3.1.38 \(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{(a g+b g x)^4 (c i+d i x)} \, dx\) [38]

Optimal. Leaf size=373 \[ -\frac {3 b B d^2 (c+d x)}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 B d (c+d x)^2}{4 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {b^3 B (c+d x)^3}{9 (b c-a d)^4 g^4 i (a+b x)^3}+\frac {B d^3 \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^4 g^4 i}-\frac {3 b d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {b^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^4 g^4 i (a+b x)^3}-\frac {d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^4 i} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.20, antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2562, 45, 2372, 12, 14, 2338} \begin {gather*} -\frac {b^3 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 g^4 i (a+b x)^3 (b c-a d)^4}+\frac {3 b^2 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^4 i (a+b x)^2 (b c-a d)^4}-\frac {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 (b c-a d)^4}-\frac {3 b d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i (a+b x) (b c-a d)^4}-\frac {b^3 B (c+d x)^3}{9 g^4 i (a+b x)^3 (b c-a d)^4}+\frac {3 b^2 B d (c+d x)^2}{4 g^4 i (a+b x)^2 (b c-a d)^4}+\frac {B d^3 \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g^4 i (b c-a d)^4}-\frac {3 b B d^2 (c+d x)}{g^4 i (a+b x) (b c-a d)^4} \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)),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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 )}{(38 c+38 d x) (a g+b g x)^4} \, dx &=\int \left (\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d) g^4 (a+b x)^4}-\frac {b d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^2 g^4 (a+b x)^3}+\frac {b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^3 g^4 (a+b x)^2}-\frac {b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4 (a+b x)}+\frac {d^4 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4 (c+d x)}\right ) \, dx\\ &=-\frac {\left (b d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{38 (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} \, dx}{38 (b c-a d)^4 g^4}+\frac {\left (b d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{38 (b c-a d)^3 g^4}-\frac {(b d) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{38 (b c-a d)^2 g^4}+\frac {b \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4} \, dx}{38 (b c-a d) g^4}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{114 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{76 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac {\left (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}{38 (b c-a d)^4 g^4}-\frac {\left (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}{38 (b c-a d)^4 g^4}+\frac {\left (B d^2\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{38 (b c-a d)^3 g^4}-\frac {(B d) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{76 (b c-a d)^2 g^4}+\frac {B \int \frac {b c-a d}{(a+b x)^4 (c+d x)} \, dx}{114 (b c-a d) g^4}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{114 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{76 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac {B \int \frac {1}{(a+b x)^4 (c+d x)} \, dx}{114 g^4}+\frac {\left (B d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{38 (b c-a d)^2 g^4}-\frac {(B d) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{76 (b c-a d) g^4}+\frac {\left (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}{38 (b c-a d)^4 e g^4}-\frac {\left (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}{38 (b c-a d)^4 e g^4}\\ &=-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{114 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{76 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac {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}{114 g^4}+\frac {\left (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}{38 (b c-a d)^2 g^4}-\frac {(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}{76 (b c-a d) g^4}+\frac {\left (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}{38 (b c-a d)^4 e g^4}-\frac {\left (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}{38 (b c-a d)^4 e g^4}\\ &=-\frac {B}{342 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d}{456 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2}{228 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 \log (a+b x)}{228 (b c-a d)^4 g^4}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{114 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{76 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4}+\frac {11 B d^3 \log (c+d x)}{228 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac {\left (b B d^3\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{38 (b c-a d)^4 g^4}-\frac {\left (b B d^3\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{38 (b c-a d)^4 g^4}-\frac {\left (B d^4\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{38 (b c-a d)^4 g^4}+\frac {\left (B d^4\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{38 (b c-a d)^4 g^4}\\ &=-\frac {B}{342 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d}{456 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2}{228 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 \log (a+b x)}{228 (b c-a d)^4 g^4}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{114 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{76 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4}+\frac {11 B d^3 \log (c+d x)}{228 (b c-a d)^4 g^4}-\frac {B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}-\frac {B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{38 (b c-a d)^4 g^4}+\frac {\left (B d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{38 (b c-a d)^4 g^4}+\frac {\left (B d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{38 (b c-a d)^4 g^4}+\frac {\left (b B d^3\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{38 (b c-a d)^4 g^4}+\frac {\left (B d^4\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{38 (b c-a d)^4 g^4}\\ &=-\frac {B}{342 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d}{456 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2}{228 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 \log (a+b x)}{228 (b c-a d)^4 g^4}+\frac {B d^3 \log ^2(a+b x)}{76 (b c-a d)^4 g^4}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{114 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{76 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4}+\frac {11 B d^3 \log (c+d x)}{228 (b c-a d)^4 g^4}-\frac {B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac {B d^3 \log ^2(c+d x)}{76 (b c-a d)^4 g^4}-\frac {B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{38 (b c-a d)^4 g^4}+\frac {\left (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 )}{38 (b c-a d)^4 g^4}+\frac {\left (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 )}{38 (b c-a d)^4 g^4}\\ &=-\frac {B}{342 (b c-a d) g^4 (a+b x)^3}+\frac {5 B d}{456 (b c-a d)^2 g^4 (a+b x)^2}-\frac {11 B d^2}{228 (b c-a d)^3 g^4 (a+b x)}-\frac {11 B d^3 \log (a+b x)}{228 (b c-a d)^4 g^4}+\frac {B d^3 \log ^2(a+b x)}{76 (b c-a d)^4 g^4}-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{114 (b c-a d) g^4 (a+b x)^3}+\frac {d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{76 (b c-a d)^2 g^4 (a+b x)^2}-\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^3 g^4 (a+b x)}-\frac {d^3 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{38 (b c-a d)^4 g^4}+\frac {11 B d^3 \log (c+d x)}{228 (b c-a d)^4 g^4}-\frac {B d^3 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{38 (b c-a d)^4 g^4}+\frac {B d^3 \log ^2(c+d x)}{76 (b c-a d)^4 g^4}-\frac {B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{38 (b c-a d)^4 g^4}-\frac {B d^3 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{38 (b c-a d)^4 g^4}-\frac {B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{38 (b c-a d)^4 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 = 0.45, size = 492, normalized size = 1.32 \begin {gather*} \frac {-\frac {12 A (b c-a d)^3}{(a+b x)^3}-\frac {4 B (b c-a d)^3}{(a+b x)^3}+\frac {18 A d (b c-a d)^2}{(a+b x)^2}+\frac {15 B d (b c-a d)^2}{(a+b x)^2}+\frac {36 A d^2 (-b c+a d)}{a+b x}+\frac {66 B d^2 (-b c+a d)}{a+b x}-36 A d^3 \log (a+b x)-66 B d^3 \log (a+b x)+18 B d^3 \log ^2(a+b x)-\frac {12 B (b c-a d)^3 \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3}+\frac {18 B d (b c-a d)^2 \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2}+\frac {36 B d^2 (-b c+a d) \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x}-36 B d^3 \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+36 A d^3 \log (c+d x)+66 B d^3 \log (c+d x)-36 B d^3 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+36 B d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)+18 B d^3 \log ^2(c+d x)-36 B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )-36 B d^3 \text {Li}_2\left (\frac {d (a+b x)}{-b c+a d}\right )-36 B d^3 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{36 (b c-a d)^4 g^4 i} \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)),x]

