3.2.90 \(\int \frac {(A+B \log (\frac {e (c+d x)}{a+b x}))^2}{(a g+b g x)^5} \, dx\) [190]

Optimal. Leaf size=498 \[ \frac {2 B^2 d^3 (c+d x)}{(b c-a d)^4 g^5 (a+b x)}-\frac {3 b B^2 d^2 (c+d x)^2}{4 (b c-a d)^4 g^5 (a+b x)^2}+\frac {2 b^2 B^2 d (c+d x)^3}{9 (b c-a d)^4 g^5 (a+b x)^3}-\frac {b^3 B^2 (c+d x)^4}{32 (b c-a d)^4 g^5 (a+b x)^4}-\frac {B^2 d^4 \log ^2\left (\frac {c+d x}{a+b x}\right )}{4 b (b c-a d)^4 g^5}-\frac {2 B d^3 (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^4 g^5 (a+b x)}+\frac {3 b B d^2 (c+d x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 (b c-a d)^4 g^5 (a+b x)^2}-\frac {2 b^2 B d (c+d x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 (b c-a d)^4 g^5 (a+b x)^3}+\frac {b^3 B (c+d x)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{8 (b c-a d)^4 g^5 (a+b x)^4}+\frac {B d^4 \log \left (\frac {c+d x}{a+b x}\right ) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4} \]

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

________________________________________________________________________________________

Rubi [A]
time = 0.20, antiderivative size = 498, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {2552, 2356, 45, 2372, 2338} \begin {gather*} \frac {b^3 B (c+d x)^4 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{8 g^5 (a+b x)^4 (b c-a d)^4}-\frac {2 b^2 B d (c+d x)^3 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{3 g^5 (a+b x)^3 (b c-a d)^4}+\frac {B d^4 \log \left (\frac {c+d x}{a+b x}\right ) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{2 b g^5 (b c-a d)^4}-\frac {2 B d^3 (c+d x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{g^5 (a+b x) (b c-a d)^4}+\frac {3 b B d^2 (c+d x)^2 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{2 g^5 (a+b x)^2 (b c-a d)^4}-\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{4 b g^5 (a+b x)^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 {B^2 d^4 \log ^2\left (\frac {c+d x}{a+b x}\right )}{4 b g^5 (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*(c + d*x))/(a + b*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) - (B^2*d^4*Log[(c + d*x)/(a + b*x)]^2)/(4*b*(b*c - a*d)^4*g^5) - (2*B*d^3*(c + d*x)*(A + B*Log[
(e*(c + d*x))/(a + b*x)]))/((b*c - a*d)^4*g^5*(a + b*x)) + (3*b*B*d^2*(c + d*x)^2*(A + B*Log[(e*(c + d*x))/(a
+ b*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*(c + d*x))/(a + b*x)]))/(3*
(b*c - a*d)^4*g^5*(a + b*x)^3) + (b^3*B*(c + d*x)^4*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(8*(b*c - a*d)^4*g^5
*(a + b*x)^4) + (B*d^4*Log[(c + d*x)/(a + b*x)]*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/(2*b*(b*c - a*d)^4*g^5)
- (A + B*Log[(e*(c + d*x))/(a + b*x)])^2/(4*b*g^5*(a + b*x)^4)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

