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

Optimal. Leaf size=515 \[ \frac{8 B^2 g^4 (b c-a d)^5 \text{PolyLog}\left (2,\frac{d (a+b x)}{b (c+d x)}\right )}{5 b d^5}-\frac{4 B g^4 (b c-a d)^5 \log \left (1-\frac{d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{5 b d^5}-\frac{4 B g^4 (c+d x) (b c-a d)^4 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{5 d^5}+\frac{2 B g^4 (a+b x)^2 (b c-a d)^3 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{5 b d^3}-\frac{4 B g^4 (a+b x)^3 (b c-a d)^2 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{15 b d^2}+\frac{B g^4 (a+b x)^4 (b c-a d) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{5 b d}+\frac{g^4 (a+b x)^5 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{5 b}+\frac{26 B^2 g^4 x (b c-a d)^4}{15 d^4}-\frac{7 B^2 g^4 (a+b x)^2 (b c-a d)^3}{15 b d^3}+\frac{2 B^2 g^4 (a+b x)^3 (b c-a d)^2}{15 b d^2}-\frac{10 B^2 g^4 (b c-a d)^5 \log (a+b x)}{3 b d^5}-\frac{26 B^2 g^4 (b c-a d)^5 \log \left (\frac{c+d x}{a+b x}\right )}{15 b d^5} \]

[Out]

(26*B^2*(b*c - a*d)^4*g^4*x)/(15*d^4) - (7*B^2*(b*c - a*d)^3*g^4*(a + b*x)^2)/(15*b*d^3) + (2*B^2*(b*c - a*d)^
2*g^4*(a + b*x)^3)/(15*b*d^2) - (10*B^2*(b*c - a*d)^5*g^4*Log[a + b*x])/(3*b*d^5) - (26*B^2*(b*c - a*d)^5*g^4*
Log[(c + d*x)/(a + b*x)])/(15*b*d^5) + (2*B*(b*c - a*d)^3*g^4*(a + b*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)
^2]))/(5*b*d^3) - (4*B*(b*c - a*d)^2*g^4*(a + b*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(15*b*d^2) + (B
*(b*c - a*d)*g^4*(a + b*x)^4*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(5*b*d) - (4*B*(b*c - a*d)^4*g^4*(c + d
*x)*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(5*d^5) + (g^4*(a + b*x)^5*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^
2])^2)/(5*b) - (4*B*(b*c - a*d)^5*g^4*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])*Log[1 - (d*(a + b*x))/(b*(c + d
*x))])/(5*b*d^5) + (8*B^2*(b*c - a*d)^5*g^4*PolyLog[2, (d*(a + b*x))/(b*(c + d*x))])/(5*b*d^5)

________________________________________________________________________________________

Rubi [A]  time = 0.864275, antiderivative size = 569, normalized size of antiderivative = 1.1, number of steps used = 28, number of rules used = 13, integrand size = 34, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.382, Rules used = {2525, 12, 2528, 2486, 31, 43, 2524, 2418, 2394, 2393, 2391, 2390, 2301} \[ \frac{8 B^2 g^4 (b c-a d)^5 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )}{5 b d^5}+\frac{4 B g^4 (b c-a d)^5 \log (c+d x) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{5 b d^5}+\frac{2 B g^4 (a+b x)^2 (b c-a d)^3 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{5 b d^3}-\frac{4 B g^4 (a+b x)^3 (b c-a d)^2 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{15 b d^2}-\frac{4 A B g^4 x (b c-a d)^4}{5 d^4}+\frac{B g^4 (a+b x)^4 (b c-a d) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{5 b d}+\frac{g^4 (a+b x)^5 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{5 b}-\frac{4 B^2 g^4 (a+b x) (b c-a d)^4 \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{5 