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

Optimal. Leaf size=487 \[ \frac{B^2 g^2 i n^2 (b c-a d)^4 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{6 b^2 d^3}+\frac{B g^2 i n (b c-a d)^4 \log \left (\frac{b c-a d}{b (c+d x)}\right ) \left (2 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+2 A+3 B n\right )}{12 b^2 d^3}+\frac{B g^2 i n (a+b x) (b c-a d)^3 \left (2 B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+2 A+B n\right )}{12 b^2 d^2}-\frac{B g^2 i n (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{12 b^2 d}+\frac{g^2 i (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{12 b^2}-\frac{B g^2 i n (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{6 b^2}+\frac{g^2 i (a+b x)^3 (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b}+\frac{B^2 g^2 i n^2 (b c-a d)^4 \log (c+d x)}{6 b^2 d^3}-\frac{B^2 g^2 i n^2 x (b c-a d)^3}{3 b d^2}+\frac{B^2 g^2 i n^2 (c+d x)^2 (b c-a d)^2}{12 d^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 1.58632, antiderivative size = 578, normalized size of antiderivative = 1.19, number of steps used = 44, number of rules used = 13, integrand size = 43, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.302, Rules used = {2528, 2525, 12, 2486, 31, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac{B^2 g^2 i n^2 (b c-a d)^4 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{6 b^2 d^3}-\frac{B g^2 i n (b c-a d)^4 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{6 b^2 d^3}-\frac{B g^2 i n (a+b x)^2 (b c-a d)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{12 b^2 d}+\frac{g^2 i (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{3 b^2}-\frac{B g^2 i n (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )}{6 b^2}+\frac{d g^2 i (a+b x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2}{4 b^2}+\frac{A B g^2 i n x (b c-a d)^3}{6 b d^2}+\frac{B^2 g^2 i n (a+b x) (b c-a d)^3 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{6 b^2 d^2}-\frac{B^2 g^2 i n^2 (b c-a d)^4 \log ^2(c+d x)}{12 b^2 d^3}-\frac{B^2 g^2 i n^2 (b c-a d)^4 \log (c+d x)}{12 b^2 d^3}+\frac{B^2 g^2 i n^2 (b c-a d)^4 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{6 b^2 d^3}+\frac{B^2 g^2 i n^2 (a+b x)^2 (b c-a d)^2}{12 b^2 d}-\frac{B^2 g^2 i n^2 x (b c-a d)^3}{12 b d^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 12

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

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 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 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 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 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 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 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 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 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]

Rubi steps

\begin{align*} \int (160 c+160 d x) (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{160 (b c-a d) (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}{b}+\frac{160 d (a g+b g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b g}\right ) \, dx\\ &=\frac{(160 (b c-a d)) \int (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}{b}+\frac{(160 d) \int (a g+b g x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2 \, dx}{b g}\\ &=\frac{160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac{40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac{(80 B d n) \int \frac{(b c-a d) g^4 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b^2 g^2}-\frac{(320 B (b c-a d) n) \int \frac{(b c-a d) g^3 (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{3 b^2 g}\\ &=\frac{160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac{40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac{\left (80 B d (b c-a d) g^2 n\right ) \int \frac{(a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{b^2}-\frac{\left (320 B (b c-a d)^2 g^2 n\right ) \int \frac{(a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{c+d x} \, dx}{3 b^2}\\ &=\frac{160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac{40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac{\left (80 B d (b c-a d) g^2 n\right ) \int \left (\frac{b (b c-a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3}-\frac{b (b c-a d) (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac{b (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac{(-b c+a d)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^3 (c+d x)}\right ) \, dx}{b^2}-\frac{\left (320 B (b c-a d)^2 g^2 n\right ) \int \left (-\frac{b (b c-a d) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2}+\frac{b (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d}+\frac{(-b c+a d)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{d^2 (c+d x)}\right ) \, dx}{3 b^2}\\ &=\frac{160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac{40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac{\left (80 B (b c-a d) g^2 n\right ) \int (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b}+\frac{\left (80 B (b c-a d)^2 g^2 n\right ) \int (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b d}-\frac{\left (320 B (b c-a d)^2 g^2 n\right ) \int (a+b x) \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{3 b d}-\frac{\left (80 B (b c-a d)^3 g^2 n\right ) \int \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \, dx}{b d^2}+\frac{\left (320 B (b c-a d)^3 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 d^2}+\frac{\left (80 B (b c-a d)^4 g^2 n\right ) \int \frac{A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{c+d x} \, dx}{b^2 d^2}-\frac{\left (320 B (b c-a d)^4 g^2 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^2 d^2}\\ &=\frac{80 A B (b c-a d)^3 g^2 n x}{3 b d^2}-\frac{40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac{80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac{160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac{40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac{80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}-\frac{\left (80 B^2 (b c-a d)^3 g^2 n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{b d^2}+\frac{\left (320 B^2 (b c-a d)^3 g^2 n\right ) \int \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right ) \, dx}{3 b d^2}+\frac{\left (80 B^2 (b c-a d) g^2 n^2\right ) \int \frac{(b c-a d) (a+b x)^2}{c+d x} \, dx}{3 b^2}-\frac{\left (40 B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac{(b c-a d) (a+b x)}{c+d x} \, dx}{b^2 d}+\frac{\left (160 B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac{(b c-a d) (a+b x)}{c+d x} \, dx}{3 b^2 d}-\frac{\left (80 B^2 (b c-a d)^4 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 (c+d x)}{a+b x} \, dx}{b^2 d^3}+\frac{\left (320 B^2 (b c-a d)^4 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 (c+d x)}{a+b x} \, dx}{3 b^2 d^3}\\ &=\frac{80 A B (b c-a d)^3 g^2 n x}{3 b d^2}+\frac{80 B^2 (b c-a d)^3 g^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^2 d^2}-\frac{40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac{80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac{160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac{40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac{80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}+\frac{\left (80 B^2 (b c-a d)^2 g^2 n^2\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{3 b^2}-\frac{\left (40 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac{a+b x}{c+d x} \, dx}{b^2 