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 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -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*} \int \frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{(a g+b g x)^5} \, dx &=-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b 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 (c+d x)}{a+b x}\right )\right )}{g^4 (a+b x)^5 (c+d x)} \, dx}{2 b g}\\ &=-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b 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 (c+d x)}{a+b x}\right )}{(a+b x)^5 (c+d x)} \, dx}{2 b g^5}\\ &=-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b 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 (c+d x)}{a+b x}\right )\right )}{(b c-a d) (a+b x)^5}-\frac {b d \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2 \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3 \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4 \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^5 (a+b x)}-\frac {d^5 \left (-A-B \log \left (\frac {e (c+d x)}{a+b 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 (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}+\frac {B \int \frac {-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x)^5} \, dx}{2 g^5}+\frac {\left (B d^4\right ) \int \frac {-A-B \log \left (\frac {e (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b x}\right )}{(a+b x)^4} \, dx}{2 (b c-a d) g^5}\\ &=\frac {B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{8 b g^5 (a+b x)^4}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\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 {(a+b x) \left (\frac {d e}{a+b x}-\frac {b e (c+d x)}{(a+b x)^2}\right ) \log (a+b x)}{e (c+d x)} \, dx}{2 b (b c-a d)^4 g^5}-\frac {\left (B^2 d^4\right ) \int \frac {(a+b x) \left (\frac {d e}{a+b x}-\frac {b e (c+d x)}{(a+b x)^2}\right ) \log (c+d x)}{e (c+d 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 (c+d x)}{a+b x}\right )\right )}{8 b g^5 (a+b x)^4}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\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 {(a+b x) \left (\frac {d e}{a+b x}-\frac {b e (c+d x)}{(a+b x)^2}\right ) \log (a+b x)}{c+d x} \, dx}{2 b (b c-a d)^4 e g^5}-\frac {\left (B^2 d^4\right ) \int \frac {(a+b x) \left (\frac {d e}{a+b x}-\frac {b e (c+d x)}{(a+b x)^2}\right ) \log (c+d x)}{c+d x} \, dx}{2 b (b c-a d)^4 e g^5}\\ &=\frac {B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{8 b g^5 (a+b x)^4}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\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 {25 B^2 d^4 \log (c+d x)}{24 b (b c-a d)^4 g^5}+\frac {B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{8 b g^5 (a+b x)^4}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\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 {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^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 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{8 b g^5 (a+b x)^4}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\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 {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^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 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{8 b g^5 (a+b x)^4}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}-\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 {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^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 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{8 b g^5 (a+b x)^4}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b 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 (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{2 b (b c-a d)^4 g^5}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{4 b g^5 (a+b x)^4}+\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*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.60, size = 748, normalized size = 1.50 \begin {gather*} \frac {-72 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2+\frac {B \left (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 )+36 (b c-a d)^4 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )+48 d (-b c+a d)^3 (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )+72 d^2 (b c-a d)^2 (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )+144 d^3 (-b c+a d) (a+b x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )-144 d^4 (a+b x)^4 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )+144 d^4 (a+b x)^4 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\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*(c + d*x))/(a + b*x)])^2/(a*g + b*g*x)^5,x]

[Out]

(-72*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2 + (B*(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]) + 36*(b*c - a*d)^4*(A + B*L
og[(e*(c + d*x))/(a + b*x)]) + 48*d*(-(b*c) + a*d)^3*(a + b*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)]) + 72*d^2*(
b*c - a*d)^2*(a + b*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]) + 144*d^3*(-(b*c) + a*d)*(a + b*x)^3*(A + B*Log[
(e*(c + d*x))/(a + b*x)]) - 144*d^4*(a + b*x)^4*Log[a + b*x]*(A + B*Log[(e*(c + d*x))/(a + b*x)]) + 144*d^4*(a
+ b*x)^4*Log[c + d*x]*(A + B*Log[(e*(c + d*x))/(a + b*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)/(288*b*g^5*(a + b*x)^4)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1421\) vs. \(2(480)=960\).
time = 0.55, size = 1422, normalized size = 2.86

