3.2.89 \(\int \frac {(A+B \log (\frac {e (c+d x)}{a+b x}))^2}{(a g+b g x)^4} \, dx\) [189]

Optimal. Leaf size=399 \[ -\frac {2 B^2 d^2 (c+d x)}{(b c-a d)^3 g^4 (a+b x)}+\frac {b B^2 d (c+d x)^2}{2 (b c-a d)^3 g^4 (a+b x)^2}-\frac {2 b^2 B^2 (c+d x)^3}{27 (b c-a d)^3 g^4 (a+b x)^3}+\frac {B^2 d^3 \log ^2\left (\frac {c+d x}{a+b x}\right )}{3 b (b c-a d)^3 g^4}+\frac {2 B d^2 (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^3 g^4 (a+b x)}-\frac {b B d (c+d x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^3 g^4 (a+b x)^2}+\frac {2 b^2 B (c+d x)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{9 (b c-a d)^3 g^4 (a+b x)^3}-\frac {2 B d^3 \log \left (\frac {c+d x}{a+b x}\right ) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{3 b g^4 (a+b x)^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.17, antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {2 b^2 B (c+d x)^3 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{9 g^4 (a+b x)^3 (b c-a d)^3}-\frac {2 B d^3 \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 )}{3 b g^4 (b c-a d)^3}+\frac {2 B d^2 (c+d x) \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{g^4 (a+b x) (b c-a d)^3}-\frac {b B d (c+d x)^2 \left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )}{g^4 (a+b x)^2 (b c-a d)^3}-\frac {\left (B \log \left (\frac {e (c+d x)}{a+b x}\right )+A\right )^2}{3 b g^4 (a+b x)^3}-\frac {2 b^2 B^2 (c+d x)^3}{27 g^4 (a+b x)^3 (b c-a d)^3}+\frac {B^2 d^3 \log ^2\left (\frac {c+d x}{a+b x}\right )}{3 b g^4 (b c-a d)^3}-\frac {2 B^2 d^2 (c+d x)}{g^4 (a+b x) (b c-a d)^3}+\frac {b B^2 d (c+d x)^2}{2 g^4 (a+b x)^2 (b c-a d)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*B^2*d^2*(c + d*x))/((b*c - a*d)^3*g^4*(a + b*x)) + (b*B^2*d*(c + d*x)^2)/(2*(b*c - a*d)^3*g^4*(a + b*x)^2)
- (2*b^2*B^2*(c + d*x)^3)/(27*(b*c - a*d)^3*g^4*(a + b*x)^3) + (B^2*d^3*Log[(c + d*x)/(a + b*x)]^2)/(3*b*(b*c
- a*d)^3*g^4) + (2*B*d^2*(c + d*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/((b*c - a*d)^3*g^4*(a + b*x)) - (b*B
*d*(c + d*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]))/((b*c - a*d)^3*g^4*(a + b*x)^2) + (2*b^2*B*(c + d*x)^3*(A
+ B*Log[(e*(c + d*x))/(a + b*x)]))/(9*(b*c - a*d)^3*g^4*(a + b*x)^3) - (2*B*d^3*Log[(c + d*x)/(a + b*x)]*(A +
B*Log[(e*(c + d*x))/(a + b*x)]))/(3*b*(b*c - a*d)^3*g^4) - (A + B*Log[(e*(c + d*x))/(a + b*x)])^2/(3*b*g^4*(a
+ b*x)^3)