d}+\frac{\left (160 B^2 (b c-a d)^3 g^2 n^2\right ) \int \frac{a+b x}{c+d x} \, dx}{3 b^2 d}-\frac{\left (80 B^2 (b c-a d)^4 g^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}{b^2 d^3}+\frac{\left (320 B^2 (b c-a d)^4 g^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 b^2 d^3}+\frac{\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac{1}{c+d x} \, dx}{b^2 d^2}-\frac{\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac{1}{c+d x} \, dx}{3 b^2 d^2}\\ &=\frac{80 A B (b c-a d)^3 g^2 n x}{3 b d^2}+\frac{80 B^2 (b c-a d)^3 g^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^2 d^2}-\frac{40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac{80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac{160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac{40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac{80 B^2 (b c-a d)^4 g^2 n^2 \log (c+d x)}{3 b^2 d^3}-\frac{80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}+\frac{\left (80 B^2 (b c-a d)^2 g^2 n^2\right ) \int \left (-\frac{b (b c-a d)}{d^2}+\frac{b (a+b x)}{d}+\frac{(-b c+a d)^2}{d^2 (c+d x)}\right ) \, dx}{3 b^2}-\frac{\left (40 B^2 (b c-a d)^3 g^2 n^2\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{b^2 d}+\frac{\left (160 B^2 (b c-a d)^3 g^2 n^2\right ) \int \left (\frac{b}{d}+\frac{-b c+a d}{d (c+d x)}\right ) \, dx}{3 b^2 d}-\frac{\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{b d^3}+\frac{\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{3 b d^3}+\frac{\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{b^2 d^2}-\frac{\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{3 b^2 d^2}\\ &=\frac{80 A B (b c-a d)^3 g^2 n x}{3 b d^2}-\frac{40 B^2 (b c-a d)^3 g^2 n^2 x}{3 b d^2}+\frac{40 B^2 (b c-a d)^2 g^2 n^2 (a+b x)^2}{3 b^2 d}+\frac{80 B^2 (b c-a d)^3 g^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^2 d^2}-\frac{40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac{80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac{160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac{40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac{40 B^2 (b c-a d)^4 g^2 n^2 \log (c+d x)}{3 b^2 d^3}+\frac{80 B^2 (b c-a d)^4 g^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 d^3}-\frac{80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}+\frac{\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{b^2 d^3}-\frac{\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{3 b^2 d^3}+\frac{\left (80 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b^2 d^2}-\frac{\left (320 B^2 (b c-a d)^4 g^2 n^2\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b^2 d^2}\\ &=\frac{80 A B (b c-a d)^3 g^2 n x}{3 b d^2}-\frac{40 B^2 (b c-a d)^3 g^2 n^2 x}{3 b d^2}+\frac{40 B^2 (b c-a d)^2 g^2 n^2 (a+b x)^2}{3 b^2 d}+\frac{80 B^2 (b c-a d)^3 g^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^2 d^2}-\frac{40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac{80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac{160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac{40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac{40 B^2 (b c-a d)^4 g^2 n^2 \log (c+d x)}{3 b^2 d^3}+\frac{80 B^2 (b c-a d)^4 g^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 d^3}-\frac{80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}-\frac{40 B^2 (b c-a d)^4 g^2 n^2 \log ^2(c+d x)}{3 b^2 d^3}+\frac{\left (80 B^2 (b c-a d)^4 g^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 )}{b^2 d^3}-\frac{\left (320 B^2 (b c-a d)^4 g^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 b^2 d^3}\\ &=\frac{80 A B (b c-a d)^3 g^2 n x}{3 b d^2}-\frac{40 B^2 (b c-a d)^3 g^2 n^2 x}{3 b d^2}+\frac{40 B^2 (b c-a d)^2 g^2 n^2 (a+b x)^2}{3 b^2 d}+\frac{80 B^2 (b c-a d)^3 g^2 n (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{3 b^2 d^2}-\frac{40 B (b c-a d)^2 g^2 n (a+b x)^2 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2 d}-\frac{80 B (b c-a d) g^2 n (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )}{3 b^2}+\frac{160 (b c-a d) g^2 (a+b x)^3 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{3 b^2}+\frac{40 d g^2 (a+b x)^4 \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right )^2}{b^2}-\frac{40 B^2 (b c-a d)^4 g^2 n^2 \log (c+d x)}{3 b^2 d^3}+\frac{80 B^2 (b c-a d)^4 g^2 n^2 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b^2 d^3}-\frac{80 B (b c-a d)^4 g^2 n \left (A+B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )\right ) \log (c+d x)}{3 b^2 d^3}-\frac{40 B^2 (b c-a d)^4 g^2 n^2 \log ^2(c+d x)}{3 b^2 d^3}+\frac{80 B^2 (b c-a d)^4 g^2 n^2 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{3 b^2 d^3}\\ \end{align*}

Mathematica [A]  time = 0.57614, size = 716, normalized size = 1.47 \[ \frac{g^2 i \left (\frac{4 B n (b c-a d)^2 \left (B n (b c-a d)^2 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-d^2 (a+b x)^2 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-2 (b c-a d)^2 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+2 A b d x (b c-a d)+2 B d (a+b x) (b c-a d) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-2 B n (b c-a d)^2 \log (c+d x)+B n (b c-a d) ((a d-b c) \log (c+d x)+b d x)\right )}{d^3}-\frac{B n (b c-a d) \left (3 B n (b c-a d)^3 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )+2 d^3 (a+b x)^3 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+3 d^2 (a+b x)^2 (a d-b c) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )-6 (b c-a d)^3 \log (c+d x) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )+6 A b d x (b c-a d)^2+B n (b c-a d) \left (2 b d x (b c-a d)-2 (b c-a d)^2 \log (c+d x)-d^2 (a+b x)^2\right )+6 B d (a+b x) (b c-a d)^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-6 B n (b c-a d)^3 \log (c+d x)+3 B n (b c-a d)^2 ((a d-b c) \log (c+d x)+b d x)\right )}{d^3}+3 d (a+b x)^4 \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2+4 (a+b x)^3 (b c-a d) \left (B \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+A\right )^2\right )}{12 b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [F]  time = 0.517, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{2} \left ( dix+ci \right ) \left ( A+B\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 3.54552, size = 3633, normalized size = 7.46 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*A*B*b^2*d*g^2*i*x^4*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 1/4*A^2*b^2*d*g^2*i*x^4 + 2/3*A*B*b^2*c*g^2*i
*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + 4/3*A*B*a*b*d*g^2*i*x^3*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) +
1/3*A^2*b^2*c*g^2*i*x^3 + 2/3*A^2*a*b*d*g^2*i*x^3 + 2*A*B*a*b*c*g^2*i*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))
^n) + A*B*a^2*d*g^2*i*x^2*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2*a*b*c*g^2*i*x^2 + 1/2*A^2*a^2*d*g^2*i*x
^2 - 1/12*A*B*b^2*d*g^2*i*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*g^2*i*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)) + 2/3*A*B*a*
b*d*g^2*i*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*g^2*i*n*(a^2*log(b*x + a)/b^2 - c^2*log(d*x + c)/d^2 + (b*c - a*d)*x/(b*d)) - A*
B*a^2*d*g^2*i*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*g^2*i*n*(a*l
og(b*x + a)/b - c*log(d*x + c)/d) + 2*A*B*a^2*c*g^2*i*x*log(e*(b*x/(d*x + c) + a/(d*x + c))^n) + A^2*a^2*c*g^2
*i*x - 1/12*(2*a^3*c*d^3*g^2*i*n^2 + (g^2*i*n^2 + 2*g^2*i*n*log(e))*b^3*c^4 - 2*(g^2*i*n^2 + 4*g^2*i*n*log(e))
*a*b^2*c^3*d - (g^2*i*n^2 - 12*g^2*i*n*log(e))*a^2*b*c^2*d^2)*B^2*log(d*x + c)/(b*d^3) - 1/6*(b^4*c^4*g^2*i*n^
