3.1.46 \(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{(a g+b g x)^4 (c i+d i x)^2} \, dx\) [46]

Optimal. Leaf size=457 \[ -\frac {B d^4 (a+b x)}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {6 b^2 B d^2 (c+d x)}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {b^3 B d (c+d x)^2}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {b^4 B (c+d x)^3}{9 (b c-a d)^5 g^4 i^2 (a+b x)^3}+\frac {2 b B d^3 \log ^2\left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^4 i^2}+\frac {d^4 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^4 i^2 (c+d x)}-\frac {6 b^2 d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)}+\frac {2 b^3 d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^4 i^2 (a+b x)^2}-\frac {b^4 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^5 g^4 i^2 (a+b x)^3}-\frac {4 b d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^4 i^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.21, antiderivative size = 457, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2562, 45, 2372, 2338} \begin {gather*} -\frac {b^4 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 g^4 i^2 (a+b x)^3 (b c-a d)^5}+\frac {2 b^3 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (a+b x)^2 (b c-a d)^5}-\frac {6 b^2 d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (a+b x) (b c-a d)^5}+\frac {d^4 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (c+d x) (b c-a d)^5}-\frac {4 b d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i^2 (b c-a d)^5}-\frac {b^4 B (c+d x)^3}{9 g^4 i^2 (a+b x)^3 (b c-a d)^5}+\frac {b^3 B d (c+d x)^2}{g^4 i^2 (a+b x)^2 (b c-a d)^5}-\frac {6 b^2 B d^2 (c+d x)}{g^4 i^2 (a+b x) (b c-a d)^5}-\frac {B d^4 (a+b x)}{g^4 i^2 (c+d x) (b c-a d)^5}+\frac {2 b B d^3 \log ^2\left (\frac {a+b x}{c+d x}\right )}{g^4 i^2 (b c-a d)^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((B*d^4*(a + b*x))/((b*c - a*d)^5*g^4*i^2*(c + d*x))) - (6*b^2*B*d^2*(c + d*x))/((b*c - a*d)^5*g^4*i^2*(a + b
*x)) + (b^3*B*d*(c + d*x)^2)/((b*c - a*d)^5*g^4*i^2*(a + b*x)^2) - (b^4*B*(c + d*x)^3)/(9*(b*c - a*d)^5*g^4*i^
2*(a + b*x)^3) + (2*b*B*d^3*Log[(a + b*x)/(c + d*x)]^2)/((b*c - a*d)^5*g^4*i^2) + (d^4*(a + b*x)*(A + B*Log[(e
*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5*g^4*i^2*(c + d*x)) - (6*b^2*d^2*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]))/((b*c - a*d)^5*g^4*i^2*(a + b*x)) + (2*b^3*d*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c
- a*d)^5*g^4*i^2*(a + b*x)^2) - (b^4*(c + d*x)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(3*(b*c - a*d)^5*g^4*i
^2*(a + b*x)^3) - (4*b*d^3*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5*g^4*i
^2)

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 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 2562

