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

Optimal. Leaf size=429 \[ \frac {2 b^3 B n \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d g^4 (b c-a d)^3}-\frac {2 b^2 B n (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 (c+d x) (b c-a d)^3}-\frac {2 B d^2 n (a+b x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 g^4 (c+d x)^3 (b c-a d)^3}+\frac {b B d n (a+b x)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{g^4 (c+d x)^2 (b c-a d)^3}-\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 d g^4 (c+d x)^3}-\frac {b^3 B^2 n^2 \log ^2\left (\frac {a+b x}{c+d x}\right )}{3 d g^4 (b c-a d)^3}+\frac {2 b^2 B^2 n^2 (a+b x)}{g^4 (c+d x) (b c-a d)^3}+\frac {2 B^2 d^2 n^2 (a+b x)^3}{27 g^4 (c+d x)^3 (b c-a d)^3}-\frac {b B^2 d n^2 (a+b x)^2}{2 g^4 (c+d x)^2 (b c-a d)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [C]  time = 1.10, antiderivative size = 736, normalized size of antiderivative = 1.72, number of steps used = 32, number of rules used = 11, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {2525, 12, 2528, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 44} \[ \frac {2 b^3 B^2 n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{3 d g^4 (b c-a d)^3}+\frac {2 b^3 B^2 n^2 \text {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{3 d g^4 (b c-a d)^3}+\frac {2 b^3 B n \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d g^4 (b c-a d)^3}-\frac {2 b^3 B n \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d g^4 (b c-a d)^3}+\frac {2 b^2 B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d g^4 (c+d x) (b c-a d)^2}+\frac {b B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{3 d g^4 (c+d x)^2 (b c-a d)}-\frac {\left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 d g^4 (c+d x)^3}+\frac {2 B n \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{9 d g^4 (c+d x)^3}-\frac {11 b^2 B^2 n^2}{9 d g^4 (c+d x) (b c-a d)^2}-\frac {b^3 B^2 n^2 \log ^2(a+b x)}{3 d g^4 (b c-a d)^3}-\frac {b^3 B^2 n^2 \log ^2(c+d x)}{3 d g^4 (b c-a d)^3}-\frac {11 b^3 B^2 n^2 \log (a+b x)}{9 d g^4 (b c-a d)^3}+\frac {11 b^3 B^2 n^2 \log (c+d x)}{9 d g^4 (b c-a d)^3}+\frac {2 b^3 B^2 n^2 \log (c+d x) \log \left (-\frac {d (a+b x)}{b c-a d}\right )}{3 d g^4 (b c-a d)^3}+\frac {2 b^3 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 d g^4 (b c-a d)^3}-\frac {5 b B^2 n^2}{18 d g^4 (c+d x)^2 (b c-a d)}-\frac {2 B^2 n^2}{27 d g^4 (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*g + d*g*x)^4,x]

[Out]