2 - 4*a*b^3*c^3*d*g^2*i*n^2 + 6*a^2*b^2*c^2*d^2*g^2*i*n^2 - 4*a^3*b*c*d^3*g^2*i*n^2 + a^4*d^4*g^2*i*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^2*d^3) + 1/12*(3*B^2*b
^4*d^4*g^2*i*x^4*log(e)^2 - 2*((g^2*i*n*log(e) - 2*g^2*i*log(e)^2)*b^4*c*d^3 - (g^2*i*n*log(e) + 4*g^2*i*log(e
)^2)*a*b^3*d^4)*B^2*x^3 + ((g^2*i*n^2 - g^2*i*n*log(e))*b^4*c^2*d^2 - 2*(g^2*i*n^2 + 2*g^2*i*n*log(e) - 6*g^2*
i*log(e)^2)*a*b^3*c*d^3 + (g^2*i*n^2 + 5*g^2*i*n*log(e) + 6*g^2*i*log(e)^2)*a^2*b^2*d^4)*B^2*x^2 - (4*a^3*b*c*
d^3*g^2*i*n^2 - a^4*d^4*g^2*i*n^2)*B^2*log(b*x + a)^2 + 2*(b^4*c^4*g^2*i*n^2 - 4*a*b^3*c^3*d*g^2*i*n^2 + 6*a^2
*b^2*c^2*d^2*g^2*i*n^2)*B^2*log(b*x + a)*log(d*x + c) - (b^4*c^4*g^2*i*n^2 - 4*a*b^3*c^3*d*g^2*i*n^2 + 6*a^2*b
^2*c^2*d^2*g^2*i*n^2)*B^2*log(d*x + c)^2 - ((g^2*i*n^2 - 2*g^2*i*n*log(e))*b^4*c^3*d - (5*g^2*i*n^2 - 8*g^2*i*
n*log(e))*a*b^3*c^2*d^2 + (7*g^2*i*n^2 - 4*g^2*i*n*log(e) - 12*g^2*i*log(e)^2)*a^2*b^2*c*d^3 - (3*g^2*i*n^2 +
2*g^2*i*n*log(e))*a^3*b*d^4)*B^2*x + (2*a*b^3*c^3*d*g^2*i*n^2 - 7*a^2*b^2*c^2*d^2*g^2*i*n^2 + 2*(3*g^2*i*n^2 +
4*g^2*i*n*log(e))*a^3*b*c*d^3 - (g^2*i*n^2 + 2*g^2*i*n*log(e))*a^4*d^4)*B^2*log(b*x + a) + (3*B^2*b^4*d^4*g^2
*i*x^4 + 12*B^2*a^2*b^2*c*d^3*g^2*i*x + 4*(b^4*c*d^3*g^2*i + 2*a*b^3*d^4*g^2*i)*B^2*x^3 + 6*(2*a*b^3*c*d^3*g^2
*i + a^2*b^2*d^4*g^2*i)*B^2*x^2)*log((b*x + a)^n)^2 + (3*B^2*b^4*d^4*g^2*i*x^4 + 12*B^2*a^2*b^2*c*d^3*g^2*i*x
+ 4*(b^4*c*d^3*g^2*i + 2*a*b^3*d^4*g^2*i)*B^2*x^3 + 6*(2*a*b^3*c*d^3*g^2*i + a^2*b^2*d^4*g^2*i)*B^2*x^2)*log((
d*x + c)^n)^2 + (6*B^2*b^4*d^4*g^2*i*x^4*log(e) - 2*((g^2*i*n - 4*g^2*i*log(e))*b^4*c*d^3 - (g^2*i*n + 8*g^2*i
*log(e))*a*b^3*d^4)*B^2*x^3 - (b^4*c^2*d^2*g^2*i*n + 4*(g^2*i*n - 6*g^2*i*log(e))*a*b^3*c*d^3 - (5*g^2*i*n + 1
2*g^2*i*log(e))*a^2*b^2*d^4)*B^2*x^2 + 2*(b^4*c^3*d*g^2*i*n - 4*a*b^3*c^2*d^2*g^2*i*n + a^3*b*d^4*g^2*i*n + 2*
(g^2*i*n + 6*g^2*i*log(e))*a^2*b^2*c*d^3)*B^2*x + 2*(4*a^3*b*c*d^3*g^2*i*n - a^4*d^4*g^2*i*n)*B^2*log(b*x + a)
- 2*(b^4*c^4*g^2*i*n - 4*a*b^3*c^3*d*g^2*i*n + 6*a^2*b^2*c^2*d^2*g^2*i*n)*B^2*log(d*x + c))*log((b*x + a)^n)
- (6*B^2*b^4*d^4*g^2*i*x^4*log(e) - 2*((g^2*i*n - 4*g^2*i*log(e))*b^4*c*d^3 - (g^2*i*n + 8*g^2*i*log(e))*a*b^3
*d^4)*B^2*x^3 - (b^4*c^2*d^2*g^2*i*n + 4*(g^2*i*n - 6*g^2*i*log(e))*a*b^3*c*d^3 - (5*g^2*i*n + 12*g^2*i*log(e)
)*a^2*b^2*d^4)*B^2*x^2 + 2*(b^4*c^3*d*g^2*i*n - 4*a*b^3*c^2*d^2*g^2*i*n + a^3*b*d^4*g^2*i*n + 2*(g^2*i*n + 6*g
^2*i*log(e))*a^2*b^2*c*d^3)*B^2*x + 2*(4*a^3*b*c*d^3*g^2*i*n - a^4*d^4*g^2*i*n)*B^2*log(b*x + a) - 2*(b^4*c^4*
g^2*i*n - 4*a*b^3*c^3*d*g^2*i*n + 6*a^2*b^2*c^2*d^2*g^2*i*n)*B^2*log(d*x + c) + 2*(3*B^2*b^4*d^4*g^2*i*x^4 + 1
2*B^2*a^2*b^2*c*d^3*g^2*i*x + 4*(b^4*c*d^3*g^2*i + 2*a*b^3*d^4*g^2*i)*B^2*x^3 + 6*(2*a*b^3*c*d^3*g^2*i + a^2*b
^2*d^4*g^2*i)*B^2*x^2)*log((b*x + a)^n))*log((d*x + c)^n))/(b^2*d^3)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (A^{2} b^{2} d g^{2} i x^{3} + A^{2} a^{2} c g^{2} i +{\left (A^{2} b^{2} c + 2 \, A^{2} a b d\right )} g^{2} i x^{2} +{\left (2 \, A^{2} a b c + A^{2} a^{2} d\right )} g^{2} i x +{\left (B^{2} b^{2} d g^{2} i x^{3} + B^{2} a^{2} c g^{2} i +{\left (B^{2} b^{2} c + 2 \, B^{2} a b d\right )} g^{2} i x^{2} +{\left (2 \, B^{2} a b c + B^{2} a^{2} d\right )} g^{2} i x\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )^{2} + 2 \,{\left (A B b^{2} d g^{2} i x^{3} + A B a^{2} c g^{2} i +{\left (A B b^{2} c + 2 \, A B a b d\right )} g^{2} i x^{2} +{\left (2 \, A B a b c + A B a^{2} d\right )} g^{2} i x\right )} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b g x + a g\right )}^{2}{\left (d i x + c i\right )}{\left (B \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right ) + A\right )}^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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