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)^4} \, dx &=-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {(2 B) \int \frac {(b c-a d) \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{g^3 (a+b x)^4 (c+d x)} \, dx}{3 b g}\\ &=-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {(2 B (b c-a d)) \int \frac {-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x)^4 (c+d x)} \, dx}{3 b g^4}\\ &=-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {(2 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)^4}-\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)^3}+\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)^2}-\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)}+\frac {d^4 \left (-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{(b c-a d)^4 (c+d x)}\right ) \, dx}{3 b g^4}\\ &=-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {(2 B) \int \frac {-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x)^4} \, dx}{3 g^4}-\frac {\left (2 B d^3\right ) \int \frac {-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )}{a+b x} \, dx}{3 (b c-a d)^3 g^4}+\frac {\left (2 B d^4\right ) \int \frac {-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}+\frac {\left (2 B d^2\right ) \int \frac {-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x)^2} \, dx}{3 (b c-a d)^2 g^4}-\frac {(2 B d) \int \frac {-A-B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x)^3} \, dx}{3 (b c-a d) g^4}\\ &=\frac {2 B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{9 b g^4 (a+b x)^3}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}+\frac {2 B d^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}+\frac {2 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {2 B d^3 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{3 b g^4 (a+b x)^3}-\frac {\left (2 B^2\right ) \int \frac {-b c+a d}{(a+b x)^4 (c+d x)} \, dx}{9 b g^4}-\frac {\left (2 B^2 d^3\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}{3 b (b c-a d)^3 g^4}+\frac {\left (2 B^2 d^3\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}{3 b (b c-a d)^3 g^4}-\frac {\left (2 B^2 d^2\right ) \int \frac {-b c+a d}{(a+b x)^2 (c+d x)} \, dx}{3 b (b c-a d)^2 g^4}+\frac {\left (B^2 d\right ) \int \frac {-b c+a d}{(a+b x)^3 (c+d x)} \, dx}{3 b (b c-a d) g^4}\\ &=\frac {2 B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{9 b g^4 (a+b x)^3}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}+\frac {2 B d^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}+\frac {2 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {2 B d^3 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{3 b g^4 (a+b x)^3}-\frac {\left (B^2 d\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{3 b g^4}+\frac {\left (2 B^2 d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{3 b (b c-a d) g^4}+\frac {\left (2 B^2 (b c-a d)\right ) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{9 b g^4}-\frac {\left (2 B^2 d^3\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}{3 b (b c-a d)^3 e g^4}+\frac {\left (2 B^2 d^3\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}{3 b (b c-a d)^3 e g^4}\\ &=\frac {2 B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{9 b g^4 (a+b x)^3}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}+\frac {2 B d^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}+\frac {2 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {2 B d^3 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{3 b g^4 (a+b x)^3}-\frac {\left (B^2 d\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}{3 b g^4}+\frac {\left (2 B^2 d^2\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}{3 b (b c-a d) g^4}+\frac {\left (2 B^2 (b c-a 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}{9 b g^4}-\frac {\left (2 B^2 d^3\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}{3 b (b c-a d)^3 e g^4}+\frac {\left (2 B^2 d^3\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}{3 b (b c-a d)^3 e g^4}\\ &=-\frac {2 B^2}{27 b g^4 (a+b x)^3}+\frac {5 B^2 d}{18 b (b c-a d) g^4 (a+b x)^2}-\frac {11 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac {11 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}+\frac {11 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}+\frac {2 B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{9 b g^4 (a+b x)^3}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}+\frac {2 B d^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}+\frac {2 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {2 B d^3 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {\left (2 B^2 d^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{3 (b c-a d)^3 g^4}-\frac {\left (2 B^2 d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{3 (b c-a d)^3 g^4}-\frac {\left (2 B^2 d^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}+\frac {\left (2 B^2 d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}\\ &=-\frac {2 B^2}{27 b g^4 (a+b x)^3}+\frac {5 B^2 d}{18 b (b c-a d) g^4 (a+b x)^2}-\frac {11 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac {11 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}+\frac {11 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac {2 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}-\frac {2 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}+\frac {2 B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{9 b g^4 (a+b x)^3}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}+\frac {2 B d^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}+\frac {2 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {2 B d^3 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {\left (2 B^2 d^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{3 (b c-a d)^3 g^4}+\frac {\left (2 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{3 b (b c-a d)^3 g^4}+\frac {\left (2 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{3 b (b c-a d)^3 g^4}+\frac {\left (2 B^2 d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{3 b (b c-a d)^3 g^4}\\ &=-\frac {2 B^2}{27 b g^4 (a+b x)^3}+\frac {5 B^2 d}{18 b (b c-a d) g^4 (a+b x)^2}-\frac {11 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac {11 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}+\frac {B^2 d^3 \log ^2(a+b x)}{3 b (b c-a d)^3 g^4}+\frac {11 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac {2 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac {B^2 d^3 \log ^2(c+d x)}{3 b (b c-a d)^3 g^4}-\frac {2 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}+\frac {2 B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{9 b g^4 (a+b x)^3}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}+\frac {2 B d^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}+\frac {2 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {2 B d^3 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{3 b g^4 (a+b x)^3}+\frac {\left (2 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{3 b (b c-a d)^3 g^4}+\frac {\left (2 B^2 d^3\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{3 b (b c-a d)^3 g^4}\\ &=-\frac {2 B^2}{27 b g^4 (a+b x)^3}+\frac {5 B^2 d}{18 b (b c-a d) g^4 (a+b x)^2}-\frac {11 B^2 d^2}{9 b (b c-a d)^2 g^4 (a+b x)}-\frac {11 B^2 d^3 \log (a+b x)}{9 b (b c-a d)^3 g^4}+\frac {B^2 d^3 \log ^2(a+b x)}{3 b (b c-a d)^3 g^4}+\frac {11 B^2 d^3 \log (c+d x)}{9 b (b c-a d)^3 g^4}-\frac {2 B^2 d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{3 b (b c-a d)^3 g^4}+\frac {B^2 d^3 \log ^2(c+d x)}{3 b (b c-a d)^3 g^4}-\frac {2 B^2 d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}+\frac {2 B \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{9 b g^4 (a+b x)^3}-\frac {B d \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d) g^4 (a+b x)^2}+\frac {2 B d^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^2 g^4 (a+b x)}+\frac {2 B d^3 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {2 B d^3 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )}{3 b (b c-a d)^3 g^4}-\frac {\left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2}{3 b g^4 (a+b x)^3}-\frac {2 B^2 d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}-\frac {2 B^2 d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{3 b (b c-a d)^3 g^4}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.45, size = 585, normalized size = 1.47 \begin {gather*} -\frac {18 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )^2+\frac {B \left (36 B d^2 (a+b x)^2 (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-9 B d (a+b x) \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 )+2 B \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 )-12 (b c-a d)^3 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )+18 d (b c-a d)^2 (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )+36 d^2 (-b c+a d) (a+b x)^2 \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )-36 d^3 (a+b x)^3 \log (a+b x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )+36 d^3 (a+b x)^3 \log (c+d x) \left (A+B \log \left (\frac {e (c+d x)}{a+b x}\right )\right )-18 B d^3 (a+b x)^3 \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 )+18 B d^3 (a+b x)^3 \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)^3}}{54 b g^4 (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/54*(18*(A + B*Log[(e*(c + d*x))/(a + b*x)])^2 + (B*(36*B*d^2*(a + b*x)^2*(b*c - a*d + d*(a + b*x)*Log[a + b
*x] - d*(a + b*x)*Log[c + d*x]) - 9*B*d*(a + b*x)*((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]) + 2*B*(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]) - 12*(b*c - a*d
)^3*(A + B*Log[(e*(c + d*x))/(a + b*x)]) + 18*d*(b*c - a*d)^2*(a + b*x)*(A + B*Log[(e*(c + d*x))/(a + b*x)]) +
36*d^2*(-(b*c) + a*d)*(a + b*x)^2*(A + B*Log[(e*(c + d*x))/(a + b*x)]) - 36*d^3*(a + b*x)^3*Log[a + b*x]*(A +
B*Log[(e*(c + d*x))/(a + b*x)]) + 36*d^3*(a + b*x)^3*Log[c + d*x]*(A + B*Log[(e*(c + d*x))/(a + b*x)]) - 18*B
*d^3*(a + b*x)^3*(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)]) + 18*B*d^3*(a + b*x)^3*((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)^3)/(b*g^4*(a + b*x)^3)

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1060\) vs. \(2(387)=774\).
time = 0.49, size = 1061, normalized size = 2.66 Too large to display

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)^4,x,method=_RETURNVERBOSE)

[Out]