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

Rubi steps

\begin {align*} \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(46 c+46 d x)^2 (a g+b g x)^4} \, dx &=\int \left (\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^2 g^4 (a+b x)^4}-\frac {b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{1058 (b c-a d)^3 g^4 (a+b x)^3}+\frac {3 b^2 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)^2}-\frac {b^2 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)^2}+\frac {b d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4 (c+d x)}\right ) \, dx\\ &=-\frac {\left (b^2 d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{529 (b c-a d)^5 g^4}+\frac {\left (b d^4\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{529 (b c-a d)^5 g^4}+\frac {\left (3 b^2 d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{2116 (b c-a d)^4 g^4}+\frac {d^4 \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{2116 (b c-a d)^4 g^4}-\frac {\left (b^2 d\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{1058 (b c-a d)^3 g^4}+\frac {b^2 \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{2116 (b c-a d)^2 g^4}\\ &=-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {\left (b B d^3\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}{529 (b c-a d)^5 g^4}-\frac {\left (b B d^3\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}{529 (b c-a d)^5 g^4}+\frac {\left (3 b B d^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{2116 (b c-a d)^4 g^4}+\frac {\left (B d^3\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{2116 (b c-a d)^4 g^4}-\frac {(b B d) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{2116 (b c-a d)^3 g^4}+\frac {(b B) \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{6348 (b c-a d)^2 g^4}\\ &=-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {\left (3 b B d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{2116 (b c-a d)^3 g^4}+\frac {\left (B d^3\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{2116 (b c-a d)^3 g^4}-\frac {(b B d) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{2116 (b c-a d)^2 g^4}+\frac {(b B) \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{6348 (b c-a d) g^4}+\frac {\left (b B d^3\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}{529 (b c-a d)^5 e g^4}-\frac {\left (b B d^3\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}{529 (b c-a d)^5 e g^4}\\ &=-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {\left (3 b B 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}{2116 (b c-a d)^3 g^4}+\frac {\left (B d^3\right ) \int \left (\frac {b^2}{(b c-a d)^2 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^2}-\frac {b d}{(b c-a d)^2 (c+d x)}\right ) \, dx}{2116 (b c-a d)^3 g^4}-\frac {(b B d) \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}{2116 (b c-a d)^2 g^4}+\frac {(b B) \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}{6348 (b c-a d) g^4}+\frac {\left (b B 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}{529 (b c-a d)^5 e g^4}-\frac {\left (b B 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}{529 (b c-a d)^5 e g^4}\\ &=-\frac {b B}{19044 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b B d}{3174 (b c-a d)^3 g^4 (a+b x)^2}-\frac {13 b B d^2}{6348 (b c-a d)^4 g^4 (a+b x)}+\frac {B d^3}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {5 b B d^3 \log (a+b x)}{3174 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac {5 b B d^3 \log (c+d x)}{3174 (b c-a d)^5 g^4}+\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {\left (b^2 B d^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{529 (b c-a d)^5 g^4}-\frac {\left (b^2 B d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{529 (b c-a d)^5 g^4}-\frac {\left (b B d^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{529 (b c-a d)^5 g^4}+\frac {\left (b B d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{529 (b c-a d)^5 g^4}\\ &=-\frac {b B}{19044 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b B d}{3174 (b c-a d)^3 g^4 (a+b x)^2}-\frac {13 b B d^2}{6348 (b c-a d)^4 g^4 (a+b x)}+\frac {B d^3}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {5 b B d^3 \log (a+b x)}{3174 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac {5 b B d^3 \log (c+d x)}{3174 (b c-a d)^5 g^4}-\frac {b B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}-\frac {b B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{529 (b c-a d)^5 g^4}+\frac {\left (b B d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{529 (b c-a d)^5 g^4}+\frac {\left (b B d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{529 (b c-a d)^5 g^4}+\frac {\left (b^2 B d^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{529 (b c-a d)^5 g^4}+\frac {\left (b B d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{529 (b c-a d)^5 g^4}\\ &=-\frac {b B}{19044 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b B d}{3174 (b c-a d)^3 g^4 (a+b x)^2}-\frac {13 b B d^2}{6348 (b c-a d)^4 g^4 (a+b x)}+\frac {B d^3}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {5 b B d^3 \log (a+b x)}{3174 (b c-a d)^5 g^4}+\frac {b B d^3 \log ^2(a+b x)}{1058 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac {5 b B d^3 \log (c+d x)}{3174 (b c-a d)^5 g^4}-\frac {b B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {b B d^3 \log ^2(c+d x)}{1058 (b c-a d)^5 g^4}-\frac {b B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{529 (b c-a d)^5 g^4}+\frac {\left (b B 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 )}{529 (b c-a d)^5 g^4}+\frac {\left (b B 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 )}{529 (b c-a d)^5 g^4}\\ &=-\frac {b B}{19044 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b B d}{3174 (b c-a d)^3 g^4 (a+b x)^2}-\frac {13 b B d^2}{6348 (b c-a d)^4 g^4 (a+b x)}+\frac {B d^3}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {5 b B d^3 \log (a+b x)}{3174 (b c-a d)^5 g^4}+\frac {b B d^3 \log ^2(a+b x)}{1058 (b c-a d)^5 g^4}-\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{6348 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^3 g^4 (a+b x)^2}-\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2116 (b c-a d)^4 g^4 (c+d x)}-\frac {b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{529 (b c-a d)^5 g^4}+\frac {5 b B d^3 \log (c+d x)}{3174 (b c-a d)^5 g^4}-\frac {b B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{529 (b c-a d)^5 g^4}+\frac {b B d^3 \log ^2(c+d x)}{1058 (b c-a d)^5 g^4}-\frac {b B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{529 (b c-a d)^5 g^4}-\frac {b B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{529 (b c-a d)^5 g^4}-\frac {b B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{529 (b c-a d)^5 g^4}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.84, size = 520, normalized size = 1.14 \begin {gather*} -\frac {\frac {b B (b c-a d)^3}{(a+b x)^3}-\frac {6 b B d (b c-a d)^2}{(a+b x)^2}+\frac {27 b^2 B c d^2}{a+b x}-\frac {27 a b B d^3}{a+b x}+\frac {12 b B d^2 (b c-a d)}{a+b x}-\frac {9 b B c d^3}{c+d x}+\frac {9 a B d^4}{c+d x}+30 b B d^3 \log (a+b x)+\frac {3 b (b c-a d)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^3}-\frac {9 b d (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(a+b x)^2}+\frac {27 b d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}-\frac {9 d^3 (-b c+a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x}+36 b d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-30 b B d^3 \log (c+d x)-36 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-18 b B d^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 B d^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 )}{9 (b c-a d)^5 g^4 i^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/9*((b*B*(b*c - a*d)^3)/(a + b*x)^3 - (6*b*B*d*(b*c - a*d)^2)/(a + b*x)^2 + (27*b^2*B*c*d^2)/(a + b*x) - (27
*a*b*B*d^3)/(a + b*x) + (12*b*B*d^2*(b*c - a*d))/(a + b*x) - (9*b*B*c*d^3)/(c + d*x) + (9*a*B*d^4)/(c + d*x) +
30*b*B*d^3*Log[a + b*x] + (3*b*(b*c - a*d)^3*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x)^3 - (9*b*d*(b*c
- a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x)^2 + (27*b*d^2*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]))/(a + b*x) - (9*d^3*(-(b*c) + a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x) + 36*b*d^3*Log[a
+ b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 30*b*B*d^3*Log[c + d*x] - 36*b*d^3*(A + B*Log[(e*(a + b*x))/(c +
d*x)])*Log[c + d*x] - 18*b*B*d^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*B*d^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)^5*g^4*i^2)

