∫ (A+B log(e(a+bx)2 (c+dx)2 ))2 ________________________________

Optimal. Leaf size=587 \[ \frac {8 B^2 d^3 (c+d x)}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B^2 d^2 (c+d x)^2}{(b c-a d)^4 g^5 (a+b x)^2}+\frac {8 b^2 B^2 d (c+d x)^3}{9 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B^2 (c+d x)^4}{8 (b c-a d)^4 g^5 (a+b x)^4}+\frac {4 B d^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B d^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^4 g^5 (a+b x)^2}+\frac {4 b^2 B d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 (b c-a d)^4 g^5 (a+b x)^4}+\frac {d^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b d^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{2 (b c-a d)^4 g^5 (a+b x)^2}+\frac {b^2 d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 (b c-a d)^4 g^5 (a+b x)^4} \]

8*B^2*d^3*(d*x+c)/(-a*d+b*c)^4/g^5/(b*x+a)-3*b*B^2*d^2*(d*x+c)^2/(-a*d+b*c)^4/g^5/(b*x+a)^2+8/9*b^2*B^2*d*(d*x

+c)^3/(-a*d+b*c)^4/g^5/(b*x+a)^3-1/8*b^3*B^2*(d*x+c)^4/(-a*d+b*c)^4/g^5/(b*x+a)^4+4*B*d^3*(d*x+c)*(A+B*ln(e*(b

*x+a)^2/(d*x+c)^2))/(-a*d+b*c)^4/g^5/(b*x+a)-3*b*B*d^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(-a*d+b*c)^4/

g^5/(b*x+a)^2+4/3*b^2*B*d*(d*x+c)^3*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(-a*d+b*c)^4/g^5/(b*x+a)^3-1/4*b^3*B*(d*x+

c)^4*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))/(-a*d+b*c)^4/g^5/(b*x+a)^4+d^3*(d*x+c)*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/

(-a*d+b*c)^4/g^5/(b*x+a)-3/2*b*d^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(-a*d+b*c)^4/g^5/(b*x+a)^2+b^2*

d*(d*x+c)^3*(A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(-a*d+b*c)^4/g^5/(b*x+a)^3-1/4*b^3*(d*x+c)^4*(A+B*ln(e*(b*x+a)^2

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Rubi [A]
time = 0.27, antiderivative size = 587, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2550, 2395, 2342, 2341} \begin {gather*} -\frac {b^3 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{4 g^5 (a+b x)^4 (b c-a d)^4}-\frac {b^3 B (c+d x)^4 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{4 g^5 (a+b x)^4 (b c-a d)^4}+\frac {b^2 d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{g^5 (a+b x)^3 (b c-a d)^4}+\frac {4 b^2 B d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{3 g^5 (a+b x)^3 (b c-a d)^4}+\frac {d^3 (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{g^5 (a+b x) (b c-a d)^4}+\frac {4 B d^3 (c+d x) \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g^5 (a+b x) (b c-a d)^4}-\frac {3 b d^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )^2}{2 g^5 (a+b x)^2 (b c-a d)^4}-\frac {3 b B d^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )+A\right )}{g^5 (a+b x)^2 (b c-a d)^4}-\frac {b^3 B^2 (c+d x)^4}{8 g^5 (a+b x)^4 (b c-a d)^4}+\frac {8 b^2 B^2 d (c+d x)^3}{9 g^5 (a+b x)^3 (b c-a d)^4}+\frac {8 B^2 d^3 (c+d x)}{g^5 (a+b x) (b c-a d)^4}-\frac {3 b B^2 d^2 (c+d x)^2}{g^5 (a+b x)^2 (b c-a d)^4} \end {gather*}

Int[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(a*g + b*g*x)^5,x]

(8*B^2*d^3*(c + d*x))/((b*c - a*d)^4*g^5*(a + b*x)) - (3*b*B^2*d^2*(c + d*x)^2)/((b*c - a*d)^4*g^5*(a + b*x)^2

) + (8*b^2*B^2*d*(c + d*x)^3)/(9*(b*c - a*d)^4*g^5*(a + b*x)^3) - (b^3*B^2*(c + d*x)^4)/(8*(b*c - a*d)^4*g^5*(

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

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

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

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

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

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

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

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^

n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Lo

g[c*x^n])^p/(d*(m + 1))), x] - Dist[b*n*(p/(m + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p - 1), x], x] /; FreeQ[{

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]

:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[

{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r

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/b)^m, Subst[Int[x^m*((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[b*f - a*g, 0] && (GtQ[p, 0] || LtQ[m, -1])