(-2*B^2*n^2)/(27*d*g^4*(c + d*x)^3) - (5*b*B^2*n^2)/(18*d*(b*c - a*d)*g^4*(c + d*x)^2) - (11*b^2*B^2*n^2)/(9*d
*(b*c - a*d)^2*g^4*(c + d*x)) - (11*b^3*B^2*n^2*Log[a + b*x])/(9*d*(b*c - a*d)^3*g^4) - (b^3*B^2*n^2*Log[a + b
*x]^2)/(3*d*(b*c - a*d)^3*g^4) + (2*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(9*d*g^4*(c + d*x)^3) + (b*B*n
*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(3*d*(b*c - a*d)*g^4*(c + d*x)^2) + (2*b^2*B*n*(A + B*Log[e*((a + b*x
)/(c + d*x))^n]))/(3*d*(b*c - a*d)^2*g^4*(c + d*x)) + (2*b^3*B*n*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x
))^n]))/(3*d*(b*c - a*d)^3*g^4) - (A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(3*d*g^4*(c + d*x)^3) + (11*b^3*B^2
*n^2*Log[c + d*x])/(9*d*(b*c - a*d)^3*g^4) + (2*b^3*B^2*n^2*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(3
*d*(b*c - a*d)^3*g^4) - (2*b^3*B*n*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x])/(3*d*(b*c - a*d)^3*g^4
) - (b^3*B^2*n^2*Log[c + d*x]^2)/(3*d*(b*c - a*d)^3*g^4) + (2*b^3*B^2*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c
- a*d)])/(3*d*(b*c - a*d)^3*g^4) + (2*b^3*B^2*n^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(3*d*(b*c - a*d)^3
*g^4) + (2*b^3*B^2*n^2*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(3*d*(b*c - a*d)^3*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 {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{(c g+d g x)^4} \, dx &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}+\frac {(2 B n) \int \frac {(b c-a d) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{g^3 (a+b x) (c+d x)^4} \, dx}{3 d g}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}+\frac {(2 B (b c-a d) n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(a+b x) (c+d x)^4} \, dx}{3 d g^4}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}+\frac {(2 B (b c-a d) n) \int \left (\frac {b^4 \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 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{(b c-a d) (c+d 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 (c+d x)^3}-\frac {b^2 d \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)^2}-\frac {b^3 d \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 d g^4}\\ &=-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}-\frac {(2 B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^4} \, dx}{3 g^4}-\frac {\left (2 b^3 B 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 c-a d)^3 g^4}+\frac {\left (2 b^4 B 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 d (b c-a d)^3 g^4}-\frac {\left (2 b^2 B n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^2} \, dx}{3 (b c-a d)^2 g^4}-\frac {(2 b B n) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(c+d x)^3} \, dx}{3 (b c-a d) g^4}\\ &=\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 d g^4 (c+d x)^3}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d) g^4 (c+d x)^2}+\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^2 g^4 (c+d x)}+\frac {2 b^3 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}-\frac {2 b^3 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}-\frac {\left (2 B^2 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^4} \, dx}{9 d g^4}-\frac {\left (2 b^3 B^2 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 d (b c-a d)^3 g^4}+\frac {\left (2 b^3 B^2 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 d (b c-a d)^3 g^4}-\frac {\left (2 b^2 B^2 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{3 d (b c-a d)^2 g^4}-\frac {\left (b B^2 n^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{3 d (b c-a d) g^4}\\ &=\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 d g^4 (c+d x)^3}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d) g^4 (c+d x)^2}+\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^2 g^4 (c+d x)}+\frac {2 b^3 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}-\frac {2 b^3 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}-\frac {\left (b B^2 n^2\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{3 d g^4}-\frac {\left (2 b^3 B^2 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 d (b c-a d)^3 g^4}+\frac {\left (2 b^3 B^2 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 d (b c-a d)^3 g^4}-\frac {\left (2 b^2 B^2 n^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{3 d (b c-a d) g^4}-\frac {\left (2 B^2 (b c-a d) n^2\right ) \int \frac {1}{(a+b x) (c+d x)^4} \, dx}{9 d g^4}\\ &=\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 d g^4 (c+d x)^3}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d) g^4 (c+d x)^2}+\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^2 g^4 (c+d x)}+\frac {2 b^3 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}-\frac {2 b^3 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}-\frac {\left (b B^2 n^2\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{3 d g^4}+\frac {\left (2 b^3 B^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 (b c-a d)^3 g^4}-\frac {\left (2 b^3 B^2 n^2\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 (b c-a d)^3 g^4}-\frac {\left (2 b^4 B^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3 d (b c-a d)^3 g^4}+\frac {\left (2 b^4 B^2 n^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 d (b c-a d)^3 g^4}-\frac {\left (2 b^2 B^2 n^2\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{3 d (b c-a d) g^4}-\frac {\left (2 B^2 (b c-a d) n^2\right ) \int \left (\frac {b^4}{(b c-a d)^4 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^4}-\frac {b d}{(b c-a d)^2 (c+d x)^3}-\frac {b^2 d}{(b c-a d)^3 (c+d x)^2}-\frac {b^3 d}{(b c-a d)^4 (c+d x)}\right ) \, dx}{9 d g^4}\\ &=-\frac {2 B^2 n^2}{27 d g^4 (c+d x)^3}-\frac {5 b B^2 n^2}{18 d (b c-a d) g^4 (c+d x)^2}-\frac {11 b^2 B^2 n^2}{9 d (b c-a d)^2 g^4 (c+d x)}-\frac {11 b^3 B^2 n^2 \log (a+b x)}{9 d (b c-a d)^3 g^4}+\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 d g^4 (c+d x)^3}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d) g^4 (c+d x)^2}+\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^2 g^4 (c+d x)}+\frac {2 b^3 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}+\frac {11 b^3 B^2 n^2 \log (c+d x)}{9 d (b c-a d)^3 g^4}+\frac {2 b^3 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}-\frac {2 b^3 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}+\frac {2 b^3 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 d (b c-a d)^3 g^4}-\frac {\left (2 b^3 B^2 n^2\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 (b c-a d)^3 g^4}-\frac {\left (2 b^3 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 d (b c-a d)^3 g^4}-\frac {\left (2 b^3 B^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 d (b c-a d)^3 g^4}-\frac {\left (2 b^4 B^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 d (b c-a d)^3 g^4}\\ &=-\frac {2 B^2 n^2}{27 d g^4 (c+d x)^3}-\frac {5 b B^2 n^2}{18 d (b c-a d) g^4 (c+d x)^2}-\frac {11 b^2 B^2 n^2}{9 d (b c-a d)^2 g^4 (c+d x)}-\frac {11 b^3 B^2 n^2 \log (a+b x)}{9 d (b c-a d)^3 g^4}-\frac {b^3 B^2 n^2 \log ^2(a+b x)}{3 d (b c-a d)^3 g^4}+\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 d g^4 (c+d x)^3}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d) g^4 (c+d x)^2}+\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^2 g^4 (c+d x)}+\frac {2 b^3 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}+\frac {11 b^3 B^2 n^2 \log (c+d x)}{9 d (b c-a d)^3 g^4}+\frac {2 b^3 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}-\frac {2 b^3 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}-\frac {b^3 B^2 n^2 \log ^2(c+d x)}{3 d (b c-a d)^3 g^4}+\frac {2 b^3 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 d (b c-a d)^3 g^4}-\frac {\left (2 b^3 B^2 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 d (b c-a d)^3 g^4}-\frac {\left (2 b^3 B^2 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 d (b c-a d)^3 g^4}\\ &=-\frac {2 B^2 n^2}{27 d g^4 (c+d x)^3}-\frac {5 b B^2 n^2}{18 d (b c-a d) g^4 (c+d x)^2}-\frac {11 b^2 B^2 n^2}{9 d (b c-a d)^2 g^4 (c+d x)}-\frac {11 b^3 B^2 n^2 \log (a+b x)}{9 d (b c-a d)^3 g^4}-\frac {b^3 B^2 n^2 \log ^2(a+b x)}{3 d (b c-a d)^3 g^4}+\frac {2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{9 d g^4 (c+d x)^3}+\frac {b B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d) g^4 (c+d x)^2}+\frac {2 b^2 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^2 g^4 (c+d x)}+\frac {2 b^3 B n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{3 d (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{3 d g^4 (c+d x)^3}+\frac {11 b^3 B^2 n^2 \log (c+d x)}{9 d (b c-a d)^3 g^4}+\frac {2 b^3 B^2 n^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}-\frac {2 b^3 B n \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 d (b c-a d)^3 g^4}-\frac {b^3 B^2 n^2 \log ^2(c+d x)}{3 d (b c-a d)^3 g^4}+\frac {2 b^3 B^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 d (b c-a d)^3 g^4}+\frac {2 b^3 B^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3 d (b c-a d)^3 g^4}+\frac {2 b^3 B^2 n^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 d (b c-a d)^3 g^4}\\ \end {align*}

