3.2.5 \(\int \frac {(A+B \log (\frac {e (a+b x)}{c+d x}))^2}{(a g+b g x)^5} \, dx\) [105]

Optimal. Leaf size=575 \[ \frac {2 B^2 d^3 (c+d x)}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B^2 d^2 (c+d x)^2}{4 (b c-a d)^4 g^5 (a+b x)^2}+\frac {2 b^2 B^2 d (c+d x)^3}{9 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B^2 (c+d x)^4}{32 (b c-a d)^4 g^5 (a+b x)^4}+\frac {2 B d^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B d^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^4 g^5 (a+b x)^2}+\frac {2 b^2 B d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 (b c-a d)^4 g^5 (a+b x)^4}+\frac {d^3 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b d^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 (b c-a d)^4 g^5 (a+b x)^2}+\frac {b^2 d (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 (c+d x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 (b c-a d)^4 g^5 (a+b x)^4} \]

[Out]

2*B^2*d^3*(d*x+c)/(-a*d+b*c)^4/g^5/(b*x+a)-3/4*b*B^2*d^2*(d*x+c)^2/(-a*d+b*c)^4/g^5/(b*x+a)^2+2/9*b^2*B^2*d*(d
*x+c)^3/(-a*d+b*c)^4/g^5/(b*x+a)^3-1/32*b^3*B^2*(d*x+c)^4/(-a*d+b*c)^4/g^5/(b*x+a)^4+2*B*d^3*(d*x+c)*(A+B*ln(e
*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^5/(b*x+a)-3/2*b*B*d^2*(d*x+c)^2*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^5
/(b*x+a)^2+2/3*b^2*B*d*(d*x+c)^3*(A+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^5/(b*x+a)^3-1/8*b^3*B*(d*x+c)^4*(A
+B*ln(e*(b*x+a)/(d*x+c)))/(-a*d+b*c)^4/g^5/(b*x+a)^4+d^3*(d*x+c)*(A+B*ln(e*(b*x+a)/(d*x+c)))^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)/(d*x+c)))^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)/(d*x+c)))^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)/(d*x+c)))^2/(-a*d+b*c)^4
/g^5/(b*x+a)^4

________________________________________________________________________________________

Rubi [A]
time = 0.27, antiderivative size = 575, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2550, 2395, 2342, 2341} \begin {gather*} -\frac {b^3 (c+d x)^4 \left (B \log \left (\frac {e (a+b x)}{c+d x}\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)}{c+d x}\right )+A\right )}{8 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)}{c+d x}\right )+A\right )^2}{g^5 (a+b x)^3 (b c-a d)^4}+\frac {2 b^2 B d (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\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)}{c+d x}\right )+A\right )^2}{g^5 (a+b x) (b c-a d)^4}+\frac {2 B d^3 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\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)}{c+d x}\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)}{c+d x}\right )+A\right )}{2 g^5 (a+b x)^2 (b c-a d)^4}-\frac {b^3 B^2 (c+d x)^4}{32 g^5 (a+b x)^4 (b c-a d)^4}+\frac {2 b^2 B^2 d (c+d x)^3}{9 g^5 (a+b x)^3 (b c-a d)^4}+\frac {2 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}{4 g^5 (a+b x)^2 (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*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)/(4*(b*c - a*d)^4*g^5*(a + b*x)
^2) + (2*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)/(32*(b*c - a*d)^4*g^
5*(a + b*x)^4) + (2*B*d^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((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))/(c + d*x)]))/(2*(b*c - a*d)^4*g^5*(a + b*x)^2) + (2*b^2*B*d*(c + d
*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(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))/(c + d*x)]))/(8*(b*c - a*d)^4*g^5*(a + b*x)^4) + (d^3*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*
x)])^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))/(c + d*x)])^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))/(c + d*x)])^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))/(c + d*x)])^2)/(4*(b*c - a*d)^4*g^5*(a + b*x)^4)

Rule 2341

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
]

Rule 2342

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[{
a, b, c, d, m, n}, x] && NeQ[m, -1] && GtQ[p, 0]

