\(\int (a g+b g x)^4 (A+B \log (\frac {e (c+d x)^2}{(a+b x)^2}))^2 \, dx\) [210]

Optimal result

Integrand size = 34, antiderivative size = 515 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\frac {26 B^2 (b c-a d)^4 g^4 x}{15 d^4}-\frac {7 B^2 (b c-a d)^3 g^4 (a+b x)^2}{15 b d^3}+\frac {2 B^2 (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}-\frac {10 B^2 (b c-a d)^5 g^4 \log (a+b x)}{3 b d^5}-\frac {26 B^2 (b c-a d)^5 g^4 \log \left (\frac {c+d x}{a+b x}\right )}{15 b d^5}+\frac {2 B (b c-a d)^3 g^4 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d^3}-\frac {4 B (b c-a d)^2 g^4 (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{15 b d^2}+\frac {B (b c-a d) g^4 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 b d}-\frac {4 B (b c-a d)^4 g^4 (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 d^5}+\frac {g^4 (a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2}{5 b}-\frac {4 B (b c-a d)^5 g^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5}+\frac {8 B^2 (b c-a d)^5 g^4 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5} \]

[Out]

26/15*B^2*(-a*d+b*c)^4*g^4*x/d^4-7/15*B^2*(-a*d+b*c)^3*g^4*(b*x+a)^2/b/d^3+2/15*B^2*(-a*d+b*c)^2*g^4*(b*x+a)^3
/b/d^2-10/3*B^2*(-a*d+b*c)^5*g^4*ln(b*x+a)/b/d^5-26/15*B^2*(-a*d+b*c)^5*g^4*ln((d*x+c)/(b*x+a))/b/d^5+2/5*B*(-
a*d+b*c)^3*g^4*(b*x+a)^2*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))/b/d^3-4/15*B*(-a*d+b*c)^2*g^4*(b*x+a)^3*(A+B*ln(e*(d*
x+c)^2/(b*x+a)^2))/b/d^2+1/5*B*(-a*d+b*c)*g^4*(b*x+a)^4*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))/b/d-4/5*B*(-a*d+b*c)^4
*g^4*(d*x+c)*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))/d^5+1/5*g^4*(b*x+a)^5*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))^2/b-4/5*B*(
-a*d+b*c)^5*g^4*(A+B*ln(e*(d*x+c)^2/(b*x+a)^2))*ln(1-d*(b*x+a)/b/(d*x+c))/b/d^5+8/5*B^2*(-a*d+b*c)^5*g^4*polyl
og(2,d*(b*x+a)/b/(d*x+c))/b/d^5

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 515, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2552, 2356, 2389, 2379, 2438, 2351, 31, 46} \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=-\frac {4 B g^4 (b c-a d)^5 \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right ) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{5 b d^5}-\frac {4 B g^4 (c+d x) (b c-a d)^4 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{5 d^5}+\frac {2 B g^4 (a+b x)^2 (b c-a d)^3 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{5 b d^3}-\frac {4 B g^4 (a+b x)^3 (b c-a d)^2 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{15 b d^2}+\frac {B g^4 (a+b x)^4 (b c-a d) \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )}{5 b d}+\frac {g^4 (a+b x)^5 \left (B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )+A\right )^2}{5 b}+\frac {8 B^2 g^4 (b c-a d)^5 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5}-\frac {10 B^2 g^4 (b c-a d)^5 \log (a+b x)}{3 b d^5}-\frac {26 B^2 g^4 (b c-a d)^5 \log \left (\frac {c+d x}{a+b x}\right )}{15 b d^5}+\frac {26 B^2 g^4 x (b c-a d)^4}{15 d^4}-\frac {7 B^2 g^4 (a+b x)^2 (b c-a d)^3}{15 b d^3}+\frac {2 B^2 g^4 (a+b x)^3 (b c-a d)^2}{15 b d^2} \]

[In]

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

[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)

Rule 31

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

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2351

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

Rule 2356

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)^(q + 1)
*((a + b*Log[c*x^n])^p/(e*(q + 1))), x] - Dist[b*n*(p/(e*(q + 1))), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rule 2379

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

Rule 2389

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_))/(x_), x_Symbol] :> Dist[1/d, Int[(d
+ e*x)^(q + 1)*((a + b*Log[c*x^n])^p/x), x], x] - Dist[e/d, Int[(d + e*x)^q*(a + b*Log[c*x^n])^p, x], x] /; F
reeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[q, -1] && IntegerQ[2*q]

Rule 2438

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 2552

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.), x_Symbol] :> Dist[(b*c - a*d)^(m + 1)*(g/d)^m, Subst[Int[(A + B*Log[e*x^n])^p/(b - d*x)^(m + 2), x],
x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, A, B, n}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ
[b*c - a*d, 0] && IntegersQ[m, p] && EqQ[d*f - c*g, 0] && (GtQ[p, 0] || LtQ[m, -1])