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Mathematica [C]  time = 0.67, size = 609, normalized size = 1.42 \[ \frac {\frac {B n \left (36 b^3 (c+d x)^3 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-36 b^3 (c+d x)^3 \log (c+d x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+36 b^2 (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+12 (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+18 b (c+d x) (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )-18 b^3 B n (c+d x)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )+18 b^3 B n (c+d x)^3 \left (2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac {d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-36 b^2 B n (c+d x)^2 (b (c+d x) \log (a+b x)-a d-b (c+d x) \log (c+d x)+b c)-9 b B n (c+d x) \left (2 b^2 (c+d x)^2 \log (a+b x)+2 b (c+d x) (b c-a d)+(b c-a d)^2-2 b^2 (c+d x)^2 \log (c+d x)\right )-2 B n \left (6 b^3 (c+d x)^3 \log (a+b x)+6 b^2 (c+d x)^2 (b c-a d)+3 b (c+d x) (b c-a d)^2+2 (b c-a d)^3-6 b^3 (c+d x)^3 \log (c+d x)\right )\right )}{(b c-a d)^3}-18 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{54 d g^4 (c+d x)^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2/(c*g + d*g*x)^4,x]

[Out]

(-18*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + (B*n*(12*(b*c - a*d)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])
+ 18*b*(b*c - a*d)^2*(c + d*x)*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 36*b^2*(b*c - a*d)*(c + d*x)^2*(A + B*
Log[e*((a + b*x)/(c + d*x))^n]) + 36*b^3*(c + d*x)^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 36*
b^3*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])*Log[c + d*x] - 36*b^2*B*n*(c + d*x)^2*(b*c - a*d + b*(c
+ d*x)*Log[a + b*x] - b*(c + d*x)*Log[c + d*x]) - 9*b*B*n*(c + d*x)*((b*c - a*d)^2 + 2*b*(b*c - a*d)*(c + d*x
) + 2*b^2*(c + d*x)^2*Log[a + b*x] - 2*b^2*(c + d*x)^2*Log[c + d*x]) - 2*B*n*(2*(b*c - a*d)^3 + 3*b*(b*c - a*d
)^2*(c + d*x) + 6*b^2*(b*c - a*d)*(c + d*x)^2 + 6*b^3*(c + d*x)^3*Log[a + b*x] - 6*b^3*(c + d*x)^3*Log[c + d*x
]) - 18*b^3*B*n*(c + d*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^3*B*n*(c + d*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*c - a*d)^3)/(54*d*g^4*(c + d*x)^3)

