3.179 \(\int (a g+b g x)^2 (c i+d i x)^3 (A+B \log (e (\frac {a+b x}{c+d x})^n))^2 \, dx\)

Optimal. Leaf size=976 \[ \frac {B g^2 i^3 n \left (2 A+3 B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \log \left (\frac {b c-a d}{b (c+d x)}\right ) (b c-a d)^6}{60 b^4 d^3}+\frac {B^2 g^2 i^3 n^2 \log \left (\frac {a+b x}{c+d x}\right ) (b c-a d)^6}{36 b^4 d^3}+\frac {11 B^2 g^2 i^3 n^2 \log (c+d x) (b c-a d)^6}{180 b^4 d^3}+\frac {B^2 g^2 i^3 n^2 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right ) (b c-a d)^6}{30 b^4 d^3}-\frac {7 B^2 g^2 i^3 n^2 x (b c-a d)^5}{180 b^3 d^2}+\frac {B g^2 i^3 n (a+b x) \left (2 A+B n+2 B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^5}{60 b^4 d^2}-\frac {7 B^2 g^2 i^3 n^2 (c+d x)^2 (b c-a d)^4}{360 b^2 d^3}-\frac {B g^2 i^3 n (a+b x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^4}{60 b^4 d}-\frac {B g^2 i^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^4}{10 b^2 d^3}-\frac {B^2 g^2 i^3 n^2 (c+d x)^3 (b c-a d)^3}{60 b d^3}+\frac {g^2 i^3 (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)^3}{60 b^4}-\frac {B g^2 i^3 n (a+b x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^3}{30 b^4}+\frac {B g^2 i^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^3}{45 b d^3}+\frac {B^2 g^2 i^3 n^2 (c+d x)^4 (b c-a d)^2}{60 d^3}+\frac {g^2 i^3 (a+b x)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)^2}{20 b^3}+\frac {7 B g^2 i^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2}{60 d^3}+\frac {g^2 i^3 (a+b x)^3 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)}{10 b^2}-\frac {b B g^2 i^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)}{15 d^3}+\frac {g^2 i^3 (a+b x)^3 (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 b} \]

[Out]

-7/180*B^2*(-a*d+b*c)^5*g^2*i^3*n^2*x/b^3/d^2-7/360*B^2*(-a*d+b*c)^4*g^2*i^3*n^2*(d*x+c)^2/b^2/d^3-1/60*B^2*(-
a*d+b*c)^3*g^2*i^3*n^2*(d*x+c)^3/b/d^3+1/60*B^2*(-a*d+b*c)^2*g^2*i^3*n^2*(d*x+c)^4/d^3-1/60*B*(-a*d+b*c)^4*g^2
*i^3*n*(b*x+a)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^4/d-1/30*B*(-a*d+b*c)^3*g^2*i^3*n*(b*x+a)^3*(A+B*ln(e*((b*x
+a)/(d*x+c))^n))/b^4-1/10*B*(-a*d+b*c)^4*g^2*i^3*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^2/d^3+1/45*B*(-
a*d+b*c)^3*g^2*i^3*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b/d^3+7/60*B*(-a*d+b*c)^2*g^2*i^3*n*(d*x+c)^4*(
A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3-1/15*b*B*(-a*d+b*c)*g^2*i^3*n*(d*x+c)^5*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/d^3
+1/60*(-a*d+b*c)^3*g^2*i^3*(b*x+a)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^4+1/20*(-a*d+b*c)^2*g^2*i^3*(b*x+a)^3
*(d*x+c)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b^3+1/10*(-a*d+b*c)*g^2*i^3*(b*x+a)^3*(d*x+c)^2*(A+B*ln(e*((b*x+a)/
(d*x+c))^n))^2/b^2+1/6*g^2*i^3*(b*x+a)^3*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))^2/b+1/60*B*(-a*d+b*c)^5*g^2
*i^3*n*(b*x+a)*(2*A+B*n+2*B*ln(e*((b*x+a)/(d*x+c))^n))/b^4/d^2+1/60*B*(-a*d+b*c)^6*g^2*i^3*n*(2*A+3*B*n+2*B*ln
(e*((b*x+a)/(d*x+c))^n))*ln((-a*d+b*c)/b/(d*x+c))/b^4/d^3+1/36*B^2*(-a*d+b*c)^6*g^2*i^3*n^2*ln((b*x+a)/(d*x+c)
)/b^4/d^3+11/180*B^2*(-a*d+b*c)^6*g^2*i^3*n^2*ln(d*x+c)/b^4/d^3+1/30*B^2*(-a*d+b*c)^6*g^2*i^3*n^2*polylog(2,d*
(b*x+a)/b/(d*x+c))/b^4/d^3

________________________________________________________________________________________

Rubi [A]  time = 3.20, antiderivative size = 886, normalized size of antiderivative = 0.91, number of steps used = 83, number of rules used = 13, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.289, Rules used = {2528, 2525, 12, 2486, 31, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 43} \[ \frac {B^2 g^2 i^3 n^2 \log ^2(a+b x) (b c-a d)^6}{60 b^4 d^3}-\frac {B^2 g^2 i^3 n^2 \log (a+b x) (b c-a d)^6}{45 b^4 d^3}-\frac {B g^2 i^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^6}{30 b^4 d^3}+\frac {B^2 g^2 i^3 n^2 \log (c+d x) (b c-a d)^6}{30 b^4 d^3}-\frac {B^2 g^2 i^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right ) (b c-a d)^6}{30 b^4 d^3}-\frac {B^2 g^2 i^3 n^2 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right ) (b c-a d)^6}{30 b^4 d^3}-\frac {B^2 g^2 i^3 n^2 x (b c-a d)^5}{45 b^3 d^2}-\frac {A B g^2 i^3 n x (b c-a d)^5}{30 b^3 d^2}-\frac {B^2 g^2 i^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) (b c-a d)^5}{30 b^4 d^2}-\frac {7 B^2 g^2 i^3 n^2 (c+d x)^2 (b c-a d)^4}{360 b^2 d^3}-\frac {B g^2 i^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^4}{60 b^2 d^3}-\frac {B^2 g^2 i^3 n^2 (c+d x)^3 (b c-a d)^3}{60 b d^3}-\frac {B g^2 i^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^3}{90 b d^3}+\frac {B^2 g^2 i^3 n^2 (c+d x)^4 (b c-a d)^2}{60 d^3}+\frac {g^2 i^3 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)^2}{4 d^3}+\frac {7 B g^2 i^3 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)^2}{60 d^3}-\frac {2 b g^2 i^3 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 (b c-a d)}{5 d^3}-\frac {b B g^2 i^3 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) (b c-a d)}{15 d^3}+\frac {b^2 g^2 i^3 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 d^3} \]

Antiderivative was successfully verified.

