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

Optimal. Leaf size=454 \[ \frac {B g^3 n (b c-a d)^4 \log \left (1-\frac {b (c+d x)}{d (a+b x)}\right ) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^4 d}-\frac {B g^3 n (a+b x) (b c-a d)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^4}-\frac {B g^3 n (c+d x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 b^2 d}-\frac {B g^3 n (c+d x)^3 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 b d}+\frac {g^3 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d}-\frac {B^2 g^3 n^2 (b c-a d)^4 \text {Li}_2\left (\frac {b (c+d x)}{d (a+b x)}\right )}{2 b^4 d}+\frac {5 B^2 g^3 n^2 (b c-a d)^4 \log \left (\frac {a+b x}{c+d x}\right )}{12 b^4 d}+\frac {11 B^2 g^3 n^2 (b c-a d)^4 \log (c+d x)}{12 b^4 d}+\frac {5 B^2 g^3 n^2 x (b c-a d)^3}{12 b^3}+\frac {B^2 g^3 n^2 (c+d x)^2 (b c-a d)^2}{12 b^2 d} \]

[Out]

5/12*B^2*(-a*d+b*c)^3*g^3*n^2*x/b^3+1/12*B^2*(-a*d+b*c)^2*g^3*n^2*(d*x+c)^2/b^2/d-1/2*B*(-a*d+b*c)^3*g^3*n*(b*
x+a)*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^4-1/4*B*(-a*d+b*c)^2*g^3*n*(d*x+c)^2*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b^
2/d-1/6*B*(-a*d+b*c)*g^3*n*(d*x+c)^3*(A+B*ln(e*((b*x+a)/(d*x+c))^n))/b/d+1/4*g^3*(d*x+c)^4*(A+B*ln(e*((b*x+a)/
(d*x+c))^n))^2/d+5/12*B^2*(-a*d+b*c)^4*g^3*n^2*ln((b*x+a)/(d*x+c))/b^4/d+11/12*B^2*(-a*d+b*c)^4*g^3*n^2*ln(d*x
+c)/b^4/d+1/2*B*(-a*d+b*c)^4*g^3*n*(A+B*ln(e*((b*x+a)/(d*x+c))^n))*ln(1-b*(d*x+c)/d/(b*x+a))/b^4/d-1/2*B^2*(-a
*d+b*c)^4*g^3*n^2*polylog(2,b*(d*x+c)/d/(b*x+a))/b^4/d

________________________________________________________________________________________

Rubi [A]  time = 0.68, antiderivative size = 544, normalized size of antiderivative = 1.20, number of steps used = 23, number of rules used = 13, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.371, Rules used = {2525, 12, 2528, 2486, 31, 2524, 2418, 2390, 2301, 2394, 2393, 2391, 43} \[ -\frac {B^2 g^3 n^2 (b c-a d)^4 \text {PolyLog}\left (2,-\frac {d (a+b x)}{b c-a d}\right )}{2 b^4 d}-\frac {B g^3 n (b c-a d)^4 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{2 b^4 d}-\frac {B g^3 n (c+d x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{4 b^2 d}-\frac {A B g^3 n x (b c-a d)^3}{2 b^3}-\frac {B g^3 n (c+d x)^3 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )}{6 b d}+\frac {g^3 (c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 d}-\frac {B^2 g^3 n (a+b x) (b c-a d)^3 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 b^4}+\frac {5 B^2 g^3 n^2 x (b c-a d)^3}{12 b^3}+\frac {B^2 g^3 n^2 (c+d x)^2 (b c-a d)^2}{12 b^2 d}+\frac {B^2 g^3 n^2 (b c-a d)^4 \log ^2(a+b x)}{4 b^4 d}+\frac {5 B^2 g^3 n^2 (b c-a d)^4 \log (a+b x)}{12 b^4 d}+\frac {B^2 g^3 n^2 (b c-a d)^4 \log (c+d x)}{2 b^4 d}-\frac {B^2 g^3 n^2 (b c-a d)^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b^4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(A*B*(b*c - a*d)^3*g^3*n*x)/(2*b^3) + (5*B^2*(b*c - a*d)^3*g^3*n^2*x)/(12*b^3) + (B^2*(b*c - a*d)^2*g^3*n^2*(
c + d*x)^2)/(12*b^2*d) + (5*B^2*(b*c - a*d)^4*g^3*n^2*Log[a + b*x])/(12*b^4*d) + (B^2*(b*c - a*d)^4*g^3*n^2*Lo
g[a + b*x]^2)/(4*b^4*d) - (B^2*(b*c - a*d)^3*g^3*n*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/(2*b^4) - (B*(b*c
- a*d)^2*g^3*n*(c + d*x)^2*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(4*b^2*d) - (B*(b*c - a*d)*g^3*n*(c + d*x)
^3*(A + B*Log[e*((a + b*x)/(c + d*x))^n]))/(6*b*d) - (B*(b*c - a*d)^4*g^3*n*Log[a + b*x]*(A + B*Log[e*((a + b*
x)/(c + d*x))^n]))/(2*b^4*d) + (g^3*(c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2)/(4*d) + (B^2*(b*c -
a*d)^4*g^3*n^2*Log[c + d*x])/(2*b^4*d) - (B^2*(b*c - a*d)^4*g^3*n^2*Log[a + b*x]*Log[(b*(c + d*x))/(b*c - a*d)
])/(2*b^4*d) - (B^2*(b*c - a*d)^4*g^3*n^2*PolyLog[2, -((d*(a + b*x))/(b*c - a*d))])/(2*b^4*d)