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fricas [B]  time = 0.85, size = 1167, normalized size = 2.72 \[ -\frac {18 \, A^{2} b^{3} c^{3} - 54 \, A^{2} a b^{2} c^{2} d + 54 \, A^{2} a^{2} b c d^{2} - 18 \, A^{2} a^{3} d^{3} + {\left (85 \, B^{2} b^{3} c^{3} - 108 \, B^{2} a b^{2} c^{2} d + 27 \, B^{2} a^{2} b c d^{2} - 4 \, B^{2} a^{3} d^{3}\right )} n^{2} + 6 \, {\left (11 \, {\left (B^{2} b^{3} c d^{2} - B^{2} a b^{2} d^{3}\right )} n^{2} - 6 \, {\left (A B b^{3} c d^{2} - A B a b^{2} d^{3}\right )} n\right )} x^{2} + 18 \, {\left (B^{2} b^{3} c^{3} - 3 \, B^{2} a b^{2} c^{2} d + 3 \, B^{2} a^{2} b c d^{2} - B^{2} a^{3} d^{3}\right )} \log \relax (e)^{2} - 18 \, {\left (B^{2} b^{3} d^{3} n^{2} x^{3} + 3 \, B^{2} b^{3} c d^{2} n^{2} x^{2} + 3 \, B^{2} b^{3} c^{2} d n^{2} x + {\left (3 \, B^{2} a b^{2} c^{2} d - 3 \, B^{2} a^{2} b c d^{2} + B^{2} a^{3} d^{3}\right )} n^{2}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - 6 \, {\left (11 \, A B b^{3} c^{3} - 18 \, A B a b^{2} c^{2} d + 9 \, A B a^{2} b c d^{2} - 2 \, A B a^{3} d^{3}\right )} n + 3 \, {\left ({\left (49 \, B^{2} b^{3} c^{2} d - 54 \, B^{2} a b^{2} c d^{2} + 5 \, B^{2} a^{2} b d^{3}\right )} n^{2} - 6 \, {\left (5 \, A B b^{3} c^{2} d - 6 \, A B a b^{2} c d^{2} + A B a^{2} b d^{3}\right )} n\right )} x + 6 \, {\left (6 \, A B b^{3} c^{3} - 18 \, A B a b^{2} c^{2} d + 18 \, A B a^{2} b c d^{2} - 6 \, A B a^{3} d^{3} - 6 \, {\left (B^{2} b^{3} c d^{2} - B^{2} a b^{2} d^{3}\right )} n x^{2} - 3 \, {\left (5 \, B^{2} b^{3} c^{2} d - 6 \, B^{2} a b^{2} c d^{2} + B^{2} a^{2} b d^{3}\right )} n x - {\left (11 \, B^{2} b^{3} c^{3} - 18 \, B^{2} a b^{2} c^{2} d + 9 \, B^{2} a^{2} b c d^{2} - 2 \, B^{2} a^{3} d^{3}\right )} n - 6 \, {\left (B^{2} b^{3} d^{3} n x^{3} + 3 \, B^{2} b^{3} c d^{2} n x^{2} + 3 \, B^{2} b^{3} c^{2} d n x + {\left (3 \, B^{2} a b^{2} c^{2} d - 3 \, B^{2} a^{2} b c d^{2} + B^{2} a^{3} d^{3}\right )} n\right )} \log \left (\frac {b x + a}{d x + c}\right )\right )} \log \relax (e) + 6 \, {\left ({\left (11 \, B^{2} b^{3} d^{3} n^{2} - 6 \, A B b^{3} d^{3} n\right )} x^{3} + {\left (18 \, B^{2} a b^{2} c^{2} d - 9 \, B^{2} a^{2} b c d^{2} + 2 \, B^{2} a^{3} d^{3}\right )} n^{2} - 3 \, {\left (6 \, A B b^{3} c d^{2} n - {\left (9 \, B^{2} b^{3} c d^{2} + 2 \, B^{2} a b^{2} d^{3}\right )} n^{2}\right )} x^{2} - 6 \, {\left (3 \, A B a b^{2} c^{2} d - 3 \, A B a^{2} b c d^{2} + A B a^{3} d^{3}\right )} n - 3 \, {\left (6 \, A B b^{3} c^{2} d n - {\left (6 \, B^{2} b^{3} c^{2} d + 6 \, B^{2} a b^{2} c d^{2} - B^{2} a^{2} b d^{3}\right )} n^{2}\right )} x\right )} \log \left (\frac {b x + a}{d x + c}\right )}{54 \, {\left ({\left (b^{3} c^{3} d^{4} - 3 \, a b^{2} c^{2} d^{5} + 3 \, a^{2} b c d^{6} - a^{3} d^{7}\right )} g^{4} x^{3} + 3 \, {\left (b^{3} c^{4} d^{3} - 3 \, a b^{2} c^{3} d^{4} + 3 \, a^{2} b c^{2} d^{5} - a^{3} c d^{6}\right )} g^{4} x^{2} + 3 \, {\left (b^{3} c^{5} d^{2} - 3 \, a b^{2} c^{4} d^{3} + 3 \, a^{2} b c^{3} d^{4} - a^{3} c^{2} d^{5}\right )} g^{4} x + {\left (b^{3} c^{6} d - 3 \, a b^{2} c^{5} d^{2} + 3 \, a^{2} b c^{4} d^{3} - a^{3} c^{3} d^{4}\right )} g^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(d*g*x+c*g)^4,x, algorithm="fricas")

[Out]