Rubi steps \begin{align*} \text {integral}& = -\left (\left ((b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {\left (A+B \log \left (e x^2\right )\right )^2}{(d-b x)^6} \, dx,x,\frac {c+d x}{a+b x}\right )\right ) \\ & = \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)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^2\right )}{x (d-b x)^5} \, dx,x,\frac {c+d x}{a+b x}\right )}{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)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^2\right )}{(d-b x)^5} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 d}+\frac {\left (4 B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^2\right )}{x (d-b x)^4} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b d} \\ & = \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 {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)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^2\right )}{(d-b x)^4} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 d^2}+\frac {\left (4 B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^2\right )}{x (d-b x)^3} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b d^2}-\frac {\left (2 B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {1}{x (d-b x)^4} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b d} \\ & = -\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 {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)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^2\right )}{(d-b x)^3} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 d^3}+\frac {\left (4 B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^2\right )}{x (d-b x)^2} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b d^3}-\frac {\left (8 B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {1}{x (d-b x)^3} \, dx,x,\frac {c+d x}{a+b x}\right )}{15 b d^2}-\frac {\left (2 B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \left (\frac {1}{d^4 x}+\frac {b}{d (d-b x)^4}+\frac {b}{d^2 (d-b x)^3}+\frac {b}{d^3 (d-b x)^2}+\frac {b}{d^4 (d-b x)}\right ) \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b d} \\ & = \frac {2 B^2 (b c-a d)^4 g^4 x}{5 d^4}-\frac {B^2 (b c-a d)^3 g^4 (a+b x)^2}{5 b d^3}+\frac {2 B^2 (b c-a d)^2 g^4 (a+b x)^3}{15 b d^2}-\frac {2 B^2 (b c-a d)^5 g^4 \log (a+b x)}{5 b d^5}-\frac {2 B^2 (b c-a d)^5 g^4 \log \left (\frac {c+d x}{a+b x}\right )}{5 b d^5}+\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 {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)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^2\right )}{(d-b x)^2} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 d^4}+\frac {\left (4 B (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {A+B \log \left (e x^2\right )}{x (d-b x)} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b d^4}-\frac {\left (4 B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {1}{x (d-b x)^2} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b d^3}-\frac {\left (8 B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \left (\frac {1}{d^3 x}+\frac {b}{d (d-b x)^3}+\frac {b}{d^2 (d-b x)^2}+\frac {b}{d^3 (d-b x)}\right ) \, dx,x,\frac {c+d x}{a+b x}\right )}{15 b d^2} \\ & = \frac {14 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 {14 B^2 (b c-a d)^5 g^4 \log (a+b x)}{15 b d^5}-\frac {14 B^2 (b c-a d)^5 g^4 \log \left (\frac {c+d x}{a+b x}\right )}{15 b d^5}+\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)^4 g^4 (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 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 {4 B (b c-a d)^5 g^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5}-\frac {\left (8 B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {1}{d-b x} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 d^5}+\frac {\left (8 B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \frac {\log \left (1-\frac {d}{b x}\right )}{x} \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b d^5}-\frac {\left (4 B^2 (b c-a d)^5 g^4\right ) \text {Subst}\left (\int \left (\frac {1}{d^2 x}+\frac {b}{d (d-b x)^2}+\frac {b}{d^2 (d-b x)}\right ) \, dx,x,\frac {c+d x}{a+b x}\right )}{5 b d^3} \\ & = \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 (a+b x)}{3 b d^5}-\frac {26 B^2 (b c-a d)^5 g^4 \log \left (\frac {c+d x}{a+b x}\right )}{15 b d^5}+\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)^4 g^4 (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )}{5 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 {4 B (b c-a d)^5 g^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right ) \log \left (1-\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5}+\frac {8 B^2 (b c-a d)^5 g^4 \text {Li}_2\left (\frac {d (a+b x)}{b (c+d x)}\right )}{5 b d^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.32 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.02 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\frac {g^4 \left ((a+b x)^5 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2-\frac {B (b c-a d) \left (12 A b d (b c-a d)^3 x+24 B (b c-a d)^4 \log (c+d x)-4 B (b c-a d)^2 \left (2 b d (b c-a d) x-d^2 (a+b x)^2-2 (b c-a d)^2 \log (c+d x)\right )-B (b c-a d) \left (6 b d (b c-a d)^2 x+3 d^2 (-b c+a d) (a+b x)^2+2 d^3 (a+b x)^3-6 (b c-a d)^3 \log (c+d x)\right )-12 B (b c-a d)^3 (b d x+(-b c+a d) \log (c+d x))+12 B d (b c-a d)^3 (a+b x) \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )-6 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )+4 d^3 (b c-a d) (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )-3 d^4 (a+b x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )-12 (b c-a d)^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )-12 B (b c-a d)^4 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{3 d^5}\right )}{5 b} \]

[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)

Maple [F]

\[\int \left (b g x +a g \right )^{4} {\left (A +B \ln \left (\frac {e \left (d x +c \right )^{2}}{\left (b x +a \right )^{2}}\right )\right )}^{2}d x\]

[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)

Fricas [F]

\[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{4} {\left (B \log \left (\frac {{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A\right )}^{2} \,d x } \]

[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)

Sympy [F(-1)]

Timed out. \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\text {Timed out} \]

[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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2660 vs. \(2 (490) = 980\).

Time = 0.38 (sec) , antiderivative size = 2660, normalized size of antiderivative = 5.17 \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\text {Too large to display} \]

[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)

Giac [F]

\[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\int { {\left (b g x + a g\right )}^{4} {\left (B \log \left (\frac {{\left (d x + c\right )}^{2} e}{{\left (b x + a\right )}^{2}}\right ) + A\right )}^{2} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int (a g+b g x)^4 \left (A+B \log \left (\frac {e (c+d x)^2}{(a+b x)^2}\right )\right )^2 \, dx=\int {\left (a\,g+b\,g\,x\right )}^4\,{\left (A+B\,\ln \left (\frac {e\,{\left (c+d\,x\right )}^2}{{\left (a+b\,x\right )}^2}\right )\right )}^2 \,d x \]

[In]

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

[Out]

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