1/b^2*e*(a*d-b*c)*(1/3*b^4/(a*d-b*c)^4/e^4/g^4*A^2*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^3-b^3/(a*d-b*c)^4/e^3/g^4*A^2
*d*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^2+b^2/(a*d-b*c)^4/e^2/g^4*A^2*d^2*(d*e/b-e*(a*d-b*c)/b/(b*x+a))+2*b^4/(a*d-b*
c)^4/e^4/g^4*A*B*(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)-4*b^3/(a*d-b*c)^4/e^3/g^4*A*B*d*(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)^4/e^2/g^4*A*B*d^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))+e*(a*d-b*c)/b/(b*x+a)-d*e/b)+b^4/(a*d-b*c)^4/e^4/g^4*B^2*(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)-2*b^3/(a*d-b*c)^4/e^3/g^4*B^2*d*(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)^4/e^2/g^4*B^2*d^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))^2-2*(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. 1426 vs. \(2 (392) = 784\).
time = 0.46, size = 1426, normalized size = 3.57 \begin {gather*} \frac {1}{54} \, {\left (6 \, {\left (\frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} g^{4} x + {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} g^{4}} + \frac {6 \, d^{3} \log \left (b x + a\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}}\right )} \log \left (\frac {d x e}{b x + a} + \frac {c e}{b x + a}\right ) - \frac {4 \, b^{3} c^{3} - 27 \, a b^{2} c^{2} d + 108 \, a^{2} b c d^{2} - 85 \, a^{3} d^{3} + 66 \, {\left (b^{3} c d^{2} - a b^{2} d^{3}\right )} x^{2} - 18 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (b x + a\right )^{2} - 18 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (d x + c\right )^{2} - 3 \, {\left (5 \, b^{3} c^{2} d - 54 \, a b^{2} c d^{2} + 49 \, a^{2} b d^{3}\right )} x + 66 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (b x + a\right ) - 6 \, {\left (11 \, b^{3} d^{3} x^{3} + 33 \, a b^{2} d^{3} x^{2} + 33 \, a^{2} b d^{3} x + 11 \, a^{3} d^{3} - 6 \, {\left (b^{3} d^{3} x^{3} + 3 \, a b^{2} d^{3} x^{2} + 3 \, a^{2} b d^{3} x + a^{3} d^{3}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{a^{3} b^{4} c^{3} g^{4} - 3 \, a^{4} b^{3} c^{2} d g^{4} + 3 \, a^{5} b^{2} c d^{2} g^{4} - a^{6} b d^{3} g^{4} + {\left (b^{7} c^{3} g^{4} - 3 \, a b^{6} c^{2} d g^{4} + 3 \, a^{2} b^{5} c d^{2} g^{4} - a^{3} b^{4} d^{3} g^{4}\right )} x^{3} + 3 \, {\left (a b^{6} c^{3} g^{4} - 3 \, a^{2} b^{5} c^{2} d g^{4} + 3 \, a^{3} b^{4} c d^{2} g^{4} - a^{4} b^{3} d^{3} g^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} g^{4} - 3 \, a^{3} b^{4} c^{2} d g^{4} + 3 \, a^{4} b^{3} c d^{2} g^{4} - a^{5} b^{2} d^{3} g^{4}\right )} x}\right )} B^{2} + \frac {1}{9} \, A B {\left (\frac {6 \, b^{2} d^{2} x^{2} + 2 \, b^{2} c^{2} - 7 \, a b c d + 11 \, a^{2} d^{2} - 3 \, {\left (b^{2} c d - 5 \, a b d^{2}\right )} x}{{\left (b^{6} c^{2} - 2 \, a b^{5} c d + a^{2} b^{4} d^{2}\right )} g^{4} x^{3} + 3 \, {\left (a b^{5} c^{2} - 2 \, a^{2} b^{4} c d + a^{3} b^{3} d^{2}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{4} c^{2} - 2 \, a^{3} b^{3} c d + a^{4} b^{2} d^{2}\right )} g^{4} x + {\left (a^{3} b^{3} c^{2} - 2 \, a^{4} b^{2} c d + a^{5} b d^{2}\right )} g^{4}} - \frac {6 \, \log \left (\frac {d x e}{b x + a} + \frac {c e}{b x + a}\right )}{b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}} + \frac {6 \, d^{3} \log \left (b x + a\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} g^{4}}\right )} - \frac {B^{2} \log \left (\frac {d x e}{b x + a} + \frac {c e}{b x + a}\right )^{2}}{3 \, {\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\right )}} - \frac {A^{2}}{3 \, {\left (b^{4} g^{4} x^{3} + 3 \, a b^{3} g^{4} x^{2} + 3 \, a^{2} b^{2} g^{4} x + a^{3} b g^{4}\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)^4,x, algorithm="maxima")

[Out]