-1/54*(18*A^2*b^3*c^3 - 54*A^2*a*b^2*c^2*d + 54*A^2*a^2*b*c*d^2 - 18*A^2*a^3*d^3 + (85*B^2*b^3*c^3 - 108*B^2*a
*b^2*c^2*d + 27*B^2*a^2*b*c*d^2 - 4*B^2*a^3*d^3)*n^2 + 6*(11*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*n^2 - 6*(A*B*b^3*
c*d^2 - A*B*a*b^2*d^3)*n)*x^2 + 18*(B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2*a^2*b*c*d^2 - B^2*a^3*d^3)*log(e)^
2 - 18*(B^2*b^3*d^3*n^2*x^3 + 3*B^2*b^3*c*d^2*n^2*x^2 + 3*B^2*b^3*c^2*d*n^2*x + (3*B^2*a*b^2*c^2*d - 3*B^2*a^2
*b*c*d^2 + B^2*a^3*d^3)*n^2)*log((b*x + a)/(d*x + c))^2 - 6*(11*A*B*b^3*c^3 - 18*A*B*a*b^2*c^2*d + 9*A*B*a^2*b
*c*d^2 - 2*A*B*a^3*d^3)*n + 3*((49*B^2*b^3*c^2*d - 54*B^2*a*b^2*c*d^2 + 5*B^2*a^2*b*d^3)*n^2 - 6*(5*A*B*b^3*c^
2*d - 6*A*B*a*b^2*c*d^2 + A*B*a^2*b*d^3)*n)*x + 6*(6*A*B*b^3*c^3 - 18*A*B*a*b^2*c^2*d + 18*A*B*a^2*b*c*d^2 - 6
*A*B*a^3*d^3 - 6*(B^2*b^3*c*d^2 - B^2*a*b^2*d^3)*n*x^2 - 3*(5*B^2*b^3*c^2*d - 6*B^2*a*b^2*c*d^2 + B^2*a^2*b*d^
3)*n*x - (11*B^2*b^3*c^3 - 18*B^2*a*b^2*c^2*d + 9*B^2*a^2*b*c*d^2 - 2*B^2*a^3*d^3)*n - 6*(B^2*b^3*d^3*n*x^3 +
3*B^2*b^3*c*d^2*n*x^2 + 3*B^2*b^3*c^2*d*n*x + (3*B^2*a*b^2*c^2*d - 3*B^2*a^2*b*c*d^2 + B^2*a^3*d^3)*n)*log((b*
x + a)/(d*x + c)))*log(e) + 6*((11*B^2*b^3*d^3*n^2 - 6*A*B*b^3*d^3*n)*x^3 + (18*B^2*a*b^2*c^2*d - 9*B^2*a^2*b*
c*d^2 + 2*B^2*a^3*d^3)*n^2 - 3*(6*A*B*b^3*c*d^2*n - (9*B^2*b^3*c*d^2 + 2*B^2*a*b^2*d^3)*n^2)*x^2 - 6*(3*A*B*a*
b^2*c^2*d - 3*A*B*a^2*b*c*d^2 + A*B*a^3*d^3)*n - 3*(6*A*B*b^3*c^2*d*n - (6*B^2*b^3*c^2*d + 6*B^2*a*b^2*c*d^2 -
B^2*a^2*b*d^3)*n^2)*x)*log((b*x + a)/(d*x + c)))/((b^3*c^3*d^4 - 3*a*b^2*c^2*d^5 + 3*a^2*b*c*d^6 - a^3*d^7)*g
^4*x^3 + 3*(b^3*c^4*d^3 - 3*a*b^2*c^3*d^4 + 3*a^2*b*c^2*d^5 - a^3*c*d^6)*g^4*x^2 + 3*(b^3*c^5*d^2 - 3*a*b^2*c^
4*d^3 + 3*a^2*b*c^3*d^4 - a^3*c^2*d^5)*g^4*x + (b^3*c^6*d - 3*a*b^2*c^5*d^2 + 3*a^2*b*c^4*d^3 - a^3*c^3*d^4)*g
^4)

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giac [A]  time = 13.81, size = 746, normalized size = 1.74 \[ \frac {1}{54} \, {\left (18 \, {\left (\frac {3 \, {\left (b x + a\right )} B^{2} b^{2} n^{2}}{{\left (b^{2} c^{2} g^{4} - 2 \, a b c d g^{4} + a^{2} d^{2} g^{4}\right )} {\left (d x + c\right )}} - \frac {3 \, {\left (b x + a\right )}^{2} B^{2} b d n^{2}}{{\left (b^{2} c^{2} g^{4} - 2 \, a b c d g^{4} + a^{2} d^{2} g^{4}\right )} {\left (d x + c\right )}^{2}} + \frac {{\left (b x + a\right )}^{3} B^{2} d^{2} n^{2}}{{\left (b^{2} c^{2} g^{4} - 2 \, a b c d g^{4} + a^{2} d^{2} g^{4}\right )} {\left (d x + c\right )}^{3}}\right )} \log \left (\frac {b x + a}{d x + c}\right )^{2} - 6 \, {\left (\frac {2 \, {\left (B^{2} d^{2} n^{2} - 3 \, A B d^{2} n - 3 \, B^{2} d^{2} n\right )} {\left (b x + a\right )}^{3}}{{\left (b^{2} c^{2} g^{4} - 2 \, a b c d g^{4} + a^{2} d^{2} g^{4}\right )} {\left (d x + c\right )}^{3}} - \frac {9 \, {\left (B^{2} b d n^{2} - 2 \, A B b d n - 2 \, B^{2} b d n\right )} {\left (b x + a\right )}^{2}}{{\left (b^{2} c^{2} g^{4} - 2 \, a b c d g^{4} + a^{2} d^{2} g^{4}\right )} {\left (d x + c\right )}^{2}} + \frac {18 \, {\left (B^{2} b^{2} n^{2} - A B b^{2} n - B^{2} b^{2} n\right )} {\left (b x + a\right )}}{{\left (b^{2} c^{2} g^{4} - 2 \, a b c d g^{4} + a^{2} d^{2} g^{4}\right )} {\left (d x + c\right )}}\right )} \log \left (\frac {b x + a}{d x + c}\right ) + \frac {2 \, {\left (2 \, B^{2} d^{2} n^{2} - 6 \, A B d^{2} n - 6 \, B^{2} d^{2} n + 9 \, A^{2} d^{2} + 18 \, A B d^{2} + 9 \, B^{2} d^{2}\right )} {\left (b x + a\right )}^{3}}{{\left (b^{2} c^{2} g^{4} - 2 \, a b c d g^{4} + a^{2} d^{2} g^{4}\right )} {\left (d x + c\right )}^{3}} - \frac {27 \, {\left (B^{2} b d n^{2} - 2 \, A B b d n - 2 \, B^{2} b d n + 2 \, A^{2} b d + 4 \, A B b d + 2 \, B^{2} b d\right )} {\left (b x + a\right )}^{2}}{{\left (b^{2} c^{2} g^{4} - 2 \, a b c d g^{4} + a^{2} d^{2} g^{4}\right )} {\left (d x + c\right )}^{2}} + \frac {54 \, {\left (2 \, B^{2} b^{2} n^{2} - 2 \, A B b^{2} n - 2 \, B^{2} b^{2} n + A^{2} b^{2} + 2 \, A B b^{2} + B^{2} b^{2}\right )} {\left (b x + a\right )}}{{\left (b^{2} c^{2} g^{4} - 2 \, a b c d g^{4} + a^{2} d^{2} g^{4}\right )} {\left (d x + c\right )}}\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((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(d*g*x+c*g)^4,x, algorithm="giac")