[In]

Int[(a*g + b*g*x)^2*(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2,x]

[Out]

-(A*B*(b*c - a*d)^5*g^2*i^3*n*x)/(30*b^3*d^2) - (B^2*(b*c - a*d)^5*g^2*i^3*n^2*x)/(45*b^3*d^2) - (7*B^2*(b*c -
a*d)^4*g^2*i^3*n^2*(c + d*x)^2)/(360*b^2*d^3) - (B^2*(b*c - a*d)^3*g^2*i^3*n^2*(c + d*x)^3)/(60*b*d^3) + (B^2
*(b*c - a*d)^2*g^2*i^3*n^2*(c + d*x)^4)/(60*d^3) - (B^2*(b*c - a*d)^6*g^2*i^3*n^2*Log[a + b*x])/(45*b^4*d^3) +
(B^2*(b*c - a*d)^6*g^2*i^3*n^2*Log[a + b*x]^2)/(60*b^4*d^3) - (B^2*(b*c - a*d)^5*g^2*i^3*n*(a + b*x)*Log[e*((
a + b*x)/(c + d*x))^n])/(30*b^4*d^2) - (B*(b*c - a*d)^4*g^2*i^3*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x
))^n]))/(60*b^2*d^3) - (B*(b*c - a*d)^3*g^2*i^3*n*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(90*b*d^
3) + (7*B*(b*c - a*d)^2*g^2*i^3*n*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(60*d^3) - (b*B*(b*c - a
*d)*g^2*i^3*n*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(15*d^3) - (B*(b*c - a*d)^6*g^2*i^3*n*Log[a
+ b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(30*b^4*d^3) + ((b*c - a*d)^2*g^2*i^3*(c + d*x)^4*(A + B*Log[e*
((a + b*x)/(c + d*x))^n])^2)/(4*d^3) - (2*b*(b*c - a*d)*g^2*i^3*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))
^n])^2)/(5*d^3) + (b^2*g^2*i^3*(c + d*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(6*d^3) + (B^2*(b*c - a*d
)^6*g^2*i^3*n^2*Log[c + d*x])/(30*b^4*d^3) - (B^2*(b*c - a*d)^6*g^2*i^3*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*
c - a*d)])/(30*b^4*d^3) - (B^2*(b*c - a*d)^6*g^2*i^3*n^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(30*b^4*d^3
)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[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 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 0]

