

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fricas [B]  time = 1.00, size = 975, normalized size = 6.21 \[ -\frac {2 \, {\left (B^{2} b^{3} c^{3} - B^{2} a^{3} d^{3}\right )} i^{2} n^{2} + 6 \, {\left (A B b^{3} c^{3} - A B a^{3} d^{3}\right )} i^{2} n + 9 \, {\left (A^{2} b^{3} c^{3} - A^{2} a^{3} d^{3}\right )} i^{2} + 3 \, {\left (2 \, {\left (B^{2} b^{3} c d^{2} - B^{2} a b^{2} d^{3}\right )} i^{2} n^{2} + 6 \, {\left (A B b^{3} c d^{2} - A B a b^{2} d^{3}\right )} i^{2} n + 9 \, {\left (A^{2} b^{3} c d^{2} - A^{2} a b^{2} d^{3}\right )} i^{2}\right )} x^{2} + 9 \, {\left (3 \, {\left (B^{2} b^{3} c d^{2} - B^{2} a b^{2} d^{3}\right )} i^{2} x^{2} + 3 \, {\left (B^{2} b^{3} c^{2} d - B^{2} a^{2} b d^{3}\right )} i^{2} x + {\left (B^{2} b^{3} c^{3} - B^{2} a^{3} d^{3}\right )} i^{2}\right )} \log \relax (e)^{2} + 9 \, {\left (B^{2} b^{3} d^{3} i^{2} n^{2} x^{3} + 3 \, B^{2} b^{3} c d^{2} i^{2} n^{2} x^{2} + 3 \, B^{2} b^{3} c^{2} d i^{2} n^{2} x + B^{2} b^{3} c^{3} i^{2} n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} + 3 \, {\left (2 \, {\left (B^{2} b^{3} c^{2} d - B^{2} a^{2} b d^{3}\right )} i^{2} n^{2} + 6 \, {\left (A B b^{3} c^{2} d - A B a^{2} b d^{3}\right )} i^{2} n + 9 \, {\left (A^{2} b^{3} c^{2} d - A^{2} a^{2} b d^{3}\right )} i^{2}\right )} x + 6 \, {\left ({\left (B^{2} b^{3} c^{3} - B^{2} a^{3} d^{3}\right )} i^{2} n + 3 \, {\left (A B b^{3} c^{3} - A B a^{3} d^{3}\right )} i^{2} + 3 \, {\left ({\left (B^{2} b^{3} c d^{2} - B^{2} a b^{2} d^{3}\right )} i^{2} n + 3 \, {\left (A B b^{3} c d^{2} - A B a b^{2} d^{3}\right )} i^{2}\right )} x^{2} + 3 \, {\left ({\left (B^{2} b^{3} c^{2} d - B^{2} a^{2} b d^{3}\right )} i^{2} n + 3 \, {\left (A B b^{3} c^{2} d - A B a^{2} b d^{3}\right )} i^{2}\right )} x + 3 \, {\left (B^{2} b^{3} d^{3} i^{2} n x^{3} + 3 \, B^{2} b^{3} c d^{2} i^{2} n x^{2} + 3 \, B^{2} b^{3} c^{2} d i^{2} n x + B^{2} b^{3} c^{3} i^{2} n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e) + 6 \, {\left (B^{2} b^{3} c^{3} i^{2} n^{2} + 3 \, A B b^{3} c^{3} i^{2} n + {\left (B^{2} b^{3} d^{3} i^{2} n^{2} + 3 \, A B b^{3} d^{3} i^{2} n\right )} x^{3} + 3 \, {\left (B^{2} b^{3} c d^{2} i^{2} n^{2} + 3 \, A B b^{3} c d^{2} i^{2} n\right )} x^{2} + 3 \, {\left (B^{2} b^{3} c^{2} d i^{2} n^{2} + 3 \, A B b^{3} c^{2} d i^{2} n\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{27 \, {\left ({\left (b^{7} c - a b^{6} d\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c - a^{2} b^{5} d\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c - a^{3} b^{4} d\right )} g^{4} x + {\left (a^{3} b^{4} c - a^{4} b^{3} d\right )} g^{4}\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)^4,x, algorithm="fricas")