\begin {align*} \int \frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{(a g+b g x)^5} \, dx &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac {B \int \frac {2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{g^4 (a+b x)^5 (c+d x)} \, dx}{2 b g}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac {(B (b c-a d)) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^5 (c+d x)} \, dx}{b g^5}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac {(B (b c-a d)) \int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d) (a+b x)^5}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^5 (a+b x)}-\frac {d^5 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{b g^5}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac {B \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^5} \, dx}{g^5}+\frac {\left (B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{a+b x} \, dx}{(b c-a d)^4 g^5}-\frac {\left (B d^5\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{c+d x} \, dx}{b (b c-a d)^4 g^5}-\frac {\left (B d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^2} \, dx}{(b c-a d)^3 g^5}+\frac {\left (B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^3} \, dx}{(b c-a d)^2 g^5}-\frac {(B d) \int \frac {A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )}{(a+b x)^4} \, dx}{(b c-a d) g^5}\\ &=-\frac {B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 b g^5 (a+b x)^4}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^5 (a+b x)^3}-\frac {B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d)^3 g^5 (a+b x)}+\frac {B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d)^4 g^5}+\frac {B^2 \int \frac {2 (b c-a d)}{(a+b x)^5 (c+d x)} \, dx}{4 b g^5}-\frac {\left (B^2 d^4\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (a+b x)}{e (a+b x)^2} \, dx}{b (b c-a d)^4 g^5}+\frac {\left (B^2 d^4\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{e (a+b x)^2} \, dx}{b (b c-a d)^4 g^5}-\frac {\left (B^2 d^3\right ) \int \frac {2 (b c-a d)}{(a+b x)^2 (c+d x)} \, dx}{b (b c-a d)^3 g^5}+\frac {\left (B^2 d^2\right ) \int \frac {2 (b c-a d)}{(a+b x)^3 (c+d x)} \, dx}{2 b (b c-a d)^2 g^5}-\frac {\left (B^2 d\right ) \int \frac {2 (b c-a d)}{(a+b x)^4 (c+d x)} \, dx}{3 b (b c-a d) g^5}\\ &=-\frac {B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 b g^5 (a+b x)^4}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^5 (a+b x)^3}-\frac {B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d)^3 g^5 (a+b x)}+\frac {B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d)^4 g^5}-\frac {\left (2 B^2 d\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{3 b g^5}-\frac {\left (2 B^2 d^3\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{b (b c-a d)^2 g^5}+\frac {\left (B^2 d^2\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{b (b c-a d) g^5}+\frac {\left (B^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{2 b g^5}-\frac {\left (B^2 d^4\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (a+b x)}{(a+b x)^2} \, dx}{b (b c-a d)^4 e g^5}+\frac {\left (B^2 d^4\right ) \int \frac {(c+d x)^2 \left (-\frac {2 d e (a+b x)^2}{(c+d x)^3}+\frac {2 b e (a+b x)}{(c+d x)^2}\right ) \log (c+d x)}{(a+b x)^2} \, dx}{b (b c-a d)^4 e g^5}\\ &=-\frac {B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 b g^5 (a+b x)^4}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^5 (a+b x)^3}-\frac {B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d)^3 g^5 (a+b x)}+\frac {B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d)^4 g^5}-\frac {\left (2 B^2 d\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^4}-\frac {b d}{(b c-a d)^2 (a+b x)^3}+\frac {b d^2}{(b c-a d)^3 (a+b x)^2}-\frac {b d^3}{(b c-a d)^4 (a+b x)}+\frac {d^4}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b g^5}-\frac {\left (2 B^2 d^3\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^2}-\frac {b d}{(b c-a d)^2 (a+b x)}+\frac {d^2}{(b c-a d)^2 (c+d x)}\right ) \, dx}{b (b c-a d)^2 g^5}+\frac {\left (B^2 d^2\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^3}-\frac {b d}{(b c-a d)^2 (a+b x)^2}+\frac {b d^2}{(b c-a d)^3 (a+b x)}-\frac {d^3}{(b c-a d)^3 (c+d x)}\right ) \, dx}{b (b c-a d) g^5}+\frac {\left (B^2 (b c-a d)\right ) \int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{2 b g^5}-\frac {\left (B^2 d^4\right ) \int \left (\frac {2 b e \log (a+b x)}{a+b x}-\frac {2 d e \log (a+b x)}{c+d x}\right ) \, dx}{b (b c-a d)^4 e g^5}+\frac {\left (B^2 d^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}{b (b c-a d)^4 e g^5}\\ &=-\frac {B^2}{8 b g^5 (a+b x)^4}+\frac {7 B^2 d}{18 b (b c-a d) g^5 (a+b x)^3}-\frac {13 B^2 d^2}{12 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {25 B^2 d^3}{6 b (b c-a d)^3 g^5 (a+b x)}+\frac {25 B^2 d^4 \log (a+b x)}{6 b (b c-a d)^4 g^5}-\frac {B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 b g^5 (a+b x)^4}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^5 (a+b x)^3}-\frac {B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d)^3 g^5 (a+b x)}+\frac {B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {25 B^2 d^4 \log (c+d x)}{6 b (b c-a d)^4 g^5}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d)^4 g^5}-\frac {\left (2 B^2 d^4\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{(b c-a d)^4 g^5}+\frac {\left (2 B^2 d^4\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{(b c-a d)^4 g^5}+\frac {\left (2 B^2 d^5\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{b (b c-a d)^4 g^5}-\frac {\left (2 B^2 d^5\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{b (b c-a d)^4 g^5}\\ &=-\frac {B^2}{8 b g^5 (a+b x)^4}+\frac {7 B^2 d}{18 b (b c-a d) g^5 (a+b x)^3}-\frac {13 B^2 d^2}{12 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {25 B^2 d^3}{6 b (b c-a d)^3 g^5 (a+b x)}+\frac {25 B^2 d^4 \log (a+b x)}{6 b (b c-a d)^4 g^5}-\frac {B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 b g^5 (a+b x)^4}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^5 (a+b x)^3}-\frac {B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d)^3 g^5 (a+b x)}+\frac {B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {25 B^2 d^4 \log (c+d x)}{6 b (b c-a d)^4 g^5}+\frac {2 B^2 d^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^4 g^5}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d)^4 g^5}+\frac {2 B^2 d^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^4 g^5}-\frac {\left (2 B^2 d^4\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{(b c-a d)^4 g^5}-\frac {\left (2 B^2 d^4\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{b (b c-a d)^4 g^5}-\frac {\left (2 B^2 d^4\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{b (b c-a d)^4 g^5}-\frac {\left (2 B^2 d^5\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{b (b c-a d)^4 g^5}\\ &=-\frac {B^2}{8 b g^5 (a+b x)^4}+\frac {7 B^2 d}{18 b (b c-a d) g^5 (a+b x)^3}-\frac {13 B^2 d^2}{12 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {25 B^2 d^3}{6 b (b c-a d)^3 g^5 (a+b x)}+\frac {25 B^2 d^4 \log (a+b x)}{6 b (b c-a d)^4 g^5}-\frac {B^2 d^4 \log ^2(a+b x)}{b (b c-a d)^4 g^5}-\frac {B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 b g^5 (a+b x)^4}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^5 (a+b x)^3}-\frac {B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d)^3 g^5 (a+b x)}+\frac {B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {25 B^2 d^4 \log (c+d x)}{6 b (b c-a d)^4 g^5}+\frac {2 B^2 d^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^4 g^5}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d)^4 g^5}-\frac {B^2 d^4 \log ^2(c+d x)}{b (b c-a d)^4 g^5}+\frac {2 B^2 d^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^4 g^5}-\frac {\left (2 B^2 d^4\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{b (b c-a d)^4 g^5}-\frac {\left (2 B^2 d^4\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{b (b c-a d)^4 g^5}\\ &=-\frac {B^2}{8 b g^5 (a+b x)^4}+\frac {7 B^2 d}{18 b (b c-a d) g^5 (a+b x)^3}-\frac {13 B^2 d^2}{12 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {25 B^2 d^3}{6 b (b c-a d)^3 g^5 (a+b x)}+\frac {25 B^2 d^4 \log (a+b x)}{6 b (b c-a d)^4 g^5}-\frac {B^2 d^4 \log ^2(a+b x)}{b (b c-a d)^4 g^5}-\frac {B \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{4 b g^5 (a+b x)^4}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{3 b (b c-a d) g^5 (a+b x)^3}-\frac {B d^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{2 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {B d^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d)^3 g^5 (a+b x)}+\frac {B d^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )}{b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {25 B^2 d^4 \log (c+d x)}{6 b (b c-a d)^4 g^5}+\frac {2 B^2 d^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{b (b c-a d)^4 g^5}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)}{b (b c-a d)^4 g^5}-\frac {B^2 d^4 \log ^2(c+d x)}{b (b c-a d)^4 g^5}+\frac {2 B^2 d^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^4 g^5}+\frac {2 B^2 d^4 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{b (b c-a d)^4 g^5}+\frac {2 B^2 d^4 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{b (b c-a d)^4 g^5}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.59, size = 762, normalized size = 1.30 \begin {gather*} -\frac {18 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )^2+\frac {B \left (18 (b c-a d)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+24 d (-b c+a d)^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+36 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+72 d^3 (-b c+a d) (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )-72 d^4 (a+b x)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right )+72 d^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)^2}{(c+d x)^2}\right )\right ) \log (c+d x)-144 B d^3 (a+b x)^3 (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))+36 B d^2 (a+b x)^2 \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )-8 B d (a+b x) \left (2 (b c-a d)^3-3 d (b c-a d)^2 (a+b x)+6 d^2 (b c-a d) (a+b x)^2+6 d^3 (a+b x)^3 \log (a+b x)-6 d^3 (a+b x)^3 \log (c+d x)\right )+3 B \left (3 (b c-a d)^4+4 d (-b c+a d)^3 (a+b x)+6 d^2 (b c-a d)^2 (a+b x)^2+12 d^3 (-b c+a d) (a+b x)^3-12 d^4 (a+b x)^4 \log (a+b x)+12 d^4 (a+b x)^4 \log (c+d x)\right )+72 B d^4 (a+b x)^4 \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)}{-b c+a d}\right )\right )-72 B d^4 (a+b x)^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 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )\right )\right )}{(b c-a d)^4}}{72 b g^5 (a+b x)^4} \end {gather*}