Rule 2395

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
]))

Rule 2550

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])

Rubi steps

\begin {align*} \int \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{(a g+b g x)^5} \, dx &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac {B \int \frac {(b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\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)}{c+d x}\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)}{c+d x}\right )}{(a+b x)^5 (c+d x)} \, dx}{2 b g^5}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\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)}{c+d x}\right )\right )}{(b c-a d) (a+b x)^5}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\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)}{c+d x}\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)}{c+d x}\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)}{c+d x}\right )\right )}{(b c-a d)^5 (a+b x)}-\frac {d^5 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 (c+d x)}\right ) \, dx}{2 b g^5}\\ &=-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac {B \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^5} \, dx}{2 g^5}+\frac {\left (B d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{2 (b c-a d)^4 g^5}-\frac {\left (B d^5\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{2 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)}{c+d x}\right )}{(a+b x)^2} \, dx}{2 (b c-a d)^3 g^5}+\frac {\left (B d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{2 (b c-a d)^2 g^5}-\frac {(B d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{2 (b c-a d) g^5}\\ &=-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 b g^5 (a+b x)^4}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 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)}{c+d x}\right )\right )}{4 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)}{c+d x}\right )\right )}{2 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)}{c+d x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\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)}{c+d x}\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}+\frac {B^2 \int \frac {b c-a d}{(a+b x)^5 (c+d x)} \, dx}{8 b g^5}-\frac {\left (B^2 d^4\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{e (a+b x)} \, dx}{2 b (b c-a d)^4 g^5}+\frac {\left (B^2 d^4\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{e (a+b x)} \, dx}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^3\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{2 b (b c-a d)^3 g^5}+\frac {\left (B^2 d^2\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{4 b (b c-a d)^2 g^5}-\frac {\left (B^2 d\right ) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{6 b (b c-a d) g^5}\\ &=-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 b g^5 (a+b x)^4}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 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)}{c+d x}\right )\right )}{4 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)}{c+d x}\right )\right )}{2 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)}{c+d x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\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)}{c+d x}\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{6 b g^5}-\frac {\left (B^2 d^3\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{2 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}{4 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}{8 b g^5}-\frac {\left (B^2 d^4\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (a+b x)}{a+b x} \, dx}{2 b (b c-a d)^4 e g^5}+\frac {\left (B^2 d^4\right ) \int \frac {(c+d x) \left (-\frac {d e (a+b x)}{(c+d x)^2}+\frac {b e}{c+d x}\right ) \log (c+d x)}{a+b x} \, dx}{2 b (b c-a d)^4 e g^5}\\ &=-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 b g^5 (a+b x)^4}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 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)}{c+d x}\right )\right )}{4 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)}{c+d x}\right )\right )}{2 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)}{c+d x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\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)}{c+d x}\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {\left (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}{6 b g^5}-\frac {\left (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}{2 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}{4 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}{8 b g^5}-\frac {\left (B^2 d^4\right ) \int \left (\frac {b e \log (a+b x)}{a+b x}-\frac {d e \log (a+b x)}{c+d x}\right ) \, dx}{2 b (b c-a d)^4 e g^5}+\frac {\left (B^2 d^4\right ) \int \left (\frac {b e \log (c+d x)}{a+b x}-\frac {d e \log (c+d x)}{c+d x}\right ) \, dx}{2 b (b c-a d)^4 e g^5}\\ &=-\frac {B^2}{32 b g^5 (a+b x)^4}+\frac {7 B^2 d}{72 b (b c-a d) g^5 (a+b x)^3}-\frac {13 B^2 d^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {25 B^2 d^3}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac {25 B^2 d^4 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 b g^5 (a+b x)^4}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 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)}{c+d x}\right )\right )}{4 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)}{c+d x}\right )\right )}{2 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)}{c+d x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {25 B^2 d^4 \log (c+d x)}{24 b (b c-a d)^4 g^5}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^4\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{2 (b c-a d)^4 g^5}+\frac {\left (B^2 d^4\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{2 (b c-a d)^4 g^5}+\frac {\left (B^2 d^5\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^5\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}\\ &=-\frac {B^2}{32 b g^5 (a+b x)^4}+\frac {7 B^2 d}{72 b (b c-a d) g^5 (a+b x)^3}-\frac {13 B^2 d^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {25 B^2 d^3}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac {25 B^2 d^4 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 b g^5 (a+b x)^4}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 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)}{c+d x}\right )\right )}{4 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)}{c+d x}\right )\right )}{2 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)}{c+d x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {25 B^2 d^4 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac {B^2 d^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}+\frac {B^2 d^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^4\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{2 (b c-a d)^4 g^5}-\frac {\left (B^2 d^4\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^4\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^5\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{2 b (b c-a d)^4 g^5}\\ &=-\frac {B^2}{32 b g^5 (a+b x)^4}+\frac {7 B^2 d}{72 b (b c-a d) g^5 (a+b x)^3}-\frac {13 B^2 d^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {25 B^2 d^3}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac {25 B^2 d^4 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac {B^2 d^4 \log ^2(a+b x)}{4 b (b c-a d)^4 g^5}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 b g^5 (a+b x)^4}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 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)}{c+d x}\right )\right )}{4 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)}{c+d x}\right )\right )}{2 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)}{c+d x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {25 B^2 d^4 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac {B^2 d^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {B^2 d^4 \log ^2(c+d x)}{4 b (b c-a d)^4 g^5}+\frac {B^2 d^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (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 )}{2 b (b c-a d)^4 g^5}-\frac {\left (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 )}{2 b (b c-a d)^4 g^5}\\ &=-\frac {B^2}{32 b g^5 (a+b x)^4}+\frac {7 B^2 d}{72 b (b c-a d) g^5 (a+b x)^3}-\frac {13 B^2 d^2}{48 b (b c-a d)^2 g^5 (a+b x)^2}+\frac {25 B^2 d^3}{24 b (b c-a d)^3 g^5 (a+b x)}+\frac {25 B^2 d^4 \log (a+b x)}{24 b (b c-a d)^4 g^5}-\frac {B^2 d^4 \log ^2(a+b x)}{4 b (b c-a d)^4 g^5}-\frac {B \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{8 b g^5 (a+b x)^4}+\frac {B d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6 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)}{c+d x}\right )\right )}{4 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)}{c+d x}\right )\right )}{2 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)}{c+d x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\frac {25 B^2 d^4 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac {B^2 d^4 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {B d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{2 b (b c-a d)^4 g^5}-\frac {B^2 d^4 \log ^2(c+d x)}{4 b (b c-a d)^4 g^5}+\frac {B^2 d^4 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}+\frac {B^2 d^4 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{2 b (b c-a d)^4 g^5}+\frac {B^2 d^4 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{2 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 = 0.63, size = 748, normalized size = 1.30 \begin {gather*} -\frac {72 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2+\frac {B \left (36 (b c-a d)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+48 d (-b c+a d)^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+72 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+144 d^3 (-b c+a d) (a+b x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-144 d^4 (a+b x)^4 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+144 d^4 (a+b x)^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\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}}{288 b g^5 (a+b x)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/288*(72*(A + B*Log[(e*(a + b*x))/(c + d*x)])^2 + (B*(36*(b*c - a*d)^4*(A + B*Log[(e*(a + b*x))/(c + d*x)])
+ 48*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 72*d^2*(b*c - a*d)^2*(a + b*x)^2*(A +
B*Log[(e*(a + b*x))/(c + d*x)]) + 144*d^3*(-(b*c) + a*d)*(a + b*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 1
44*d^4*(a + b*x)^4*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 144*d^4*(a + b*x)^4*(A + B*Log[(e*(a +
b*x))/(c + d*x)])*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 - 1
2*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. \(1392\) vs. \(2(559)=1118\).
time = 0.44, size = 1393, normalized size = 2.42