[Out]

((-12*A*(b*c - a*d)^3)/(a + b*x)^3 - (4*B*(b*c - a*d)^3)/(a + b*x)^3 + (18*A*d*(b*c - a*d)^2)/(a + b*x)^2 + (1
5*B*d*(b*c - a*d)^2)/(a + b*x)^2 + (36*A*d^2*(-(b*c) + a*d))/(a + b*x) + (66*B*d^2*(-(b*c) + a*d))/(a + b*x) -
36*A*d^3*Log[a + b*x] - 66*B*d^3*Log[a + b*x] + 18*B*d^3*Log[a + b*x]^2 - (12*B*(b*c - a*d)^3*Log[(e*(a + b*x
))/(c + d*x)])/(a + b*x)^3 + (18*B*d*(b*c - a*d)^2*Log[(e*(a + b*x))/(c + d*x)])/(a + b*x)^2 + (36*B*d^2*(-(b*
c) + a*d)*Log[(e*(a + b*x))/(c + d*x)])/(a + b*x) - 36*B*d^3*Log[a + b*x]*Log[(e*(a + b*x))/(c + d*x)] + 36*A*
d^3*Log[c + d*x] + 66*B*d^3*Log[c + d*x] - 36*B*d^3*Log[(d*(a + b*x))/(-(b*c) + a*d)]*Log[c + d*x] + 36*B*d^3*
Log[(e*(a + b*x))/(c + d*x)]*Log[c + d*x] + 18*B*d^3*Log[c + d*x]^2 - 36*B*d^3*Log[a + b*x]*Log[(b*(c + d*x))/
(b*c - a*d)] - 36*B*d^3*PolyLog[2, (d*(a + b*x))/(-(b*c) + a*d)] - 36*B*d^3*PolyLog[2, (b*(c + d*x))/(b*c - a*
d)])/(36*(b*c - a*d)^4*g^4*i)