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Maple [A]
time = 0.91, size = 803, normalized size = 1.76 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 2424 vs. \(2 (429) = 858\).
time = 0.57, size = 2424, normalized size = 5.30 \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)))/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x, algorithm="maxima")

[Out]

1/3*B*((12*b^3*d^3*x^3 + b^3*c^3 - 5*a*b^2*c^2*d + 13*a^2*b*c*d^2 + 3*a^3*d^3 + 6*(b^3*c*d^2 + 5*a*b^2*d^3)*x^
2 - 2*(b^3*c^2*d - 8*a*b^2*c*d^2 - 11*a^2*b*d^3)*x)/((b^7*c^4*d - 4*a*b^6*c^3*d^2 + 6*a^2*b^5*c^2*d^3 - 4*a^3*
b^4*c*d^4 + a^4*b^3*d^5)*g^4*x^4 + (b^7*c^5 - a*b^6*c^4*d - 6*a^2*b^5*c^3*d^2 + 14*a^3*b^4*c^2*d^3 - 11*a^4*b^
3*c*d^4 + 3*a^5*b^2*d^5)*g^4*x^3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^4*d + 2*a^3*b^4*c^3*d^2 + 2*a^4*b^3*c^2*d^3 - 3*
a^5*b^2*c*d^4 + a^6*b*d^5)*g^4*x^2 + (3*a^2*b^5*c^5 - 11*a^3*b^4*c^4*d + 14*a^4*b^3*c^3*d^2 - 6*a^5*b^2*c^2*d^
3 - a^6*b*c*d^4 + a^7*d^5)*g^4*x + (a^3*b^4*c^5 - 4*a^4*b^3*c^4*d + 6*a^5*b^2*c^3*d^2 - 4*a^6*b*c^2*d^3 + a^7*
c*d^4)*g^4) + 12*b*d^3*log(b*x + a)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^
4*b*c*d^4 - a^5*d^5)*g^4) - 12*b*d^3*log(d*x + c)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*
c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4))*log(b*x*e/(d*x + c) + a*e/(d*x + c)) + 1/3*A*((12*b^3*d^3*x^3 + b^3*c
^3 - 5*a*b^2*c^2*d + 13*a^2*b*c*d^2 + 3*a^3*d^3 + 6*(b^3*c*d^2 + 5*a*b^2*d^3)*x^2 - 2*(b^3*c^2*d - 8*a*b^2*c*d
^2 - 11*a^2*b*d^3)*x)/((b^7*c^4*d - 4*a*b^6*c^3*d^2 + 6*a^2*b^5*c^2*d^3 - 4*a^3*b^4*c*d^4 + a^4*b^3*d^5)*g^4*x
^4 + (b^7*c^5 - a*b^6*c^4*d - 6*a^2*b^5*c^3*d^2 + 14*a^3*b^4*c^2*d^3 - 11*a^4*b^3*c*d^4 + 3*a^5*b^2*d^5)*g^4*x
^3 + 3*(a*b^6*c^5 - 3*a^2*b^5*c^4*d + 2*a^3*b^4*c^3*d^2 + 2*a^4*b^3*c^2*d^3 - 3*a^5*b^2*c*d^4 + a^6*b*d^5)*g^4
*x^2 + (3*a^2*b^5*c^5 - 11*a^3*b^4*c^4*d + 14*a^4*b^3*c^3*d^2 - 6*a^5*b^2*c^2*d^3 - a^6*b*c*d^4 + a^7*d^5)*g^4
*x + (a^3*b^4*c^5 - 4*a^4*b^3*c^4*d + 6*a^5*b^2*c^3*d^2 - 4*a^6*b*c^2*d^3 + a^7*c*d^4)*g^4) + 12*b*d^3*log(b*x
+ a)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*d^5)*g^4) - 12
*b*d^3*log(d*x + c)/((b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5*a^4*b*c*d^4 - a^5*
d^5)*g^4)) + 1/9*(b^4*c^4 - 9*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 - 55*a^3*b*c*d^3 + 9*a^4*d^4 + 30*(b^4*c*d^3 -
a*b^3*d^4)*x^3 + 3*(11*b^4*c^2*d^2 + 8*a*b^3*c*d^3 - 19*a^2*b^2*d^4)*x^2 - 18*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^
4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a)
^2 - 18*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^
2*b^2*c*d^3 + a^3*b*d^4)*x)*log(d*x + c)^2 - (5*b^4*c^3*d - 81*a*b^3*c^2*d^2 + 57*a^2*b^2*c*d^3 + 19*a^3*b*d^4
)*x + 30*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a
^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a) - 6*(5*b^4*d^4*x^4 + 5*a^3*b*c*d^3 + 5*(b^4*c*d^3 + 3*a*b^3*d^4)*x^3
+ 15*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + 5*(3*a^2*b^2*c*d^3 + a^3*b*d^4)*x - 6*(b^4*d^4*x^4 + a^3*b*c*d^3 + (b^
4*c*d^3 + 3*a*b^3*d^4)*x^3 + 3*(a*b^3*c*d^3 + a^2*b^2*d^4)*x^2 + (3*a^2*b^2*c*d^3 + a^3*b*d^4)*x)*log(b*x + a)
)*log(d*x + c))*B/(a^3*b^5*c^6*g^4 - 5*a^4*b^4*c^5*d*g^4 + 10*a^5*b^3*c^4*d^2*g^4 - 10*a^6*b^2*c^3*d^3*g^4 + 5
*a^7*b*c^2*d^4*g^4 - a^8*c*d^5*g^4 + (b^8*c^5*d*g^4 - 5*a*b^7*c^4*d^2*g^4 + 10*a^2*b^6*c^3*d^3*g^4 - 10*a^3*b^
5*c^2*d^4*g^4 + 5*a^4*b^4*c*d^5*g^4 - a^5*b^3*d^6*g^4)*x^4 + (b^8*c^6*g^4 - 2*a*b^7*c^5*d*g^4 - 5*a^2*b^6*c^4*