b d^4}+\frac{26 B^2 g^4 x (b c-a d)^4}{15 d^4}-\frac{7 B^2 g^4 (a+b x)^2 (b c-a d)^3}{15 b d^3}+\frac{2 B^2 g^4 (a+b x)^3 (b c-a d)^2}{15 b d^2}-\frac{4 B^2 g^4 (b c-a d)^5 \log ^2(c+d x)}{5 b d^5}-\frac{10 B^2 g^4 (b c-a d)^5 \log (c+d x)}{3 b d^5}+\frac{8 B^2 g^4 (b c-a d)^5 \log (c+d x) \log \left (-\frac{d (a+b x)}{b c-a d}\right )}{5 b d^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*A*B*(b*c - a*d)^4*g^4*x)/(5*d^4) + (26*B^2*(b*c - a*d)^4*g^4*x)/(15*d^4) - (7*B^2*(b*c - a*d)^3*g^4*(a + b
*x)^2)/(15*b*d^3) + (2*B^2*(b*c - a*d)^2*g^4*(a + b*x)^3)/(15*b*d^2) - (10*B^2*(b*c - a*d)^5*g^4*Log[c + d*x])
/(3*b*d^5) + (8*B^2*(b*c - a*d)^5*g^4*Log[-((d*(a + b*x))/(b*c - a*d))]*Log[c + d*x])/(5*b*d^5) - (4*B^2*(b*c
- a*d)^5*g^4*Log[c + d*x]^2)/(5*b*d^5) - (4*B^2*(b*c - a*d)^4*g^4*(a + b*x)*Log[(e*(c + d*x)^2)/(a + b*x)^2])/
(5*b*d^4) + (2*B*(b*c - a*d)^3*g^4*(a + b*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(5*b*d^3) - (4*B*(b*c
- a*d)^2*g^4*(a + b*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(15*b*d^2) + (B*(b*c - a*d)*g^4*(a + b*x)^
4*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]))/(5*b*d) + (4*B*(b*c - a*d)^5*g^4*Log[c + d*x]*(A + B*Log[(e*(c + d
*x)^2)/(a + b*x)^2]))/(5*b*d^5) + (g^4*(a + b*x)^5*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2)/(5*b) + (8*B^2*
(b*c - a*d)^5*g^4*PolyLog[2, (b*(c + d*x))/(b*c - a*d)])/(5*b*d^5)

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 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 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 (a g+b g x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx &=\frac{g^4 (a+b x)^5 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{5 b}-\frac{(2 B) \int \frac{2 (b c-a d) g^5 (a+b x)^4 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{c+d x} \, dx}{5 b g}\\ &=\frac{g^4 (a+b x)^5 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{5 b}-\frac{\left (4 B (b c-a d) g^4\right ) \int \frac{(a+b x)^4 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{c+d x} \, dx}{5 b}\\ &=\frac{g^4 (a+b x)^5 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{5 b}-\frac{\left (4 B (b c-a d) g^4\right ) \int \left (-\frac{b (b c-a d)^3 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{d^4}+\frac{b (b c-a d)^2 (a+b x) \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{d^3}-\frac{b (b c-a d) (a+b x)^2 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{d^2}+\frac{b (a+b x)^3 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{d}+\frac{(-b c+a d)^4 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{d^4 (c+d x)}\right ) \, dx}{5 b}\\ &=\frac{g^4 (a+b x)^5 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{5 b}-\frac{\left (4 B (b c-a d) g^4\right ) \int (a+b x)^3 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx}{5 d}+\frac{\left (4 B (b c-a d)^2 g^4\right ) \int (a+b x)^2 \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx}{5 d^2}-\frac{\left (4 B (b c-a d)^3 g^4\right ) \int (a+b x) \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx}{5 