Integrate[(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2/(a*g + b*g*x)^5,x]

-1/72*(18*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])^2 + (B*(18*(b*c - a*d)^4*(A + B*Log[(e*(a + b*x)^2)/(c + d*

x)^2]) + 24*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + 36*d^2*(b*c - a*d)^2*(a +

b*x)^2*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + 72*d^3*(-(b*c) + a*d)*(a + b*x)^3*(A + B*Log[(e*(a + b*x)^2)

/(c + d*x)^2]) - 72*d^4*(a + b*x)^4*Log[a + b*x]*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2]) + 72*d^4*(a + b*x)^4

*(A + B*Log[(e*(a + b*x)^2)/(c + d*x)^2])*Log[c + d*x] - 144*B*d^3*(a + b*x)^3*(b*c - a*d + d*(a + b*x)*Log[a

+ b*x] - d*(a + b*x)*Log[c + d*x]) + 36*B*d^2*(a + b*x)^2*((b*c - a*d)^2 + 2*d*(-(b*c) + a*d)*(a + b*x) - 2*d^

2*(a + b*x)^2*Log[a + b*x] + 2*d^2*(a + b*x)^2*Log[c + d*x]) - 8*B*d*(a + b*x)*(2*(b*c - a*d)^3 - 3*d*(b*c - a

*d)^2*(a + b*x) + 6*d^2*(b*c - a*d)*(a + b*x)^2 + 6*d^3*(a + b*x)^3*Log[a + b*x] - 6*d^3*(a + b*x)^3*Log[c + d

*x]) + 3*B*(3*(b*c - a*d)^4 + 4*d*(-(b*c) + a*d)^3*(a + b*x) + 6*d^2*(b*c - a*d)^2*(a + b*x)^2 + 12*d^3*(-(b*c

) + a*d)*(a + b*x)^3 - 12*d^4*(a + b*x)^4*Log[a + b*x] + 12*d^4*(a + b*x)^4*Log[c + d*x]) + 72*B*d^4*(a + b*x)

^4*(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)

]) - 72*B*d^4*(a + b*x)^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)])))/(b*c - a*d)^4)/(b*g^5*(a + b*x)^4)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1485\) vs. \(2(575)=1150\).
time = 1.17, size = 1486, normalized size = 2.53


method	result	size

derivativedivides	\(\text {Expression too large to display}\)	\(1486\)
default	\(\text {Expression too large to display}\)	\(1486\)
norman	\(\text {Expression too large to display}\)	\(1816\)
risch	\(\text {Expression too large to display}\)	\(2235\)

int((A+B*ln(e*(b*x+a)^2/(d*x+c)^2))^2/(b*g*x+a*g)^5,x,method=_RETURNVERBOSE)

-1/d*(d^5/g^5*A^2*(3/2*b/(a*d-b*c)^4/(a/(d*x+c)*d-b*c/(d*x+c)+b)^2-b^2/(a*d-b*c)^4/(a/(d*x+c)*d-b*c/(d*x+c)+b)

^3-1/(a*d-b*c)^4/(a/(d*x+c)*d-b*c/(d*x+c)+b)+1/4*b^3/(a*d-b*c)^4/(a/(d*x+c)*d-b*c/(d*x+c)+b)^4)+(415/72*B^2/b*

d^5/g/(d*x+c)^4-25/12*b^3*B^2*d^5/g/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*a*b^3*c^3*d+b^4*c^4)*ln(e*(a/(d

*x+c)*d-b*c/(d*x+c)+b)^2/d^2)+25/6*B^2*b^2*d^5/g/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(d*x+c)+163/12*

B^2*b*d^5/g/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)^2+271/18*B^2*d^5/g/(a*d-b*c)/(d*x+c)^3-4*B^2*d^5/g/(a*d-b*c)/(

d*x+c)^3*ln(e*(a/(d*x+c)*d-b*c/(d*x+c)+b)^2/d^2)-1/4*B^2*b^3*d^5/g/(a^4*d^4-4*a^3*b*c*d^3+6*a^2*b^2*c^2*d^2-4*

a*b^3*c^3*d+b^4*c^4)*ln(e*(a/(d*x+c)*d-b*c/(d*x+c)+b)^2/d^2)^2-B^2*d^5/g/(a*d-b*c)/(d*x+c)^3*ln(e*(a/(d*x+c)*d