d^2*g^4 + 20*a^3*b^5*c^3*d^3*g^4 - 25*a^4*b^4*c^2*d^4*g^4 + 14*a^5*b^3*c*d^5*g^4 - 3*a^6*b^2*d^6*g^4)*x^3 + 3*
(a*b^7*c^6*g^4 - 4*a^2*b^6*c^5*d*g^4 + 5*a^3*b^5*c^4*d^2*g^4 - 5*a^5*b^3*c^2*d^4*g^4 + 4*a^6*b^2*c*d^5*g^4 - a
^7*b*d^6*g^4)*x^2 + (3*a^2*b^6*c^6*g^4 - 14*a^3*b^5*c^5*d*g^4 + 25*a^4*b^4*c^4*d^2*g^4 - 20*a^5*b^3*c^3*d^3*g^
4 + 5*a^6*b^2*c^2*d^4*g^4 + 2*a^7*b*c*d^5*g^4 - a^8*d^6*g^4)*x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1002 vs. \(2 (429) = 858\).
time = 0.43, size = 1002, normalized size = 2.19 \begin {gather*} \frac {{\left (3 \, A + B\right )} b^{4} c^{4} - 9 \, {\left (2 \, A + B\right )} a b^{3} c^{3} d + 54 \, {\left (A + B\right )} a^{2} b^{2} c^{2} d^{2} - 5 \, {\left (6 \, A + 11 \, B\right )} a^{3} b c d^{3} - 9 \, {\left (A - B\right )} a^{4} d^{4} + 6 \, {\left ({\left (6 \, A + 5 \, B\right )} b^{4} c d^{3} - {\left (6 \, A + 5 \, B\right )} a b^{3} d^{4}\right )} x^{3} + 3 \, {\left ({\left (6 \, A + 11 \, B\right )} b^{4} c^{2} d^{2} + 8 \, {\left (3 \, A + B\right )} a b^{3} c d^{3} - {\left (30 \, A + 19 \, B\right )} a^{2} b^{2} d^{4}\right )} x^{2} + 18 \, {\left (B b^{4} d^{4} x^{4} + B a^{3} b c d^{3} + {\left (B b^{4} c d^{3} + 3 \, B a b^{3} d^{4}\right )} x^{3} + 3 \, {\left (B a b^{3} c d^{3} + B a^{2} b^{2} d^{4}\right )} x^{2} + {\left (3 \, B a^{2} b^{2} c d^{3} + B a^{3} b d^{4}\right )} x\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )^{2} - {\left ({\left (6 \, A + 5 \, B\right )} b^{4} c^{3} d - 27 \, {\left (2 \, A + 3 \, B\right )} a b^{3} c^{2} d^{2} - 3 \, {\left (6 \, A - 19 \, B\right )} a^{2} b^{2} c d^{3} + {\left (66 \, A + 19 \, B\right )} a^{3} b d^{4}\right )} x + 3 \, {\left (2 \, {\left (6 \, A + 5 \, B\right )} b^{4} d^{4} x^{4} + B b^{4} c^{4} - 6 \, B a b^{3} c^{3} d + 18 \, B a^{2} b^{2} c^{2} d^{2} + 12 \, A a^{3} b c d^{3} - 3 \, B a^{4} d^{4} + 2 \, {\left ({\left (6 \, A + 11 \, B\right )} b^{4} c d^{3} + 9 \, {\left (2 \, A + B\right )} a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (B b^{4} c^{2} d^{2} + 3 \, {\left (2 \, A + 3 \, B\right )} a b^{3} c d^{3} + 6 \, A a^{2} b^{2} d^{4}\right )} x^{2} - 2 \, {\left (B b^{4} c^{3} d - 9 \, B a b^{3} c^{2} d^{2} - 18 \, {\left (A + B\right )} a^{2} b^{2} c d^{3} - 6 \, {\left (A - B\right )} a^{3} b d^{4}\right )} x\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{9 \, {\left ({\left (b^{8} c^{5} d - 5 \, a b^{7} c^{4} d^{2} + 10 \, a^{2} b^{6} c^{3} d^{3} - 10 \, a^{3} b^{5} c^{2} d^{4} + 5 \, a^{4} b^{4} c d^{5} - a^{5} b^{3} d^{6}\right )} g^{4} x^{4} + {\left (b^{8} c^{6} - 2 \, a b^{7} c^{5} d - 5 \, a^{2} b^{6} c^{4} d^{2} + 20 \, a^{3} b^{5} c^{3} d^{3} - 25 \, a^{4} b^{4} c^{2} d^{4} + 14 \, a^{5} b^{3} c d^{5} - 3 \, a^{6} b^{2} d^{6}\right )} g^{4} x^{3} + 3 \, {\left (a b^{7} c^{6} - 4 \, a^{2} b^{6} c^{5} d + 5 \, a^{3} b^{5} c^{4} d^{2} - 5 \, a^{5} b^{3} c^{2} d^{4} + 4 \, a^{6} b^{2} c d^{5} - a^{7} b d^{6}\right )} g^{4} x^{2} + {\left (3 \, a^{2} b^{6} c^{6} - 14 \, a^{3} b^{5} c^{5} d + 25 \, a^{4} b^{4} c^{4} d^{2} - 20 \, a^{5} b^{3} c^{3} d^{3} + 5 \, a^{6} b^{2} c^{2} d^{4} + 2 \, a^{7} b c d^{5} - a^{8} d^{6}\right )} g^{4} x + {\left (a^{3} b^{5} c^{6} - 5 \, a^{4} b^{4} c^{5} d + 10 \, a^{5} b^{3} c^{4} d^{2} - 10 \, a^{6} b^{2} c^{3} d^{3} + 5 \, a^{7} b c^{2} d^{4} - a^{8} c d^{5}\right )} g^{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)))/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x, algorithm="fricas")