d^3}+\frac{\left (4 B (b c-a d)^4 g^4\right ) \int \left (-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right ) \, dx}{5 d^4}-\frac{\left (4 B (b c-a d)^5 g^4\right ) \int \frac{-A-B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{c+d x} \, dx}{5 b d^4}\\ &=-\frac{4 A B (b c-a d)^4 g^4 x}{5 d^4}+\frac{2 B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d^3}-\frac{4 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{15 b d^2}+\frac{B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d}+\frac{4 B (b c-a d)^5 g^4 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d^5}+\frac{g^4 (a+b x)^5 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{5 b}-\frac{\left (B^2 (b c-a d) g^4\right ) \int \frac{2 (-b c+a d) (a+b x)^3}{c+d x} \, dx}{5 b d}+\frac{\left (4 B^2 (b c-a d)^2 g^4\right ) \int \frac{2 (-b c+a d) (a+b x)^2}{c+d x} \, dx}{15 b d^2}-\frac{\left (2 B^2 (b c-a d)^3 g^4\right ) \int \frac{2 (b c-a d) (-a-b x)}{c+d x} \, dx}{5 b d^3}-\frac{\left (4 B^2 (b c-a d)^4 g^4\right ) \int \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right ) \, dx}{5 d^4}-\frac{\left (4 B^2 (b c-a d)^5 g^4\right ) \int \frac{(a+b x)^2 \left (\frac{2 d e (c+d x)}{(a+b x)^2}-\frac{2 b e (c+d x)^2}{(a+b x)^3}\right ) \log (c+d x)}{e (c+d x)^2} \, dx}{5 b d^5}\\ &=-\frac{4 A B (b c-a d)^4 g^4 x}{5 d^4}-\frac{4 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{5 b d^4}+\frac{2 B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d^3}-\frac{4 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{15 b d^2}+\frac{B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d}+\frac{4 B (b c-a d)^5 g^4 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d^5}+\frac{g^4 (a+b x)^5 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{5 b}+\frac{\left (2 B^2 (b c-a d)^2 g^4\right ) \int \frac{(a+b x)^3}{c+d x} \, dx}{5 b d}-\frac{\left (8 B^2 (b c-a d)^3 g^4\right ) \int \frac{(a+b x)^2}{c+d x} \, dx}{15 b d^2}-\frac{\left (4 B^2 (b c-a d)^4 g^4\right ) \int \frac{-a-b x}{c+d x} \, dx}{5 b d^3}-\frac{\left (8 B^2 (b c-a d)^5 g^4\right ) \int \frac{1}{c+d x} \, dx}{5 b d^4}-\frac{\left (4 B^2 (b c-a d)^5 g^4\right ) \int \frac{(a+b x)^2 \left (\frac{2 d e (c+d x)}{(a+b x)^2}-\frac{2 b e (c+d x)^2}{(a+b x)^3}\right ) \log (c+d x)}{(c+d x)^2} \, dx}{5 b d^5 e}\\ &=-\frac{4 A B (b c-a d)^4 g^4 x}{5 d^4}-\frac{8 B^2 (b c-a d)^5 g^4 \log (c+d x)}{5 b d^5}-\frac{4 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{5 b d^4}+\frac{2 B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d^3}-\frac{4 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{15 b d^2}+\frac{B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d}+\frac{4 B (b c-a d)^5 g^4 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d^5}+\frac{g^4 (a+b x)^5 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{5 b}+\frac{\left (2 B^2 (b c-a d)^2 g^4\right ) \int \left (\frac{b (b c-a d)^2}{d^3}-\frac{b (b c-a d) (a+b x)}{d^2}+\frac{b (a+b x)^2}{d}+\frac{(-b c+a d)^3}{d^3 (c+d x)}\right ) \, dx}{5 b d}-\frac{\left (8 B^2 (b c-a d)^3 g^4\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}{15 