-b*c/(d*x+c)+b)^2/d^2)^2-9*B^2*d^5*b/g/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)^2*ln(e*(a/(d*x+c)*d-b*c/(d*x+c)+b)^

2/d^2)-22/3*B^2*d^5*b^2/g/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(d*x+c)*ln(e*(a/(d*x+c)*d-b*c/(d*x+c)+

b)^2/d^2)-3/2*B^2*b*d^5/g/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)^2*ln(e*(a/(d*x+c)*d-b*c/(d*x+c)+b)^2/d^2)^2-B^2*

b^2*d^5/g/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(d*x+c)*ln(e*(a/(d*x+c)*d-b*c/(d*x+c)+b)^2/d^2)^2)/(a/

(d*x+c)*d-b*c/(d*x+c)+b)^4/g^4+(A*B*b^2*d^5/g/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(d*x+c)+25/12*A*B/

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

*x+c)*d-b*c/(d*x+c)+b)^2/d^2)+7/2*A*B*b*d^5/g/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(d*x+c)^2+13/3*A*B*d^5/g/(a*d-b*c)/(

d*x+c)^3-2*A*B*d^5/g/(a*d-b*c)/(d*x+c)^3*ln(e*(a/(d*x+c)*d-b*c/(d*x+c)+b)^2/d^2)-3*A*B*d^5*b/g/(a^2*d^2-2*a*b*

c*d+b^2*c^2)/(d*x+c)^2*ln(e*(a/(d*x+c)*d-b*c/(d*x+c)+b)^2/d^2)-2*A*B*d^5*b^2/g/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*

c^2*d-b^3*c^3)/(d*x+c)*ln(e*(a/(d*x+c)*d-b*c/(d*x+c)+b)^2/d^2))/(a/(d*x+c)*d-b*c/(d*x+c)+b)^4/g^4)

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2288 vs. \(2 (583) = 1166\).
time = 0.57, size = 2288, normalized size = 3.90 \begin {gather*} \text {Too large to display} \end {gather*}

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2/(b*g*x+a*g)^5,x, algorithm="maxima")

1/72*(6*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d

^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^

5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*

a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 -

a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) + 12*d^4*log(b*x + a)/

((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*

c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5))*log(b^2*x^2*e/(d^2*x^2 + 2*c*d*x

+ c^2) + 2*a*b*x*e/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2)) - (9*b^4*c^4 - 64*a*b^3*c^3*d

+ 216*a^2*b^2*c^2*d^2 - 576*a^3*b*c*d^3 + 415*a^4*d^4 - 300*(b^4*c*d^3 - a*b^3*d^4)*x^3 + 6*(13*b^4*c^2*d^2 -

176*a*b^3*c*d^3 + 163*a^2*b^2*d^4)*x^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x

+ a^4*d^4)*log(b*x + a)^2 + 72*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*

log(d*x + c)^2 - 4*(7*b^4*c^3*d - 60*a*b^3*c^2*d^2 + 324*a^2*b^2*c*d^3 - 271*a^3*b*d^4)*x - 300*(b^4*d^4*x^4 +

4*a*b^3*d^4*x^3 + 6*a^2*b^2*d^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a) + 12*(25*b^4*d^4*x^4 + 100*a*b^3*

d^4*x^3 + 150*a^2*b^2*d^4*x^2 + 100*a^3*b*d^4*x + 25*a^4*d^4 - 12*(b^4*d^4*x^4 + 4*a*b^3*d^4*x^3 + 6*a^2*b^2*d

^4*x^2 + 4*a^3*b*d^4*x + a^4*d^4)*log(b*x + a))*log(d*x + c))/(a^4*b^5*c^4*g^5 - 4*a^5*b^4*c^3*d*g^5 + 6*a^6*b

^3*c^2*d^2*g^5 - 4*a^7*b^2*c*d^3*g^5 + a^8*b*d^4*g^5 + (b^9*c^4*g^5 - 4*a*b^8*c^3*d*g^5 + 6*a^2*b^7*c^2*d^2*g^

5 - 4*a^3*b^6*c*d^3*g^5 + a^4*b^5*d^4*g^5)*x^4 + 4*(a*b^8*c^4*g^5 - 4*a^2*b^7*c^3*d*g^5 + 6*a^3*b^6*c^2*d^2*g^

5 - 4*a^4*b^5*c*d^3*g^5 + a^5*b^4*d^4*g^5)*x^3 + 6*(a^2*b^7*c^4*g^5 - 4*a^3*b^6*c^3*d*g^5 + 6*a^4*b^5*c^2*d^2*

g^5 - 4*a^5*b^4*c*d^3*g^5 + a^6*b^3*d^4*g^5)*x^2 + 4*(a^3*b^6*c^4*g^5 - 4*a^4*b^5*c^3*d*g^5 + 6*a^5*b^4*c^2*d^

2*g^5 - 4*a^6*b^3*c*d^3*g^5 + a^7*b^2*d^4*g^5)*x))*B^2 + 1/12*A*B*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*

d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^

3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*

a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x

^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 - a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d

+ 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) - 6*log(b^2*x^2*e/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*x*e/(d^2*x^2 + 2*c*d*x

+ c^2) + a^2*e/(d^2*x^2 + 2*c*d*x + c^2))/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*

x + a^4*b*g^5) + 12*d^4*log(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d

^4)*g^5) - 12*d^4*log(d*x + c)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^

5)) - 1/4*B^2*log(b^2*x^2*e/(d^2*x^2 + 2*c*d*x + c^2) + 2*a*b*x*e/(d^2*x^2 + 2*c*d*x + c^2) + a^2*e/(d^2*x^2 +

2*c*d*x + c^2))^2/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) - 1/4*A^2

/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5)