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1393\)
default	\(\text {Expression too large to display}\)	\(1393\)
norman	\(\text {Expression too large to display}\)	\(1796\)
risch	\(\text {Expression too large to display}\)	\(3080\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/d^2*e*(a*d-b*c)*(1/4*d^2/(a*d-b*c)^5/g^5*A^2*b^3*e^3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4-d^3/(a*d-b*c)^5/g^5*A^
2*b^2*e^2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3+3/2*d^4/(a*d-b*c)^5/g^5*A^2*b*e/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-d^5/
(a*d-b*c)^5/g^5*A^2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))-2*d^2/(a*d-b*c)^5/g^5*A*B*b^3*e^3*(-1/4/(b*e/d+(a*d-b*c)*e/d
/(d*x+c))^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/16/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4)+6*d^3/(a*d-b*c)^5/g^5*A*B*b^
2*e^2*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/9/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^
3)-6*d^4/(a*d-b*c)^5/g^5*A*B*b*e*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b*
e/d+(a*d-b*c)*e/d/(d*x+c))^2)+2*d^5/(a*d-b*c)^5/g^5*A*B*(-1/(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e
/d/(d*x+c))-1/(b*e/d+(a*d-b*c)*e/d/(d*x+c)))-d^2/(a*d-b*c)^5/g^5*B^2*b^3*e^3*(-1/4/(b*e/d+(a*d-b*c)*e/d/(d*x+c
))^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-1/8/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/3
2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^4)+3*d^3/(a*d-b*c)^5/g^5*B^2*b^2*e^2*(-1/3/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(
b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-2/9/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^3*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-2/27/(b*e/d
+(a*d-b*c)*e/d/(d*x+c))^3)-3*d^4/(a*d-b*c)^5/g^5*B^2*b*e*(-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+(a*d-b
*c)*e/d/(d*x+c))^2-1/2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-1/4/(b*e/d+(a*d-b*c)*e/
d/(d*x+c))^2)+d^5/(a*d-b*c)^5/g^5*B^2*(-1/(b*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))^2-2/(b
*e/d+(a*d-b*c)*e/d/(d*x+c))*ln(b*e/d+(a*d-b*c)*e/d/(d*x+c))-2/(b*e/d+(a*d-b*c)*e/d/(d*x+c))))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2129 vs. \(2 (567) = 1134\).
time = 0.51, size = 2129, normalized size = 3.70 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/288*(12*((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*x*e/(d*x + c) + a*e/(d*x
+ c)) - (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 - 2
71*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 + 1/24*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) - 12*log(b*x*e/(d*x + c) + a*e/(d*x +
c))/(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*x*e/(d*x + c) + a*e
/(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) - 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 = 0.37, size = 1033, normalized size = 1.80 \begin {gather*} -\frac {9 \, {\left (8 \, A^{2} + 4 \, A B + B^{2}\right )} b^{4} c^{4} - 32 \, {\left (9 \, A^{2} + 6 \, A B + 2 \, B^{2}\right )} a b^{3} c^{3} d + 216 \, {\left (2 \, A^{2} + 2 \, A B + B^{2}\right )} a^{2} b^{2} c^{2} d^{2} - 288 \, {\left (A^{2} + 2 \, A B + 2 \, B^{2}\right )} a^{3} b c d^{3} + {\left (72 \, A^{2} + 300 \, A B + 415 \, B^{2}\right )} a^{4} d^{4} - 12 \, {\left ({\left (12 \, A B + 25 \, B^{2}\right )} b^{4} c d^{3} - {\left (12 \, A B + 25 \, B^{2}\right )} a