b d^2}-\frac{\left (4 B^2 (b c-a d)^4 g^4\right ) \int \left (-\frac{b}{d}+\frac{b c-a d}{d (c+d x)}\right ) \, dx}{5 b d^3}-\frac{\left (4 B^2 (b c-a d)^5 g^4\right ) \int \left (-\frac{2 b e \log (c+d x)}{a+b x}+\frac{2 d e \log (c+d x)}{c+d x}\right ) \, dx}{5 b d^5 e}\\ &=-\frac{4 A B (b c-a d)^4 g^4 x}{5 d^4}+\frac{26 B^2 (b c-a d)^4 g^4 x}{15 d^4}-\frac{7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{15 b d^3}+\frac{2 B^2 (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}-\frac{10 B^2 (b c-a d)^5 g^4 \log (c+d x)}{3 b d^5}-\frac{4 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{5 b d^4}+\frac{2 B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d^3}-\frac{4 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{15 b d^2}+\frac{B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d}+\frac{4 B (b c-a d)^5 g^4 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d^5}+\frac{g^4 (a+b x)^5 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{5 b}+\frac{\left (8 B^2 (b c-a d)^5 g^4\right ) \int \frac{\log (c+d x)}{a+b x} \, dx}{5 d^5}-\frac{\left (8 B^2 (b c-a d)^5 g^4\right ) \int \frac{\log (c+d x)}{c+d x} \, dx}{5 b d^4}\\ &=-\frac{4 A B (b c-a d)^4 g^4 x}{5 d^4}+\frac{26 B^2 (b c-a d)^4 g^4 x}{15 d^4}-\frac{7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{15 b d^3}+\frac{2 B^2 (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}-\frac{10 B^2 (b c-a d)^5 g^4 \log (c+d x)}{3 b d^5}+\frac{8 B^2 (b c-a d)^5 g^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b d^5}-\frac{4 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{5 b d^4}+\frac{2 B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d^3}-\frac{4 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{15 b d^2}+\frac{B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d}+\frac{4 B (b c-a d)^5 g^4 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d^5}+\frac{g^4 (a+b x)^5 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{5 b}-\frac{\left (8 B^2 (b c-a d)^5 g^4\right ) \operatorname{Subst}\left (\int \frac{\log (x)}{x} \, dx,x,c+d x\right )}{5 b d^5}-\frac{\left (8 B^2 (b c-a d)^5 g^4\right ) \int \frac{\log \left (\frac{d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{5 b d^4}\\ &=-\frac{4 A B (b c-a d)^4 g^4 x}{5 d^4}+\frac{26 B^2 (b c-a d)^4 g^4 x}{15 d^4}-\frac{7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{15 b d^3}+\frac{2 B^2 (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}-\frac{10 B^2 (b c-a d)^5 g^4 \log (c+d x)}{3 b d^5}+\frac{8 B^2 (b c-a d)^5 g^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b d^5}-\frac{4 B^2 (b c-a d)^5 g^4 \log ^2(c+d x)}{5 b d^5}-\frac{4 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{5 b d^4}+\frac{2 B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d^3}-\frac{4 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{15 b d^2}+\frac{B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d}+\frac{4 B (b c-a d)^5 g^4 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d^5}+\frac{g^4 (a+b x)^5 