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Maple [A]
time = 0.84, size = 637, normalized size = 1.71 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),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1456 vs. \(2 (343) = 686\).
time = 0.54, size = 1456, normalized size = 3.90 \begin {gather*} -\frac {1}{6} \, 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 (i \, b^{6} c^{3} - 3 i \, a b^{5} c^{2} d + 3 i \, a^{2} b^{4} c d^{2} - i \, a^{3} b^{3} d^{3}\right )} g^{4} x^{3} + 3 \, {\left (i \, a b^{5} c^{3} - 3 i \, a^{2} b^{4} c^{2} d + 3 i \, a^{3} b^{3} c d^{2} - i \, a^{4} b^{2} d^{3}\right )} g^{4} x^{2} + 3 \, {\left (i \, a^{2} b^{4} c^{3} - 3 i \, a^{3} b^{3} c^{2} d + 3 i \, a^{4} b^{2} c d^{2} - i \, a^{5} b d^{3}\right )} g^{4} x + {\left (i \, a^{3} b^{3} c^{3} - 3 i \, a^{4} b^{2} c^{2} d + 3 i \, a^{5} b c d^{2} - i \, a^{6} d^{3}\right )} g^{4}} + \frac {6 \, d^{3} \log \left (b x + a\right )}{{\left (i \, b^{4} c^{4} - 4 i \, a b^{3} c^{3} d + 6 i \, a^{2} b^{2} c^{2} d^{2} - 4 i \, a^{3} b c d^{3} + i \, a^{4} d^{4}\right )} g^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{{\left (i \, b^{4} c^{4} - 4 i \, a b^{3} c^{3} d + 6 i \, a^{2} b^{2} c^{2} d^{2} - 4 i \, a^{3} b c d^{3} + i \, a^{4} d^{4}\right )} g^{4}}\right )} \log \left (\frac {b x e}{d x + c} + \frac {a e}{d x + c}\right ) - \frac {1}{6} \, A {\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 (i \, b^{6} c^{3} - 3 i \, a b^{5} c^{2} d + 3 i \, a^{2} b^{4} c d^{2} - i \, a^{3} b^{3} d^{3}\right )} g^{4} x^{3} + 3 \, {\left (i \, a b^{5} c^{3} - 3 i \, a^{2} b^{4} c^{2} d + 3 i \, a^{3} b^{3} c d^{2} - i \, a^{4} b^{2} d^{3}\right )} g^{4} x^{2} + 3 \, {\left (i \, a^{2} b^{4} c^{3} - 3 i \, a^{3} b^{3} c^{2} d + 3 i \, a^{4} b^{2} c d^{2} - i \, a^{5} b d^{3}\right )} g^{4} x + {\left (i \, a^{3} b^{3} c^{3} - 3 i \, a^{4} b^{2} c^{2} d + 3 i \, a^{5} b c d^{2} - i \, a^{6} d^{3}\right )} g^{4}} + \frac {6 \, d^{3} \log \left (b x + a\right )}{{\left (i \, b^{4} c^{4} - 4 i \, a b^{3} c^{3} d + 6 i \, a^{2} b^{2} c^{2} d^{2} - 4 i \, a^{3} b c d^{3} + i \, a^{4} d^{4}\right )} g^{4}} - \frac {6 \, d^{3} \log \left (d x + c\right )}{{\left (i \, b^{4} c^{4} - 4 i \, a b^{3} c^{3} d + 6 i \, a^{2} b^{2} c^{2} d^{2} - 4 i \, a^{3} b c d^{3} + i \, a^{4} d^{4}\right )} g^{4}}\right )} - \frac {{\left (-4 i \, b^{3} c^{3} + 27 i \, a b^{2} c^{2} d - 108 i \, a^{2} b c d^{2} + 85 i \, a^{3} d^{3} - 66 \, {\left (i \, b^{3} c d^{2} - i \, a b^{2} d^{3}\right )} x^{2} - 18 \, {\left (-i \, b^{3} d^{3} x^{3} - 3 i \, a b^{2} d^{3} x^{2} - 3 i \, a^{2} b d^{3} x - i \, a^{3} d^{3}\right )} \log \left (b x + a\right )^{2} - 18 \, {\left (-i \, b^{3} d^{3} x^{3} - 3 i \, a b^{2} d^{3} x^{2} - 3 i \, a^{2} b d^{3} x - i \, a^{3} d^{3}\right )} \log \left (d x + c\right )^{2} - 3 \, {\left (-5 i \, b^{3} c^{2} d + 54 i \, a b^{2} c d^{2} - 49 i \, a^{2} b d^{3}\right )} x - 66 \, {\left (i \, b^{3} d^{3} x^{3} + 3 i \, a b^{2} d^{3} x^{2} + 3 i \, a^{2} b d^{3} x + i \, a^{3} d^{3}\right )} \log \left (b x + a\right ) - 6 \, {\left (-11 i \, b^{3} d^{3} x^{3} - 33 i \, a b^{2} d^{3} x^{2} - 33 i \, a^{2} b d^{3} x - 11 i \, a^{3} d^{3} + 6 \, {\left (i \, b^{3} d^{3} x^{3} + 3 i \, a b^{2} d^{3} x^{2} + 3 i \, a^{2} b d^{3} x + i \, a^{3} d^{3}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B}{36 \, {\left (a^{3} b^{4} c^{4} g^{4} - 4 \, a^{4} b^{3} c^{3} d g^{4} + 6 \, a^{5} b^{2} c^{2} d^{2} g^{4} - 4 \, a^{6} b c d^{3} g^{4} + a^{7} d^{4} g^{4} + {\left (b^{7} c^{4} g^{4} - 4 \, a b^{6} c^{3} d g^{4} + 6 \, a^{2} b^{5} c^{2} d^{2} g^{4} - 4 \, a^{3} b^{4} c d^{3} g^{4} + a^{4} b^{3} d^{4} g^{4}\right )} x^{3} + 3 \, {\left (a b^{6} c^{4} g^{4} - 4 \, a^{2} b^{5} c^{3} d g^{4} + 6 \, a^{3} b^{4} c^{2} d^{2} g^{4} - 4 \, a^{4} b^{3} c d^{3} g^{4} + a^{5} b^{2} d^{4} g^{4}\right )} x^{2} + 3 \, {\left (a^{2} b^{5} c^{4} g^{4} - 4 \, a^{3} b^{4} c^{3} d g^{4} + 6 \, a^{4} b^{3} c^{2} d^{2} g^{4} - 4 \, a^{5} b^{2} c d^{3} g^{4} + a^{6} b d^{4} g^{4}\right )} x\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),x, algorithm="maxima")