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 (c g+d g x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )^2 \, dx &=\frac {g^3 (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}-\frac {(B n) \int \frac {(b c-a d) g^4 (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 g}\\ &=\frac {g^3 (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}-\frac {\left (B (b c-a d) g^3 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}\\ &=\frac {g^3 (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}-\frac {\left (B (b c-a d) g^3 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}\\ &=\frac {g^3 (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}-\frac {\left (B (b c-a d) g^3 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}-\frac {\left (B (b c-a d)^2 g^3 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}-\frac {\left (B (b c-a d)^3 g^3 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}-\frac {\left (B (b c-a d)^4 g^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}{2 b^3 d}\\ &=-\frac {A B (b c-a d)^3 g^3 n x}{2 b^3}-\frac {B (b c-a d)^2 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d}-\frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b d}-\frac {B (b c-a d)^4 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d}+\frac {g^3 (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}-\frac {\left (B^2 (b c-a d)^3 g^3 n\right ) \int \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \, dx}{2 b^3}+\frac {\left (B^2 (b c-a d) g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)^2}{a+b x} \, dx}{6 b d}+\frac {\left (B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac {(b c-a d) (c+d x)}{a+b x} \, dx}{4 b^2 d}+\frac {\left (B^2 (b c-a d)^4 g^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}{2 b^4 d}\\ &=-\frac {A B (b c-a d)^3 g^3 n x}{2 b^3}-\frac {B^2 (b c-a d)^3 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 b^4}-\frac {B (b c-a d)^2 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d}-\frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b d}-\frac {B (b c-a d)^4 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d}+\frac {g^3 (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}+\frac {\left (B^2 (b c-a d)^2 g^3 n^2\right ) \int \frac {(c+d x)^2}{a+b x} \, dx}{6 b d}+\frac {\left (B^2 (b c-a d)^3 g^3 n^2\right ) \int \frac {c+d x}{a+b x} \, dx}{4 b^2 d}+\frac {\left (B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {1}{c+d x} \, dx}{2 b^4}+\frac {\left (B^2 (b c-a d)^4 g^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}{2 b^4 d}\\ &=-\frac {A B (b c-a d)^3 g^3 n x}{2 b^3}-\frac {B^2 (b c-a d)^3 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 b^4}-\frac {B (b c-a d)^2 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d}-\frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b d}-\frac {B (b c-a d)^4 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d}+\frac {g^3 (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}+\frac {B^2 (b c-a d)^4 g^3 n^2 \log (c+d x)}{2 b^4 d}+\frac {\left (B^2 (b c-a d)^2 g^3 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}+\frac {\left (B^2 (b c-a d)^3 g^3 n^2\right ) \int \left (\frac {d}{b}+\frac {b c-a d}{b (a+b x)}\right ) \, dx}{4 b^2 d}-\frac {\left (B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{2 b^4}+\frac {\left (B^2 (b c-a d)^4 g^3 n^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{2 b^3 d}\\ &=-\frac {A B (b c-a d)^3 g^3 n x}{2 b^3}+\frac {5 B^2 (b c-a d)^3 g^3 n^2 x}{12 b^3}+\frac {B^2 (b c-a d)^2 g^3 n^2 (c+d x)^2}{12 b^2 d}+\frac {5 B^2 (b c-a d)^4 g^3 n^2 \log (a+b x)}{12 b^4 