[Out]

-1/27*(2*(B^2*b^3*c^3 - B^2*a^3*d^3)*i^2*n^2 + 6*(A*B*b^3*c^3 - A*B*a^3*d^3)*i^2*n + 9*(A^2*b^3*c^3 - A^2*a^3*
d^3)*i^2 + 3*(2*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*i^2*n^2 + 6*(A*B*b^3*c*d^2 - A*B*a*b^2*d^3)*i^2*n + 9*(A^2*b^3
*c*d^2 - A^2*a*b^2*d^3)*i^2)*x^2 + 9*(3*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*i^2*x^2 + 3*(B^2*b^3*c^2*d - B^2*a^2*b
*d^3)*i^2*x + (B^2*b^3*c^3 - B^2*a^3*d^3)*i^2)*log(e)^2 + 9*(B^2*b^3*d^3*i^2*n^2*x^3 + 3*B^2*b^3*c*d^2*i^2*n^2
*x^2 + 3*B^2*b^3*c^2*d*i^2*n^2*x + B^2*b^3*c^3*i^2*n^2)*log((b*x + a)/(d*x + c))^2 + 3*(2*(B^2*b^3*c^2*d - B^2
*a^2*b*d^3)*i^2*n^2 + 6*(A*B*b^3*c^2*d - A*B*a^2*b*d^3)*i^2*n + 9*(A^2*b^3*c^2*d - A^2*a^2*b*d^3)*i^2)*x + 6*(
(B^2*b^3*c^3 - B^2*a^3*d^3)*i^2*n + 3*(A*B*b^3*c^3 - A*B*a^3*d^3)*i^2 + 3*((B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*i^2
*n + 3*(A*B*b^3*c*d^2 - A*B*a*b^2*d^3)*i^2)*x^2 + 3*((B^2*b^3*c^2*d - B^2*a^2*b*d^3)*i^2*n + 3*(A*B*b^3*c^2*d
- A*B*a^2*b*d^3)*i^2)*x + 3*(B^2*b^3*d^3*i^2*n*x^3 + 3*B^2*b^3*c*d^2*i^2*n*x^2 + 3*B^2*b^3*c^2*d*i^2*n*x + B^2
*b^3*c^3*i^2*n)*log((b*x + a)/(d*x + c)))*log(e) + 6*(B^2*b^3*c^3*i^2*n^2 + 3*A*B*b^3*c^3*i^2*n + (B^2*b^3*d^3
*i^2*n^2 + 3*A*B*b^3*d^3*i^2*n)*x^3 + 3*(B^2*b^3*c*d^2*i^2*n^2 + 3*A*B*b^3*c*d^2*i^2*n)*x^2 + 3*(B^2*b^3*c^2*d
*i^2*n^2 + 3*A*B*b^3*c^2*d*i^2*n)*x)*log((b*x + a)/(d*x + c)))/((b^7*c - a*b^6*d)*g^4*x^3 + 3*(a*b^6*c - a^2*b
^5*d)*g^4*x^2 + 3*(a^2*b^5*c - a^3*b^4*d)*g^4*x + (a^3*b^4*c - a^4*b^3*d)*g^4)

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giac [A]  time = 110.18, size = 176, normalized size = 1.12 \[ \frac {1}{27} \, {\left (\frac {9 \, {\left (d x + c\right )}^{3} B^{2} n^{2} \log \left (\frac {b x + a}{d x + c}\right )^{2}}{{\left (b x + a\right )}^{3} g^{4}} + \frac {6 \, {\left (B^{2} n^{2} + 3 \, A B n + 3 \, B^{2} n\right )} {\left (d x + c\right )}^{3} \log \left (\frac {b x + a}{d x + c}\right )}{{\left (b x + a\right )}^{3} g^{4}} + \frac {{\left (2 \, B^{2} n^{2} + 6 \, A B n + 6 \, B^{2} n + 9 \, A^{2} + 18 \, A B + 9 \, B^{2}\right )} {\left (d x + c\right )}^{3}}{{\left (b x + a\right )}^{3} g^{4}}\right )} {\left (\frac {b c}{{\left (b c - a d\right )}^{2}} - \frac {a d}{{\left (b c - a d\right )}^{2}}\right )} \]