[Out]

1/9*((3*A + B)*b^4*c^4 - 9*(2*A + B)*a*b^3*c^3*d + 54*(A + B)*a^2*b^2*c^2*d^2 - 5*(6*A + 11*B)*a^3*b*c*d^3 - 9
*(A - B)*a^4*d^4 + 6*((6*A + 5*B)*b^4*c*d^3 - (6*A + 5*B)*a*b^3*d^4)*x^3 + 3*((6*A + 11*B)*b^4*c^2*d^2 + 8*(3*
A + B)*a*b^3*c*d^3 - (30*A + 19*B)*a^2*b^2*d^4)*x^2 + 18*(B*b^4*d^4*x^4 + B*a^3*b*c*d^3 + (B*b^4*c*d^3 + 3*B*a
*b^3*d^4)*x^3 + 3*(B*a*b^3*c*d^3 + B*a^2*b^2*d^4)*x^2 + (3*B*a^2*b^2*c*d^3 + B*a^3*b*d^4)*x)*log((b*x + a)*e/(
d*x + c))^2 - ((6*A + 5*B)*b^4*c^3*d - 27*(2*A + 3*B)*a*b^3*c^2*d^2 - 3*(6*A - 19*B)*a^2*b^2*c*d^3 + (66*A + 1
9*B)*a^3*b*d^4)*x + 3*(2*(6*A + 5*B)*b^4*d^4*x^4 + B*b^4*c^4 - 6*B*a*b^3*c^3*d + 18*B*a^2*b^2*c^2*d^2 + 12*A*a
^3*b*c*d^3 - 3*B*a^4*d^4 + 2*((6*A + 11*B)*b^4*c*d^3 + 9*(2*A + B)*a*b^3*d^4)*x^3 + 6*(B*b^4*c^2*d^2 + 3*(2*A
+ 3*B)*a*b^3*c*d^3 + 6*A*a^2*b^2*d^4)*x^2 - 2*(B*b^4*c^3*d - 9*B*a*b^3*c^2*d^2 - 18*(A + B)*a^2*b^2*c*d^3 - 6*
(A - B)*a^3*b*d^4)*x)*log((b*x + a)*e/(d*x + c)))/((b^8*c^5*d - 5*a*b^7*c^4*d^2 + 10*a^2*b^6*c^3*d^3 - 10*a^3*
b^5*c^2*d^4 + 5*a^4*b^4*c*d^5 - a^5*b^3*d^6)*g^4*x^4 + (b^8*c^6 - 2*a*b^7*c^5*d - 5*a^2*b^6*c^4*d^2 + 20*a^3*b
^5*c^3*d^3 - 25*a^4*b^4*c^2*d^4 + 14*a^5*b^3*c*d^5 - 3*a^6*b^2*d^6)*g^4*x^3 + 3*(a*b^7*c^6 - 4*a^2*b^6*c^5*d +
5*a^3*b^5*c^4*d^2 - 5*a^5*b^3*c^2*d^4 + 4*a^6*b^2*c*d^5 - a^7*b*d^6)*g^4*x^2 + (3*a^2*b^6*c^6 - 14*a^3*b^5*c^
5*d + 25*a^4*b^4*c^4*d^2 - 20*a^5*b^3*c^3*d^3 + 5*a^6*b^2*c^2*d^4 + 2*a^7*b*c*d^5 - a^8*d^6)*g^4*x + (a^3*b^5*
c^6 - 5*a^4*b^4*c^5*d + 10*a^5*b^3*c^4*d^2 - 10*a^6*b^2*c^3*d^3 + 5*a^7*b*c^2*d^4 - a^8*c*d^5)*g^4)