d}-\frac {B^2 (b c-a d)^3 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 b^4}-\frac {B (b c-a d)^2 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d}-\frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b d}-\frac {B (b c-a d)^4 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d}+\frac {g^3 (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}+\frac {B^2 (b c-a d)^4 g^3 n^2 \log (c+d x)}{2 b^4 d}-\frac {B^2 (b c-a d)^4 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b^4 d}+\frac {\left (B^2 (b c-a d)^4 g^3 n^2\right ) \operatorname {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{2 b^4 d}+\frac {\left (B^2 (b c-a d)^4 g^3 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}\\ &=-\frac {A B (b c-a d)^3 g^3 n x}{2 b^3}+\frac {5 B^2 (b c-a d)^3 g^3 n^2 x}{12 b^3}+\frac {B^2 (b c-a d)^2 g^3 n^2 (c+d x)^2}{12 b^2 d}+\frac {5 B^2 (b c-a d)^4 g^3 n^2 \log (a+b x)}{12 b^4 d}+\frac {B^2 (b c-a d)^4 g^3 n^2 \log ^2(a+b x)}{4 b^4 d}-\frac {B^2 (b c-a d)^3 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 b^4}-\frac {B (b c-a d)^2 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d}-\frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b d}-\frac {B (b c-a d)^4 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d}+\frac {g^3 (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}+\frac {B^2 (b c-a d)^4 g^3 n^2 \log (c+d x)}{2 b^4 d}-\frac {B^2 (b c-a d)^4 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b^4 d}+\frac {\left (B^2 (b c-a d)^4 g^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 )}{2 b^4 d}\\ &=-\frac {A B (b c-a d)^3 g^3 n x}{2 b^3}+\frac {5 B^2 (b c-a d)^3 g^3 n^2 x}{12 b^3}+\frac {B^2 (b c-a d)^2 g^3 n^2 (c+d x)^2}{12 b^2 d}+\frac {5 B^2 (b c-a d)^4 g^3 n^2 \log (a+b x)}{12 b^4 d}+\frac {B^2 (b c-a d)^4 g^3 n^2 \log ^2(a+b x)}{4 b^4 d}-\frac {B^2 (b c-a d)^3 g^3 n (a+b x) \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{2 b^4}-\frac {B (b c-a d)^2 g^3 n (c+d x)^2 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{4 b^2 d}-\frac {B (b c-a d) g^3 n (c+d x)^3 \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{6 b d}-\frac {B (b c-a d)^4 g^3 n \log (a+b x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )}{2 b^4 d}+\frac {g^3 (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}+\frac {B^2 (b c-a d)^4 g^3 n^2 \log (c+d x)}{2 b^4 d}-\frac {B^2 (b c-a d)^4 g^3 n^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b^4 d}-\frac {B^2 (b c-a d)^4 g^3 n^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2 b^4 d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.34, size = 409, normalized size = 0.90 \[ \frac {g^3 \left ((c+d x)^4 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )^2-\frac {B n (b c-a d) \left (2 b^3 (c+d x)^3 \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+3 b^2 (c+d x)^2 (b c-a d) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+6 (b c-a d)^3 \log (a+b x) \left (B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+A\right )+6 A b d x (b c-a d)^2-B n (b c-a d) \left (2 b d x (b c-a d)+2 (b c-a d)^2 \log (a+b x)+b^2 (c+d x)^2\right )+6 B d (a+b x) (b c-a d)^2 \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-3 B n (b c-a d)^3 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \text {Li}_2\left (\frac {d (a+b x)}{a d-b c}\right )\right )-6 B n (b c-a d)^3 \log (c+d x)-3 B n (b c-a d)^2 ((b c-a d) \log (a+b x)+b d x)\right )}{3 b^4}\right )}{4 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^3*((c + d*x)^4*(A + B*Log[e*((a + b*x)/(c + d*x))^n])^2 - (B*(b*c - a*d)*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)])))/(3*b^4)))/(4*d)