method	result	size
derivativedivides	\(\text {Expression too large to display}\)	\(1422\)
default	\(\text {Expression too large to display}\)	\(1422\)
norman	\(\text {Expression too large to display}\)	\(1796\)
risch	\(\text {Expression too large to display}\)	\(2601\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^2*e*(a*d-b*c)*(-1/4*b^5/(a*d-b*c)^5/e^5/g^5*A^2*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^4+b^4/(a*d-b*c)^5/e^4/g^5*A^
2*d*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^3-3/2*b^3/(a*d-b*c)^5/e^3/g^5*A^2*d^2*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^2+b^2/(a
*d-b*c)^5/e^2/g^5*A^2*d^3*(d*e/b-e*(a*d-b*c)/b/(b*x+a))-2*b^5/(a*d-b*c)^5/e^5/g^5*A*B*(1/4*(d*e/b-e*(a*d-b*c)/
b/(b*x+a))^4*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))-1/16*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^4)+6*b^4/(a*d-b*c)^5/e^4/g^5*A
*B*d*(1/3*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^3*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))-1/9*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^3)
-6*b^3/(a*d-b*c)^5/e^3/g^5*A*B*d^2*(1/2*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^2*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))-1/4*(d
*e/b-e*(a*d-b*c)/b/(b*x+a))^2)+2*b^2/(a*d-b*c)^5/e^2/g^5*A*B*d^3*((d*e/b-e*(a*d-b*c)/b/(b*x+a))*ln(d*e/b-e*(a*
d-b*c)/b/(b*x+a))+e*(a*d-b*c)/b/(b*x+a)-d*e/b)-b^5/(a*d-b*c)^5/e^5/g^5*B^2*(1/4*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^
4*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))^2-1/8*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^4*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))+1/32*(
d*e/b-e*(a*d-b*c)/b/(b*x+a))^4)+3*b^4/(a*d-b*c)^5/e^4/g^5*B^2*d*(1/3*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^3*ln(d*e/b-
e*(a*d-b*c)/b/(b*x+a))^2-2/9*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^3*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))+2/27*(d*e/b-e*(a*
d-b*c)/b/(b*x+a))^3)-3*b^3/(a*d-b*c)^5/e^3/g^5*B^2*d^2*(1/2*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^2*ln(d*e/b-e*(a*d-b*
c)/b/(b*x+a))^2-1/2*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^2*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))+1/4*(d*e/b-e*(a*d-b*c)/b/(
b*x+a))^2)+b^2/(a*d-b*c)^5/e^2/g^5*B^2*d^3*((d*e/b-e*(a*d-b*c)/b/(b*x+a))*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))^2-2*
(d*e/b-e*(a*d-b*c)/b/(b*x+a))*ln(d*e/b-e*(a*d-b*c)/b/(b*x+a))-2*e*(a*d-b*c)/b/(b*x+a)+2*d*e/b))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2128 vs. \(2 (486) = 972\).
time = 0.53, size = 2128, normalized size = 4.27 \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*(d*x+c)/(b*x+a)))^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(d*x*e/(b*x + a) + c*e/(b*x
+ a)) + (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 - 1/24*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(d*x*e/(b*x + a) + c*e/(b*x +
a))/(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(d*x*e/(b*x + a) + c*
e/(b*x + 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) - 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)

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1043 vs. \(2 (486) = 972\).
time = 0.44, size = 1043, normalized size = 2.09 \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 (d x + c\right )} e}{b x + a}\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 (d x + c\right )} e}{b x + a}\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*(d*x+c)/(b*x+a)))^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
((d*x + c)*e/(b*x + a))^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((d*x + c)*e/(b*x + a)))/((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)