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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)))/(b*g*x+a*g)**4/(d*i*x+c*i)**2,x)

[Out]

Timed out

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Giac [A]
time = 74.76, size = 384, normalized size = 0.84 \begin {gather*} \frac {{\left (6 \, B b^{2} e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) - \frac {18 \, {\left (b x e + a e\right )} B b d e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} + \frac {18 \, {\left (b x e + a e\right )}^{2} B d^{2} e^{2} \log \left (\frac {b x e + a e}{d x + c}\right )}{{\left (d x + c\right )}^{2}} + 6 \, A b^{2} e^{4} + 2 \, B b^{2} e^{4} - \frac {18 \, {\left (b x e + a e\right )} A b d e^{3}}{d x + c} - \frac {9 \, {\left (b x e + a e\right )} B b d e^{3}}{d x + c} + \frac {18 \, {\left (b x e + a e\right )}^{2} A d^{2} e^{2}}{{\left (d x + c\right )}^{2}} + \frac {18 \, {\left (b x e + a e\right )}^{2} B d^{2} e^{2}}{{\left (d x + c\right )}^{2}}\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 )}^{2}}{18 \, {\left (\frac {{\left (b x e + a e\right )}^{3} b^{2} c^{2} g^{4}}{{\left (d x + c\right )}^{3}} - \frac {2 \, {\left (b x e + a e\right )}^{3} a b c d g^{4}}{{\left (d x + c\right )}^{3}} + \frac {{\left (b x e + a e\right )}^{3} a^{2} d^{2} g^{4}}{{\left (d x + c\right )}^{3}}\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)))/(b*g*x+a*g)^4/(d*i*x+c*i)^2,x, algorithm="giac")

[Out]

1/18*(6*B*b^2*e^4*log((b*x*e + a*e)/(d*x + c)) - 18*(b*x*e + a*e)*B*b*d*e^3*log((b*x*e + a*e)/(d*x + c))/(d*x
+ c) + 18*(b*x*e + a*e)^2*B*d^2*e^2*log((b*x*e + a*e)/(d*x + c))/(d*x + c)^2 + 6*A*b^2*e^4 + 2*B*b^2*e^4 - 18*
(b*x*e + a*e)*A*b*d*e^3/(d*x + c) - 9*(b*x*e + a*e)*B*b*d*e^3/(d*x + c) + 18*(b*x*e + a*e)^2*A*d^2*e^2/(d*x +
c)^2 + 18*(b*x*e + a*e)^2*B*d^2*e^2/(d*x + c)^2)*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*
c - a*d)))^2/((b*x*e + a*e)^3*b^2*c^2*g^4/(d*x + c)^3 - 2*(b*x*e + a*e)^3*a*b*c*d*g^4/(d*x + c)^3 + (b*x*e + a
*e)^3*a^2*d^2*g^4/(d*x + c)^3)