b^{3} d^{4}\right )} x^{3} + 6 \, {\left ({\left (12 \, A B + 13 \, B^{2}\right )} b^{4} c^{2} d^{2} - 16 \, {\left (6 \, A B + 11 \, B^{2}\right )} a b^{3} c d^{3} + {\left (84 \, A B + 163 \, B^{2}\right )} a^{2} b^{2} d^{4}\right )} x^{2} - 72 \, {\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 x + a\right )} e}{d x + c}\right )^{2} - 4 \, {\left ({\left (12 \, A B + 7 \, B^{2}\right )} b^{4} c^{3} d - 12 \, {\left (6 \, A B + 5 \, B^{2}\right )} a b^{3} c^{2} d^{2} + 108 \, {\left (2 \, A B + 3 \, B^{2}\right )} a^{2} b^{2} c d^{3} - {\left (156 \, A B + 271 \, B^{2}\right )} a^{3} b d^{4}\right )} x - 12 \, {\left ({\left (12 \, A B + 25 \, B^{2}\right )} b^{4} d^{4} x^{4} - 3 \, {\left (4 \, A B + B^{2}\right )} b^{4} c^{4} + 16 \, {\left (3 \, A B + B^{2}\right )} a b^{3} c^{3} d - 36 \, {\left (2 \, A B + B^{2}\right )} a^{2} b^{2} c^{2} d^{2} + 48 \, {\left (A B + B^{2}\right )} a^{3} b c d^{3} + 4 \, {\left (3 \, B^{2} b^{4} c d^{3} + 2 \, {\left (6 \, 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 (2 \, 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} + 12 \, {\left (A B + B^{2}\right )} a^{3} b d^{4}\right )} x\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{288 \, {\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*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/288*(9*(8*A^2 + 4*A*B + B^2)*b^4*c^4 - 32*(9*A^2 + 6*A*B + 2*B^2)*a*b^3*c^3*d + 216*(2*A^2 + 2*A*B + B^2)*a
^2*b^2*c^2*d^2 - 288*(A^2 + 2*A*B + 2*B^2)*a^3*b*c*d^3 + (72*A^2 + 300*A*B + 415*B^2)*a^4*d^4 - 12*((12*A*B +
25*B^2)*b^4*c*d^3 - (12*A*B + 25*B^2)*a*b^3*d^4)*x^3 + 6*((12*A*B + 13*B^2)*b^4*c^2*d^2 - 16*(6*A*B + 11*B^2)*
a*b^3*c*d^3 + (84*A*B + 163*B^2)*a^2*b^2*d^4)*x^2 - 72*(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*x + a)*e/(d*x + c))^2 - 4*((12*A*B + 7*B^2)*b^4*c^3*d - 12*(6*A*B + 5*B^2)*a*b^3*c^2*d^2 + 108*(2*A*B + 3*
B^2)*a^2*b^2*c*d^3 - (156*A*B + 271*B^2)*a^3*b*d^4)*x - 12*((12*A*B + 25*B^2)*b^4*d^4*x^4 - 3*(4*A*B + B^2)*b^
4*c^4 + 16*(3*A*B + B^2)*a*b^3*c^3*d - 36*(2*A*B + B^2)*a^2*b^2*c^2*d^2 + 48*(A*B + B^2)*a^3*b*c*d^3 + 4*(3*B^
2*b^4*c*d^3 + 2*(6*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*(2*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 + 12*(A*B + B^2)*a^3*b*d^4)*x)
*log((b*x + a)*e/(d*x + c)))/((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*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 3.64, size = 995, normalized size = 1.73 \begin {gather*} -\frac {{\left (72 \, B^{2} b^{3} e^{5} \log \left (\frac {b x e + a e}{d x + c}\right )^{2} - \frac {288 \, {\left (b x e + a e\right )} B^{2} b^{2} d e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{d x + c} + \frac {432 \, {\left (b x e + a e\right )}^{2} B^{2} b d^{2} e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{{\left (d x + c\right )}^{2}} - \frac {288 \, {\left (b x e + a e\right )}^{3} B^{2} d^{3} e^{2} \log \left (\frac {b x e + a e}{d x + c}\right )^{2}}{{\left (d x + c\right )}^{3}} + 144 \, A B b^{3} e^{5} \log \left (\frac {b x e + a e}{d x + c}\right ) + 36 \, B^{2} b^{3} e^{5} \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {576 \, {\left (b x e + a e\right )} A B b^{2} d e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} - \frac {192 \, {\left (b x e + a e\right )} B^{2} b^{2} d e^{4} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + \frac {864 \, {\left (b x e + a e\right )}^{2} A B b