[Out]

-1/6*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)/((I*b^6*c^3 - 3*I*a*b
^5*c^2*d + 3*I*a^2*b^4*c*d^2 - I*a^3*b^3*d^3)*g^4*x^3 + 3*(I*a*b^5*c^3 - 3*I*a^2*b^4*c^2*d + 3*I*a^3*b^3*c*d^2
- I*a^4*b^2*d^3)*g^4*x^2 + 3*(I*a^2*b^4*c^3 - 3*I*a^3*b^3*c^2*d + 3*I*a^4*b^2*c*d^2 - I*a^5*b*d^3)*g^4*x + (I
*a^3*b^3*c^3 - 3*I*a^4*b^2*c^2*d + 3*I*a^5*b*c*d^2 - I*a^6*d^3)*g^4) + 6*d^3*log(b*x + a)/((I*b^4*c^4 - 4*I*a*
b^3*c^3*d + 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*g^4) - 6*d^3*log(d*x + c)/((I*b^4*c^4 - 4*I*a*b
^3*c^3*d + 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*g^4))*log(b*x*e/(d*x + c) + a*e/(d*x + c)) - 1/6
*A*((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)/((I*b^6*c^3 - 3*I*a*b^5*c
^2*d + 3*I*a^2*b^4*c*d^2 - I*a^3*b^3*d^3)*g^4*x^3 + 3*(I*a*b^5*c^3 - 3*I*a^2*b^4*c^2*d + 3*I*a^3*b^3*c*d^2 - I
*a^4*b^2*d^3)*g^4*x^2 + 3*(I*a^2*b^4*c^3 - 3*I*a^3*b^3*c^2*d + 3*I*a^4*b^2*c*d^2 - I*a^5*b*d^3)*g^4*x + (I*a^3
*b^3*c^3 - 3*I*a^4*b^2*c^2*d + 3*I*a^5*b*c*d^2 - I*a^6*d^3)*g^4) + 6*d^3*log(b*x + a)/((I*b^4*c^4 - 4*I*a*b^3*
c^3*d + 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*g^4) - 6*d^3*log(d*x + c)/((I*b^4*c^4 - 4*I*a*b^3*c
^3*d + 6*I*a^2*b^2*c^2*d^2 - 4*I*a^3*b*c*d^3 + I*a^4*d^4)*g^4)) - 1/36*(-4*I*b^3*c^3 + 27*I*a*b^2*c^2*d - 108*
I*a^2*b*c*d^2 + 85*I*a^3*d^3 - 66*(I*b^3*c*d^2 - I*a*b^2*d^3)*x^2 - 18*(-I*b^3*d^3*x^3 - 3*I*a*b^2*d^3*x^2 - 3
*I*a^2*b*d^3*x - I*a^3*d^3)*log(b*x + a)^2 - 18*(-I*b^3*d^3*x^3 - 3*I*a*b^2*d^3*x^2 - 3*I*a^2*b*d^3*x - I*a^3*
d^3)*log(d*x + c)^2 - 3*(-5*I*b^3*c^2*d + 54*I*a*b^2*c*d^2 - 49*I*a^2*b*d^3)*x - 66*(I*b^3*d^3*x^3 + 3*I*a*b^2
*d^3*x^2 + 3*I*a^2*b*d^3*x + I*a^3*d^3)*log(b*x + a) - 6*(-11*I*b^3*d^3*x^3 - 33*I*a*b^2*d^3*x^2 - 33*I*a^2*b*
d^3*x - 11*I*a^3*d^3 + 6*(I*b^3*d^3*x^3 + 3*I*a*b^2*d^3*x^2 + 3*I*a^2*b*d^3*x + I*a^3*d^3)*log(b*x + a))*log(d
*x + c))*B/(a^3*b^4*c^4*g^4 - 4*a^4*b^3*c^3*d*g^4 + 6*a^5*b^2*c^2*d^2*g^4 - 4*a^6*b*c*d^3*g^4 + a^7*d^4*g^4 +
(b^7*c^4*g^4 - 4*a*b^6*c^3*d*g^4 + 6*a^2*b^5*c^2*d^2*g^4 - 4*a^3*b^4*c*d^3*g^4 + a^4*b^3*d^4*g^4)*x^3 + 3*(a*b
^6*c^4*g^4 - 4*a^2*b^5*c^3*d*g^4 + 6*a^3*b^4*c^2*d^2*g^4 - 4*a^4*b^3*c*d^3*g^4 + a^5*b^2*d^4*g^4)*x^2 + 3*(a^2
*b^5*c^4*g^4 - 4*a^3*b^4*c^3*d*g^4 + 6*a^4*b^3*c^2*d^2*g^4 - 4*a^5*b^2*c*d^3*g^4 + a^6*b*d^4*g^4)*x)