________________________________________________________________________________________

fricas [F]  time = 1.04, size = 0, normalized size = 0.00 \[ {\rm integral}\left (A^{2} d^{3} g^{3} x^{3} + 3 \, A^{2} c d^{2} g^{3} x^{2} + 3 \, A^{2} c^{2} d g^{3} x + A^{2} c^{3} g^{3} + {\left (B^{2} d^{3} g^{3} x^{3} + 3 \, B^{2} c d^{2} g^{3} x^{2} + 3 \, B^{2} c^{2} d g^{3} x + B^{2} c^{3} g^{3}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \, {\left (A B d^{3} g^{3} x^{3} + 3 \, A B c d^{2} g^{3} x^{2} + 3 \, A B c^{2} d g^{3} x + A B c^{3} g^{3}\right )} \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right ), x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

maple [F]  time = 0.27, size = 0, normalized size = 0.00 \[ \int \left (d g x +c g \right )^{3} \left (B \ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )+A \right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 4.85, size = 2129, normalized size = 4.69 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*A*B*d^3*g^3*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A^2*d^3*g^3*x^4 + 2*A*B*c*d^2*g^3*x^3*log(e*(
b*x/(d*x + c) + a/(d*x + c))^n) + A^2*c*d^2*g^3*x^3 + 3*A*B*c^2*d*g^3*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^
n) + 3/2*A^2*c^2*d*g^3*x^2 - 1/12*A*B*d^3*g^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)) + A*B*c*d^2*g^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)
) - 3*A*B*c^2*d*g^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*c^3*g^3*n*(a
*log(b*x + a)/b - c*log(d*x + c)/d) + 2*A*B*c^3*g^3*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2*c^3*g^3*x -
1/12*(26*a*b^2*c^3*d*g^3*n^2 - 21*a^2*b*c^2*d^2*g^3*n^2 + 6*a^3*c*d^3*g^3*n^2 - (11*g^3*n^2 - 6*g^3*n*log(e))
*b^3*c^4)*B^2*log(d*x + c)/(b^3*d) - 1/2*(b^4*c^4*g^3*n^2 - 4*a*b^3*c^3*d*g^3*n^2 + 6*a^2*b^2*c^2*d^2*g^3*n^2
- 4*a^3*b*c*d^3*g^3*n^2 + a^4*d^4*g^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) + 1/12*(3*B^2*b^4*d^4*g^3*x^4*log(e)^2 + 6*B^2*b^4*c^4*g^3*n^2*log(b*x + a)*log
(d*x + c) - 3*B^2*b^4*c^4*g^3*n^2*log(d*x + c)^2 + 2*(a*b^3*d^4*g^3*n*log(e) - (g^3*n*log(e) - 6*g^3*log(e)^2)