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Fricas [A]
time = 0.41, size = 1080, normalized size = 1.84 \begin {gather*} -\frac {9 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} b^{4} c^{4} - 8 \, {\left (9 \, A^{2} + 12 \, A B + 8 \, B^{2}\right )} a b^{3} c^{3} d + 108 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a^{2} b^{2} c^{2} d^{2} - 72 \, {\left (A^{2} + 4 \, A B + 8 \, B^{2}\right )} a^{3} b c d^{3} + {\left (18 \, A^{2} + 150 \, A B + 415 \, B^{2}\right )} a^{4} d^{4} - 12 \, {\left ({\left (6 \, A B + 25 \, B^{2}\right )} b^{4} c d^{3} - {\left (6 \, A B + 25 \, B^{2}\right )} a b^{3} d^{4}\right )} x^{3} + 6 \, {\left ({\left (6 \, A B + 13 \, B^{2}\right )} b^{4} c^{2} d^{2} - 16 \, {\left (3 \, A B + 11 \, B^{2}\right )} a b^{3} c d^{3} + {\left (42 \, A B + 163 \, B^{2}\right )} a^{2} b^{2} d^{4}\right )} x^{2} - 18 \, {\left (B^{2} b^{4} d^{4} x^{4} + 4 \, B^{2} a b^{3} d^{4} x^{3} + 6 \, B^{2} a^{2} b^{2} d^{4} x^{2} + 4 \, B^{2} a^{3} b d^{4} x - B^{2} b^{4} c^{4} + 4 \, B^{2} a b^{3} c^{3} d - 6 \, B^{2} a^{2} b^{2} c^{2} d^{2} + 4 \, B^{2} a^{3} b c d^{3}\right )} \log \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )^{2} - 4 \, {\left ({\left (6 \, A B + 7 \, B^{2}\right )} b^{4} c^{3} d - 12 \, {\left (3 \, A B + 5 \, B^{2}\right )} a b^{3} c^{2} d^{2} + 108 \, {\left (A B + 3 \, B^{2}\right )} a^{2} b^{2} c d^{3} - {\left (78 \, A B + 271 \, B^{2}\right )} a^{3} b d^{4}\right )} x - 6 \, {\left ({\left (6 \, A B + 25 \, B^{2}\right )} b^{4} d^{4} x^{4} - 3 \, {\left (2 \, A B + B^{2}\right )} b^{4} c^{4} + 8 \, {\left (3 \, A B + 2 \, B^{2}\right )} a b^{3} c^{3} d - 36 \, {\left (A B + B^{2}\right )} a^{2} b^{2} c^{2} d^{2} + 24 \, {\left (A B + 2 \, B^{2}\right )} a^{3} b c d^{3} + 4 \, {\left (3 \, B^{2} b^{4} c d^{3} + 2 \, {\left (3 \, A B + 11 \, B^{2}\right )} a b^{3} d^{4}\right )} x^{3} - 6 \, {\left (B^{2} b^{4} c^{2} d^{2} - 8 \, B^{2} a b^{3} c d^{3} - 6 \, {\left (A B + 3 \, B^{2}\right )} a^{2} b^{2} d^{4}\right )} x^{2} + 4 \, {\left (B^{2} b^{4} c^{3} d - 6 \, B^{2} a b^{3} c^{2} d^{2} + 18 \, B^{2} a^{2} b^{2} c d^{3} + 6 \, {\left (A B + 2 \, B^{2}\right )} a^{3} b d^{4}\right )} x\right )} \log \left (\frac {{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} e}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right )}{72 \, {\left ({\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} g^{5} x + {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} g^{5}\right )}} \end {gather*}

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2/(b*g*x+a*g)^5,x, algorithm="fricas")

-1/72*(9*(2*A^2 + 2*A*B + B^2)*b^4*c^4 - 8*(9*A^2 + 12*A*B + 8*B^2)*a*b^3*c^3*d + 108*(A^2 + 2*A*B + 2*B^2)*a^

2*b^2*c^2*d^2 - 72*(A^2 + 4*A*B + 8*B^2)*a^3*b*c*d^3 + (18*A^2 + 150*A*B + 415*B^2)*a^4*d^4 - 12*((6*A*B + 25*

B^2)*b^4*c*d^3 - (6*A*B + 25*B^2)*a*b^3*d^4)*x^3 + 6*((6*A*B + 13*B^2)*b^4*c^2*d^2 - 16*(3*A*B + 11*B^2)*a*b^3

*c*d^3 + (42*A*B + 163*B^2)*a^2*b^2*d^4)*x^2 - 18*(B^2*b^4*d^4*x^4 + 4*B^2*a*b^3*d^4*x^3 + 6*B^2*a^2*b^2*d^4*x

^2 + 4*B^2*a^3*b*d^4*x - B^2*b^4*c^4 + 4*B^2*a*b^3*c^3*d - 6*B^2*a^2*b^2*c^2*d^2 + 4*B^2*a^3*b*c*d^3)*log((b^2

*x^2 + 2*a*b*x + a^2)*e/(d^2*x^2 + 2*c*d*x + c^2))^2 - 4*((6*A*B + 7*B^2)*b^4*c^3*d - 12*(3*A*B + 5*B^2)*a*b^3

*c^2*d^2 + 108*(A*B + 3*B^2)*a^2*b^2*c*d^3 - (78*A*B + 271*B^2)*a^3*b*d^4)*x - 6*((6*A*B + 25*B^2)*b^4*d^4*x^4

- 3*(2*A*B + B^2)*b^4*c^4 + 8*(3*A*B + 2*B^2)*a*b^3*c^3*d - 36*(A*B + B^2)*a^2*b^2*c^2*d^2 + 24*(A*B + 2*B^2)

*a^3*b*c*d^3 + 4*(3*B^2*b^4*c*d^3 + 2*(3*A*B + 11*B^2)*a*b^3*d^4)*x^3 - 6*(B^2*b^4*c^2*d^2 - 8*B^2*a*b^3*c*d^3

- 6*(A*B + 3*B^2)*a^2*b^2*d^4)*x^2 + 4*(B^2*b^4*c^3*d - 6*B^2*a*b^3*c^2*d^2 + 18*B^2*a^2*b^2*c*d^3 + 6*(A*B +

2*B^2)*a^3*b*d^4)*x)*log((b^2*x^2 + 2*a*b*x + a^2)*e/(d^2*x^2 + 2*c*d*x + c^2)))/((b^9*c^4 - 4*a*b^8*c^3*d +

6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*d^3 + a^4*b^5*d^4)*g^5*x^4 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^

2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^4)*g^5*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a^5*b^4*

c*d^3 + a^6*b^3*d^4)*g^5*x^2 + 4*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7*b^

2*d^4)*g^5*x + (a^4*b^5*c^4 - 4*a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^8*b*d^4)*g^5)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

integrate((A+B*ln(e*(b*x+a)**2/(d*x+c)**2))**2/(b*g*x+a*g)**5,x)