1/54*(6*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*
c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d + a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*
c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d + a^5*b*d^2)*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a
*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log(d*x + c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^
2 - a^3*b*d^3)*g^4))*log(d*x*e/(b*x + a) + c*e/(b*x + a)) - (4*b^3*c^3 - 27*a*b^2*c^2*d + 108*a^2*b*c*d^2 - 85
*a^3*d^3 + 66*(b^3*c*d^2 - a*b^2*d^3)*x^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b
*x + a)^2 - 18*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(d*x + c)^2 - 3*(5*b^3*c^2*d - 54*
a*b^2*c*d^2 + 49*a^2*b*d^3)*x + 66*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*d^3*x + a^3*d^3)*log(b*x + a) - 6*
(11*b^3*d^3*x^3 + 33*a*b^2*d^3*x^2 + 33*a^2*b*d^3*x + 11*a^3*d^3 - 6*(b^3*d^3*x^3 + 3*a*b^2*d^3*x^2 + 3*a^2*b*
d^3*x + a^3*d^3)*log(b*x + a))*log(d*x + c))/(a^3*b^4*c^3*g^4 - 3*a^4*b^3*c^2*d*g^4 + 3*a^5*b^2*c*d^2*g^4 - a^
6*b*d^3*g^4 + (b^7*c^3*g^4 - 3*a*b^6*c^2*d*g^4 + 3*a^2*b^5*c*d^2*g^4 - a^3*b^4*d^3*g^4)*x^3 + 3*(a*b^6*c^3*g^4
- 3*a^2*b^5*c^2*d*g^4 + 3*a^3*b^4*c*d^2*g^4 - a^4*b^3*d^3*g^4)*x^2 + 3*(a^2*b^5*c^3*g^4 - 3*a^3*b^4*c^2*d*g^4
+ 3*a^4*b^3*c*d^2*g^4 - a^5*b^2*d^3*g^4)*x))*B^2 + 1/9*A*B*((6*b^2*d^2*x^2 + 2*b^2*c^2 - 7*a*b*c*d + 11*a^2*d
^2 - 3*(b^2*c*d - 5*a*b*d^2)*x)/((b^6*c^2 - 2*a*b^5*c*d + a^2*b^4*d^2)*g^4*x^3 + 3*(a*b^5*c^2 - 2*a^2*b^4*c*d
+ a^3*b^3*d^2)*g^4*x^2 + 3*(a^2*b^4*c^2 - 2*a^3*b^3*c*d + a^4*b^2*d^2)*g^4*x + (a^3*b^3*c^2 - 2*a^4*b^2*c*d +
a^5*b*d^2)*g^4) - 6*log(d*x*e/(b*x + a) + c*e/(b*x + a))/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^
3*b*g^4) + 6*d^3*log(b*x + a)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4) - 6*d^3*log(d*x +
c)/((b^4*c^3 - 3*a*b^3*c^2*d + 3*a^2*b^2*c*d^2 - a^3*b*d^3)*g^4)) - 1/3*B^2*log(d*x*e/(b*x + a) + c*e/(b*x + a
))^2/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 + 3*a^2*b^2*g^4*x + a^3*b*g^4) - 1/3*A^2/(b^4*g^4*x^3 + 3*a*b^3*g^4*x^2 +
3*a^2*b^2*g^4*x + a^3*b*g^4)

________________________________________________________________________________________

Fricas [A]
time = 0.46, size = 678, normalized size = 1.70 \begin {gather*} -\frac {2 \, {\left (9 \, A^{2} - 6 \, A B + 2 \, B^{2}\right )} b^{3} c^{3} - 27 \, {\left (2 \, A^{2} - 2 \, A B + B^{2}\right )} a b^{2} c^{2} d + 54 \, {\left (A^{2} - 2 \, A B + 2 \, B^{2}\right )} a^{2} b c d^{2} - {\left (18 \, A^{2} - 66 \, A B + 85 \, B^{2}\right )} a^{3} d^{3} - 6 \, {\left ({\left (6 \, A B - 11 \, B^{2}\right )} b^{3} c d^{2} - {\left (6 \, A B - 11 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} + 18 \, {\left (B^{2} b^{3} d^{3} x^{3} + 3 \, B^{2} a b^{2} d^{3} x^{2} + 3 \, B^{2} a^{2} b d^{3} x + B^{2} b^{3} c^{3} - 3 \, B^{2} a b^{2} c^{2} d + 3 \, B^{2} a^{2} b c d^{2}\right )} \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right )^{2} + 3 \, {\left ({\left (6 \, A B - 5 \, B^{2}\right )} b^{3} c^{2} d - 18 \, {\left (2 \, A B - 3 \, B^{2}\right )} a b^{2} c d^{2} + {\left (30 \, A B - 49 \, B^{2}\right )} a^{2} b d^{3}\right )} x + 6 \, {\left ({\left (6 \, A B - 11 \, B^{2}\right )} b^{3} d^{3} x^{3} + 2 \, {\left (3 \, A B - B^{2}\right )} b^{3} c^{3} - 9 \, {\left (2 \, A B - B^{2}\right )} a b^{2} c^{2} d + 18 \, {\left (A B - B^{2}\right )} a^{2} b c d^{2} - 3 \, {\left (2 \, B^{2} b^{3} c d^{2} - 3 \, {\left (2 \, A B - 3 \, B^{2}\right )} a b^{2} d^{3}\right )} x^{2} + 3 \, {\left (B^{2} b^{3} c^{2} d - 6 \, B^{2} a b^{2} c d^{2} + 6 \, {\left (A B - B^{2}\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac {{\left (d x + c\right )} e}{b x + a}\right )}{54 \, {\left ({\left (b^{7} c^{3} - 3 \, a b^{6} c^{2} d + 3 \, a^{2} b^{5} c d^{2} - a^{3} b^{4} d^{3}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{3} - 3 \, a^{2} b^{5} c^{2} d + 3 \, a^{3} b^{4} c d^{2} - a^{4} b^{3} d^{3}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{3} - 3 \, a^{3} b^{4} c^{2} d + 3 \, a^{4} b^{3} c d^{2} - a^{5} b^{2} d^{3}\right )} g^{4} x + {\left (a^{3} b^{4} c^{3} - 3 \, a^{4} b^{3} c^{2} d + 3 \, a^{5} b^{2} c d^{2} - a^{6} b d^{3}\right )} g^{4}\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)^4,x, algorithm="fricas")

[Out]

-1/54*(2*(9*A^2 - 6*A*B + 2*B^2)*b^3*c^3 - 27*(2*A^2 - 2*A*B + B^2)*a*b^2*c^2*d + 54*(A^2 - 2*A*B + 2*B^2)*a^2
*b*c*d^2 - (18*A^2 - 66*A*B + 85*B^2)*a^3*d^3 - 6*((6*A*B - 11*B^2)*b^3*c*d^2 - (6*A*B - 11*B^2)*a*b^2*d^3)*x^
2 + 18*(B^2*b^3*d^3*x^3 + 3*B^2*a*b^2*d^3*x^2 + 3*B^2*a^2*b*d^3*x + B^2*b^3*c^3 - 3*B^2*a*b^2*c^2*d + 3*B^2*a^
2*b*c*d^2)*log((d*x + c)*e/(b*x + a))^2 + 3*((6*A*B - 5*B^2)*b^3*c^2*d - 18*(2*A*B - 3*B^2)*a*b^2*c*d^2 + (30*
A*B - 49*B^2)*a^2*b*d^3)*x + 6*((6*A*B - 11*B^2)*b^3*d^3*x^3 + 2*(3*A*B - B^2)*b^3*c^3 - 9*(2*A*B - B^2)*a*b^2
*c^2*d + 18*(A*B - B^2)*a^2*b*c*d^2 - 3*(2*B^2*b^3*c*d^2 - 3*(2*A*B - 3*B^2)*a*b^2*d^3)*x^2 + 3*(B^2*b^3*c^2*d
- 6*B^2*a*b^2*c*d^2 + 6*(A*B - B^2)*a^2*b*d^3)*x)*log((d*x + c)*e/(b*x + a)))/((b^7*c^3 - 3*a*b^6*c^2*d + 3*a
^2*b^5*c*d^2 - a^3*b^4*d^3)*g^4*x^3 + 3*(a*b^6*c^3 - 3*a^2*b^5*c^2*d + 3*a^3*b^4*c*d^2 - a^4*b^3*d^3)*g^4*x^2
+ 3*(a^2*b^5*c^3 - 3*a^3*b^4*c^2*d + 3*a^4*b^3*c*d^2 - a^5*b^2*d^3)*g^4*x + (a^3*b^4*c^3 - 3*a^4*b^3*c^2*d + 3
*a^5*b^2*c*d^2 - a^6*b*d^3)*g^4)