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Fricas [A]
time = 0.38, size = 619, normalized size = 1.66 \begin {gather*} -\frac {4 \, {\left (-3 i \, A - i \, B\right )} b^{3} c^{3} + 27 \, {\left (2 i \, A + i \, B\right )} a b^{2} c^{2} d + 108 \, {\left (-i \, A - i \, B\right )} a^{2} b c d^{2} - {\left (-66 i \, A - 85 i \, B\right )} a^{3} d^{3} + 6 \, {\left ({\left (-6 i \, A - 11 i \, B\right )} b^{3} c d^{2} + {\left (6 i \, A + 11 i \, B\right )} a b^{2} d^{3}\right )} x^{2} + 18 \, {\left (-i \, B b^{3} d^{3} x^{3} - 3 i \, B a b^{2} d^{3} x^{2} - 3 i \, B a^{2} b d^{3} x - i \, B a^{3} d^{3}\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )^{2} + 3 \, {\left ({\left (6 i \, A + 5 i \, B\right )} b^{3} c^{2} d + 18 \, {\left (-2 i \, A - 3 i \, B\right )} a b^{2} c d^{2} + {\left (30 i \, A + 49 i \, B\right )} a^{2} b d^{3}\right )} x + 6 \, {\left ({\left (-6 i \, A - 11 i \, B\right )} b^{3} d^{3} x^{3} - 2 i \, B b^{3} c^{3} + 9 i \, B a b^{2} c^{2} d - 18 i \, B a^{2} b c d^{2} - 6 i \, A a^{3} d^{3} + 3 \, {\left (-2 i \, B b^{3} c d^{2} + 3 \, {\left (-2 i \, A - 3 i \, B\right )} a b^{2} d^{3}\right )} x^{2} + 3 \, {\left (i \, B b^{3} c^{2} d - 6 i \, B a b^{2} c d^{2} + 6 \, {\left (-i \, A - i \, B\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{36 \, {\left ({\left (b^{7} c^{4} - 4 \, a b^{6} c^{3} d + 6 \, a^{2} b^{5} c^{2} d^{2} - 4 \, a^{3} b^{4} c d^{3} + a^{4} b^{3} d^{4}\right )} g^{4} x^{3} + 3 \, {\left (a b^{6} c^{4} - 4 \, a^{2} b^{5} c^{3} d + 6 \, a^{3} b^{4} c^{2} d^{2} - 4 \, a^{4} b^{3} c d^{3} + a^{5} b^{2} d^{4}\right )} g^{4} x^{2} + 3 \, {\left (a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}\right )} g^{4} x + {\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\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),x, algorithm="fricas")

[Out]