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 (179 c+179 d x)^3 (a g+b g x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\int \left (\frac {(-b c+a d)^2 g^2 (179 c+179 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{d^2}-\frac {2 b (b c-a d) g^2 (179 c+179 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{179 d^2}+\frac {b^2 g^2 (179 c+179 d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{32041 d^2}\right ) \, dx\\ &=\frac {\left (b^2 g^2\right ) \int (179 c+179 d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{32041 d^2}-\frac {\left (2 b (b c-a d) g^2\right ) \int (179 c+179 d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{179 d^2}+\frac {\left ((b c-a d)^2 g^2\right ) \int (179 c+179 d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{d^2}\\ &=\frac {5735339 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 d^3}-\frac {11470678 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac {5735339 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 d^3}-\frac {\left (b^2 B g^2 n\right ) \int \frac {32894113444921 (b c-a d) (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{17206017 d^3}+\frac {\left (4 b B (b c-a d) g^2 n\right ) \int \frac {183765996899 (b c-a d) (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{160205 d^3}-\frac {\left (B (b c-a d)^2 g^2 n\right ) \int \frac {1026625681 (b c-a d) (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{358 d^3}\\ &=\frac {5735339 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 d^3}-\frac {11470678 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac {5735339 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 d^3}-\frac {\left (5735339 b^2 B (b c-a d) g^2 n\right ) \int \frac {(c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{3 d^3}+\frac {\left (22941356 b B (b c-a d)^2 g^2 n\right ) \int \frac {(c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{5 d^3}-\frac {\left (5735339 B (b c-a d)^3 g^2 n\right ) \int \frac {(c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{a+b x} \, dx}{2 d^3}\\ &=\frac {5735339 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 d^3}-\frac {11470678 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac {5735339 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 d^3}-\frac {\left (5735339 b^2 B (b c-a d) g^2 n\right ) \int \left (\frac {d (b c-a d)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^5}+\frac {(b c-a d)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^5 (a+b x)}+\frac {d (b c-a d)^3 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4}+\frac {d (b c-a d)^2 (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3}+\frac {d (b c-a d) (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac {d (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{3 d^3}+\frac {\left (22941356 b B (b c-a d)^2 g^2 n\right ) \int \left (\frac {d (b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4}+\frac {(b c-a d)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^4 (a+b x)}+\frac {d (b c-a d)^2 (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3}+\frac {d (b c-a d) (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac {d (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{5 d^3}-\frac {\left (5735339 B (b c-a d)^3 g^2 n\right ) \int \left (\frac {d (b c-a d)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3}+\frac {(b c-a d)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b^2}+\frac {d (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{b}\right ) \, dx}{2 d^3}\\ &=\frac {5735339 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 d^3}-\frac {11470678 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac {5735339 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 d^3}-\frac {\left (5735339 b B (b c-a d) g^2 n\right ) \int (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3 d^2}-\frac {\left (5735339 B (b c-a d)^2 g^2 n\right ) \int (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3 d^2}+\frac {\left (22941356 B (b c-a d)^2 g^2 n\right ) \int (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 d^2}-\frac {\left (5735339 B (b c-a d)^3 g^2 n\right ) \int (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3 b d^2}-\frac {\left (5735339 B (b c-a d)^3 g^2 n\right ) \int (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{2 b d^2}+\frac {\left (22941356 B (b c-a d)^3 g^2 n\right ) \int (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b d^2}-\frac {\left (5735339 B (b c-a d)^4 g^2 n\right ) \int (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3 b^2 d^2}-\frac {\left (5735339 B (b c-a d)^4 g^2 n\right ) \int (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{2 b^2 d^2}+\frac {\left (22941356 B (b c-a d)^4 g^2 n\right ) \int (c+d x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b^2 d^2}-\frac {\left (5735339 B (b c-a d)^5 g^2 n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3 b^3 d^2}-\frac {\left (5735339 B (b c-a d)^5 g^2 n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{2 b^3 d^2}+\frac {\left (22941356 B (b c-a d)^5 g^2 n\right ) \int \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right ) \, dx}{5 b^3 d^2}-\frac {\left (5735339 B (b c-a d)^6 g^2 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^3 d^3}-\frac {\left (5735339 B (b c-a d)^6 g^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{2 b^3 d^3}+\frac {\left (22941356 B (b c-a d)^6 g^2 n\right ) \int \frac {A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{a+b x} \, dx}{5 b^3 d^3}\\ &=-\frac {5735339 A B (b c-a d)^5 g^2 n x}{30 b^3 d^2}-\frac {5735339 B (b c-a d)^4 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{60 b^2 d^3}-\frac {5735339 B (b c-a d)^3 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{90 b d^3}+\frac {40147373 B (b c-a d)^2 g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{60 d^3}-\frac {5735339 b B (b c-a d) g^2 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 d^3}-\frac {5735339 B (b c-a d)^6 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b^4 d^3}+\frac {5735339 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 d^3}-\frac {11470678 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac {5735339 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 d^3}-\frac {\left (5735339 B^2 (b c-a d)^5 g^2 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{3 b^3 d^2}-\frac {\left (5735339 B^2 (b c-a d)^5 g^2 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{2 b^3 d^2}+\frac {\left (22941356 B^2 (b c-a d)^5 g^2 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{5 b^3 d^2}+\frac {\left (5735339 b B^2 (b c-a d) g^2 n^2\right ) \int \frac {(b c-a d) (c+d x)^4}{a+b x} \, dx}{15 d^3}+\frac {\left (5735339 B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {(b c-a d) (c+d x)^3}{a+b x} \, dx}{12 d^3}-\frac {\left (5735339 B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {(b c-a d) (c+d x)^3}{a+b x} \, dx}{5 d^3}+\frac {\left (5735339 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{9 b d^3}+\frac {\left (5735339 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{6 b d^3}-\frac {\left (22941356 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{15 b d^3}+\frac {\left (5735339 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{6 b^2 d^3}+\frac {\left (5735339 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{4 b^2 d^3}-\frac {\left (11470678 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{5 b^2 d^3}+\frac {\left (5735339 B^2 (b c-a d)^6 g^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 b^4 d^3}+\frac {\left (5735339 B^2 (b c-a d)^6 g^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}{2 b^4 d^3}-\frac {\left (22941356 B^2 (b c-a d)^6 g^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}{5 b^4 d^3}\\ &=-\frac {5735339 A B (b c-a d)^5 g^2 n x}{30 b^3 d^2}-\frac {5735339 B^2 (b c-a d)^5 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{30 b^4 d^2}-\frac {5735339 B (b c-a d)^4 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{60 b^2 d^3}-\frac {5735339 B (b c-a d)^3 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{90 b d^3}+\frac {40147373 B (b c-a d)^2 g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{60 d^3}-\frac {5735339 b B (b c-a d) g^2 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 d^3}-\frac {5735339 B (b c-a d)^6 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b^4 d^3}+\frac {5735339 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 d^3}-\frac {11470678 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac {5735339 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 d^3}+\frac {\left (5735339 b B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac {(c+d x)^4}{a+b x} \, dx}{15 d^3}+\frac {\left (5735339 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{12 d^3}-\frac {\left (5735339 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac {(c+d x)^3}{a+b x} \, dx}{5 d^3}+\frac {\left (5735339 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{9 b d^3}+\frac {\left (5735339 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{6 b d^3}-\frac {\left (22941356 