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1544 vs. \(2 (362) = 724\).
time = 18.41, size = 1544, normalized size = 3.87 \begin {gather*} \frac {B d^{3} \cdot \left (6 A - 11 B\right ) \log {\left (x + \frac {6 A B a d^{4} + 6 A B b c d^{3} - 11 B^{2} a d^{4} - 11 B^{2} b c d^{3} - \frac {B a^{4} d^{7} \cdot \left (6 A - 11 B\right )}{\left (a d - b c\right )^{3}} + \frac {4 B a^{3} b c d^{6} \cdot \left (6 A - 11 B\right )}{\left (a d - b c\right )^{3}} - \frac {6 B a^{2} b^{2} c^{2} d^{5} \cdot \left (6 A - 11 B\right )}{\left (a d - b c\right )^{3}} + \frac {4 B a b^{3} c^{3} d^{4} \cdot \left (6 A - 11 B\right )}{\left (a d - b c\right )^{3}} - \frac {B b^{4} c^{4} d^{3} \cdot \left (6 A - 11 B\right )}{\left (a d - b c\right )^{3}}}{12 A B b d^{4} - 22 B^{2} b d^{4}} \right )}}{9 b g^{4} \left (a d - b c\right )^{3}} - \frac {B d^{3} \cdot \left (6 A - 11 B\right ) \log {\left (x + \frac {6 A B a d^{4} + 6 A B b c d^{3} - 11 B^{2} a d^{4} - 11 B^{2} b c d^{3} + \frac {B a^{4} d^{7} \cdot \left (6 A - 11 B\right )}{\left (a d - b c\right )^{3}} - \frac {4 B a^{3} b c d^{6} \cdot \left (6 A - 11 B\right )}{\left (a d - b c\right )^{3}} + \frac {6 B a^{2} b^{2} c^{2} d^{5} \cdot \left (6 A - 11 B\right )}{\left (a d - b c\right )^{3}} - \frac {4 B a b^{3} c^{3} d^{4} \cdot \left (6 A - 11 B\right )}{\left (a d - b c\right )^{3}} + \frac {B b^{4} c^{4} d^{3} \cdot \left (6 A - 11 B\right )}{\left (a d - b c\right )^{3}}}{12 A B b d^{4} - 22 B^{2} b d^{4}} \right )}}{9 b g^{4} \left (a d - b c\right )^{3}} + \frac {\left (3 B^{2} a^{2} c d^{2} + 3 B^{2} a^{2} d^{3} x - 3 B^{2} a b c^{2} d + 3 B^{2} a b d^{3} x^{2} + B^{2} b^{2} c^{3} + B^{2} b^{2} d^{3} x^{3}\right ) \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )}^{2}}{3 a^{6} d^{3} g^{4} - 9 a^{5} b c d^{2} g^{4} + 9 a^{5} b d^{3} g^{4} x + 9 a^{4} b^{2} c^{2} d g^{4} - 27 a^{4} b^{2} c d^{2} g^{4} x + 9 a^{4} b^{2} d^{3} g^{4} x^{2} - 3 a^{3} b^{3} c^{3} g^{4} + 27 a^{3} b^{3} c^{2} d g^{4} x - 27 a^{3} b^{3} c d^{2} g^{4} x^{2} + 3 a^{3} b^{3} d^{3} g^{4} x^{3} - 9 a^{2} b^{4} c^{3} g^{4} x + 27 a^{2} b^{4} c^{2} d g^{4} x^{2} - 9 a^{2} b^{4} c d^{2} g^{4} x^{3} - 9 a b^{5} c^{3} g^{4} x^{2} + 9 a b^{5} c^{2} d g^{4} x^{3} - 3 b^{6} c^{3} g^{4} x^{3}} + \frac {\left (- 6 A B a^{2} d^{2} + 12 A B a b c d - 6 A B b^{2} c^{2} + 11 B^{2} a^{2} d^{2} - 7 B^{2} a b c d + 15 B^{2} a b d^{2} x + 2 B^{2} b^{2} c^{2} - 3 B^{2} b^{2} c d x + 6 B^{2} b^{2} d^{2} x^{2}\right ) \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )}}{9 a^{5} b d^{2} g^{4} - 18 a^{4} b^{2} c d g^{4} + 27 a^{4} b^{2} d^{2} g^{4} x + 9 a^{3} b^{3} c^{2} g^{4} - 54 a^{3} b^{3} c d g^{4} x + 27 a^{3} b^{3} d^{2} g^{4} x^{2} + 27 a^{2} b^{4} c^{2} g^{4} x - 54 a^{2} b^{4} c d g^{4} x^{2} + 9 a^{2} b^{4} d^{2} g^{4} x^{3} + 27 a b^{5} c^{2} g^{4} x^{2} - 18 a b^{5} c d g^{4} x^{3} + 9 b^{6} c^{2} g^{4} x^{3}} - \frac {18 A^{2} a^{2} d^{2} - 36 A^{2} a b c d + 18 A^{2} b^{2} c^{2} - 66 A B a^{2} d^{2} + 42 A B a b c d - 12 A B b^{2} c^{2} + 85 B^{2} a^{2} d^{2} - 23 B^{2} a b c d + 4 B^{2} b^{2} c^{2} + x^{2} \left (- 36 A B b^{2} d^{2} + 66 B^{2} b^{2} d^{2}\right ) + x \left (- 90 A B a b d^{2} + 18 A B b^{2} c d + 147 B^{2} a b d^{2} - 15 B^{2} b^{2} c d\right )}{54 a^{5} b d^{2} g^{4} - 108 a^{4} b^{2} c d g^{4} + 54 a^{3} b^{3} c^{2} g^{4} + x^{3} \cdot \left (54 a^{2} b^{4} d^{2} g^{4} - 108 a b^{5} c d g^{4} + 54 b^{6} c^{2} g^{4}\right ) + x^{2} \cdot \left (162 a^{3} b^{3} d^{2} g^{4} - 324 a^{2} b^{4} c d g^{4} + 162 a b^{5} c^{2} g^{4}\right ) + x \left (162 a^{4} b^{2} d^{2} g^{4} - 324 a^{3} b^{3} c d g^{4} + 162 a^{2} b^{4} c^{2} g^{4}\right )} \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)**4,x)