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{5 b}-\frac{\left (8 B^2 (b c-a d)^5 g^4\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{5 b d^5}\\ &=-\frac{4 A B (b c-a d)^4 g^4 x}{5 d^4}+\frac{26 B^2 (b c-a d)^4 g^4 x}{15 d^4}-\frac{7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{15 b d^3}+\frac{2 B^2 (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}-\frac{10 B^2 (b c-a d)^5 g^4 \log (c+d x)}{3 b d^5}+\frac{8 B^2 (b c-a d)^5 g^4 \log \left (-\frac{d (a+b x)}{b c-a d}\right ) \log (c+d x)}{5 b d^5}-\frac{4 B^2 (b c-a d)^5 g^4 \log ^2(c+d x)}{5 b d^5}-\frac{4 B^2 (b c-a d)^4 g^4 (a+b x) \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )}{5 b d^4}+\frac{2 B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d^3}-\frac{4 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{15 b d^2}+\frac{B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d}+\frac{4 B (b c-a d)^5 g^4 \log (c+d x) \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d^5}+\frac{g^4 (a+b x)^5 \left (A+B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{5 b}+\frac{8 B^2 (b c-a d)^5 g^4 \text{Li}_2\left (\frac{b (c+d x)}{b c-a d}\right )}{5 b d^5}\\ \end{align*}

Mathematica [A]  time = 0.481292, size = 524, normalized size = 1.02 \[ \frac{g^4 \left ((a+b x)^5 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2-\frac{B (b c-a d) \left (-12 B (b c-a d)^4 \left (2 \text{PolyLog}\left (2,\frac{b (c+d x)}{b c-a d}\right )+\log (c+d x) \left (2 \log \left (\frac{d (a+b x)}{a d-b c}\right )-\log (c+d x)\right )\right )-6 d^2 (a+b x)^2 (b c-a d)^2 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )+4 d^3 (a+b x)^3 (b c-a d) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )-3 d^4 (a+b x)^4 \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )-12 (b c-a d)^4 \log (c+d x) \left (B \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+A\right )+12 A b d x (b c-a d)^3-4 B (b c-a d)^2 \left (2 b d x (b c-a d)-2 (b c-a d)^2 \log (c+d x)-d^2 (a+b x)^2\right )-B (b c-a d) \left (3 d^2 (a+b x)^2 (a d-b c)+6 b d x (b c-a d)^2-6 (b c-a d)^3 \log (c+d x)+2 d^3 (a+b x)^3\right )+12 B d (a+b x) (b c-a d)^3 \log \left (\frac{e (c+d x)^2}{(a+b x)^2}\right )+24 B (b c-a d)^4 \log (c+d x)-12 B (b c-a d)^3 ((a d-b c) \log (c+d x)+b d x)\right )}{3 d^5}\right )}{5 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(g^4*((a + b*x)^5*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2])^2 - (B*(b*c - a*d)*(12*A*b*d*(b*c - a*d)^3*x + 24*B
*(b*c - a*d)^4*Log[c + d*x] - 4*B*(b*c - a*d)^2*(2*b*d*(b*c - a*d)*x - d^2*(a + b*x)^2 - 2*(b*c - a*d)^2*Log[c
+ d*x]) - B*(b*c - a*d)*(6*b*d*(b*c - a*d)^2*x + 3*d^2*(-(b*c) + a*d)*(a + b*x)^2 + 2*d^3*(a + b*x)^3 - 6*(b*
c - a*d)^3*Log[c + d*x]) - 12*B*(b*c - a*d)^3*(b*d*x + (-(b*c) + a*d)*Log[c + d*x]) + 12*B*d*(b*c - a*d)^3*(a
+ b*x)*Log[(e*(c + d*x)^2)/(a + b*x)^2] - 6*d^2*(b*c - a*d)^2*(a + b*x)^2*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)
^2]) + 4*d^3*(b*c - a*d)*(a + b*x)^3*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]) - 3*d^4*(a + b*x)^4*(A + B*Log[(