[Out]

1/54*(18*(3*(b*x + a)*B^2*b^2*n^2/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2*g^4)*(d*x + c)) - 3*(b*x + a)^2*B^2*
b*d*n^2/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2*g^4)*(d*x + c)^2) + (b*x + a)^3*B^2*d^2*n^2/((b^2*c^2*g^4 - 2*
a*b*c*d*g^4 + a^2*d^2*g^4)*(d*x + c)^3))*log((b*x + a)/(d*x + c))^2 - 6*(2*(B^2*d^2*n^2 - 3*A*B*d^2*n - 3*B^2*
d^2*n)*(b*x + a)^3/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2*g^4)*(d*x + c)^3) - 9*(B^2*b*d*n^2 - 2*A*B*b*d*n -
2*B^2*b*d*n)*(b*x + a)^2/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2*g^4)*(d*x + c)^2) + 18*(B^2*b^2*n^2 - A*B*b^2
*n - B^2*b^2*n)*(b*x + a)/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2*g^4)*(d*x + c)))*log((b*x + a)/(d*x + c)) +
2*(2*B^2*d^2*n^2 - 6*A*B*d^2*n - 6*B^2*d^2*n + 9*A^2*d^2 + 18*A*B*d^2 + 9*B^2*d^2)*(b*x + a)^3/((b^2*c^2*g^4 -
2*a*b*c*d*g^4 + a^2*d^2*g^4)*(d*x + c)^3) - 27*(B^2*b*d*n^2 - 2*A*B*b*d*n - 2*B^2*b*d*n + 2*A^2*b*d + 4*A*B*b
*d + 2*B^2*b*d)*(b*x + a)^2/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2*g^4)*(d*x + c)^2) + 54*(2*B^2*b^2*n^2 - 2*
A*B*b^2*n - 2*B^2*b^2*n + A^2*b^2 + 2*A*B*b^2 + B^2*b^2)*(b*x + a)/((b^2*c^2*g^4 - 2*a*b*c*d*g^4 + a^2*d^2*g^4
)*(d*x + c)))*(b*c/(b*c - a*d)^2 - a*d/(b*c - a*d)^2)

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maple [F]  time = 0.28, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}}{\left (d g x +c g \right )^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*ln(e*((b*x+a)/(d*x+c))^n)+A)^2/(d*g*x+c*g)^4,x)

[Out]

int((B*ln(e*((b*x+a)/(d*x+c))^n)+A)^2/(d*g*x+c*g)^4,x)

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maxima [B]  time = 1.29, size = 1435, normalized size = 3.34 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*log(e*((b*x+a)/(d*x+c))^n))^2/(d*g*x+c*g)^4,x, algorithm="maxima")

[Out]

1/9*A*B*n*((6*b^2*d^2*x^2 + 11*b^2*c^2 - 7*a*b*c*d + 2*a^2*d^2 + 3*(5*b^2*c*d - a*b*d^2)*x)/((b^2*c^2*d^4 - 2*
a*b*c*d^5 + a^2*d^6)*g^4*x^3 + 3*(b^2*c^3*d^3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*g^4*x^2 + 3*(b^2*c^4*d^2 - 2*a*b*c^
3*d^3 + a^2*c^2*d^4)*g^4*x + (b^2*c^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3)*g^4) + 6*b^3*log(b*x + a)/((b^3*c^3*d -
3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*g^4) - 6*b^3*log(d*x + c)/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*
c*d^3 - a^3*d^4)*g^4)) + 1/54*(6*n*((6*b^2*d^2*x^2 + 11*b^2*c^2 - 7*a*b*c*d + 2*a^2*d^2 + 3*(5*b^2*c*d - a*b*d
^2)*x)/((b^2*c^2*d^4 - 2*a*b*c*d^5 + a^2*d^6)*g^4*x^3 + 3*(b^2*c^3*d^3 - 2*a*b*c^2*d^4 + a^2*c*d^5)*g^4*x^2 +
3*(b^2*c^4*d^2 - 2*a*b*c^3*d^3 + a^2*c^2*d^4)*g^4*x + (b^2*c^5*d - 2*a*b*c^4*d^2 + a^2*c^3*d^3)*g^4) + 6*b^3*l
og(b*x + a)/((b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*g^4) - 6*b^3*log(d*x + c)/((b^3*c^3*d - 3
*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*g^4))*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) - (85*b^3*c^3 - 108*a*b
^2*c^2*d + 27*a^2*b*c*d^2 - 4*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*b^3*c*d^2*x^2 + 3
*b^3*c^2*d*x + b^3*c^3)*log(b*x + a)^2 + 18*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*log(d*x
+ c)^2 + 3*(49*b^3*c^2*d - 54*a*b^2*c*d^2 + 5*a^2*b*d^3)*x + 66*(b^3*d^3*x^3 + 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x
+ b^3*c^3)*log(b*x + a) - 6*(11*b^3*d^3*x^3 + 33*b^3*c*d^2*x^2 + 33*b^3*c^2*d*x + 11*b^3*c^3 + 6*(b^3*d^3*x^3
+ 3*b^3*c*d^2*x^2 + 3*b^3*c^2*d*x + b^3*c^3)*log(b*x + a))*log(d*x + c))*n^2/(b^3*c^6*d*g^4 - 3*a*b^2*c^5*d^2
*g^4 + 3*a^2*b*c^4*d^3*g^4 - a^3*c^3*d^4*g^4 + (b^3*c^3*d^4*g^4 - 3*a*b^2*c^2*d^5*g^4 + 3*a^2*b*c*d^6*g^4 - a^
3*d^7*g^4)*x^3 + 3*(b^3*c^4*d^3*g^4 - 3*a*b^2*c^3*d^4*g^4 + 3*a^2*b*c^2*d^5*g^4 - a^3*c*d^6*g^4)*x^2 + 3*(b^3*
c^5*d^2*g^4 - 3*a*b^2*c^4*d^3*g^4 + 3*a^2*b*c^3*d^4*g^4 - a^3*c^2*d^5*g^4)*x))*B^2 - 1/3*B^2*log(e*(b*x/(d*x +
c) + a/(d*x + c))^n)^2/(d^4*g^4*x^3 + 3*c*d^3*g^4*x^2 + 3*c^2*d^2*g^4*x + c^3*d*g^4) - 2/3*A*B*log(e*(b*x/(d*
x + c) + a/(d*x + c))^n)/(d^4*g^4*x^3 + 3*c*d^3*g^4*x^2 + 3*c^2*d^2*g^4*x + c^3*d*g^4) - 1/3*A^2/(d^4*g^4*x^3
+ 3*c*d^3*g^4*x^2 + 3*c^2*d^2*g^4*x + c^3*d*g^4)