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)^4,x, algorithm="giac")

[Out]

1/27*(9*(d*x + c)^3*B^2*n^2*log((b*x + a)/(d*x + c))^2/((b*x + a)^3*g^4) + 6*(B^2*n^2 + 3*A*B*n + 3*B^2*n)*(d*
x + c)^3*log((b*x + a)/(d*x + c))/((b*x + a)^3*g^4) + (2*B^2*n^2 + 6*A*B*n + 6*B^2*n + 9*A^2 + 18*A*B + 9*B^2)
*(d*x + c)^3/((b*x + a)^3*g^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 \[ \int \frac {\left (d i x +c i \right )^{2} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (b g x +a g \right )^{4}}\, dx \]

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)^4,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)^4,x)

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maxima [B]  time = 4.97, size = 5588, normalized size = 35.59 \[ \text {result too large to display} \]

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)^4,x, algorithm="maxima")

[Out]

-1/9*A*B*d^2*i^2*n*((11*a^2*b^2*c^2 - 7*a^3*b*c*d + 2*a^4*d^2 + 6*(3*b^4*c^2 - 3*a*b^3*c*d + a^2*b^2*d^2)*x^2
+ 3*(9*a*b^3*c^2 - 7*a^2*b^2*c*d + 2*a^3*b*d^2)*x)/((b^8*c^2 - 2*a*b^7*c*d + a^2*b^6*d^2)*g^4*x^3 + 3*(a*b^7*c
^2 - 2*a^2*b^6*c*d + a^3*b^5*d^2)*g^4*x^2 + 3*(a^2*b^6*c^2 - 2*a^3*b^5*c*d + a^4*b^4*d^2)*g^4*x + (a^3*b^5*c^2
- 2*a^4*b^4*c*d + a^5*b^3*d^2)*g^4) + 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(b*x + a)/((b^6*c^3 - 3*a*b^
5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4) - 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(d*x + c)/((b^6*c^3
- 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4)) - 1/9*A*B*c^2*i^2*n*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*
b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2
- 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 -
2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*
g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4)) - 1/9*A*B*c*d*i^2*n*(
(5*a*b^2*c^2 - 22*a^2*b*c*d + 5*a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 16*a*b^2*c*d + 5*a^2*
b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*b^4*d^2)*g^4*x^2
+ 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^2*d^2)*g^4) - 6*
(3*b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4) + 6*(3*b*c*d^