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{15 b d^3}+\frac {\left (5735339 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{6 b^2 d^3}+\frac {\left (5735339 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{4 b^2 d^3}-\frac {\left (11470678 B^2 (b c-a d)^5 g^2 n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{5 b^2 d^3}+\frac {\left (5735339 B^2 (b c-a d)^6 g^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 b^4 d^3}+\frac {\left (5735339 B^2 (b c-a d)^6 g^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}{2 b^4 d^3}-\frac {\left (22941356 B^2 (b c-a d)^6 g^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}{5 b^4 d^3}+\frac {\left (5735339 B^2 (b c-a d)^6 g^2 n^2\right ) \int \frac {1}{c+d x} \, dx}{3 b^4 d^2}+\frac {\left (5735339 B^2 (b c-a d)^6 g^2 n^2\right ) \int \frac {1}{c+d x} \, dx}{2 b^4 d^2}-\frac {\left (22941356 B^2 (b c-a d)^6 g^2 n^2\right ) \int \frac {1}{c+d x} \, dx}{5 b^4 d^2}\\ &=-\frac {5735339 A B (b c-a d)^5 g^2 n x}{30 b^3 d^2}-\frac {5735339 B^2 (b c-a d)^5 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{30 b^4 d^2}-\frac {5735339 B (b c-a d)^4 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{60 b^2 d^3}-\frac {5735339 B (b c-a d)^3 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{90 b d^3}+\frac {40147373 B (b c-a d)^2 g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{60 d^3}-\frac {5735339 b B (b c-a d) g^2 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 d^3}-\frac {5735339 B (b c-a d)^6 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b^4 d^3}+\frac {5735339 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 d^3}-\frac {11470678 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac {5735339 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 d^3}+\frac {5735339 B^2 (b c-a d)^6 g^2 n^2 \log (c+d x)}{30 b^4 d^3}+\frac {\left (5735339 b B^2 (b c-a d)^2 g^2 n^2\right ) \int \left (\frac {d (b c-a d)^3}{b^4}+\frac {(b c-a d)^4}{b^4 (a+b x)}+\frac {d (b c-a d)^2 (c+d x)}{b^3}+\frac {d (b c-a d) (c+d x)^2}{b^2}+\frac {d (c+d x)^3}{b}\right ) \, dx}{15 d^3}+\frac {\left (5735339 B^2 (b c-a d)^3 g^2 n^2\right ) \int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx}{12 d^3}-\frac {\left (5735339 B^2 (b c-a d)^3 g^2 n^2\right ) \int \left (\frac {d (b c-a d)^2}{b^3}+\frac {(b c-a d)^3}{b^3 (a+b x)}+\frac {d (b c-a d) (c+d x)}{b^2}+\frac {d (c+d x)^2}{b}\right ) \, dx}{5 d^3}+\frac {\left (5735339 B^2 (b c-a d)^4 g^2 n^2\right ) \int \left (\frac {d (b c-a d)}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)}+\frac {d (c+d x)}{b}\right ) \, dx}{9 b d^3}+\frac {\left (5735339 B^2 (b c-a d)^4 g^2 n^2\right ) \int \left (\frac {d (b c-a d)}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)}+\frac {d (c+d x)}{b}\right ) \, dx}{6 b d^3}-\frac {\left (22941356 B^2 (b c-a d)^4 g^2 n^2\right ) \int \left (\frac {d (b c-a d)}{b^2}+\frac {(b c-a d)^2}{b^2 (a+b x)}+\frac {d (c+d x)}{b}\right ) \, dx}{15 b d^3}+\frac {\left (5735339 B^2 (b c-a d)^5 g^2 n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{6 b^2 d^3}+\frac {\left (5735339 B^2 (b c-a d)^5 g^2 n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{4 b^2 d^3}-\frac {\left (11470678 B^2 (b c-a d)^5 g^2 n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{5 b^2 d^3}+\frac {\left (5735339 B^2 (b c-a d)^6 g^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3 b^3 d^3}+\frac {\left (5735339 B^2 (b c-a d)^6 g^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{2 b^3 d^3}-\frac {\left (22941356 B^2 (b c-a d)^6 g^2 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{5 b^3 d^3}-\frac {\left (5735339 B^2 (b c-a d)^6 g^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 b^4 d^2}-\frac {\left (5735339 B^2 (b c-a d)^6 g^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{2 b^4 d^2}+\frac {\left (22941356 B^2 (b c-a d)^6 g^2 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{5 b^4 d^2}\\ &=-\frac {5735339 A B (b c-a d)^5 g^2 n x}{30 b^3 d^2}-\frac {5735339 B^2 (b c-a d)^5 g^2 n^2 x}{45 b^3 d^2}-\frac {40147373 B^2 (b c-a d)^4 g^2 n^2 (c+d x)^2}{360 b^2 d^3}-\frac {5735339 B^2 (b c-a d)^3 g^2 n^2 (c+d x)^3}{60 b d^3}+\frac {5735339 B^2 (b c-a d)^2 g^2 n^2 (c+d x)^4}{60 d^3}-\frac {5735339 B^2 (b c-a d)^6 g^2 n^2 \log (a+b x)}{45 b^4 d^3}-\frac {5735339 B^2 (b c-a d)^5 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{30 b^4 d^2}-\frac {5735339 B (b c-a d)^4 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{60 b^2 d^3}-\frac {5735339 B (b c-a d)^3 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{90 b d^3}+\frac {40147373 B (b c-a d)^2 g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{60 d^3}-\frac {5735339 b B (b c-a d) g^2 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 d^3}-\frac {5735339 B (b c-a d)^6 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b^4 d^3}+\frac {5735339 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 d^3}-\frac {11470678 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac {5735339 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 d^3}+\frac {5735339 B^2 (b c-a d)^6 g^2 n^2 \log (c+d x)}{30 b^4 d^3}-\frac {5735339 B^2 (b c-a d)^6 g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{30 b^4 d^3}+\frac {\left (5735339 B^2 (b c-a d)^6 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 b^4 d^3}+\frac {\left (5735339 B^2 (b c-a d)^6 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{2 b^4 d^3}-\frac {\left (22941356 B^2 (b c-a d)^6 g^2 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{5 b^4 d^3}+\frac {\left (5735339 B^2 (b c-a d)^6 g^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 b^3 d^3}+\frac {\left (5735339 B^2 (b c-a d)^6 g^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{2 b^3 d^3}-\frac {\left (22941356 B^2 (b c-a d)^6 g^2 n^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{5 b^3 d^3}\\ &=-\frac {5735339 A B (b c-a d)^5 g^2 n x}{30 b^3 d^2}-\frac {5735339 B^2 (b c-a d)^5 g^2 n^2 x}{45 b^3 d^2}-\frac {40147373 B^2 (b c-a d)^4 g^2 n^2 (c+d x)^2}{360 b^2 d^3}-\frac {5735339 B^2 (b c-a d)^3 g^2 n^2 (c+d x)^3}{60 b d^3}+\frac {5735339 B^2 (b c-a d)^2 g^2 n^2 (c+d x)^4}{60 d^3}-\frac {5735339 B^2 (b c-a d)^6 g^2 n^2 \log (a+b x)}{45 b^4 d^3}+\frac {5735339 B^2 (b c-a d)^6 g^2 n^2 \log ^2(a+b x)}{60 b^4 d^3}-\frac {5735339 B^2 (b c-a d)^5 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{30 b^4 d^2}-\frac {5735339 B (b c-a d)^4 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{60 b^2 d^3}-\frac {5735339 B (b c-a d)^3 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{90 b d^3}+\frac {40147373 B (b c-a d)^2 g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{60 d^3}-\frac {5735339 b B (b c-a d) g^2 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 d^3}-\frac {5735339 B (b c-a d)^6 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b^4 d^3}+\frac {5735339 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 d^3}-\frac {11470678 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac {5735339 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 d^3}+\frac {5735339 B^2 (b c-a d)^6 g^2 n^2 \log (c+d x)}{30 b^4 d^3}-\frac {5735339 B^2 (b c-a d)^6 g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{30 b^4 d^3}+\frac {\left (5735339 B^2 (b c-a d)^6 g^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 b^4 d^3}+\frac {\left (5735339 B^2 (b c-a d)^6 g^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 )}{2 b^4 d^3}-\frac {\left (22941356 B^2 (b c-a d)^6 g^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 )}{5 b^4 d^3}\\ &=-\frac {5735339 A B (b c-a d)^5 g^2 n x}{30 b^3 d^2}-\frac {5735339 B^2 (b c-a d)^5 g^2 n^2 x}{45 b^3 d^2}-\frac {40147373 B^2 (b c-a d)^4 g^2 n^2 (c+d x)^2}{360 b^2 d^3}-\frac {5735339 B^2 (b c-a d)^3 g^2 n^2 (c+d x)^3}{60 b d^3}+\frac {5735339 B^2 (b c-a d)^2 g^2 n^2 (c+d x)^4}{60 d^3}-\frac {5735339 B^2 (b c-a d)^6 g^2 n^2 \log (a+b x)}{45 b^4 d^3}+\frac {5735339 B^2 (b c-a d)^6 g^2 n^2 \log ^2(a+b x)}{60 b^4 d^3}-\frac {5735339 B^2 (b c-a d)^5 g^2 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{30 b^4 d^2}-\frac {5735339 B (b c-a d)^4 g^2 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{60 b^2 d^3}-\frac {5735339 B (b c-a d)^3 g^2 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{90 b d^3}+\frac {40147373 B (b c-a d)^2 g^2 n (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{60 d^3}-\frac {5735339 b B (b c-a d) g^2 n (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{15 d^3}-\frac {5735339 B (b c-a d)^6 g^2 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{30 b^4 d^3}+\frac {5735339 (b c-a d)^2 g^2 (c+d x)^4 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{4 d^3}-\frac {11470678 b (b c-a d) g^2 (c+d x)^5 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{5 d^3}+\frac {5735339 b^2 g^2 (c+d x)^6 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2}{6 d^3}+\frac {5735339 B^2 (b c-a d)^6 g^2 n^2 \log (c+d x)}{30 b^4 d^3}-\frac {5735339 B^2 (b c-a d)^6 g^2 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{30 b^4 d^3}-\frac {5735339 B^2 (b c-a d)^6 g^2 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{30 b^4 d^3}\\ \end {align*}