d^{2} e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + \frac {432 \, {\left (b x e + a e\right )}^{2} B^{2} b d^{2} e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} - \frac {576 \, {\left (b x e + a e\right )}^{3} A B d^{3} e^{2} \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{3}} - \frac {576 \, {\left (b x e + a e\right )}^{3} B^{2} d^{3} e^{2} \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{3}} + 72 \, A^{2} b^{3} e^{5} + 36 \, A B b^{3} e^{5} + 9 \, B^{2} b^{3} e^{5} - \frac {288 \, {\left (b x e + a e\right )} A^{2} b^{2} d e^{4}}{d x + c} - \frac {192 \, {\left (b x e + a e\right )} A B b^{2} d e^{4}}{d x + c} - \frac {64 \, {\left (b x e + a e\right )} B^{2} b^{2} d e^{4}}{d x + c} + \frac {432 \, {\left (b x e + a e\right )}^{2} A^{2} b d^{2} e^{3}}{{\left (d x + c\right )}^{2}} + \frac {432 \, {\left (b x e + a e\right )}^{2} A B b d^{2} e^{3}}{{\left (d x + c\right )}^{2}} + \frac {216 \, {\left (b x e + a e\right )}^{2} B^{2} b d^{2} e^{3}}{{\left (d x + c\right )}^{2}} - \frac {288 \, {\left (b x e + a e\right )}^{3} A^{2} d^{3} e^{2}}{{\left (d x + c\right )}^{3}} - \frac {576 \, {\left (b x e + a e\right )}^{3} A B d^{3} e^{2}}{{\left (d x + c\right )}^{3}} - \frac {576 \, {\left (b x e + a e\right )}^{3} B^{2} d^{3} e^{2}}{{\left (d x + c\right )}^{3}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}}{288 \, {\left (\frac {{\left (b x e + a e\right )}^{4} b^{3} c^{3} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {3 \, {\left (b x e + a e\right )}^{4} a b^{2} c^{2} d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {3 \, {\left (b x e + a e\right )}^{4} a^{2} b c d^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b x e + a e\right )}^{4} a^{3} d^{3} g^{5}}{{\left (d x + c\right )}^{4}}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/288*(72*B^2*b^3*e^5*log((b*x*e + a*e)/(d*x + c))^2 - 288*(b*x*e + a*e)*B^2*b^2*d*e^4*log((b*x*e + a*e)/(d*x
+ c))^2/(d*x + c) + 432*(b*x*e + a*e)^2*B^2*b*d^2*e^3*log((b*x*e + a*e)/(d*x + c))^2/(d*x + c)^2 - 288*(b*x*e
+ a*e)^3*B^2*d^3*e^2*log((b*x*e + a*e)/(d*x + c))^2/(d*x + c)^3 + 144*A*B*b^3*e^5*log((b*x*e + a*e)/(d*x + c)
) + 36*B^2*b^3*e^5*log((b*x*e + a*e)/(d*x + c)) - 576*(b*x*e + a*e)*A*B*b^2*d*e^4*log((b*x*e + a*e)/(d*x + c))
/(d*x + c) - 192*(b*x*e + a*e)*B^2*b^2*d*e^4*log((b*x*e + a*e)/(d*x + c))/(d*x + c) + 864*(b*x*e + a*e)^2*A*B*
b*d^2*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 432*(b*x*e + a*e)^2*B^2*b*d^2*e^3*log((b*x*e + a*e)/(d*x
+ c))/(d*x + c)^2 - 576*(b*x*e + a*e)^3*A*B*d^3*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 - 576*(b*x*e + a*
e)^3*B^2*d^3*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^3 + 72*A^2*b^3*e^5 + 36*A*B*b^3*e^5 + 9*B^2*b^3*e^5 -
288*(b*x*e + a*e)*A^2*b^2*d*e^4/(d*x + c) - 192*(b*x*e + a*e)*A*B*b^2*d*e^4/(d*x + c) - 64*(b*x*e + a*e)*B^2*b
^2*d*e^4/(d*x + c) + 432*(b*x*e + a*e)^2*A^2*b*d^2*e^3/(d*x + c)^2 + 432*(b*x*e + a*e)^2*A*B*b*d^2*e^3/(d*x +
c)^2 + 216*(b*x*e + a*e)^2*B^2*b*d^2*e^3/(d*x + c)^2 - 288*(b*x*e + a*e)^3*A^2*d^3*e^2/(d*x + c)^3 - 576*(b*x*
e + a*e)^3*A*B*d^3*e^2/(d*x + c)^3 - 576*(b*x*e + a*e)^3*B^2*d^3*e^2/(d*x + c)^3)*(b*c/((b*c*e - a*d*e)*(b*c -
a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))/((b*x*e + a*e)^4*b^3*c^3*g^5/(d*x + c)^4 - 3*(b*x*e + a*e)^4*a*b^2
*c^2*d*g^5/(d*x + c)^4 + 3*(b*x*e + a*e)^4*a^2*b*c*d^2*g^5/(d*x + c)^4 - (b*x*e + a*e)^4*a^3*d^3*g^5/(d*x + c)
^4)