2 - a*d^3)*log(d*x + c)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4)) - 1/3*(3*b*x + a)*B^2
*c*d*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)^2/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g
^4) - 1/3*(3*b^2*x^2 + 3*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^6*g^4*x^3 + 3*a*
b^5*g^4*x^2 + 3*a^2*b^4*g^4*x + a^3*b^3*g^4) - 1/54*(6*n*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2
- 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a
^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5
*b*d^2)*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log(d*
x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) +
(4*b^3*c^3 - 27*a*b^2*c^2*d + 108*a^2*b*c*d^2 - 85*a^3*d^3 + 66*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 18*(b^3*d^3*x^3
+ 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a)^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3
*x + a^3*d^3)*log(d*x + c)^2 - 3*(5*b^3*c^2*d - 54*a*b^2*c*d^2 + 49*a^2*b*d^3)*x + 66*(b^3*d^3*x^3 + 3*a*b^2*d
^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a) - 6*(11*b^3*d^3*x^3 + 33*a*b^2*d^3*x^2 + 33*a^2*b*d^3*x + 11*a^
3*d^3 - 6*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a))*log(d*x + c))*n^2/(a^3*b^4*c
^3*g^4 - 3*a^4*b^3*c^2*d*g^4 + 3*a^5*b^2*c*d^2*g^4 - a^6*b*d^3*g^4 + (b^7*c^3*g^4 - 3*a*b^6*c^2*d*g^4 + 3*a^2*
b^5*c*d^2*g^4 - a^3*b^4*d^3*g^4)*x^3 + 3*(a*b^6*c^3*g^4 - 3*a^2*b^5*c^2*d*g^4 + 3*a^3*b^4*c*d^2*g^4 - a^4*b^3*
d^3*g^4)*x^2 + 3*(a^2*b^5*c^3*g^4 - 3*a^3*b^4*c^2*d*g^4 + 3*a^4*b^3*c*d^2*g^4 - a^5*b^2*d^3*g^4)*x))*B^2*c^2*i
^2 - 1/54*(6*n*((5*a*b^2*c^2 - 22*a^2*b*c*d + 5*a^3*d^2 - 6*(3*b^3*c*d - a*b^2*d^2)*x^2 + 3*(3*b^3*c^2 - 16*a*
b^2*c*d + 5*a^2*b*d^2)*x)/((b^7*c^2 - 2*a*b^6*c*d + a^2*b^5*d^2)*g^4*x^3 + 3*(a*b^6*c^2 - 2*a^2*b^5*c*d + a^3*
b^4*d^2)*g^4*x^2 + 3*(a^2*b^5*c^2 - 2*a^3*b^4*c*d + a^4*b^3*d^2)*g^4*x + (a^3*b^4*c^2 - 2*a^4*b^3*c*d + a^5*b^
2*d^2)*g^4) - 6*(3*b*c*d^2 - a*d^3)*log(b*x + a)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^
4) + 6*(3*b*c*d^2 - a*d^3)*log(d*x + c)/((b^5*c^3 - 3*a*b^4*c^2*d + 3*a^2*b^3*c*d^2 - a^3*b^2*d^3)*g^4))*log(e
*(b*x/(d*x + c) + a/(d*x + c))^n) + (19*a*b^3*c^3 - 189*a^2*b^2*c^2*d + 189*a^3*b*c*d^2 - 19*a^4*d^3 - 6*(27*b