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Mupad [B]
time = 12.44, size = 1679, normalized size = 3.67 \begin {gather*} \frac {2\,B\,b\,d^3\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{g^4\,i^2\,{\left (a\,d-b\,c\right )}^2\,\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 {\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (x\,\left (\frac {4\,B}{3\,g^4\,i^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}+\frac {4\,B\,b\,d^3\,\left (\left (\frac {2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2}{2\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,b\,d^2}\right )\,\left (a\,d+b\,c\right )+\frac {a\,c\,\left (a\,d-b\,c\right )}{d^2}\right )}{g^4\,i^2\,{\left (a\,d-b\,c\right )}^2\,\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 {B\,\left (3\,a\,d+b\,c\right )}{3\,g^4\,i^2\,\left (a^2\,b\,d^3-2\,a\,b^2\,c\,d^2+b^3\,c^2\,d\right )}+\frac {4\,B\,b^2\,d^2\,x^3}{g^4\,i^2\,\left (a\,d-b\,c\right )\,\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 {4\,B\,b\,d^3\,x^2\,\left (b\,d\,\left (\frac {2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2}{2\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,b\,d^2}\right )+\frac {\left (a\,d+b\,c\right )\,\left (a\,d-b\,c\right )}{d^2}\right )}{g^4\,i^2\,{\left (a\,d-b\,c\right )}^2\,\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 {4\,B\,a\,b\,c\,d^3\,\left (\frac {2\,a^2\,d^2-3\,a\,b\,c\,d+b^2\,c^2}{2\,b\,d^3}+\frac {a\,\left (a\,d-b\,c\right )}{2\,b\,d^2}\right )}{g^4\,i^2\,{\left (a\,d-b\,c\right )}^2\,\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 )}{b^2\,x^4+\frac {a^3\,c}{b\,d}+\frac {x\,\left (d\,a^3+3\,b\,c\,a^2\right )}{b\,d}+\frac {x^3\,\left (c\,b^3+3\,a\,d\,b^2\right )}{b\,d}+\frac {x^2\,\left (3\,d\,a^2\,b+3\,c\,a\,b^2\right )}{b\,d}}-\frac {\frac {9\,A\,a^3\,d^3+3\,A\,b^3\,c^3-9\,B\,a^3\,d^3+B\,b^3\,c^3-15\,A\,a\,b^2\,c^2\,d+39\,A\,a^2\,b\,c\,d^2-8\,B\,a\,b^2\,c^2\,d+46\,B\,a^2\,b\,c\,d^2}{3\,\left (a\,d-b\,c\right )}+\frac {x\,\left (66\,A\,a^2\,b\,d^3+19\,B\,a^2\,b\,d^3-6\,A\,b^3\,c^2\,d-5\,B\,b^3\,c^2\,d+48\,A\,a\,b^2\,c\,d^2+76\,B\,a\,b^2\,c\,d^2\right )}{3\,\left (a\,d-b\,c\right )}+\frac {x^2\,\left (30\,A\,a\,b^2\,d^3+19\,B\,a\,b^2\,d^3+6\,A\,b^3\,c\,d^2+11\,B\,b^3\,c\,d^2\right )}{a\,d-b\,c}+\frac {2\,x^3\,\left (6\,A\,b^3\,d^3+5\,B\,b^3\,d^3\right )}{a\,d-b\,c}}{x\,\left (3\,a^6\,d^4\,g^4\,i^2-18\,a^4\,b^2\,c^2\,d^2\,g^4\,i^2+24\,a^3\,b^3\,c^3\,d\,g^4\,i^2-9\,a^2\,b^4\,c^4\,g^4\,i^2\right )-x^2\,\left (-9\,a^5\,b\,d^4\,g^4\,i^2+18\,a^4\,b^2\,c\,d^3\,g^4\,i^2-18\,a^2\,b^4\,c^3\,d\,g^4\,i^2+9\,a\,b^5\,c^4\,g^4\,i^2\right )-x^3\,\left (-9\,a^4\,b^2\,d^4\,g^4\,i^2+24\,a^3\,b^3\,c\,d^3\,g^4\,i^2-18\,a^2\,b^4\,c^2\,d^2\,g^4\,i^2+3\,b^6\,c^4\,g^4\,i^2\right )+x^4\,\left (3\,a^3\,b^3\,d^4\,g^4\,i^2-9\,a^2\,b^4\,c\,d^3\,g^4\,i^2+9\,a\,b^5\,c^2\,d^2\,g^4\,i^2-3\,b^6\,c^3\,d\,g^4\,i^2\right )-3\,a^3\,b^3\,c^4\,g^4\,i^2+3\,a^6\,c\,d^3\,g^4\,i^2+9\,a^4\,b^2\,c^3\,d\,g^4\,i^2-9\,a^5\,b\,c^2\,d^2\,g^4\,i^2}-\frac {b\,d^3\,\mathrm {atan}\left (\frac {b\,d^3\,\left (\frac {a^5\,d^5\,g^4\,i^2-3\,a^4\,b\,c\,d^4\,g^4\,i^2+2\,a^3\,b^2\,c^2\,d^3\,g^4\,i^2+2\,a^2\,b^3\,c^3\,d^2\,g^4\,i^2-3\,a\,b^4\,c^4\,d\,g^4\,i^2+b^5\,c^5\,g^4\,i^2}{a^4\,d^4\,g^4\,i^2-4\,a^3\,b\,c\,d^3\,g^4\,i^2+6\,a^2\,b^2\,c^2\,d^2\,g^4\,i^2-4\,a\,b^3\,c^3\,d\,g^4\,i^2+b^4\,c^4\,g^4\,i^2}+2\,b\,d\,x\right )\,\left (6\,A+5\,B\right )\,\left (a^4\,d^4\,g^4\,i^2-4\,a^3\,b\,c\,d^3\,g^4\,i^2+6\,a^2\,b^2\,c^2\,d^2\,g^4\,i^2-4\,a\,b^3\,c^3\,d\,g^4\,i^2+b^4\,c^4\,g^4\,i^2\right )\,2{}\mathrm {i}}{g^4\,i^2\,{\left (a\,d-b\,c\right )}^5\,\left (12\,A\,b\,d^3+10\,B\,b\,d^3\right )}\right )\,\left (6\,A+5\,B\right )\,4{}\mathrm {i}}{3\,g^4\,i^2\,{\left (a\,d-b\,c\right )}^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*B*b*d^3*log((e*(a + b*x))/(c + d*x))^2)/(g^4*i^2*(a*d - b*c)^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b
*c*d^2)) - (log((e*(a + b*x))/(c + d*x))*(x*((4*B)/(3*g^4*i^2*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (4*B*b*d^3*((
(2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)/(2*b*d^3) + (a*(a*d - b*c))/(2*b*d^2))*(a*d + b*c) + (a*c*(a*d - b*c))/d^2))
/(g^4*i^2*(a*d - b*c)^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))) + (B*(3*a*d + b*c))/(3*g^4*i^2*(
a^2*b*d^3 + b^3*c^2*d - 2*a*b^2*c*d^2)) + (4*B*b^2*d^2*x^3)/(g^4*i^2*(a*d - b*c)*(a^3*d^3 - b^3*c^3 + 3*a*b^2*
c^2*d - 3*a^2*b*c*d^2)) + (4*B*b*d^3*x^2*(b*d*((2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)/(2*b*d^3) + (a*(a*d - b*c))/(
2*b*d^2)) + ((a*d + b*c)*(a*d - b*c))/d^2))/(g^4*i^2*(a*d - b*c)^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*
b*c*d^2)) + (4*B*a*b*c*d^3*((2*a^2*d^2 + b^2*c^2 - 3*a*b*c*d)/(2*b*d^3) + (a*(a*d - b*c))/(2*b*d^2)))/(g^4*i^2
*(a*d - b*c)^2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))))/(b^2*x^4 + (a^3*c)/(b*d) + (x*(a^3*d + 3
*a^2*b*c))/(b*d) + (x^3*(b^3*c + 3*a*b^2*d))/(b*d) + (x^2*(3*a*b^2*c + 3*a^2*b*d))/(b*d)) - (b*d^3*atan((b*d^3
*((a^5*d^5*g^4*i^2 + b^5*c^5*g^4*i^2 - 3*a*b^4*c^4*d*g^4*i^2 - 3*a^4*b*c*d^4*g^4*i^2 + 2*a^2*b^3*c^3*d^2*g^4*i
^2 + 2*a^3*b^2*c^2*d^3*g^4*i^2)/(a^4*d^4*g^4*i^2 + b^4*c^4*g^4*i^2 - 4*a*b^3*c^3*d*g^4*i^2 - 4*a^3*b*c*d^3*g^4
*i^2 + 6*a^2*b^2*c^2*d^2*g^4*i^2) + 2*b*d*x)*(6*A + 5*B)*(a^4*d^4*g^4*i^2 + b^4*c^4*g^4*i^2 - 4*a*b^3*c^3*d*g^
4*i^2 - 4*a^3*b*c*d^3*g^4*i^2 + 6*a^2*b^2*c^2*d^2*g^4*i^2)*2i)/(g^4*i^2*(a*d - b*c)^5*(12*A*b*d^3 + 10*B*b*d^3
)))*(6*A + 5*B)*4i)/(3*g^4*i^2*(a*d - b*c)^5) - ((9*A*a^3*d^3 + 3*A*b^3*c^3 - 9*B*a^3*d^3 + B*b^3*c^3 - 15*A*a
*b^2*c^2*d + 39*A*a^2*b*c*d^2 - 8*B*a*b^2*c^2*d + 46*B*a^2*b*c*d^2)/(3*(a*d - b*c)) + (x*(66*A*a^2*b*d^3 + 19*
B*a^2*b*d^3 - 6*A*b^3*c^2*d - 5*B*b^3*c^2*d + 48*A*a*b^2*c*d^2 + 76*B*a*b^2*c*d^2))/(3*(a*d - b*c)) + (x^2*(30
*A*a*b^2*d^3 + 19*B*a*b^2*d^3 + 6*A*b^3*c*d^2 + 11*B*b^3*c*d^2))/(a*d - b*c) + (2*x^3*(6*A*b^3*d^3 + 5*B*b^3*d
^3))/(a*d - b*c))/(x*(3*a^6*d^4*g^4*i^2 - 9*a^2*b^4*c^4*g^4*i^2 + 24*a^3*b^3*c^3*d*g^4*i^2 - 18*a^4*b^2*c^2*d^
2*g^4*i^2) - x^2*(9*a*b^5*c^4*g^4*i^2 - 9*a^5*b*d^4*g^4*i^2 - 18*a^2*b^4*c^3*d*g^4*i^2 + 18*a^4*b^2*c*d^3*g^4*
i^2) - x^3*(3*b^6*c^4*g^4*i^2 - 9*a^4*b^2*d^4*g^4*i^2 + 24*a^3*b^3*c*d^3*g^4*i^2 - 18*a^2*b^4*c^2*d^2*g^4*i^2)
+ x^4*(3*a^3*b^3*d^4*g^4*i^2 - 3*b^6*c^3*d*g^4*i^2 + 9*a*b^5*c^2*d^2*g^4*i^2 - 9*a^2*b^4*c*d^3*g^4*i^2) - 3*a
^3*b^3*c^4*g^4*i^2 + 3*a^6*c*d^3*g^4*i^2 + 9*a^4*b^2*c^3*d*g^4*i^2 - 9*a^5*b*c^2*d^2*g^4*i^2)

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