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Mathematica [A]  time = 1.56, size = 1627, normalized size = 1.67 \[ \text {result too large to display} \]

Antiderivative was successfully verified.

[In]

Integrate[(a*g + b*g*x)^2*(c*i + d*i*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2,x]

[Out]

(g^2*i^3*(15*(b*c - a*d)^2*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - 24*b*(b*c - a*d)*(c + d*x)^5
*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 + 10*b^2*(c + d*x)^6*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - (5*B
*(b*c - a*d)^3*n*(6*A*b*d*(b*c - a*d)^2*x - 3*B*(b*c - a*d)^2*n*(b*d*x + (b*c - a*d)*Log[a + b*x]) - B*(b*c -
a*d)*n*(2*b*d*(b*c - a*d)*x + b^2*(c + d*x)^2 + 2*(b*c - a*d)^2*Log[a + b*x]) + 6*B*d*(b*c - a*d)^2*(a + b*x)*
Log[e*((a + b*x)/(c + d*x))^n] + 3*b^2*(b*c - a*d)*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 2*b^3*
(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 6*(b*c - a*d)^3*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c +
d*x))^n]) - 6*B*(b*c - a*d)^3*n*Log[c + d*x] - 3*B*(b*c - a*d)^3*n*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c
+ d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)])))/b^4 + (2*B*(b*c - a*d)^2*n*(24*A*b*d*(b*
c - a*d)^3*x - 12*B*(b*c - a*d)^3*n*(b*d*x + (b*c - a*d)*Log[a + b*x]) - 4*B*(b*c - a*d)^2*n*(2*b*d*(b*c - a*d
)*x + b^2*(c + d*x)^2 + 2*(b*c - a*d)^2*Log[a + b*x]) - B*(b*c - a*d)*n*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c -
a*d)*(c + d*x)^2 + 2*b^3*(c + d*x)^3 + 6*(b*c - a*d)^3*Log[a + b*x]) + 24*B*d*(b*c - a*d)^3*(a + b*x)*Log[e*((
a + b*x)/(c + d*x))^n] + 12*b^2*(b*c - a*d)^2*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 8*b^3*(b*c
- a*d)*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 6*b^4*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x
))^n]) + 24*(b*c - a*d)^4*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 24*B*(b*c - a*d)^4*n*Log[c + d
*x] - 12*B*(b*c - a*d)^4*n*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(
a + b*x))/(-(b*c) + a*d)])))/b^4 - (B*(b*c - a*d)*n*(120*A*b*d*(b*c - a*d)^4*x - 60*B*(b*c - a*d)^4*n*(b*d*x +
(b*c - a*d)*Log[a + b*x]) - 20*B*(b*c - a*d)^3*n*(2*b*d*(b*c - a*d)*x + b^2*(c + d*x)^2 + 2*(b*c - a*d)^2*Log
[a + b*x]) - 5*B*(b*c - a*d)^2*n*(6*b*d*(b*c - a*d)^2*x + 3*b^2*(b*c - a*d)*(c + d*x)^2 + 2*b^3*(c + d*x)^3 +
6*(b*c - a*d)^3*Log[a + b*x]) - 2*B*(b*c - a*d)*n*(12*b*d*(b*c - a*d)^3*x + 6*b^2*(b*c - a*d)^2*(c + d*x)^2 +
4*b^3*(b*c - a*d)*(c + d*x)^3 + 3*b^4*(c + d*x)^4 + 12*(b*c - a*d)^4*Log[a + b*x]) + 120*B*d*(b*c - a*d)^4*(a
+ b*x)*Log[e*((a + b*x)/(c + d*x))^n] + 60*b^2*(b*c - a*d)^3*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]
) + 40*b^3*(b*c - a*d)^2*(c + d*x)^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 30*b^4*(b*c - a*d)*(c + d*x)^4*(
A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 24*b^5*(c + d*x)^5*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) + 120*(b*c -
a*d)^5*Log[a + b*x]*(A + B*Log[e*((a + b*x)/(c + d*x))^n]) - 120*B*(b*c - a*d)^5*n*Log[c + d*x] - 60*B*(b*c -
a*d)^5*n*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*PolyLog[2, (d*(a + b*x))/(-(b*c)
+ a*d)])))/(6*b^4)))/(60*d^3)

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fricas [F]  time = 1.05, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} b^{2} d^{3} g^{2} i^{3} x^{5} + A^{2} a^{2} c^{3} g^{2} i^{3} + {\left (3 \, A^{2} b^{2} c d^{2} + 2 \, A^{2} a b d^{3}\right )} g^{2} i^{3} x^{4} + {\left (3 \, A^{2} b^{2} c^{2} d + 6 \, A^{2} a b c d^{2} + A^{2} a^{2} d^{3}\right )} g^{2} i^{3} x^{3} + {\left (A^{2} b^{2} c^{3} + 6 \, A^{2} a b c^{2} d + 3 \, A^{2} a^{2} c d^{2}\right )} g^{2} i^{3} x^{2} + {\left (2 \, A^{2} a b c^{3} + 3 \, A^{2} a^{2} c^{2} d\right )} g^{2} i^{3} x + {\left (B^{2} b^{2} d^{3} g^{2} i^{3} x^{5} + B^{2} a^{2} c^{3} g^{2} i^{3} + {\left (3 \, B^{2} b^{2} c d^{2} + 2 \, B^{2} a b d^{3}\right )} g^{2} i^{3} x^{4} + {\left (3 \, B^{2} b^{2} c^{2} d + 6 \, B^{2} a b c d^{2} + B^{2} a^{2} d^{3}\right )} g^{2} i^{3} x^{3} + {\left (B^{2} b^{2} c^{3} + 6 \, B^{2} a b c^{2} d + 3 \, B^{2} a^{2} c d^{2}\right )} g^{2} i^{3} x^{2} + {\left (2 \, B^{2} a b c^{3} + 3 \, B^{2} a^{2} c^{2} d\right )} g^{2} i^{3} x\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B b^{2} d^{3} g^{2} i^{3} x^{5} + A B a^{2} c^{3} g^{2} i^{3} + {\left (3 \, A B b^{2} c d^{2} + 2 \, A B a b d^{3}\right )} g^{2} i^{3} x^{4} + {\left (3 \, A B b^{2} c^{2} d + 6 \, A B a b c d^{2} + A B a^{2} d^{3}\right )} g^{2} i^{3} x^{3} + {\left (A B b^{2} c^{3} + 6 \, A B a b c^{2} d + 3 \, A B a^{2} c d^{2}\right )} g^{2} i^{3} x^{2} + {\left (2 \, A B a b c^{3} + 3 \, A B a^{2} c^{2} d\right )} g^{2} i^{3} x\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="fricas")

[Out]

integral(A^2*b^2*d^3*g^2*i^3*x^5 + A^2*a^2*c^3*g^2*i^3 + (3*A^2*b^2*c*d^2 + 2*A^2*a*b*d^3)*g^2*i^3*x^4 + (3*A^
2*b^2*c^2*d + 6*A^2*a*b*c*d^2 + A^2*a^2*d^3)*g^2*i^3*x^3 + (A^2*b^2*c^3 + 6*A^2*a*b*c^2*d + 3*A^2*a^2*c*d^2)*g
^2*i^3*x^2 + (2*A^2*a*b*c^3 + 3*A^2*a^2*c^2*d)*g^2*i^3*x + (B^2*b^2*d^3*g^2*i^3*x^5 + B^2*a^2*c^3*g^2*i^3 + (3
*B^2*b^2*c*d^2 + 2*B^2*a*b*d^3)*g^2*i^3*x^4 + (3*B^2*b^2*c^2*d + 6*B^2*a*b*c*d^2 + B^2*a^2*d^3)*g^2*i^3*x^3 +
(B^2*b^2*c^3 + 6*B^2*a*b*c^2*d + 3*B^2*a^2*c*d^2)*g^2*i^3*x^2 + (2*B^2*a*b*c^3 + 3*B^2*a^2*c^2*d)*g^2*i^3*x)*l
og(e*((b*x + a)/(d*x + c))^n)^2 + 2*(A*B*b^2*d^3*g^2*i^3*x^5 + A*B*a^2*c^3*g^2*i^3 + (3*A*B*b^2*c*d^2 + 2*A*B*
a*b*d^3)*g^2*i^3*x^4 + (3*A*B*b^2*c^2*d + 6*A*B*a*b*c*d^2 + A*B*a^2*d^3)*g^2*i^3*x^3 + (A*B*b^2*c^3 + 6*A*B*a*
b*c^2*d + 3*A*B*a^2*c*d^2)*g^2*i^3*x^2 + (2*A*B*a*b*c^3 + 3*A*B*a^2*c^2*d)*g^2*i^3*x)*log(e*((b*x + a)/(d*x +
c))^n), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="giac")