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Mupad [B]
time = 10.30, size = 1881, normalized size = 3.27 \begin {gather*} -\frac {\frac {72\,A^2\,a^3\,d^3-216\,A^2\,a^2\,b\,c\,d^2+216\,A^2\,a\,b^2\,c^2\,d-72\,A^2\,b^3\,c^3+300\,A\,B\,a^3\,d^3-276\,A\,B\,a^2\,b\,c\,d^2+156\,A\,B\,a\,b^2\,c^2\,d-36\,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^3-12\,A\,c\,B\,b^3\,d^2+84\,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+156\,A\,B\,a^2\,b\,d^3-60\,A\,B\,a\,b^2\,c\,d^2+12\,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+12\,A\,B\,b^3\,d^2\right )}{a\,d-b\,c}}{x\,\left (96\,a^5\,b^2\,d^2\,g^5-192\,a^4\,b^3\,c\,d\,g^5+96\,a^3\,b^4\,c^2\,g^5\right )+x^3\,\left (96\,a^3\,b^4\,d^2\,g^5-192\,a^2\,b^5\,c\,d\,g^5+96\,a\,b^6\,c^2\,g^5\right )+x^4\,\left (24\,a^2\,b^5\,d^2\,g^5-48\,a\,b^6\,c\,d\,g^5+24\,b^7\,c^2\,g^5\right )+x^2\,\left (144\,a^4\,b^3\,d^2\,g^5-288\,a^3\,b^4\,c\,d\,g^5+144\,a^2\,b^5\,c^2\,g^5\right )+24\,a^6\,b\,d^2\,g^5+24\,a^4\,b^3\,c^2\,g^5-48\,a^5\,b^2\,c\,d\,g^5}-{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\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 )}{c+d\,x}\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}{12\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{4\,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}{12\,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}{4\,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}{12\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{4\,b\,d^2}\right )+\frac {4\,a^2\,d^2-5\,a\,b\,c\,d+b^2\,c^2}{6\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,d^2}\right )-a\,\left (\frac {b^2\,c-a\,b\,d}{4\,d^2}-\frac {b\,\left (a\,d-b\,c\right )}{2\,d^2}\right )+\frac {4\,a^2\,b\,d^2-5\,a\,b^2\,c\,d+b^3\,c^2}{4\,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}{4\,d^2}-\frac {b\,\left (a\,d-b\,c\right )}{2\,d^2}\right )+\frac {b^3\,c-a\,b^2\,d}{4\,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}{12\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{4\,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}{12\,b\,d^4}\right )+a\,\left (b\,\left (\frac {4\,a^2\,d^2-5\,a\,b\,c\,d+b^2\,c^2}{12\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{4\,b\,d^2}\right )+\frac {4\,a^2\,d^2-5\,a\,b\,c\,d+b^2\,c^2}{6\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,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}{4\,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 (12\,A+25\,B\right )\,\left (-24\,a^4\,b\,d^4\,g^5+48\,a^3\,b^2\,c\,d^3\,g^5-48\,a\,b^4\,c^3\,d\,g^5+24\,b^5\,c^4\,g^5\right )\,1{}\mathrm {i}}{24\,b\,g^5\,{\left (a\,d-b\,c\right )}^4\,\left (25\,B^2\,d^4+12\,A\,B\,d^4\right )}+\frac {B\,d^5\,x\,\left (12\,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+12\,A\,B\,d^4\right )}\right )\,\left (12\,A+25\,B\right )\,1{}\mathrm {i}}{12\,b\,g^5\,{\left (a\,d-b\,c\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*d^4*atan((B*d^4*(12*A + 25*B)*(24*b^5*c^4*g^5 - 24*a^4*b*d^4*g^5 - 48*a*b^4*c^3*d*g^5 + 48*a^3*b^2*c*d^3*g^
5)*1i)/(24*b*g^5*(a*d - b*c)^4*(25*B^2*d^4 + 12*A*B*d^4)) + (B*d^5*x*(12*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 + 12*A*B*d^4)))*(12*A + 25*B)*
1i)/(12*b*g^5*(a*d - b*c)^4) - log((e*(a + b*x))/(c + d*x))^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))/(c + d*x))*((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)/(12*b*d^3) + (a*(a*d - b*c))/(4*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)/(12*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)/(4*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)/(12*b*d^3) + (a*(a*d - b*c))/(4*b*d^2)) + (4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(6*d^3) + (a*(a*d - b*
c))/(2*d^2)) - a*((b^2*c - a*b*d)/(4*d^2) - (b*(a*d - b*c))/(2*d^2)) + (b^3*c^2 + 4*a^2*b*d^2 - 5*a*b^2*c*d)/(
4*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)/(4*d^2) - (b*(a*d - b*c))/(2*d^2)) + (b^3*c - a*b^2*d)/(4*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)/(12
*b*d^3) + (a*(a*d - b*c))/(4*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)/(12*b*d^4)) + a*
(b*((4*a^2*d^2 + b^2*c^2 - 5*a*b*c*d)/(12*b*d^3) + (a*(a*d - b*c))/(4*b*d^2)) + (4*a^2*d^2 + b^2*c^2 - 5*a*b*c
*d)/(6*d^3) + (a*(a*d - b*c))/(2*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)/(4*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) - ((72*A^2*a^3*d^3 - 72*A^2*b^3*c^3 + 415*B^2*a^3*d^3 - 9*B^2*b^
3*c^3 + 300*A*B*a^3*d^3 - 36*A*B*b^3*c^3 + 216*A^2*a*b^2*c^2*d - 216*A^2*a^2*b*c*d^2 + 55*B^2*a*b^2*c^2*d - 16
1*B^2*a^2*b*c*d^2 + 156*A*B*a*b^2*c^2*d - 276*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 + 84*A*B*a*b^2*d^3 - 12*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 + 156*A*B*a^2*b*d^3 + 12*A*B*b^3*c^2*d - 60*A*B*a*b^2*c*d^2))/(3*(a*d - b*c)) + (d*x^3
*(25*B^2*b^3*d^2 + 12*A*B*b^3*d^2))/(a*d - b*c))/(x*(96*a^3*b^4*c^2*g^5 + 96*a^5*b^2*d^2*g^5 - 192*a^4*b^3*c*d
*g^5) + x^3*(96*a*b^6*c^2*g^5 + 96*a^3*b^4*d^2*g^5 - 192*a^2*b^5*c*d*g^5) + x^4*(24*b^7*c^2*g^5 + 24*a^2*b^5*d
^2*g^5 - 48*a*b^6*c*d*g^5) + x^2*(144*a^2*b^5*c^2*g^5 + 144*a^4*b^3*d^2*g^5 - 288*a^3*b^4*c*d*g^5) + 24*a^6*b*
d^2*g^5 + 24*a^4*b^3*c^2*g^5 - 48*a^5*b^2*c*d*g^5)

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