-1/36*(4*(-3*I*A - I*B)*b^3*c^3 + 27*(2*I*A + I*B)*a*b^2*c^2*d + 108*(-I*A - I*B)*a^2*b*c*d^2 - (-66*I*A - 85*
I*B)*a^3*d^3 + 6*((-6*I*A - 11*I*B)*b^3*c*d^2 + (6*I*A + 11*I*B)*a*b^2*d^3)*x^2 + 18*(-I*B*b^3*d^3*x^3 - 3*I*B
*a*b^2*d^3*x^2 - 3*I*B*a^2*b*d^3*x - I*B*a^3*d^3)*log((b*x + a)*e/(d*x + c))^2 + 3*((6*I*A + 5*I*B)*b^3*c^2*d
+ 18*(-2*I*A - 3*I*B)*a*b^2*c*d^2 + (30*I*A + 49*I*B)*a^2*b*d^3)*x + 6*((-6*I*A - 11*I*B)*b^3*d^3*x^3 - 2*I*B*
b^3*c^3 + 9*I*B*a*b^2*c^2*d - 18*I*B*a^2*b*c*d^2 - 6*I*A*a^3*d^3 + 3*(-2*I*B*b^3*c*d^2 + 3*(-2*I*A - 3*I*B)*a*
b^2*d^3)*x^2 + 3*(I*B*b^3*c^2*d - 6*I*B*a*b^2*c*d^2 + 6*(-I*A - I*B)*a^2*b*d^3)*x)*log((b*x + a)*e/(d*x + c)))
/((b^7*c^4 - 4*a*b^6*c^3*d + 6*a^2*b^5*c^2*d^2 - 4*a^3*b^4*c*d^3 + a^4*b^3*d^4)*g^4*x^3 + 3*(a*b^6*c^4 - 4*a^2
*b^5*c^3*d + 6*a^3*b^4*c^2*d^2 - 4*a^4*b^3*c*d^3 + a^5*b^2*d^4)*g^4*x^2 + 3*(a^2*b^5*c^4 - 4*a^3*b^4*c^3*d + 6
*a^4*b^3*c^2*d^2 - 4*a^5*b^2*c*d^3 + a^6*b*d^4)*g^4*x + (a^3*b^4*c^4 - 4*a^4*b^3*c^3*d + 6*a^5*b^2*c^2*d^2 - 4
*a^6*b*c*d^3 + a^7*d^4)*g^4)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1392 vs. \(2 (332) = 664\).
time = 12.04, size = 1392, normalized size = 3.73 \begin {gather*} - \frac {B d^{3} \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{4} d^{4} g^{4} i - 8 a^{3} b c d^{3} g^{4} i + 12 a^{2} b^{2} c^{2} d^{2} g^{4} i - 8 a b^{3} c^{3} d g^{4} i + 2 b^{4} c^{4} g^{4} i} + \frac {d^{3} \cdot \left (6 A + 11 B\right ) \log {\left (x + \frac {6 A a d^{4} + 6 A b c d^{3} + 11 B a d^{4} + 11 B b c d^{3} - \frac {a^{5} d^{8} \cdot \left (6 A + 11 B\right )}{\left (a d - b c\right )^{4}} + \frac {5 a^{4} b c d^{7} \cdot \left (6 A + 11 B\right )}{\left (a d - b c\right )^{4}} - \frac {10 a^{3} b^{2} c^{2} d^{6} \cdot \left (6 A + 11 B\right )}{\left (a d - b c\right )^{4}} + \frac {10 a^{2} b^{3} c^{3} d^{5} \cdot \left (6 A + 11 B\right )}{\left (a d - b c\right )^{4}} - \frac {5 a b^{4} c^{4} d^{4} \cdot \left (6 A + 11 B\right )}{\left (a d - b c\right )^{4}} + \frac {b^{5} c^{5} d^{3} \cdot \left (6 A + 11 B\right )}{\left (a d - b c\right )^{4}}}{12 A b d^{4} + 22 B b d^{4}} \right )}}{6 g^{4} i \left (a d - b c\right )^{4}} - \frac {d^{3} \cdot \left (6 A + 11 B\right ) \log {\left (x + \frac {6 A a d^{4} + 6 A b c d^{3} + 11 B a