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Giac [A]
time = 4.72, size = 874, normalized size = 1.49 \begin {gather*} \frac {1}{4} \, {\left (\frac {B^{2} d^{4}}{b^{5} c^{4} g^{5} - 4 \, a b^{4} c^{3} d g^{5} + 6 \, a^{2} b^{3} c^{2} d^{2} g^{5} - 4 \, a^{3} b^{2} c d^{3} g^{5} + a^{4} b d^{4} g^{5}} - \frac {B^{2}}{{\left (b g x + a g\right )}^{4} b g}\right )} \log \left (\frac {b^{2}}{\frac {b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac {2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {2 \, b c d g}{b g x + a g} - \frac {2 \, a d^{2} g}{b g x + a g} + d^{2}}\right )^{2} + \frac {1}{12} \, {\left (\frac {12 \, B^{2} d^{3}}{{\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )} {\left (b g x + a g\right )} b g} - \frac {6 \, B^{2} d^{2}}{{\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (b g x + a g\right )}^{2} b g^{2}} + \frac {4 \, B^{2} d}{{\left (b g x + a g\right )}^{3} {\left (b c - a d\right )} b g^{2}} - \frac {3 \, {\left (2 \, A B b^{3} g^{3} + 3 \, B^{2} b^{3} g^{3}\right )}}{{\left (b g x + a g\right )}^{4} b^{4} g^{4}}\right )} \log \left (\frac {b^{2}}{\frac {b^{2} c^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} - \frac {2 \, a b c d g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {a^{2} d^{2} g^{2}}{{\left (b g x + a g\right )}^{2}} + \frac {2 \, b c d g}{b g x + a g} - \frac {2 \, a d^{2} g}{b g x + a g} + d^{2}}\right ) - \frac {{\left (6 \, A B d^{4} + 31 \, B^{2} d^{4}\right )} \log \left (-\frac {b c g}{b g x + a g} + \frac {a d g}{b g x + a g} - d\right )}{6 \, {\left (b^{5} c^{4} g^{5} - 4 \, a b^{4} c^{3} d g^{5} + 6 \, a^{2} b^{3} c^{2} d^{2} g^{5} - 4 \, a^{3} b^{2} c d^{3} g^{5} + a^{4} b d^{4} g^{5}\right )}} + \frac {6 \, A B d^{3} + 31 \, B^{2} d^{3}}{6 \, {\left (b^{3} c^{3} g^{3} - 3 \, a b^{2} c^{2} d g^{3} + 3 \, a^{2} b c d^{2} g^{3} - a^{3} d^{3} g^{3}\right )} {\left (b g x + a g\right )} b g} - \frac {6 \, A B b d^{2} + 19 \, B^{2} b d^{2}}{12 \, {\left (b^{2} c^{2} g - 2 \, a b c d g + a^{2} d^{2} g\right )} {\left (b g x + a g\right )}^{2} b^{2} g^{2}} + \frac {6 \, A B b^{2} d g + 13 \, B^{2} b^{2} d g}{18 \, {\left (b g x + a g\right )}^{3} {\left (b c - a d\right )} b^{3} g^{3}} - \frac {2 \, A^{2} b^{3} g^{3} + 6 \, A B b^{3} g^{3} + 5 \, B^{2} b^{3} g^{3}}{8 \, {\left (b g x + a g\right )}^{4} b^{4} g^{4}} \end {gather*}

integrate((A+B*log(e*(b*x+a)^2/(d*x+c)^2))^2/(b*g*x+a*g)^5,x, algorithm="giac")

1/4*(B^2*d^4/(b^5*c^4*g^5 - 4*a*b^4*c^3*d*g^5 + 6*a^2*b^3*c^2*d^2*g^5 - 4*a^3*b^2*c*d^3*g^5 + a^4*b*d^4*g^5) -

B^2/((b*g*x + a*g)^4*b*g))*log(b^2/(b^2*c^2*g^2/(b*g*x + a*g)^2 - 2*a*b*c*d*g^2/(b*g*x + a*g)^2 + a^2*d^2*g^2

/(b*g*x + a*g)^2 + 2*b*c*d*g/(b*g*x + a*g) - 2*a*d^2*g/(b*g*x + a*g) + d^2))^2 + 1/12*(12*B^2*d^3/((b^3*c^3*g^

3 - 3*a*b^2*c^2*d*g^3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^3)*(b*g*x + a*g)*b*g) - 6*B^2*d^2/((b^2*c^2*g - 2*a*b*c*

d*g + a^2*d^2*g)*(b*g*x + a*g)^2*b*g^2) + 4*B^2*d/((b*g*x + a*g)^3*(b*c - a*d)*b*g^2) - 3*(2*A*B*b^3*g^3 + 3*B

^2*b^3*g^3)/((b*g*x + a*g)^4*b^4*g^4))*log(b^2/(b^2*c^2*g^2/(b*g*x + a*g)^2 - 2*a*b*c*d*g^2/(b*g*x + a*g)^2 +

a^2*d^2*g^2/(b*g*x + a*g)^2 + 2*b*c*d*g/(b*g*x + a*g) - 2*a*d^2*g/(b*g*x + a*g) + d^2)) - 1/6*(6*A*B*d^4 + 31*

B^2*d^4)*log(-b*c*g/(b*g*x + a*g) + a*d*g/(b*g*x + a*g) - d)/(b^5*c^4*g^5 - 4*a*b^4*c^3*d*g^5 + 6*a^2*b^3*c^2*

d^2*g^5 - 4*a^3*b^2*c*d^3*g^5 + a^4*b*d^4*g^5) + 1/6*(6*A*B*d^3 + 31*B^2*d^3)/((b^3*c^3*g^3 - 3*a*b^2*c^2*d*g^

3 + 3*a^2*b*c*d^2*g^3 - a^3*d^3*g^3)*(b*g*x + a*g)*b*g) - 1/12*(6*A*B*b*d^2 + 19*B^2*b*d^2)/((b^2*c^2*g - 2*a*

b*c*d*g + a^2*d^2*g)*(b*g*x + a*g)^2*b^2*g^2) + 1/18*(6*A*B*b^2*d*g + 13*B^2*b^2*d*g)/((b*g*x + a*g)^3*(b*c -

a*d)*b^3*g^3) - 1/8*(2*A^2*b^3*g^3 + 6*A*B*b^3*g^3 + 5*B^2*b^3*g^3)/((b*g*x + a*g)^4*b^4*g^4)