e*(c + d*x)^2)/(a + b*x)^2]) - 12*(b*c - a*d)^4*Log[c + d*x]*(A + B*Log[(e*(c + d*x)^2)/(a + b*x)^2]) - 12*B*(
b*c - a*d)^4*((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)])))/(3*d^5)))/(5*b)

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Maple [F]  time = 1.913, size = 0, normalized size = 0. \begin{align*} \int \left ( bgx+ag \right ) ^{4} \left ( A+B\ln \left ({\frac{e \left ( dx+c \right ) ^{2}}{ \left ( bx+a \right ) ^{2}}} \right ) \right ) ^{2}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [B]  time = 2.02881, size = 3591, normalized size = 6.97 \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)^4*(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x, algorithm="maxima")

[Out]

1/5*A^2*b^4*g^4*x^5 + A^2*a*b^3*g^4*x^4 + 2*A^2*a^2*b^2*g^4*x^3 + 2*A^2*a^3*b*g^4*x^2 + 2*(x*log(d^2*e*x^2/(b^
2*x^2 + 2*a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x + a^2) + c^2*e/(b^2*x^2 + 2*a*b*x + a^2)) - 2*a*log(b*x
+ a)/b + 2*c*log(d*x + c)/d)*A*B*a^4*g^4 + 4*(x^2*log(d^2*e*x^2/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2
+ 2*a*b*x + a^2) + c^2*e/(b^2*x^2 + 2*a*b*x + a^2)) + 2*a^2*log(b*x + a)/b^2 - 2*c^2*log(d*x + c)/d^2 + 2*(b*
c - a*d)*x/(b*d))*A*B*a^3*b*g^4 + 4*(x^3*log(d^2*e*x^2/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*
x + a^2) + c^2*e/(b^2*x^2 + 2*a*b*x + a^2)) - 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*b^2*g^4 + 2/3*(3*x^4*log(d^2*e*x^2/(b^2*x^2 + 2*a*b*x
+ a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b*x + a^2) + c^2*e/(b^2*x^2 + 2*a*b*x + a^2)) + 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*a*b^3*g^4 + 1/15*(6*x^5*log(d^2*e*x^2/(b^2*x^2 + 2*a*b*x + a^2) + 2*c*d*e*x/(b^2*x^2 + 2*a*b
*x + a^2) + c^2*e/(b^2*x^2 + 2*a*b*x + a^2)) - 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))*A*B*b^4*g^4 + A^2*a^4*g^4*x + 2/15*((6*g^4*log(e) - 25*g^4)*b^4*c^5 - (30*g^4*log(e) - 113*g^
4)*a*b^3*c^4*d + 4*(15*g^4*log(e) - 49*g^4)*a^2*b^2*c^3*d^2 - 12*(5*g^4*log(e) - 13*g^4)*a^3*b*c^2*d^3 + 6*(5*
g^4*log(e) - 8*g^4)*a^4*c*d^4)*B^2*log(d*x + c)/d^5 - 8/5*(b^5*c^5*g^4 - 5*a*b^4*c^4*d*g^4 + 10*a^2*b^3*c^3*d^
2*g^4 - 10*a^3*b^2*c^2*d^3*g^4 + 5*a^4*b*c*d^4*g^4 - a^5*d^5*g^4)*(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*d^5) + 1/15*(3*B^2*b^5*d^5*g^4*x^5*log(e)^2 + 3*(b^5*c*d^4*g^
4*log(e) + (5*g^4*log(e)^2 - g^4*log(e))*a*b^4*d^5)*B^2*x^4 - 2*((2*g^4*log(e) - g^4)*b^5*c^2*d^3 - 2*(5*g^4*l
og(e) - g^4)*a*b^4*c*d^4 - (15*g^4*log(e)^2 - 8*g^4*log(e) + g^4)*a^2*b^3*d^5)*B^2*x^3 + ((6*g^4*log(e) - 7*g^
4)*b^5*c^3*d^2 - 3*(10*g^4*log(e) - 9*g^4)*a*b^4*c^2*d^3 + 3*(20*g^4*log(e) - 11*g^4)*a^2*b^3*c*d^4 + (30*g^4*
log(e)^2 - 36*g^4*log(e) + 13*g^4)*a^3*b^2*d^5)*B^2*x^2 - (2*(6*g^4*log(e) - 13*g^4)*b^5*c^4*d - 2*(30*g^4*log
(e) - 59*g^4)*a*b^4*c^3*d^2 + 12*(10*g^4*log(e) - 17*g^4)*a^2*b^3*c^2*d^3 - 2*(60*g^4*log(e) - 79*g^4)*a^3*b^2
*c*d^4 - (15*g^4*log(e)^2 - 48*g^4*log(e) + 46*g^4)*a^4*b*d^5)*B^2*x + 12*(B^2*b^5*d^5*g^4*x^5 + 5*B^2*a*b^4*d
^5*g^4*x^4 + 10*B^2*a^2*b^3*d^5*g^4*x^3 + 10*B^2*a^3*b^2*d^5*g^4*x^2 + 5*B^2*a^4*b*d^5*g^4*x + B^2*a^5*d^5*g^4