d^{4} + 11 B b c d^{3} + \frac {a^{5} d^{8} \cdot \left (6 A + 11 B\right )}{\left (a d - b c\right )^{4}} - \frac {5 a^{4} b c d^{7} \cdot \left (6 A + 11 B\right )}{\left (a d - b c\right )^{4}} + \frac {10 a^{3} b^{2} c^{2} d^{6} \cdot \left (6 A + 11 B\right )}{\left (a d - b c\right )^{4}} - \frac {10 a^{2} b^{3} c^{3} d^{5} \cdot \left (6 A + 11 B\right )}{\left (a d - b c\right )^{4}} + \frac {5 a b^{4} c^{4} d^{4} \cdot \left (6 A + 11 B\right )}{\left (a d - b c\right )^{4}} - \frac {b^{5} c^{5} d^{3} \cdot \left (6 A + 11 B\right )}{\left (a d - b c\right )^{4}}}{12 A b d^{4} + 22 B b d^{4}} \right )}}{6 g^{4} i \left (a d - b c\right )^{4}} + \frac {\left (11 B a^{2} d^{2} - 7 B a b c d + 15 B a b d^{2} x + 2 B b^{2} c^{2} - 3 B b^{2} c d x + 6 B b^{2} d^{2} x^{2}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{6 a^{6} d^{3} g^{4} i - 18 a^{5} b c d^{2} g^{4} i + 18 a^{5} b d^{3} g^{4} i x + 18 a^{4} b^{2} c^{2} d g^{4} i - 54 a^{4} b^{2} c d^{2} g^{4} i x + 18 a^{4} b^{2} d^{3} g^{4} i x^{2} - 6 a^{3} b^{3} c^{3} g^{4} i + 54 a^{3} b^{3} c^{2} d g^{4} i x - 54 a^{3} b^{3} c d^{2} g^{4} i x^{2} + 6 a^{3} b^{3} d^{3} g^{4} i x^{3} - 18 a^{2} b^{4} c^{3} g^{4} i x + 54 a^{2} b^{4} c^{2} d g^{4} i x^{2} - 18 a^{2} b^{4} c d^{2} g^{4} i x^{3} - 18 a b^{5} c^{3} g^{4} i x^{2} + 18 a b^{5} c^{2} d g^{4} i x^{3} - 6 b^{6} c^{3} g^{4} i x^{3}} + \frac {66 A a^{2} d^{2} - 42 A a b c d + 12 A b^{2} c^{2} + 85 B a^{2} d^{2} - 23 B a b c d + 4 B b^{2} c^{2} + x^{2} \cdot \left (36 A b^{2} d^{2} + 66 B b^{2} d^{2}\right ) + x \left (90 A a b d^{2} - 18 A b^{2} c d + 147 B a b d^{2} - 15 B b^{2} c d\right )}{36 a^{6} d^{3} g^{4} i - 108 a^{5} b c d^{2} g^{4} i + 108 a^{4} b^{2} c^{2} d g^{4} i - 36 a^{3} b^{3} c^{3} g^{4} i + x^{3} \cdot \left (36 a^{3} b^{3} d^{3} g^{4} i - 108 a^{2} b^{4} c d^{2} g^{4} i + 108 a b^{5} c^{2} d g^{4} i - 36 b^{6} c^{3} g^{4} i\right ) + x^{2} \cdot \left (108 a^{4} b^{2} d^{3} g^{4} i - 324 a^{3} b^{3} c d^{2} g^{4} i + 324 a^{2} b^{4} c^{2} d g^{4} i - 108 a b^{5} c^{3} g^{4} i\right ) + x \left (108 a^{5} b d^{3} g^{4} i - 324 a^{4} b^{2} c d^{2} g^{4} i + 324 a^{3} b^{3} c^{2} d g^{4} i - 108 a^{2} b^{4} c^{3} g^{4} i\right )} \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),x)