[Out]

Timed out

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maple [F]  time = 0.46, size = 0, normalized size = 0.00 \[ \int \left (b g x +a g \right )^{2} \left (d i x +c i \right )^{3} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^2*(d*i*x+c*i)^3*(B*ln(e*((b*x+a)/(d*x+c))^n)+A)^2,x)

[Out]

int((b*g*x+a*g)^2*(d*i*x+c*i)^3*(B*ln(e*((b*x+a)/(d*x+c))^n)+A)^2,x)

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maxima [B]  time = 5.59, size = 5931, normalized size = 6.08 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^2*(d*i*x+c*i)^3*(A+B*log(e*((b*x+a)/(d*x+c))^n))^2,x, algorithm="maxima")

[Out]

1/3*A*B*b^2*d^3*g^2*i^3*x^6*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/6*A^2*b^2*d^3*g^2*i^3*x^6 + 6/5*A*B*b^2
*c*d^2*g^2*i^3*x^5*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 4/5*A*B*a*b*d^3*g^2*i^3*x^5*log(e*(b*x/(d*x + c) +
a/(d*x + c))^n) + 3/5*A^2*b^2*c*d^2*g^2*i^3*x^5 + 2/5*A^2*a*b*d^3*g^2*i^3*x^5 + 3/2*A*B*b^2*c^2*d*g^2*i^3*x^4
*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3*A*B*a*b*c*d^2*g^2*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) +
1/2*A*B*a^2*d^3*g^2*i^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3/4*A^2*b^2*c^2*d*g^2*i^3*x^4 + 3/2*A^2*
a*b*c*d^2*g^2*i^3*x^4 + 1/4*A^2*a^2*d^3*g^2*i^3*x^4 + 2/3*A*B*b^2*c^3*g^2*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*
x + c))^n) + 4*A*B*a*b*c^2*d*g^2*i^3*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 2*A*B*a^2*c*d^2*g^2*i^3*x^3*
log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/3*A^2*b^2*c^3*g^2*i^3*x^3 + 2*A^2*a*b*c^2*d*g^2*i^3*x^3 + A^2*a^2*c
*d^2*g^2*i^3*x^3 + 2*A*B*a*b*c^3*g^2*i^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 3*A*B*a^2*c^2*d*g^2*i^3*
x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2*a*b*c^3*g^2*i^3*x^2 + 3/2*A^2*a^2*c^2*d*g^2*i^3*x^2 - 1/180*A
*B*b^2*d^3*g^2*i^3*n*(60*a^6*log(b*x + a)/b^6 - 60*c^6*log(d*x + c)/d^6 + (12*(b^5*c*d^4 - a*b^4*d^5)*x^5 - 15
*(b^5*c^2*d^3 - a^2*b^3*d^5)*x^4 + 20*(b^5*c^3*d^2 - a^3*b^2*d^5)*x^3 - 30*(b^5*c^4*d - a^4*b*d^5)*x^2 + 60*(b
^5*c^5 - a^5*d^5)*x)/(b^5*d^5)) + 1/10*A*B*b^2*c*d^2*g^2*i^3*n*(12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/
d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*
(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4)) + 1/15*A*B*a*b*d^3*g^2*i^3*n*(12*a^5*log(b*x + a)/b^5 - 12*c^5*log(d*x + c)/
d^5 - (3*(b^4*c*d^3 - a*b^3*d^4)*x^4 - 4*(b^4*c^2*d^2 - a^2*b^2*d^4)*x^3 + 6*(b^4*c^3*d - a^3*b*d^4)*x^2 - 12*
(b^4*c^4 - a^4*d^4)*x)/(b^4*d^4)) - 1/4*A*B*b^2*c^2*d*g^2*i^3*n*(6*a^4*log(b*x + a)/b^4 - 6*c^4*log(d*x + c)/d
^4 + (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3)) - 1/
2*A*B*a*b*c*d^2*g^2*i^3*n*(6*a^4*log(b*x + a)/b^4 - 6*c^4*log(d*x + c)/d^4 + (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 -
3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*c^3 - a^3*d^3)*x)/(b^3*d^3)) - 1/12*A*B*a^2*d^3*g^2*i^3*n*(6*a^4*log(b*
x + a)/b^4 - 6*c^4*log(d*x + c)/d^4 + (2*(b^3*c*d^2 - a*b^2*d^3)*x^3 - 3*(b^3*c^2*d - a^2*b*d^3)*x^2 + 6*(b^3*
c^3 - a^3*d^3)*x)/(b^3*d^3)) + 1/3*A*B*b^2*c^3*g^2*i^3*n*(2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((
b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2)) + 2*A*B*a*b*c^2*d*g^2*i^3*n*(2*a^3*log(b*x + a)/b
^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x)/(b^2*d^2)) + A*B*a^2*c*d^2*g
^2*i^3*n*(2*a^3*log(b*x + a)/b^3 - 2*c^3*log(d*x + c)/d^3 - ((b^2*c*d - a*b*d^2)*x^2 - 2*(b^2*c^2 - a^2*d^2)*x