________________________________________________________________________________________

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*(d*x+c)/(b*x+a)))**2/(b*g*x+a*g)**5,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1029 vs. \(2 (486) = 972\).
time = 3.13, size = 1029, normalized size = 2.07 \begin {gather*} \frac {{\left (\frac {288 \, {\left (d x e + c e\right )} B^{2} d^{3} e^{3} \log \left (\frac {d x e + c e}{b x + a}\right )^{2}}{b x + a} - \frac {432 \, {\left (d x e + c e\right )}^{2} B^{2} b d^{2} e^{2} \log \left (\frac {d x e + c e}{b x + a}\right )^{2}}{{\left (b x + a\right )}^{2}} + \frac {288 \, {\left (d x e + c e\right )}^{3} B^{2} b^{2} d e \log \left (\frac {d x e + c e}{b x + a}\right )^{2}}{{\left (b x + a\right )}^{3}} + \frac {576 \, {\left (d x e + c e\right )} A B d^{3} e^{3} \log \left (\frac {d x e + c e}{b x + a}\right )}{b x + a} - \frac {576 \, {\left (d x e + c e\right )} B^{2} d^{3} e^{3} \log \left (\frac {d x e + c e}{b x + a}\right )}{b x + a} - \frac {864 \, {\left (d x e + c e\right )}^{2} A B b d^{2} e^{2} \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{2}} + \frac {432 \, {\left (d x e + c e\right )}^{2} B^{2} b d^{2} e^{2} \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{2}} + \frac {576 \, {\left (d x e + c e\right )}^{3} A B b^{2} d e \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{3}} - \frac {192 \, {\left (d x e + c e\right )}^{3} B^{2} b^{2} d e \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{3}} - \frac {72 \, {\left (d x e + c e\right )}^{4} B^{2} b^{3} \log \left (\frac {d x e + c e}{b x + a}\right )^{2}}{{\left (b x + a\right )}^{4}} + \frac {288 \, {\left (d x e + c e\right )} A^{2} d^{3} e^{3}}{b x + a} - \frac {576 \, {\left (d x e + c e\right )} A B d^{3} e^{3}}{b x + a} + \frac {576 \, {\left (d x e + c e\right )} B^{2} d^{3} e^{3}}{b x + a} - \frac {432 \, {\left (d x e + c e\right )}^{2} A^{2} b d^{2} e^{2}}{{\left (b x + a\right )}^{2}} + \frac {432 \, {\left (d x e + c e\right )}^{2} A B b d^{2} e^{2}}{{\left (b x + a\right )}^{2}} - \frac {216 \, {\left (d x e + c e\right )}^{2} B^{2} b d^{2} e^{2}}{{\left (b x + a\right )}^{2}} + \frac {288 \, {\left (d x e + c e\right )}^{3} A^{2} b^{2} d e}{{\left (b x + a\right )}^{3}} - \frac {192 \, {\left (d x e + c e\right )}^{3} A B b^{2} d e}{{\left (b x + a\right )}^{3}} + \frac {64 \, {\left (d x e + c e\right )}^{3} B^{2} b^{2} d e}{{\left (b x + a\right )}^{3}} - \frac {144 \, {\left (d x e + c e\right )}^{4} A B b^{3} \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{4}} + \frac {36 \, {\left (d x e + c e\right )}^{4} B^{2} b^{3} \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{4}} - \frac {72 \, {\left (d x e + c e\right )}^{4} A^{2} b^{3}}{{\left (b x + a\right )}^{4}} + \frac {36 \, {\left (d x e + c e\right )}^{4} A B b^{3}}{{\left (b x + a\right )}^{4}} - \frac {9 \, {\left (d x e + c e\right )}^{4} B^{2} b^{3}}{{\left (b x + a\right )}^{4}}\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 (b^{3} c^{3} g^{5} e^{3} - 3 \, a b^{2} c^{2} d g^{5} e^{3} + 3 \, a^{2} b c d^{2} g^{5} e^{3} - a^{3} d^{3} g^{5} e^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [B]
time = 10.94, size = 1880, normalized size = 3.78 \begin {gather*} \frac {\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\,\left (\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 {A\,B}{2\,b^2\,d\,g^5}+\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}}-{\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,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 {\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}+\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*(c + d*x))/(a + b*x)))^2/(a*g + b*g*x)^5,x)

[Out]

(log((e*(c + d*x))/(a + b*x))*((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)) - (A*B)/(2*b^2*d*g^5) + (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) - log((e*(c + d*x))/(a + b*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))) - ((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 - 161*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) + (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)

________________________________________________________________________________________