[Out]

B*d**3*(6*A - 11*B)*log(x + (6*A*B*a*d**4 + 6*A*B*b*c*d**3 - 11*B**2*a*d**4 - 11*B**2*b*c*d**3 - B*a**4*d**7*(
6*A - 11*B)/(a*d - b*c)**3 + 4*B*a**3*b*c*d**6*(6*A - 11*B)/(a*d - b*c)**3 - 6*B*a**2*b**2*c**2*d**5*(6*A - 11
*B)/(a*d - b*c)**3 + 4*B*a*b**3*c**3*d**4*(6*A - 11*B)/(a*d - b*c)**3 - B*b**4*c**4*d**3*(6*A - 11*B)/(a*d - b
*c)**3)/(12*A*B*b*d**4 - 22*B**2*b*d**4))/(9*b*g**4*(a*d - b*c)**3) - B*d**3*(6*A - 11*B)*log(x + (6*A*B*a*d**
4 + 6*A*B*b*c*d**3 - 11*B**2*a*d**4 - 11*B**2*b*c*d**3 + B*a**4*d**7*(6*A - 11*B)/(a*d - b*c)**3 - 4*B*a**3*b*
c*d**6*(6*A - 11*B)/(a*d - b*c)**3 + 6*B*a**2*b**2*c**2*d**5*(6*A - 11*B)/(a*d - b*c)**3 - 4*B*a*b**3*c**3*d**
4*(6*A - 11*B)/(a*d - b*c)**3 + B*b**4*c**4*d**3*(6*A - 11*B)/(a*d - b*c)**3)/(12*A*B*b*d**4 - 22*B**2*b*d**4)
)/(9*b*g**4*(a*d - b*c)**3) + (3*B**2*a**2*c*d**2 + 3*B**2*a**2*d**3*x - 3*B**2*a*b*c**2*d + 3*B**2*a*b*d**3*x
**2 + B**2*b**2*c**3 + B**2*b**2*d**3*x**3)*log(e*(c + d*x)/(a + b*x))**2/(3*a**6*d**3*g**4 - 9*a**5*b*c*d**2*
g**4 + 9*a**5*b*d**3*g**4*x + 9*a**4*b**2*c**2*d*g**4 - 27*a**4*b**2*c*d**2*g**4*x + 9*a**4*b**2*d**3*g**4*x**
2 - 3*a**3*b**3*c**3*g**4 + 27*a**3*b**3*c**2*d*g**4*x - 27*a**3*b**3*c*d**2*g**4*x**2 + 3*a**3*b**3*d**3*g**4
*x**3 - 9*a**2*b**4*c**3*g**4*x + 27*a**2*b**4*c**2*d*g**4*x**2 - 9*a**2*b**4*c*d**2*g**4*x**3 - 9*a*b**5*c**3
*g**4*x**2 + 9*a*b**5*c**2*d*g**4*x**3 - 3*b**6*c**3*g**4*x**3) + (-6*A*B*a**2*d**2 + 12*A*B*a*b*c*d - 6*A*B*b
**2*c**2 + 11*B**2*a**2*d**2 - 7*B**2*a*b*c*d + 15*B**2*a*b*d**2*x + 2*B**2*b**2*c**2 - 3*B**2*b**2*c*d*x + 6*
B**2*b**2*d**2*x**2)*log(e*(c + d*x)/(a + b*x))/(9*a**5*b*d**2*g**4 - 18*a**4*b**2*c*d*g**4 + 27*a**4*b**2*d**
2*g**4*x + 9*a**3*b**3*c**2*g**4 - 54*a**3*b**3*c*d*g**4*x + 27*a**3*b**3*d**2*g**4*x**2 + 27*a**2*b**4*c**2*g
**4*x - 54*a**2*b**4*c*d*g**4*x**2 + 9*a**2*b**4*d**2*g**4*x**3 + 27*a*b**5*c**2*g**4*x**2 - 18*a*b**5*c*d*g**
4*x**3 + 9*b**6*c**2*g**4*x**3) - (18*A**2*a**2*d**2 - 36*A**2*a*b*c*d + 18*A**2*b**2*c**2 - 66*A*B*a**2*d**2
+ 42*A*B*a*b*c*d - 12*A*B*b**2*c**2 + 85*B**2*a**2*d**2 - 23*B**2*a*b*c*d + 4*B**2*b**2*c**2 + x**2*(-36*A*B*b
**2*d**2 + 66*B**2*b**2*d**2) + x*(-90*A*B*a*b*d**2 + 18*A*B*b**2*c*d + 147*B**2*a*b*d**2 - 15*B**2*b**2*c*d))
/(54*a**5*b*d**2*g**4 - 108*a**4*b**2*c*d*g**4 + 54*a**3*b**3*c**2*g**4 + x**3*(54*a**2*b**4*d**2*g**4 - 108*a
*b**5*c*d*g**4 + 54*b**6*c**2*g**4) + x**2*(162*a**3*b**3*d**2*g**4 - 324*a**2*b**4*c*d*g**4 + 162*a*b**5*c**2
*g**4) + x*(162*a**4*b**2*d**2*g**4 - 324*a**3*b**3*c*d*g**4 + 162*a**2*b**4*c**2*g**4))