^4*c^2*d - 32*a*b^3*c*d^2 + 5*a^2*b^2*d^3)*x^2 + 18*(3*a^3*b*c*d^2 - a^4*d^3 + (3*b^4*c*d^2 - a*b^3*d^3)*x^3 +
3*(3*a*b^3*c*d^2 - a^2*b^2*d^3)*x^2 + 3*(3*a^2*b^2*c*d^2 - a^3*b*d^3)*x)*log(b*x + a)^2 + 18*(3*a^3*b*c*d^2 -
a^4*d^3 + (3*b^4*c*d^2 - a*b^3*d^3)*x^3 + 3*(3*a*b^3*c*d^2 - a^2*b^2*d^3)*x^2 + 3*(3*a^2*b^2*c*d^2 - a^3*b*d^
3)*x)*log(d*x + c)^2 + 3*(9*b^4*c^3 - 125*a*b^3*c^2*d + 135*a^2*b^2*c*d^2 - 19*a^3*b*d^3)*x - 6*(27*a^3*b*c*d^
2 - 5*a^4*d^3 + (27*b^4*c*d^2 - 5*a*b^3*d^3)*x^3 + 3*(27*a*b^3*c*d^2 - 5*a^2*b^2*d^3)*x^2 + 3*(27*a^2*b^2*c*d^
2 - 5*a^3*b*d^3)*x)*log(b*x + a) + 6*(27*a^3*b*c*d^2 - 5*a^4*d^3 + (27*b^4*c*d^2 - 5*a*b^3*d^3)*x^3 + 3*(27*a*
b^3*c*d^2 - 5*a^2*b^2*d^3)*x^2 + 3*(27*a^2*b^2*c*d^2 - 5*a^3*b*d^3)*x - 6*(3*a^3*b*c*d^2 - a^4*d^3 + (3*b^4*c*
d^2 - a*b^3*d^3)*x^3 + 3*(3*a*b^3*c*d^2 - a^2*b^2*d^3)*x^2 + 3*(3*a^2*b^2*c*d^2 - a^3*b*d^3)*x)*log(b*x + a))*
log(d*x + c))*n^2/(a^3*b^5*c^3*g^4 - 3*a^4*b^4*c^2*d*g^4 + 3*a^5*b^3*c*d^2*g^4 - a^6*b^2*d^3*g^4 + (b^8*c^3*g^
4 - 3*a*b^7*c^2*d*g^4 + 3*a^2*b^6*c*d^2*g^4 - a^3*b^5*d^3*g^4)*x^3 + 3*(a*b^7*c^3*g^4 - 3*a^2*b^6*c^2*d*g^4 +
3*a^3*b^5*c*d^2*g^4 - a^4*b^4*d^3*g^4)*x^2 + 3*(a^2*b^6*c^3*g^4 - 3*a^3*b^5*c^2*d*g^4 + 3*a^4*b^4*c*d^2*g^4 -
a^5*b^3*d^3*g^4)*x))*B^2*c*d*i^2 - 1/54*(6*n*((11*a^2*b^2*c^2 - 7*a^3*b*c*d + 2*a^4*d^2 + 6*(3*b^4*c^2 - 3*a*b
^3*c*d + a^2*b^2*d^2)*x^2 + 3*(9*a*b^3*c^2 - 7*a^2*b^2*c*d + 2*a^3*b*d^2)*x)/((b^8*c^2 - 2*a*b^7*c*d + a^2*b^6
*d^2)*g^4*x^3 + 3*(a*b^7*c^2 - 2*a^2*b^6*c*d + a^3*b^5*d^2)*g^4*x^2 + 3*(a^2*b^6*c^2 - 2*a^3*b^5*c*d + a^4*b^4
*d^2)*g^4*x + (a^3*b^5*c^2 - 2*a^4*b^4*c*d + a^5*b^3*d^2)*g^4) + 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d^3)*log(b
*x + a)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4) - 6*(3*b^2*c^2*d - 3*a*b*c*d^2 + a^2*d
^3)*log(d*x + c)/((b^6*c^3 - 3*a*b^5*c^2*d + 3*a^2*b^4*c*d^2 - a^3*b^3*d^3)*g^4))*log(e*(b*x/(d*x + c) + a/(d*
x + c))^n) + (85*a^2*b^3*c^3 - 108*a^3*b^2*c^2*d + 27*a^4*b*c*d^2 - 4*a^5*d^3 + 6*(18*b^5*c^3 - 27*a*b^4*c^2*d
+ 11*a^2*b^3*c*d^2 - 2*a^3*b^2*d^3)*x^2 - 18*(3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2 + a^5*d^3 + (3*b^5*c^2*d - 3*a*
b^4*c*d^2 + a^2*b^3*d^3)*x^3 + 3*(3*a*b^4*c^2*d - 3*a^2*b^3*c*d^2 + a^3*b^2*d^3)*x^2 + 3*(3*a^2*b^3*c^2*d - 3*