*b^4*c*d^3)*B^2*x^3 + ((g^3*n^2 - 9*g^3*n*log(e) + 18*g^3*log(e)^2)*b^4*c^2*d^2 - 2*(g^3*n^2 - 6*g^3*n*log(e))
*a*b^3*c*d^3 + (g^3*n^2 - 3*g^3*n*log(e))*a^2*b^2*d^4)*B^2*x^2 - 3*(4*a*b^3*c^3*d*g^3*n^2 - 6*a^2*b^2*c^2*d^2*
g^3*n^2 + 4*a^3*b*c*d^3*g^3*n^2 - a^4*d^4*g^3*n^2)*B^2*log(b*x + a)^2 + ((7*g^3*n^2 - 18*g^3*n*log(e) + 12*g^3
*log(e)^2)*b^4*c^3*d - (19*g^3*n^2 - 36*g^3*n*log(e))*a*b^3*c^2*d^2 + (17*g^3*n^2 - 24*g^3*n*log(e))*a^2*b^2*c
*d^3 - (5*g^3*n^2 - 6*g^3*n*log(e))*a^3*b*d^4)*B^2*x - (6*(3*g^3*n^2 - 4*g^3*n*log(e))*a*b^3*c^3*d - 9*(5*g^3*
n^2 - 4*g^3*n*log(e))*a^2*b^2*c^2*d^2 + 2*(19*g^3*n^2 - 12*g^3*n*log(e))*a^3*b*c*d^3 - (11*g^3*n^2 - 6*g^3*n*l
og(e))*a^4*d^4)*B^2*log(b*x + a) + 3*(B^2*b^4*d^4*g^3*x^4 + 4*B^2*b^4*c*d^3*g^3*x^3 + 6*B^2*b^4*c^2*d^2*g^3*x^
2 + 4*B^2*b^4*c^3*d*g^3*x)*log((b*x + a)^n)^2 + 3*(B^2*b^4*d^4*g^3*x^4 + 4*B^2*b^4*c*d^3*g^3*x^3 + 6*B^2*b^4*c
^2*d^2*g^3*x^2 + 4*B^2*b^4*c^3*d*g^3*x)*log((d*x + c)^n)^2 + (6*B^2*b^4*d^4*g^3*x^4*log(e) - 6*B^2*b^4*c^4*g^3
*n*log(d*x + c) + 2*(a*b^3*d^4*g^3*n - (g^3*n - 12*g^3*log(e))*b^4*c*d^3)*B^2*x^3 + 3*(4*a*b^3*c*d^3*g^3*n - a
^2*b^2*d^4*g^3*n - 3*(g^3*n - 4*g^3*log(e))*b^4*c^2*d^2)*B^2*x^2 + 6*(6*a*b^3*c^2*d^2*g^3*n - 4*a^2*b^2*c*d^3*
g^3*n + a^3*b*d^4*g^3*n - (3*g^3*n - 4*g^3*log(e))*b^4*c^3*d)*B^2*x + 6*(4*a*b^3*c^3*d*g^3*n - 6*a^2*b^2*c^2*d
^2*g^3*n + 4*a^3*b*c*d^3*g^3*n - a^4*d^4*g^3*n)*B^2*log(b*x + a))*log((b*x + a)^n) - (6*B^2*b^4*d^4*g^3*x^4*lo
g(e) - 6*B^2*b^4*c^4*g^3*n*log(d*x + c) + 2*(a*b^3*d^4*g^3*n - (g^3*n - 12*g^3*log(e))*b^4*c*d^3)*B^2*x^3 + 3*
(4*a*b^3*c*d^3*g^3*n - a^2*b^2*d^4*g^3*n - 3*(g^3*n - 4*g^3*log(e))*b^4*c^2*d^2)*B^2*x^2 + 6*(6*a*b^3*c^2*d^2*
g^3*n - 4*a^2*b^2*c*d^3*g^3*n + a^3*b*d^4*g^3*n - (3*g^3*n - 4*g^3*log(e))*b^4*c^3*d)*B^2*x + 6*(4*a*b^3*c^3*d
*g^3*n - 6*a^2*b^2*c^2*d^2*g^3*n + 4*a^3*b*c*d^3*g^3*n - a^4*d^4*g^3*n)*B^2*log(b*x + a) + 6*(B^2*b^4*d^4*g^3*
x^4 + 4*B^2*b^4*c*d^3*g^3*x^3 + 6*B^2*b^4*c^2*d^2*g^3*x^2 + 4*B^2*b^4*c^3*d*g^3*x)*log((b*x + a)^n))*log((d*x
+ c)^n))/(b^4*d)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________