________________________________________________________________________________________

Giac [A]
time = 3.12, size = 760, normalized size = 1.90 \begin {gather*} -\frac {{\left (\frac {54 \, {\left (d x e + c e\right )} B^{2} d^{2} e^{2} \log \left (\frac {d x e + c e}{b x + a}\right )^{2}}{b x + a} - \frac {54 \, {\left (d x e + c e\right )}^{2} B^{2} b d e \log \left (\frac {d x e + c e}{b x + a}\right )^{2}}{{\left (b x + a\right )}^{2}} + \frac {108 \, {\left (d x e + c e\right )} A B d^{2} e^{2} \log \left (\frac {d x e + c e}{b x + a}\right )}{b x + a} - \frac {108 \, {\left (d x e + c e\right )} B^{2} d^{2} e^{2} \log \left (\frac {d x e + c e}{b x + a}\right )}{b x + a} - \frac {108 \, {\left (d x e + c e\right )}^{2} A B b d e \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{2}} + \frac {54 \, {\left (d x e + c e\right )}^{2} B^{2} b d e \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{2}} + \frac {18 \, {\left (d x e + c e\right )}^{3} B^{2} b^{2} \log \left (\frac {d x e + c e}{b x + a}\right )^{2}}{{\left (b x + a\right )}^{3}} + \frac {54 \, {\left (d x e + c e\right )} A^{2} d^{2} e^{2}}{b x + a} - \frac {108 \, {\left (d x e + c e\right )} A B d^{2} e^{2}}{b x + a} + \frac {108 \, {\left (d x e + c e\right )} B^{2} d^{2} e^{2}}{b x + a} - \frac {54 \, {\left (d x e + c e\right )}^{2} A^{2} b d e}{{\left (b x + a\right )}^{2}} + \frac {54 \, {\left (d x e + c e\right )}^{2} A B b d e}{{\left (b x + a\right )}^{2}} - \frac {27 \, {\left (d x e + c e\right )}^{2} B^{2} b d e}{{\left (b x + a\right )}^{2}} + \frac {36 \, {\left (d x e + c e\right )}^{3} A B b^{2} \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{3}} - \frac {12 \, {\left (d x e + c e\right )}^{3} B^{2} b^{2} \log \left (\frac {d x e + c e}{b x + a}\right )}{{\left (b x + a\right )}^{3}} + \frac {18 \, {\left (d x e + c e\right )}^{3} A^{2} b^{2}}{{\left (b x + a\right )}^{3}} - \frac {12 \, {\left (d x e + c e\right )}^{3} A B b^{2}}{{\left (b x + a\right )}^{3}} + \frac {4 \, {\left (d x e + c e\right )}^{3} B^{2} b^{2}}{{\left (b x + a\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 )}}{54 \, {\left (b^{2} c^{2} g^{4} e^{2} - 2 \, a b c d g^{4} e^{2} + a^{2} d^{2} g^{4} e^{2}\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)^4,x, algorithm="giac")

[Out]

-1/54*(54*(d*x*e + c*e)*B^2*d^2*e^2*log((d*x*e + c*e)/(b*x + a))^2/(b*x + a) - 54*(d*x*e + c*e)^2*B^2*b*d*e*lo
g((d*x*e + c*e)/(b*x + a))^2/(b*x + a)^2 + 108*(d*x*e + c*e)*A*B*d^2*e^2*log((d*x*e + c*e)/(b*x + a))/(b*x + a
) - 108*(d*x*e + c*e)*B^2*d^2*e^2*log((d*x*e + c*e)/(b*x + a))/(b*x + a) - 108*(d*x*e + c*e)^2*A*B*b*d*e*log((
d*x*e + c*e)/(b*x + a))/(b*x + a)^2 + 54*(d*x*e + c*e)^2*B^2*b*d*e*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^2 +
18*(d*x*e + c*e)^3*B^2*b^2*log((d*x*e + c*e)/(b*x + a))^2/(b*x + a)^3 + 54*(d*x*e + c*e)*A^2*d^2*e^2/(b*x + a)
- 108*(d*x*e + c*e)*A*B*d^2*e^2/(b*x + a) + 108*(d*x*e + c*e)*B^2*d^2*e^2/(b*x + a) - 54*(d*x*e + c*e)^2*A^2*
b*d*e/(b*x + a)^2 + 54*(d*x*e + c*e)^2*A*B*b*d*e/(b*x + a)^2 - 27*(d*x*e + c*e)^2*B^2*b*d*e/(b*x + a)^2 + 36*(
d*x*e + c*e)^3*A*B*b^2*log((d*x*e + c*e)/(b*x + a))/(b*x + a)^3 - 12*(d*x*e + c*e)^3*B^2*b^2*log((d*x*e + c*e)
/(b*x + a))/(b*x + a)^3 + 18*(d*x*e + c*e)^3*A^2*b^2/(b*x + a)^3 - 12*(d*x*e + c*e)^3*A*B*b^2/(b*x + a)^3 + 4*
(d*x*e + c*e)^3*B^2*b^2/(b*x + a)^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^2*c^2*g^4*e^2 - 2*a*b*c*d*g^4*e^2 + a^2*d^2*g^4*e^2)