a^3*b^2*c*d^2 + a^4*b*d^3)*x)*log(b*x + a)^2 - 18*(3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2 + a^5*d^3 + (3*b^5*c^2*d -
3*a*b^4*c*d^2 + a^2*b^3*d^3)*x^3 + 3*(3*a*b^4*c^2*d - 3*a^2*b^3*c*d^2 + a^3*b^2*d^3)*x^2 + 3*(3*a^2*b^3*c^2*d
- 3*a^3*b^2*c*d^2 + a^4*b*d^3)*x)*log(d*x + c)^2 + 3*(63*a*b^4*c^3 - 86*a^2*b^3*c^2*d + 27*a^3*b^2*c*d^2 - 4*a
^4*b*d^3)*x + 6*(18*a^3*b^2*c^2*d - 9*a^4*b*c*d^2 + 2*a^5*d^3 + (18*b^5*c^2*d - 9*a*b^4*c*d^2 + 2*a^2*b^3*d^3)
*x^3 + 3*(18*a*b^4*c^2*d - 9*a^2*b^3*c*d^2 + 2*a^3*b^2*d^3)*x^2 + 3*(18*a^2*b^3*c^2*d - 9*a^3*b^2*c*d^2 + 2*a^
4*b*d^3)*x)*log(b*x + a) - 6*(18*a^3*b^2*c^2*d - 9*a^4*b*c*d^2 + 2*a^5*d^3 + (18*b^5*c^2*d - 9*a*b^4*c*d^2 + 2
*a^2*b^3*d^3)*x^3 + 3*(18*a*b^4*c^2*d - 9*a^2*b^3*c*d^2 + 2*a^3*b^2*d^3)*x^2 + 3*(18*a^2*b^3*c^2*d - 9*a^3*b^2
*c*d^2 + 2*a^4*b*d^3)*x - 6*(3*a^3*b^2*c^2*d - 3*a^4*b*c*d^2 + a^5*d^3 + (3*b^5*c^2*d - 3*a*b^4*c*d^2 + a^2*b^
3*d^3)*x^3 + 3*(3*a*b^4*c^2*d - 3*a^2*b^3*c*d^2 + a^3*b^2*d^3)*x^2 + 3*(3*a^2*b^3*c^2*d - 3*a^3*b^2*c*d^2 + a^
4*b*d^3)*x)*log(b*x + a))*log(d*x + c))*n^2/(a^3*b^6*c^3*g^4 - 3*a^4*b^5*c^2*d*g^4 + 3*a^5*b^4*c*d^2*g^4 - a^6
*b^3*d^3*g^4 + (b^9*c^3*g^4 - 3*a*b^8*c^2*d*g^4 + 3*a^2*b^7*c*d^2*g^4 - a^3*b^6*d^3*g^4)*x^3 + 3*(a*b^8*c^3*g^
4 - 3*a^2*b^7*c^2*d*g^4 + 3*a^3*b^6*c*d^2*g^4 - a^4*b^5*d^3*g^4)*x^2 + 3*(a^2*b^7*c^3*g^4 - 3*a^3*b^6*c^2*d*g^
4 + 3*a^4*b^5*c*d^2*g^4 - a^5*b^4*d^3*g^4)*x))*B^2*d^2*i^2 - 2/3*(3*b*x + a)*A*B*c*d*i^2*log(e*(b*x/(d*x + c)
+ a/(d*x + c))^n)/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2*g^4) - 2/3*(3*b^2*x^2 + 3*a*b*x +
a^2)*A*B*d^2*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^6*g^4*x^3 + 3*a*b^5*g^4*x^2 + 3*a^2*b^4*g^4*x + a^
3*b^3*g^4) - 1/3*B^2*c^2*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)^2/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b
^2*g^4*x + a^3*b*g^4) - 1/3*(3*b*x + a)*A^2*c*d*i^2/(b^5*g^4*x^3 + 3*a*b^4*g^4*x^2 + 3*a^2*b^3*g^4*x + a^3*b^2
*g^4) - 1/3*(3*b^2*x^2 + 3*a*b*x + a^2)*A^2*d^2*i^2/(b^6*g^4*x^3 + 3*a*b^5*g^4*x^2 + 3*a^2*b^4*g^4*x + a^3*b^3
*g^4) - 2/3*A*B*c^2*i^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n)/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*
x + a^3*b*g^4) - 1/3*A^2*c^2*i^2/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4)