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Mupad [B]
time = 10.55, size = 1883, normalized size = 3.21 \begin {gather*} -\frac {\frac {18\,A^2\,a^3\,d^3-54\,A^2\,a^2\,b\,c\,d^2+54\,A^2\,a\,b^2\,c^2\,d-18\,A^2\,b^3\,c^3+150\,A\,B\,a^3\,d^3-138\,A\,B\,a^2\,b\,c\,d^2+78\,A\,B\,a\,b^2\,c^2\,d-18\,A\,B\,b^3\,c^3+415\,B^2\,a^3\,d^3-161\,B^2\,a^2\,b\,c\,d^2+55\,B^2\,a\,b^2\,c^2\,d-9\,B^2\,b^3\,c^3}{12\,\left (a\,d-b\,c\right )}+\frac {x^2\,\left (-13\,c\,B^2\,b^3\,d^2+163\,a\,B^2\,b^2\,d^3-6\,A\,c\,B\,b^3\,d^2+42\,A\,a\,B\,b^2\,d^3\right )}{2\,\left (a\,d-b\,c\right )}+\frac {x\,\left (271\,B^2\,a^2\,b\,d^3-53\,B^2\,a\,b^2\,c\,d^2+7\,B^2\,b^3\,c^2\,d+78\,A\,B\,a^2\,b\,d^3-30\,A\,B\,a\,b^2\,c\,d^2+6\,A\,B\,b^3\,c^2\,d\right )}{3\,\left (a\,d-b\,c\right )}+\frac {d\,x^3\,\left (25\,B^2\,b^3\,d^2+6\,A\,B\,b^3\,d^2\right )}{a\,d-b\,c}}{x\,\left (24\,a^5\,b^2\,d^2\,g^5-48\,a^4\,b^3\,c\,d\,g^5+24\,a^3\,b^4\,c^2\,g^5\right )+x^3\,\left (24\,a^3\,b^4\,d^2\,g^5-48\,a^2\,b^5\,c\,d\,g^5+24\,a\,b^6\,c^2\,g^5\right )+x^4\,\left (6\,a^2\,b^5\,d^2\,g^5-12\,a\,b^6\,c\,d\,g^5+6\,b^7\,c^2\,g^5\right )+x^2\,\left (36\,a^4\,b^3\,d^2\,g^5-72\,a^3\,b^4\,c\,d\,g^5+36\,a^2\,b^5\,c^2\,g^5\right )+6\,a^6\,b\,d^2\,g^5+6\,a^4\,b^3\,c^2\,g^5-12\,a^5\,b^2\,c\,d\,g^5}-{\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )}^2\,\left (\frac {B^2}{4\,b^2\,g^5\,\left (4\,a^3\,x+\frac {a^4}{b}+b^3\,x^4+6\,a^2\,b\,x^2+4\,a\,b^2\,x^3\right )}-\frac {B^2\,d^4}{4\,b\,g^5\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}\right )-\frac {\ln \left (\frac {e\,{\left (a+b\,x\right )}^2}{{\left (c+d\,x\right )}^2}\right )\,\left (\frac {A\,B}{2\,b^2\,d\,g^5}+\frac {B^2\,d^4\,\left (a\,\left (a\,\left (\frac {4\,a^2\,d^2-5\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,b\,d^2}\right )+\frac {6\,a^3\,d^3-10\,a^2\,b\,c\,d^2+5\,a\,b^2\,c^2\,d-b^3\,c^3}{6\,b\,d^4}\right )+\frac {4\,a^4\,d^4-10\,a^3\,b\,c\,d^3+10\,a^2\,b^2\,c^2\,d^2-5\,a\,b^3\,c^3\,d+b^4\,c^4}{2\,b\,d^5}\right )}{2\,b\,g^5\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {B^2\,d^4\,x^2\,\left (b\,\left (b\,\left (\frac {4\,a^2\,d^2-5\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,b\,d^2}\right )+\frac {4\,a^2\,d^2-5\,a\,b\,c\,d+b^2\,c^2}{3\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{d^2}\right )-a\,\left (\frac {b^2\,c-a\,b\,d}{2\,d^2}-\frac {b\,\left (a\,d-b\,c\right )}{d^2}\right )+\frac {4\,a^2\,b\,d^2-5\,a\,b^2\,c\,d+b^3\,c^2}{2\,d^3}\right )}{2\,b\,g^5\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}-\frac {B^2\,d^4\,x^3\,\left (b\,\left (\frac {b^2\,c-a\,b\,d}{2\,d^2}-\frac {b\,\left (a\,d-b\,c\right )}{d^2}\right )+\frac {b^3\,c-a\,b^2\,d}{2\,d^2}\right )}{2\,b\,g^5\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}+\frac {B^2\,d^4\,x\,\left (b\,\left (a\,\left (\frac {4\,a^2\,d^2-5\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,b\,d^2}\right )+\frac {6\,a^3\,d^3-10\,a^2\,b\,c\,d^2+5\,a\,b^2\,c^2\,d-b^3\,c^3}{6\,b\,d^4}\right )+a\,\left (b\,\left (\frac {4\,a^2\,d^2-5\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,b\,d^2}\right )+\frac {4\,a^2\,d^2-5\,a\,b\,c\,d+b^2\,c^2}{3\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{d^2}\right )+\frac {6\,a^3\,d^3-10\,a^2\,b\,c\,d^2+5\,a\,b^2\,c^2\,d-b^3\,c^3}{2\,d^4}\right )}{2\,b\,g^5\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}\right )}{\frac {4\,a^3\,x}{d}+\frac {a^4}{b\,d}+\frac {b^3\,x^4}{d}+\frac {6\,a^2\,b\,x^2}{d}+\frac {4\,a\,b^2\,x^3}{d}}+\frac {B\,d^4\,\mathrm {atan}\left (\frac {B\,d^4\,\left (6\,A+25\,B\right )\,\left (-6\,a^4\,b\,d^4\,g^5+12\,a^3\,b^2\,c\,d^3\,g^5-12\,a\,b^4\,c^3\,d\,g^5+6\,b^5\,c^4\,g^5\right )\,1{}\mathrm {i}}{6\,b\,g^5\,{\left (a\,d-b\,c\right )}^4\,\left (25\,B^2\,d^4+6\,A\,B\,d^4\right )}+\frac {B\,d^5\,x\,\left (6\,A+25\,B\right )\,\left (-a^3\,b\,d^3\,g^5+3\,a^2\,b^2\,c\,d^2\,g^5-3\,a\,b^3\,c^2\,d\,g^5+b^4\,c^3\,g^5\right )\,2{}\mathrm {i}}{g^5\,{\left (a\,d-b\,c\right )}^4\,\left (25\,B^2\,d^4+6\,A\,B\,d^4\right )}\right )\,\left (6\,A+25\,B\right )\,1{}\mathrm {i}}{3\,b\,g^5\,{\left (a\,d-b\,c\right )}^4} \end {gather*}

int((A + B*log((e*(a + b*x)^2)/(c + d*x)^2))^2/(a*g + b*g*x)^5,x)