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mupad [B]  time = 7.16, size = 1040, normalized size = 2.42 \[ -{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^2\,\left (\frac {B^2}{3\,d\,\left (c^3\,g^4+3\,c^2\,d\,g^4\,x+3\,c\,d^2\,g^4\,x^2+d^3\,g^4\,x^3\right )}+\frac {B^2\,b^3}{3\,d\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )-\frac {\frac {18\,A^2\,a^2\,d^2-36\,A^2\,a\,b\,c\,d+18\,A^2\,b^2\,c^2-12\,A\,B\,a^2\,d^2\,n+42\,A\,B\,a\,b\,c\,d\,n-66\,A\,B\,b^2\,c^2\,n+4\,B^2\,a^2\,d^2\,n^2-23\,B^2\,a\,b\,c\,d\,n^2+85\,B^2\,b^2\,c^2\,n^2}{6\,\left (a\,d-b\,c\right )}-\frac {x\,\left (-49\,c\,B^2\,b^2\,d\,n^2+5\,a\,B^2\,b\,d^2\,n^2+30\,A\,c\,B\,b^2\,d\,n-6\,A\,a\,B\,b\,d^2\,n\right )}{2\,\left (a\,d-b\,c\right )}+\frac {b\,x^2\,\left (11\,B^2\,b\,d^2\,n^2-6\,A\,B\,b\,d^2\,n\right )}{a\,d-b\,c}}{x\,\left (27\,a\,c^2\,d^3\,g^4-27\,b\,c^3\,d^2\,g^4\right )-x^2\,\left (27\,b\,c^2\,d^3\,g^4-27\,a\,c\,d^4\,g^4\right )+x^3\,\left (9\,a\,d^5\,g^4-9\,b\,c\,d^4\,g^4\right )+9\,a\,c^3\,d^2\,g^4-9\,b\,c^4\,d\,g^4}-\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\,\left (\frac {2\,A\,B}{3\,c^3\,d\,g^4+9\,c^2\,d^2\,g^4\,x+9\,c\,d^3\,g^4\,x^2+3\,d^4\,g^4\,x^3}+\frac {2\,B^2\,b^3\,\left (x\,\left (d\,\left (\frac {d\,g^4\,n\,\left (a\,d-b\,c\right )\,\left (a\,d-3\,b\,c\right )}{2\,b^2}-\frac {c\,d\,g^4\,n\,\left (a\,d-b\,c\right )}{b}\right )-\frac {2\,c\,d^2\,g^4\,n\,\left (a\,d-b\,c\right )}{b}+\frac {d^2\,g^4\,n\,\left (a\,d-b\,c\right )\,\left (a\,d-3\,b\,c\right )}{b^2}\right )+c\,\left (\frac {d\,g^4\,n\,\left (a\,d-b\,c\right )\,\left (a\,d-3\,b\,c\right )}{2\,b^2}-\frac {c\,d\,g^4\,n\,\left (a\,d-b\,c\right )}{b}\right )-\frac {d\,g^4\,n\,\left (a\,d-b\,c\right )\,\left (a^2\,d^2-3\,a\,b\,c\,d+3\,b^2\,c^2\right )}{b^3}-\frac {3\,d^3\,g^4\,n\,x^2\,\left (a\,d-b\,c\right )}{b}\right )}{3\,d\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )\,\left (3\,c^3\,d\,g^4+9\,c^2\,d^2\,g^4\,x+9\,c\,d^3\,g^4\,x^2+3\,d^4\,g^4\,x^3\right )}\right )-\frac {B\,b^3\,n\,\mathrm {atan}\left (\frac {B\,b^3\,n\,\left (6\,A-11\,B\,n\right )\,\left (\frac {a^3\,d^4\,g^4-a^2\,b\,c\,d^3\,g^4-a\,b^2\,c^2\,d^2\,g^4+b^3\,c^3\,d\,g^4}{a^2\,d^3\,g^4-2\,a\,b\,c\,d^2\,g^4+b^2\,c^2\,d\,g^4}+2\,b\,d\,x\right )\,\left (a^2\,d^3\,g^4-2\,a\,b\,c\,d^2\,g^4+b^2\,c^2\,d\,g^4\right )\,1{}\mathrm {i}}{d\,g^4\,\left (11\,B^2\,b^3\,n^2-6\,A\,B\,b^3\,n\right )\,{\left (a\,d-b\,c\right )}^3}\right )\,\left (6\,A-11\,B\,n\right )\,2{}\mathrm {i}}{9\,d\,g^4\,{\left (a\,d-b\,c\right )}^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*log(e*((a + b*x)/(c + d*x))^n))^2/(c*g + d*g*x)^4,x)