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mupad [B]  time = 7.48, size = 1195, normalized size = 7.61 \[ -\frac {x\,\left (9\,c\,A^2\,b^2\,d\,i^2+9\,a\,A^2\,b\,d^2\,i^2+6\,c\,A\,B\,b^2\,d\,i^2\,n+6\,a\,A\,B\,b\,d^2\,i^2\,n+2\,c\,B^2\,b^2\,d\,i^2\,n^2+2\,a\,B^2\,b\,d^2\,i^2\,n^2\right )+x^2\,\left (9\,A^2\,b^2\,d^2\,i^2+6\,A\,B\,b^2\,d^2\,i^2\,n+2\,B^2\,b^2\,d^2\,i^2\,n^2\right )+3\,A^2\,a^2\,d^2\,i^2+3\,A^2\,b^2\,c^2\,i^2+\frac {2\,B^2\,a^2\,d^2\,i^2\,n^2}{3}+\frac {2\,B^2\,b^2\,c^2\,i^2\,n^2}{3}+3\,A^2\,a\,b\,c\,d\,i^2+2\,A\,B\,a^2\,d^2\,i^2\,n+2\,A\,B\,b^2\,c^2\,i^2\,n+\frac {2\,B^2\,a\,b\,c\,d\,i^2\,n^2}{3}+2\,A\,B\,a\,b\,c\,d\,i^2\,n}{9\,a^3\,b^3\,g^4+27\,a^2\,b^4\,g^4\,x+27\,a\,b^5\,g^4\,x^2+9\,b^6\,g^4\,x^3}-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {a\,\left (-a\,n\,B^2\,d^2\,i^2+b\,c\,n\,B^2\,d\,i^2+2\,A\,a\,B\,d^2\,i^2+2\,A\,b\,c\,B\,d\,i^2\right )+x\,\left (b\,\left (-a\,n\,B^2\,d^2\,i^2+b\,c\,n\,B^2\,d\,i^2+2\,A\,a\,B\,d^2\,i^2+2\,A\,b\,c\,B\,d\,i^2\right )+4\,A\,B\,a\,b\,d^2\,i^2+4\,A\,B\,b^2\,c\,d\,i^2-2\,B^2\,a\,b\,d^2\,i^2\,n+2\,B^2\,b^2\,c\,d\,i^2\,n\right )+2\,A\,B\,b^2\,c^2\,i^2-2\,B^2\,a^2\,d^2\,i^2\,n+6\,A\,B\,b^2\,d^2\,i^2\,x^2+2\,B^2\,a\,b\,c\,d\,i^2\,n}{3\,a^3\,b^3\,g^4+9\,a^2\,b^4\,g^4\,x+9\,a\,b^5\,g^4\,x^2+3\,b^6\,g^4\,x^3}+\frac {2\,B^2\,d^3\,i^2\,\left (x\,\left (b\,\left (\frac {a\,b^3\,g^4\,n\,\left (a\,d-b\,c\right )}{d}+\frac {b^3\,g^4\,n\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-b\,c\right )}{2\,d^2}\right )+\frac {2\,a\,b^4\,g^4\,n\,\left (a\,d-b\,c\right )}{d}+\frac {b^4\,g^4\,n\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-b\,c\right )}{d^2}\right )+a\,\left (\frac {a\,b^3\,g^4\,n\,\left (a\,d-b\,c\right )}{d}+\frac {b^3\,g^4\,n\,\left (a\,d-b\,c\right )\,\left (3\,a\,d-b\,c\right )}{2\,d^2}\right )+\frac {3\,b^5\,g^4\,n\,x^2\,\left (a\,d-b\,c\right )}{d}+\frac {b^3\,g^4\,n\,\left (a\,d-b\,c\right )\,\left (3\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2\right )}{d^3}\right )}{3\,b^3\,g^4\,\left (a\,d-b\,c\right )\,\left (3\,a^3\,b^3\,g^4+9\,a^2\,b^4\,g^4\,x+9\,a\,b^5\,g^4\,x^2+3\,b^6\,g^4\,x^3\right )}\right )-{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {a\,\left (\frac {B^2\,c\,d\,i^2}{3\,b^2}+\frac {B^2\,a\,d^2\,i^2}{3\,b^3}\right )+x\,\left (b\,\left (\frac {B^2\,c\,d\,i^2}{3\,b^2}+\frac {B^2\,a\,d^2\,i^2}{3\,b^3}\right )+\frac {2\,B^2\,c\,d\,i^2}{3\,b}+\frac {2\,B^2\,a\,d^2\,i^2}{3\,b^2}\right )+\frac {B^2\,c^2\,i^2}{3\,b}+\frac {B^2\,d^2\,i^2\,x^2}{b}}{a^3\,g^4+3\,a^2\,b\,g^4\,x+3\,a\,b^2\,g^4\,x^2+b^3\,g^4\,x^3}-\frac {B^2\,d^3\,i^2}{3\,b^3\,g^4\,\left (a\,d-b\,c\right )}\right )-\frac {B\,d^3\,i^2\,n\,\mathrm {atan}\left (\frac {\left (\frac {9\,c\,b^4\,g^4+9\,a\,d\,b^3\,g^4}{9\,b^3\,g^4}+2\,b\,d\,x\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (3\,A+B\,n\right )\,4{}\mathrm {i}}{9\,b^3\,g^4\,\left (a\,d-b\,c\right )} \]