(B*d^4*atan((B*d^4*(6*A + 25*B)*(6*b^5*c^4*g^5 - 6*a^4*b*d^4*g^5 - 12*a*b^4*c^3*d*g^5 + 12*a^3*b^2*c*d^3*g^5)*

1i)/(6*b*g^5*(a*d - b*c)^4*(25*B^2*d^4 + 6*A*B*d^4)) + (B*d^5*x*(6*A + 25*B)*(b^4*c^3*g^5 - a^3*b*d^3*g^5 - 3*

a*b^3*c^2*d*g^5 + 3*a^2*b^2*c*d^2*g^5)*2i)/(g^5*(a*d - b*c)^4*(25*B^2*d^4 + 6*A*B*d^4)))*(6*A + 25*B)*1i)/(3*b

*g^5*(a*d - b*c)^4) - log((e*(a + b*x)^2)/(c + d*x)^2)^2*(B^2/(4*b^2*g^5*(4*a^3*x + a^4/b + b^3*x^4 + 6*a^2*b*

x^2 + 4*a*b^2*x^3)) - (B^2*d^4)/(4*b*g^5*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^

3))) - (log((e*(a + b*x)^2)/(c + d*x)^2)*((A*B)/(2*b^2*d*g^5) + (B^2*d^4*(a*(a*((4*a^2*d^2 + b^2*c^2 - 5*a*b*c

*d)/(6*b*d^3) + (a*(a*d - b*c))/(2*b*d^2)) + (6*a^3*d^3 - b^3*c^3 + 5*a*b^2*c^2*d - 10*a^2*b*c*d^2)/(6*b*d^4))

+ (4*a^4*d^4 + b^4*c^4 + 10*a^2*b^2*c^2*d^2 - 5*a*b^3*c^3*d - 10*a^3*b*c*d^3)/(2*b*d^5)))/(2*b*g^5*(a^4*d^4 +

b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (B^2*d^4*x^2*(b*(b*((4*a^2*d^2 + b^2*c^2 - 5*

a*b*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(2*b*d^2)) + (4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(3*d^3) + (a*(a*d - b*c))/

d^2) - a*((b^2*c - a*b*d)/(2*d^2) - (b*(a*d - b*c))/d^2) + (b^3*c^2 + 4*a^2*b*d^2 - 5*a*b^2*c*d)/(2*d^3)))/(2*

b*g^5*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) - (B^2*d^4*x^3*(b*((b^2*c - a*b

*d)/(2*d^2) - (b*(a*d - b*c))/d^2) + (b^3*c - a*b^2*d)/(2*d^2)))/(2*b*g^5*(a^4*d^4 + b^4*c^4 + 6*a^2*b^2*c^2*d

^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3)) + (B^2*d^4*x*(b*(a*((4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(6*b*d^3) + (a*(a*d

- b*c))/(2*b*d^2)) + (6*a^3*d^3 - b^3*c^3 + 5*a*b^2*c^2*d - 10*a^2*b*c*d^2)/(6*b*d^4)) + a*(b*((4*a^2*d^2 + b

^2*c^2 - 5*a*b*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(2*b*d^2)) + (4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(3*d^3) + (a*(a

*d - b*c))/d^2) + (6*a^3*d^3 - b^3*c^3 + 5*a*b^2*c^2*d - 10*a^2*b*c*d^2)/(2*d^4)))/(2*b*g^5*(a^4*d^4 + b^4*c^4

+ 6*a^2*b^2*c^2*d^2 - 4*a*b^3*c^3*d - 4*a^3*b*c*d^3))))/((4*a^3*x)/d + a^4/(b*d) + (b^3*x^4)/d + (6*a^2*b*x^2

)/d + (4*a*b^2*x^3)/d) - ((18*A^2*a^3*d^3 - 18*A^2*b^3*c^3 + 415*B^2*a^3*d^3 - 9*B^2*b^3*c^3 + 150*A*B*a^3*d^3

- 18*A*B*b^3*c^3 + 54*A^2*a*b^2*c^2*d - 54*A^2*a^2*b*c*d^2 + 55*B^2*a*b^2*c^2*d - 161*B^2*a^2*b*c*d^2 + 78*A*

B*a*b^2*c^2*d - 138*A*B*a^2*b*c*d^2)/(12*(a*d - b*c)) + (x^2*(163*B^2*a*b^2*d^3 - 13*B^2*b^3*c*d^2 + 42*A*B*a*

b^2*d^3 - 6*A*B*b^3*c*d^2))/(2*(a*d - b*c)) + (x*(271*B^2*a^2*b*d^3 + 7*B^2*b^3*c^2*d - 53*B^2*a*b^2*c*d^2 + 7

8*A*B*a^2*b*d^3 + 6*A*B*b^3*c^2*d - 30*A*B*a*b^2*c*d^2))/(3*(a*d - b*c)) + (d*x^3*(25*B^2*b^3*d^2 + 6*A*B*b^3*

d^2))/(a*d - b*c))/(x*(24*a^3*b^4*c^2*g^5 + 24*a^5*b^2*d^2*g^5 - 48*a^4*b^3*c*d*g^5) + x^3*(24*a*b^6*c^2*g^5 +

24*a^3*b^4*d^2*g^5 - 48*a^2*b^5*c*d*g^5) + x^4*(6*b^7*c^2*g^5 + 6*a^2*b^5*d^2*g^5 - 12*a*b^6*c*d*g^5) + x^2*(

36*a^2*b^5*c^2*g^5 + 36*a^4*b^3*d^2*g^5 - 72*a^3*b^4*c*d*g^5) + 6*a^6*b*d^2*g^5 + 6*a^4*b^3*c^2*g^5 - 12*a^5*b

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3.2.36 \(\int \frac {(A+B \log (\frac {e (a+b x)^2}{(c+d x)^2}))^2}{(a g+b g x)^5} \, dx\) [136]