)/(b^2*d^2)) - 2*A*B*a*b*c^3*g^2*i^3*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) - 3
*A*B*a^2*c^2*d*g^2*i^3*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) + 2*A*B*a^2*c^3*g
^2*i^3*n*(a*log(b*x + a)/b - c*log(d*x + c)/d) + 2*A*B*a^2*c^3*g^2*i^3*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n
) + A^2*a^2*c^3*g^2*i^3*x - 1/180*(74*a^3*b^2*c^3*d^3*g^2*i^3*n^2 - 33*a^4*b*c^2*d^4*g^2*i^3*n^2 + 6*a^5*c*d^5
*g^2*i^3*n^2 - 2*(g^2*i^3*n^2 - 3*g^2*i^3*n*log(e))*b^5*c^6 + 18*(g^2*i^3*n^2 - 2*g^2*i^3*n*log(e))*a*b^4*c^5*
d - 9*(7*g^2*i^3*n^2 - 10*g^2*i^3*n*log(e))*a^2*b^3*c^4*d^2)*B^2*log(d*x + c)/(b^3*d^3) - 1/30*(b^6*c^6*g^2*i^
3*n^2 - 6*a*b^5*c^5*d*g^2*i^3*n^2 + 15*a^2*b^4*c^4*d^2*g^2*i^3*n^2 - 20*a^3*b^3*c^3*d^3*g^2*i^3*n^2 + 15*a^4*b
^2*c^2*d^4*g^2*i^3*n^2 - 6*a^5*b*c*d^5*g^2*i^3*n^2 + a^6*d^6*g^2*i^3*n^2)*(log(b*x + a)*log((b*d*x + a*d)/(b*c
- a*d) + 1) + dilog(-(b*d*x + a*d)/(b*c - a*d)))*B^2/(b^4*d^3) + 1/360*(60*B^2*b^6*d^6*g^2*i^3*x^6*log(e)^2 -
24*((g^2*i^3*n*log(e) - 9*g^2*i^3*log(e)^2)*b^6*c*d^5 - (g^2*i^3*n*log(e) + 6*g^2*i^3*log(e)^2)*a*b^5*d^6)*B^
2*x^5 + 6*((g^2*i^3*n^2 - 13*g^2*i^3*n*log(e) + 45*g^2*i^3*log(e)^2)*b^6*c^2*d^4 - 2*(g^2*i^3*n^2 - 3*g^2*i^3*
n*log(e) - 45*g^2*i^3*log(e)^2)*a*b^5*c*d^5 + (g^2*i^3*n^2 + 7*g^2*i^3*n*log(e) + 15*g^2*i^3*log(e)^2)*a^2*b^4
*d^6)*B^2*x^4 + 2*((9*g^2*i^3*n^2 - 38*g^2*i^3*n*log(e) + 60*g^2*i^3*log(e)^2)*b^6*c^3*d^3 - 3*(5*g^2*i^3*n^2
+ 14*g^2*i^3*n*log(e) - 120*g^2*i^3*log(e)^2)*a*b^5*c^2*d^4 + 3*(g^2*i^3*n^2 + 26*g^2*i^3*n*log(e) + 60*g^2*i^
3*log(e)^2)*a^2*b^4*c*d^5 + (3*g^2*i^3*n^2 + 2*g^2*i^3*n*log(e))*a^3*b^3*d^6)*B^2*x^3 + ((11*g^2*i^3*n^2 - 6*g
^2*i^3*n*log(e))*b^6*c^4*d^2 + 2*(5*g^2*i^3*n^2 - 102*g^2*i^3*n*log(e) + 180*g^2*i^3*log(e)^2)*a*b^5*c^3*d^3 -
60*(g^2*i^3*n^2 - 3*g^2*i^3*n*log(e) - 9*g^2*i^3*log(e)^2)*a^2*b^4*c^2*d^4 + 2*(23*g^2*i^3*n^2 + 18*g^2*i^3*n
*log(e))*a^3*b^3*c*d^5 - (7*g^2*i^3*n^2 + 6*g^2*i^3*n*log(e))*a^4*b^2*d^6)*B^2*x^2 - 6*(20*a^3*b^3*c^3*d^3*g^2
*i^3*n^2 - 15*a^4*b^2*c^2*d^4*g^2*i^3*n^2 + 6*a^5*b*c*d^5*g^2*i^3*n^2 - a^6*d^6*g^2*i^3*n^2)*B^2*log(b*x + a)^
2 + 12*(b^6*c^6*g^2*i^3*n^2 - 6*a*b^5*c^5*d*g^2*i^3*n^2 + 15*a^2*b^4*c^4*d^2*g^2*i^3*n^2)*B^2*log(b*x + a)*log
(d*x + c) - 6*(b^6*c^6*g^2*i^3*n^2 - 6*a*b^5*c^5*d*g^2*i^3*n^2 + 15*a^2*b^4*c^4*d^2*g^2*i^3*n^2)*B^2*log(d*x +
c)^2 - 2*(2*(4*g^2*i^3*n^2 - 3*g^2*i^3*n*log(e))*b^6*c^5*d - 3*(17*g^2*i^3*n^2 - 12*g^2*i^3*n*log(e))*a*b^5*c
^4*d^2 + (97*g^2*i^3*n^2 + 30*g^2*i^3*n*log(e) - 180*g^2*i^3*log(e)^2)*a^2*b^4*c^3*d^3 - (77*g^2*i^3*n^2 + 90*
g^2*i^3*n*log(e))*a^3*b^3*c^2*d^4 + 9*(3*g^2*i^3*n^2 + 4*g^2*i^3*n*log(e))*a^4*b^2*c*d^5 - 2*(2*g^2*i^3*n^2 +
3*g^2*i^3*n*log(e))*a^5*b*d^6)*B^2*x + 2*(6*a*b^5*c^5*d*g^2*i^3*n^2 - 33*a^2*b^4*c^4*d^2*g^2*i^3*n^2 + 2*(17*g
^2*i^3*n^2 + 60*g^2*i^3*n*log(e))*a^3*b^3*c^3*d^3 - 3*(g^2*i^3*n^2 + 30*g^2*i^3*n*log(e))*a^4*b^2*c^2*d^4 - 6*
(g^2*i^3*n^2 - 6*g^2*i^3*n*log(e))*a^5*b*c*d^5 + 2*(g^2*i^3*n^2 - 3*g^2*i^3*n*log(e))*a^6*d^6)*B^2*log(b*x + a
) + 6*(10*B^2*b^6*d^6*g^2*i^3*x^6 + 60*B^2*a^2*b^4*c^3*d^3*g^2*i^3*x + 12*(3*b^6*c*d^5*g^2*i^3 + 2*a*b^5*d^6*g
^2*i^3)*B^2*x^5 + 15*(3*b^6*c^2*d^4*g^2*i^3 + 6*a*b^5*c*d^5*g^2*i^3 + a^2*b^4*d^6*g^2*i^3)*B^2*x^4 + 20*(b^6*c
^3*d^3*g^2*i^3 + 6*a*b^5*c^2*d^4*g^2*i^3 + 3*a^2*b^4*c*d^5*g^2*i^3)*B^2*x^3 + 30*(2*a*b^5*c^3*d^3*g^2*i^3 + 3*
a^2*b^4*c^2*d^4*g^2*i^3)*B^2*x^2)*log((b*x + a)^n)^2 + 6*(10*B^2*b^6*d^6*g^2*i^3*x^6 + 60*B^2*a^2*b^4*c^3*d^3*
g^2*i^3*x + 12*(3*b^6*c*d^5*g^2*i^3 + 2*a*b^5*d^6*g^2*i^3)*B^2*x^5 + 15*(3*b^6*c^2*d^4*g^2*i^3 + 6*a*b^5*c*d^5