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)^4,x)

[Out]

- (x*(9*A^2*a*b*d^2*i^2 + 9*A^2*b^2*c*d*i^2 + 2*B^2*a*b*d^2*i^2*n^2 + 2*B^2*b^2*c*d*i^2*n^2 + 6*A*B*a*b*d^2*i^
2*n + 6*A*B*b^2*c*d*i^2*n) + x^2*(9*A^2*b^2*d^2*i^2 + 2*B^2*b^2*d^2*i^2*n^2 + 6*A*B*b^2*d^2*i^2*n) + 3*A^2*a^2
*d^2*i^2 + 3*A^2*b^2*c^2*i^2 + (2*B^2*a^2*d^2*i^2*n^2)/3 + (2*B^2*b^2*c^2*i^2*n^2)/3 + 3*A^2*a*b*c*d*i^2 + 2*A
*B*a^2*d^2*i^2*n + 2*A*B*b^2*c^2*i^2*n + (2*B^2*a*b*c*d*i^2*n^2)/3 + 2*A*B*a*b*c*d*i^2*n)/(9*a^3*b^3*g^4 + 9*b
^6*g^4*x^3 + 27*a^2*b^4*g^4*x + 27*a*b^5*g^4*x^2) - log(e*((a + b*x)/(c + d*x))^n)*((a*(2*A*B*a*d^2*i^2 - B^2*
a*d^2*i^2*n + B^2*b*c*d*i^2*n + 2*A*B*b*c*d*i^2) + x*(b*(2*A*B*a*d^2*i^2 - B^2*a*d^2*i^2*n + B^2*b*c*d*i^2*n +
2*A*B*b*c*d*i^2) + 4*A*B*a*b*d^2*i^2 + 4*A*B*b^2*c*d*i^2 - 2*B^2*a*b*d^2*i^2*n + 2*B^2*b^2*c*d*i^2*n) + 2*A*B
*b^2*c^2*i^2 - 2*B^2*a^2*d^2*i^2*n + 6*A*B*b^2*d^2*i^2*x^2 + 2*B^2*a*b*c*d*i^2*n)/(3*a^3*b^3*g^4 + 3*b^6*g^4*x
^3 + 9*a^2*b^4*g^4*x + 9*a*b^5*g^4*x^2) + (2*B^2*d^3*i^2*(x*(b*((a*b^3*g^4*n*(a*d - b*c))/d + (b^3*g^4*n*(a*d
- b*c)*(3*a*d - b*c))/(2*d^2)) + (2*a*b^4*g^4*n*(a*d - b*c))/d + (b^4*g^4*n*(a*d - b*c)*(3*a*d - b*c))/d^2) +
a*((a*b^3*g^4*n*(a*d - b*c))/d + (b^3*g^4*n*(a*d - b*c)*(3*a*d - b*c))/(2*d^2)) + (3*b^5*g^4*n*x^2*(a*d - b*c)
)/d + (b^3*g^4*n*(a*d - b*c)*(3*a^2*d^2 + b^2*c^2 - 3*a*b*c*d))/d^3))/(3*b^3*g^4*(a*d - b*c)*(3*a^3*b^3*g^4 +
3*b^6*g^4*x^3 + 9*a^2*b^4*g^4*x + 9*a*b^5*g^4*x^2))) - log(e*((a + b*x)/(c + d*x))^n)^2*((a*((B^2*c*d*i^2)/(3*
b^2) + (B^2*a*d^2*i^2)/(3*b^3)) + x*(b*((B^2*c*d*i^2)/(3*b^2) + (B^2*a*d^2*i^2)/(3*b^3)) + (2*B^2*c*d*i^2)/(3*
b) + (2*B^2*a*d^2*i^2)/(3*b^2)) + (B^2*c^2*i^2)/(3*b) + (B^2*d^2*i^2*x^2)/b)/(a^3*g^4 + b^3*g^4*x^3 + 3*a*b^2*
g^4*x^2 + 3*a^2*b*g^4*x) - (B^2*d^3*i^2)/(3*b^3*g^4*(a*d - b*c))) - (B*d^3*i^2*n*atan((((9*b^4*c*g^4 + 9*a*b^3
*d*g^4)/(9*b^3*g^4) + 2*b*d*x)*1i)/(a*d - b*c))*(3*A + B*n)*4i)/(9*b^3*g^4*(a*d - b*c))

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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)**4,x)

[Out]

Timed out

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