[Out]

- log(e*((a + b*x)/(c + d*x))^n)^2*(B^2/(3*d*(c^3*g^4 + d^3*g^4*x^3 + 3*c*d^2*g^4*x^2 + 3*c^2*d*g^4*x)) + (B^2
*b^3)/(3*d*g^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) - ((18*A^2*a^2*d^2 + 18*A^2*b^2*c^2 + 4*B
^2*a^2*d^2*n^2 + 85*B^2*b^2*c^2*n^2 - 36*A^2*a*b*c*d - 12*A*B*a^2*d^2*n - 66*A*B*b^2*c^2*n - 23*B^2*a*b*c*d*n^
2 + 42*A*B*a*b*c*d*n)/(6*(a*d - b*c)) - (x*(5*B^2*a*b*d^2*n^2 - 49*B^2*b^2*c*d*n^2 - 6*A*B*a*b*d^2*n + 30*A*B*
b^2*c*d*n))/(2*(a*d - b*c)) + (b*x^2*(11*B^2*b*d^2*n^2 - 6*A*B*b*d^2*n))/(a*d - b*c))/(x*(27*a*c^2*d^3*g^4 - 2
7*b*c^3*d^2*g^4) - x^2*(27*b*c^2*d^3*g^4 - 27*a*c*d^4*g^4) + x^3*(9*a*d^5*g^4 - 9*b*c*d^4*g^4) + 9*a*c^3*d^2*g
^4 - 9*b*c^4*d*g^4) - log(e*((a + b*x)/(c + d*x))^n)*((2*A*B)/(3*c^3*d*g^4 + 3*d^4*g^4*x^3 + 9*c^2*d^2*g^4*x +
9*c*d^3*g^4*x^2) + (2*B^2*b^3*(x*(d*((d*g^4*n*(a*d - b*c)*(a*d - 3*b*c))/(2*b^2) - (c*d*g^4*n*(a*d - b*c))/b)
- (2*c*d^2*g^4*n*(a*d - b*c))/b + (d^2*g^4*n*(a*d - b*c)*(a*d - 3*b*c))/b^2) + c*((d*g^4*n*(a*d - b*c)*(a*d -
3*b*c))/(2*b^2) - (c*d*g^4*n*(a*d - b*c))/b) - (d*g^4*n*(a*d - b*c)*(a^2*d^2 + 3*b^2*c^2 - 3*a*b*c*d))/b^3 -
(3*d^3*g^4*n*x^2*(a*d - b*c))/b))/(3*d*g^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(3*c^3*d*g^4 +
3*d^4*g^4*x^3 + 9*c^2*d^2*g^4*x + 9*c*d^3*g^4*x^2))) - (B*b^3*n*atan((B*b^3*n*(6*A - 11*B*n)*((a^3*d^4*g^4 + b
^3*c^3*d*g^4 - a^2*b*c*d^3*g^4 - a*b^2*c^2*d^2*g^4)/(a^2*d^3*g^4 + b^2*c^2*d*g^4 - 2*a*b*c*d^2*g^4) + 2*b*d*x)
*(a^2*d^3*g^4 + b^2*c^2*d*g^4 - 2*a*b*c*d^2*g^4)*1i)/(d*g^4*(11*B^2*b^3*n^2 - 6*A*B*b^3*n)*(a*d - b*c)^3))*(6*
A - 11*B*n)*2i)/(9*d*g^4*(a*d - b*c)^3)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {A^{2}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {B^{2} \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}^{2}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx + \int \frac {2 A B \log {\left (e \left (\frac {a}{c + d x} + \frac {b x}{c + d x}\right )^{n} \right )}}{c^{4} + 4 c^{3} d x + 6 c^{2} d^{2} x^{2} + 4 c d^{3} x^{3} + d^{4} x^{4}}\, dx}{g^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*ln(e*((b*x+a)/(d*x+c))**n))**2/(d*g*x+c*g)**4,x)

[Out]

(Integral(A**2/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x) + Integral(B**2*log(e*(a
/(c + d*x) + b*x/(c + d*x))**n)**2/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d**4*x**4), x) + In
tegral(2*A*B*log(e*(a/(c + d*x) + b*x/(c + d*x))**n)/(c**4 + 4*c**3*d*x + 6*c**2*d**2*x**2 + 4*c*d**3*x**3 + d
**4*x**4), x))/g**4

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