________________________________________________________________________________________

Mupad [B]
time = 7.70, size = 1064, normalized size = 2.67 \begin {gather*} \frac {\frac {18\,A^2\,a^2\,d^2-36\,A^2\,a\,b\,c\,d+18\,A^2\,b^2\,c^2-66\,A\,B\,a^2\,d^2+42\,A\,B\,a\,b\,c\,d-12\,A\,B\,b^2\,c^2+85\,B^2\,a^2\,d^2-23\,B^2\,a\,b\,c\,d+4\,B^2\,b^2\,c^2}{6\,\left (a\,d-b\,c\right )}+\frac {x\,\left (-5\,c\,B^2\,b^2\,d+49\,a\,B^2\,b\,d^2+6\,A\,c\,B\,b^2\,d-30\,A\,a\,B\,b\,d^2\right )}{2\,\left (a\,d-b\,c\right )}+\frac {d\,x^2\,\left (11\,B^2\,b^2\,d-6\,A\,B\,b^2\,d\right )}{a\,d-b\,c}}{x\,\left (27\,a^2\,b^3\,c\,g^4-27\,a^3\,b^2\,d\,g^4\right )-x^2\,\left (27\,a^2\,b^3\,d\,g^4-27\,a\,b^4\,c\,g^4\right )+x^3\,\left (9\,b^5\,c\,g^4-9\,a\,b^4\,d\,g^4\right )+9\,a^3\,b^2\,c\,g^4-9\,a^4\,b\,d\,g^4}-{\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )}^2\,\left (\frac {B^2}{3\,b^2\,g^4\,\left (3\,a^2\,x+\frac {a^3}{b}+b^2\,x^3+3\,a\,b\,x^2\right )}-\frac {B^2\,d^3}{3\,b\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )-\frac {\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )\,\left (\frac {2\,A\,B}{3\,b^2\,d\,g^4}-\frac {2\,B^2\,d^3\,\left (a\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {3\,a^3\,d^3-6\,a^2\,b\,c\,d^2+4\,a\,b^2\,c^2\,d-b^3\,c^3}{3\,b\,d^4}\right )}{3\,b\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {2\,B^2\,d^3\,x^2\,\left (\frac {b^2\,c-a\,b\,d}{3\,d^2}-\frac {2\,b\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,b\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {2\,B^2\,d^3\,x\,\left (b\,\left (\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{6\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{3\,b\,d^2}\right )+\frac {3\,a^2\,d^2-4\,a\,b\,c\,d+b^2\,c^2}{3\,d^3}+\frac {2\,a\,\left (a\,d-b\,c\right )}{3\,d^2}\right )}{3\,b\,g^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}\right )}{\frac {3\,a^2\,x}{d}+\frac {a^3}{b\,d}+\frac {b^2\,x^3}{d}+\frac {3\,a\,b\,x^2}{d}}-\frac {B\,d^3\,\mathrm {atan}\left (\frac {B\,d^3\,\left (\frac {a^3\,b\,d^3\,g^4-a^2\,b^2\,c\,d^2\,g^4-a\,b^3\,c^2\,d\,g^4+b^4\,c^3\,g^4}{a^2\,b\,d^2\,g^4-2\,a\,b^2\,c\,d\,g^4+b^3\,c^2\,g^4}+2\,b\,d\,x\right )\,\left (6\,A-11\,B\right )\,\left (a^2\,b\,d^2\,g^4-2\,a\,b^2\,c\,d\,g^4+b^3\,c^2\,g^4\right )\,1{}\mathrm {i}}{b\,g^4\,{\left (a\,d-b\,c\right )}^3\,\left (11\,B^2\,d^3-6\,A\,B\,d^3\right )}\right )\,\left (6\,A-11\,B\right )\,2{}\mathrm {i}}{9\,b\,g^4\,{\left (a\,d-b\,c\right )}^3} \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)^4,x)

[Out]

((18*A^2*a^2*d^2 + 18*A^2*b^2*c^2 + 85*B^2*a^2*d^2 + 4*B^2*b^2*c^2 - 66*A*B*a^2*d^2 - 12*A*B*b^2*c^2 - 36*A^2*
a*b*c*d - 23*B^2*a*b*c*d + 42*A*B*a*b*c*d)/(6*(a*d - b*c)) + (x*(49*B^2*a*b*d^2 - 5*B^2*b^2*c*d - 30*A*B*a*b*d
^2 + 6*A*B*b^2*c*d))/(2*(a*d - b*c)) + (d*x^2*(11*B^2*b^2*d - 6*A*B*b^2*d))/(a*d - b*c))/(x*(27*a^2*b^3*c*g^4
- 27*a^3*b^2*d*g^4) - x^2*(27*a^2*b^3*d*g^4 - 27*a*b^4*c*g^4) + x^3*(9*b^5*c*g^4 - 9*a*b^4*d*g^4) + 9*a^3*b^2*
c*g^4 - 9*a^4*b*d*g^4) - log((e*(c + d*x))/(a + b*x))^2*(B^2/(3*b^2*g^4*(3*a^2*x + a^3/b + b^2*x^3 + 3*a*b*x^2
)) - (B^2*d^3)/(3*b*g^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) - (log((e*(c + d*x))/(a + b*x))*
((2*A*B)/(3*b^2*d*g^4) - (2*B^2*d^3*(a*((3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(6*b*d^3) + (a*(a*d - b*c))/(3*b*d^2
)) + (3*a^3*d^3 - b^3*c^3 + 4*a*b^2*c^2*d - 6*a^2*b*c*d^2)/(3*b*d^4)))/(3*b*g^4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c
^2*d - 3*a^2*b*c*d^2)) + (2*B^2*d^3*x^2*((b^2*c - a*b*d)/(3*d^2) - (2*b*(a*d - b*c))/(3*d^2)))/(3*b*g^4*(a^3*d
^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (2*B^2*d^3*x*(b*((3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(6*b*d^3)
+ (a*(a*d - b*c))/(3*b*d^2)) + (3*a^2*d^2 + b^2*c^2 - 4*a*b*c*d)/(3*d^3) + (2*a*(a*d - b*c))/(3*d^2)))/(3*b*g^
4*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))))/((3*a^2*x)/d + a^3/(b*d) + (b^2*x^3)/d + (3*a*b*x^2)/
d) - (B*d^3*atan((B*d^3*((b^4*c^3*g^4 + a^3*b*d^3*g^4 - a*b^3*c^2*d*g^4 - a^2*b^2*c*d^2*g^4)/(b^3*c^2*g^4 + a^
2*b*d^2*g^4 - 2*a*b^2*c*d*g^4) + 2*b*d*x)*(6*A - 11*B)*(b^3*c^2*g^4 + a^2*b*d^2*g^4 - 2*a*b^2*c*d*g^4)*1i)/(b*
g^4*(a*d - b*c)^3*(11*B^2*d^3 - 6*A*B*d^3)))*(6*A - 11*B)*2i)/(9*b*g^4*(a*d - b*c)^3)

________________________________________________________________________________________