*g^2*i^3 + a^2*b^4*d^6*g^2*i^3)*B^2*x^4 + 20*(b^6*c^3*d^3*g^2*i^3 + 6*a*b^5*c^2*d^4*g^2*i^3 + 3*a^2*b^4*c*d^5*
g^2*i^3)*B^2*x^3 + 30*(2*a*b^5*c^3*d^3*g^2*i^3 + 3*a^2*b^4*c^2*d^4*g^2*i^3)*B^2*x^2)*log((d*x + c)^n)^2 + 2*(6
0*B^2*b^6*d^6*g^2*i^3*x^6*log(e) - 12*((g^2*i^3*n - 18*g^2*i^3*log(e))*b^6*c*d^5 - (g^2*i^3*n + 12*g^2*i^3*log
(e))*a*b^5*d^6)*B^2*x^5 - 3*((13*g^2*i^3*n - 90*g^2*i^3*log(e))*b^6*c^2*d^4 - 6*(g^2*i^3*n + 30*g^2*i^3*log(e)
)*a*b^5*c*d^5 - (7*g^2*i^3*n + 30*g^2*i^3*log(e))*a^2*b^4*d^6)*B^2*x^4 + 2*(a^3*b^3*d^6*g^2*i^3*n - (19*g^2*i^
3*n - 60*g^2*i^3*log(e))*b^6*c^3*d^3 - 3*(7*g^2*i^3*n - 120*g^2*i^3*log(e))*a*b^5*c^2*d^4 + 3*(13*g^2*i^3*n +
60*g^2*i^3*log(e))*a^2*b^4*c*d^5)*B^2*x^3 - 3*(b^6*c^4*d^2*g^2*i^3*n - 6*a^3*b^3*c*d^5*g^2*i^3*n + a^4*b^2*d^6
*g^2*i^3*n + 2*(17*g^2*i^3*n - 60*g^2*i^3*log(e))*a*b^5*c^3*d^3 - 30*(g^2*i^3*n + 6*g^2*i^3*log(e))*a^2*b^4*c^
2*d^4)*B^2*x^2 + 6*(b^6*c^5*d*g^2*i^3*n - 6*a*b^5*c^4*d^2*g^2*i^3*n + 15*a^3*b^3*c^2*d^4*g^2*i^3*n - 6*a^4*b^2
*c*d^5*g^2*i^3*n + a^5*b*d^6*g^2*i^3*n - 5*(g^2*i^3*n - 12*g^2*i^3*log(e))*a^2*b^4*c^3*d^3)*B^2*x + 6*(20*a^3*
b^3*c^3*d^3*g^2*i^3*n - 15*a^4*b^2*c^2*d^4*g^2*i^3*n + 6*a^5*b*c*d^5*g^2*i^3*n - a^6*d^6*g^2*i^3*n)*B^2*log(b*
x + a) - 6*(b^6*c^6*g^2*i^3*n - 6*a*b^5*c^5*d*g^2*i^3*n + 15*a^2*b^4*c^4*d^2*g^2*i^3*n)*B^2*log(d*x + c))*log(
(b*x + a)^n) - 2*(60*B^2*b^6*d^6*g^2*i^3*x^6*log(e) - 12*((g^2*i^3*n - 18*g^2*i^3*log(e))*b^6*c*d^5 - (g^2*i^3
*n + 12*g^2*i^3*log(e))*a*b^5*d^6)*B^2*x^5 - 3*((13*g^2*i^3*n - 90*g^2*i^3*log(e))*b^6*c^2*d^4 - 6*(g^2*i^3*n
+ 30*g^2*i^3*log(e))*a*b^5*c*d^5 - (7*g^2*i^3*n + 30*g^2*i^3*log(e))*a^2*b^4*d^6)*B^2*x^4 + 2*(a^3*b^3*d^6*g^2
*i^3*n - (19*g^2*i^3*n - 60*g^2*i^3*log(e))*b^6*c^3*d^3 - 3*(7*g^2*i^3*n - 120*g^2*i^3*log(e))*a*b^5*c^2*d^4 +
3*(13*g^2*i^3*n + 60*g^2*i^3*log(e))*a^2*b^4*c*d^5)*B^2*x^3 - 3*(b^6*c^4*d^2*g^2*i^3*n - 6*a^3*b^3*c*d^5*g^2*
i^3*n + a^4*b^2*d^6*g^2*i^3*n + 2*(17*g^2*i^3*n - 60*g^2*i^3*log(e))*a*b^5*c^3*d^3 - 30*(g^2*i^3*n + 6*g^2*i^3
*log(e))*a^2*b^4*c^2*d^4)*B^2*x^2 + 6*(b^6*c^5*d*g^2*i^3*n - 6*a*b^5*c^4*d^2*g^2*i^3*n + 15*a^3*b^3*c^2*d^4*g^
2*i^3*n - 6*a^4*b^2*c*d^5*g^2*i^3*n + a^5*b*d^6*g^2*i^3*n - 5*(g^2*i^3*n - 12*g^2*i^3*log(e))*a^2*b^4*c^3*d^3)
*B^2*x + 6*(20*a^3*b^3*c^3*d^3*g^2*i^3*n - 15*a^4*b^2*c^2*d^4*g^2*i^3*n + 6*a^5*b*c*d^5*g^2*i^3*n - a^6*d^6*g^
2*i^3*n)*B^2*log(b*x + a) - 6*(b^6*c^6*g^2*i^3*n - 6*a*b^5*c^5*d*g^2*i^3*n + 15*a^2*b^4*c^4*d^2*g^2*i^3*n)*B^2
*log(d*x + c) + 6*(10*B^2*b^6*d^6*g^2*i^3*x^6 + 60*B^2*a^2*b^4*c^3*d^3*g^2*i^3*x + 12*(3*b^6*c*d^5*g^2*i^3 + 2
*a*b^5*d^6*g^2*i^3)*B^2*x^5 + 15*(3*b^6*c^2*d^4*g^2*i^3 + 6*a*b^5*c*d^5*g^2*i^3 + a^2*b^4*d^6*g^2*i^3)*B^2*x^4
+ 20*(b^6*c^3*d^3*g^2*i^3 + 6*a*b^5*c^2*d^4*g^2*i^3 + 3*a^2*b^4*c*d^5*g^2*i^3)*B^2*x^3 + 30*(2*a*b^5*c^3*d^3*
g^2*i^3 + 3*a^2*b^4*c^2*d^4*g^2*i^3)*B^2*x^2)*log((b*x + a)^n))*log((d*x + c)^n))/(b^4*d^3)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a\,g+b\,g\,x\right )}^2\,{\left (c\,i+d\,i\,x\right )}^3\,{\left (A+B\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*g + b*g*x)^2*(c*i + d*i*x)^3*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2,x)

[Out]

int((a*g + b*g*x)^2*(c*i + d*i*x)^3*(A + B*log(e*((a + b*x)/(c + d*x))^n))^2, x)

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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((b*g*x+a*g)**2*(d*i*x+c*i)**3*(A+B*ln(e*((b*x+a)/(d*x+c))**n))**2,x)

[Out]

Timed out

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