)*log(b*x + a)^2 + 12*(B^2*b^5*d^5*g^4*x^5 + 5*B^2*a*b^4*d^5*g^4*x^4 + 10*B^2*a^2*b^3*d^5*g^4*x^3 + 10*B^2*a^3
*b^2*d^5*g^4*x^2 + 5*B^2*a^4*b*d^5*g^4*x + (b^5*c^5*g^4 - 5*a*b^4*c^4*d*g^4 + 10*a^2*b^3*c^3*d^2*g^4 - 10*a^3*
b^2*c^2*d^3*g^4 + 5*a^4*b*c*d^4*g^4)*B^2)*log(d*x + c)^2 - 2*(6*B^2*b^5*d^5*g^4*x^5*log(e) + 3*(b^5*c*d^4*g^4
+ (10*g^4*log(e) - g^4)*a*b^4*d^5)*B^2*x^4 - 4*(b^5*c^2*d^3*g^4 - 5*a*b^4*c*d^4*g^4 - (15*g^4*log(e) - 4*g^4)*
a^2*b^3*d^5)*B^2*x^3 + 6*(b^5*c^3*d^2*g^4 - 5*a*b^4*c^2*d^3*g^4 + 10*a^2*b^3*c*d^4*g^4 + 2*(5*g^4*log(e) - 3*g
^4)*a^3*b^2*d^5)*B^2*x^2 - 6*(2*b^5*c^4*d*g^4 - 10*a*b^4*c^3*d^2*g^4 + 20*a^2*b^3*c^2*d^3*g^4 - 20*a^3*b^2*c*d
^4*g^4 - (5*g^4*log(e) - 8*g^4)*a^4*b*d^5)*B^2*x - (12*a*b^4*c^4*d*g^4 - 54*a^2*b^3*c^3*d^2*g^4 + 94*a^3*b^2*c
^2*d^3*g^4 - 77*a^4*b*c*d^4*g^4 - (6*g^4*log(e) - 25*g^4)*a^5*d^5)*B^2)*log(b*x + a) + 2*(6*B^2*b^5*d^5*g^4*x^
5*log(e) + 3*(b^5*c*d^4*g^4 + (10*g^4*log(e) - g^4)*a*b^4*d^5)*B^2*x^4 - 4*(b^5*c^2*d^3*g^4 - 5*a*b^4*c*d^4*g^
4 - (15*g^4*log(e) - 4*g^4)*a^2*b^3*d^5)*B^2*x^3 + 6*(b^5*c^3*d^2*g^4 - 5*a*b^4*c^2*d^3*g^4 + 10*a^2*b^3*c*d^4
*g^4 + 2*(5*g^4*log(e) - 3*g^4)*a^3*b^2*d^5)*B^2*x^2 - 6*(2*b^5*c^4*d*g^4 - 10*a*b^4*c^3*d^2*g^4 + 20*a^2*b^3*
c^2*d^3*g^4 - 20*a^3*b^2*c*d^4*g^4 - (5*g^4*log(e) - 8*g^4)*a^4*b*d^5)*B^2*x - 12*(B^2*b^5*d^5*g^4*x^5 + 5*B^2
*a*b^4*d^5*g^4*x^4 + 10*B^2*a^2*b^3*d^5*g^4*x^3 + 10*B^2*a^3*b^2*d^5*g^4*x^2 + 5*B^2*a^4*b*d^5*g^4*x + B^2*a^5
*d^5*g^4)*log(b*x + a))*log(d*x + c))/(b*d^5)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (A^{2} b^{4} g^{4} x^{4} + 4 \, A^{2} a b^{3} g^{4} x^{3} + 6 \, A^{2} a^{2} b^{2} g^{4} x^{2} + 4 \, A^{2} a^{3} b g^{4} x + A^{2} a^{4} g^{4} +{\left (B^{2} b^{4} g^{4} x^{4} + 4 \, B^{2} a b^{3} g^{4} x^{3} + 6 \, B^{2} a^{2} b^{2} g^{4} x^{2} + 4 \, B^{2} a^{3} b g^{4} x + B^{2} a^{4} g^{4}\right )} \log \left (\frac{d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right )^{2} + 2 \,{\left (A B b^{4} g^{4} x^{4} + 4 \, A B a b^{3} g^{4} x^{3} + 6 \, A B a^{2} b^{2} g^{4} x^{2} + 4 \, A B a^{3} b g^{4} x + A B a^{4} g^{4}\right )} \log \left (\frac{d^{2} e x^{2} + 2 \, c d e x + c^{2} e}{b^{2} x^{2} + 2 \, a b x + a^{2}}\right ), x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(A^2*b^4*g^4*x^4 + 4*A^2*a*b^3*g^4*x^3 + 6*A^2*a^2*b^2*g^4*x^2 + 4*A^2*a^3*b*g^4*x + A^2*a^4*g^4 + (B^
2*b^4*g^4*x^4 + 4*B^2*a*b^3*g^4*x^3 + 6*B^2*a^2*b^2*g^4*x^2 + 4*B^2*a^3*b*g^4*x + B^2*a^4*g^4)*log((d^2*e*x^2
+ 2*c*d*e*x + c^2*e)/(b^2*x^2 + 2*a*b*x + a^2))^2 + 2*(A*B*b^4*g^4*x^4 + 4*A*B*a*b^3*g^4*x^3 + 6*A*B*a^2*b^2*g
^4*x^2 + 4*A*B*a^3*b*g^4*x + A*B*a^4*g^4)*log((d^2*e*x^2 + 2*c*d*e*x + c^2*e)/(b^2*x^2 + 2*a*b*x + a^2)), 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)**4*(A+B*ln(e*(d*x+c)**2/(b*x+a)**2))**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 )}^{4}{\left (B \log \left (\frac{{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\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)^4*(A+B*log(e*(d*x+c)^2/(b*x+a)^2))^2,x, algorithm="giac")

[Out]

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