[Out]

-B*d**3*log(e*(a + b*x)/(c + d*x))**2/(2*a**4*d**4*g**4*i - 8*a**3*b*c*d**3*g**4*i + 12*a**2*b**2*c**2*d**2*g*
*4*i - 8*a*b**3*c**3*d*g**4*i + 2*b**4*c**4*g**4*i) + d**3*(6*A + 11*B)*log(x + (6*A*a*d**4 + 6*A*b*c*d**3 + 1
1*B*a*d**4 + 11*B*b*c*d**3 - a**5*d**8*(6*A + 11*B)/(a*d - b*c)**4 + 5*a**4*b*c*d**7*(6*A + 11*B)/(a*d - b*c)*
*4 - 10*a**3*b**2*c**2*d**6*(6*A + 11*B)/(a*d - b*c)**4 + 10*a**2*b**3*c**3*d**5*(6*A + 11*B)/(a*d - b*c)**4 -
5*a*b**4*c**4*d**4*(6*A + 11*B)/(a*d - b*c)**4 + b**5*c**5*d**3*(6*A + 11*B)/(a*d - b*c)**4)/(12*A*b*d**4 + 2
2*B*b*d**4))/(6*g**4*i*(a*d - b*c)**4) - d**3*(6*A + 11*B)*log(x + (6*A*a*d**4 + 6*A*b*c*d**3 + 11*B*a*d**4 +
11*B*b*c*d**3 + a**5*d**8*(6*A + 11*B)/(a*d - b*c)**4 - 5*a**4*b*c*d**7*(6*A + 11*B)/(a*d - b*c)**4 + 10*a**3*
b**2*c**2*d**6*(6*A + 11*B)/(a*d - b*c)**4 - 10*a**2*b**3*c**3*d**5*(6*A + 11*B)/(a*d - b*c)**4 + 5*a*b**4*c**
4*d**4*(6*A + 11*B)/(a*d - b*c)**4 - b**5*c**5*d**3*(6*A + 11*B)/(a*d - b*c)**4)/(12*A*b*d**4 + 22*B*b*d**4))/
(6*g**4*i*(a*d - b*c)**4) + (11*B*a**2*d**2 - 7*B*a*b*c*d + 15*B*a*b*d**2*x + 2*B*b**2*c**2 - 3*B*b**2*c*d*x +
6*B*b**2*d**2*x**2)*log(e*(a + b*x)/(c + d*x))/(6*a**6*d**3*g**4*i - 18*a**5*b*c*d**2*g**4*i + 18*a**5*b*d**3
*g**4*i*x + 18*a**4*b**2*c**2*d*g**4*i - 54*a**4*b**2*c*d**2*g**4*i*x + 18*a**4*b**2*d**3*g**4*i*x**2 - 6*a**3
*b**3*c**3*g**4*i + 54*a**3*b**3*c**2*d*g**4*i*x - 54*a**3*b**3*c*d**2*g**4*i*x**2 + 6*a**3*b**3*d**3*g**4*i*x
**3 - 18*a**2*b**4*c**3*g**4*i*x + 54*a**2*b**4*c**2*d*g**4*i*x**2 - 18*a**2*b**4*c*d**2*g**4*i*x**3 - 18*a*b*
*5*c**3*g**4*i*x**2 + 18*a*b**5*c**2*d*g**4*i*x**3 - 6*b**6*c**3*g**4*i*x**3) + (66*A*a**2*d**2 - 42*A*a*b*c*d
+ 12*A*b**2*c**2 + 85*B*a**2*d**2 - 23*B*a*b*c*d + 4*B*b**2*c**2 + x**2*(36*A*b**2*d**2 + 66*B*b**2*d**2) + x
*(90*A*a*b*d**2 - 18*A*b**2*c*d + 147*B*a*b*d**2 - 15*B*b**2*c*d))/(36*a**6*d**3*g**4*i - 108*a**5*b*c*d**2*g*
*4*i + 108*a**4*b**2*c**2*d*g**4*i - 36*a**3*b**3*c**3*g**4*i + x**3*(36*a**3*b**3*d**3*g**4*i - 108*a**2*b**4
*c*d**2*g**4*i + 108*a*b**5*c**2*d*g**4*i - 36*b**6*c**3*g**4*i) + x**2*(108*a**4*b**2*d**3*g**4*i - 324*a**3*
b**3*c*d**2*g**4*i + 324*a**2*b**4*c**2*d*g**4*i - 108*a*b**5*c**3*g**4*i) + x*(108*a**5*b*d**3*g**4*i - 324*a
**4*b**2*c*d**2*g**4*i + 324*a**3*b**3*c**2*d*g**4*i - 108*a**2*b**4*c**3*g**4*i))

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Giac [A]
time = 62.32, size = 240, normalized size = 0.64 \begin {gather*} -\frac {{\left (-12 i \, B b e^{4} \log \left (\frac {b x e + a e}{d x + c}\right ) + \frac {18 i \, {\left (b x e + a e\right )} B d e^{3} \log \left (\frac {b x e + a e}{d x + c}\right )}{d x + c} - 12 i \, A b e^{4} - 4 i \, B b e^{4} + \frac {18 i \, {\left (b x e + a e\right )} A d e^{3}}{d x + c} + \frac {9 i \, {\left (b x e + a e\right )} B d e^{3}}{d x + c}\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}}{36 \, {\left (\frac {{\left (b x e + a e\right )}^{3} b c g^{4}}{{\left (d x + c\right )}^{3}} - \frac {{\left (b x e + a e\right )}^{3} a d 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),x, algorithm="giac")

[Out]

-1/36*(-12*I*B*b*e^4*log((b*x*e + a*e)/(d*x + c)) + 18*I*(b*x*e + a*e)*B*d*e^3*log((b*x*e + a*e)/(d*x + c))/(d
*x + c) - 12*I*A*b*e^4 - 4*I*B*b*e^4 + 18*I*(b*x*e + a*e)*A*d*e^3/(d*x + c) + 9*I*(b*x*e + a*e)*B*d*e^3/(d*x +
c))*(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*c*g^4/(d*x +
c)^3 - (b*x*e + a*e)^3*a*d*g^4/(d*x + c)^3)

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Mupad [B]
time = 9.51, size = 970, normalized size = 2.60 \begin {gather*} \frac {11\,A\,a^2\,d^2}{6\,g^4\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^3}-\frac {B\,d^3\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{2\,g^4\,i\,{\left (a\,d-b\,c\right )}^4}+\frac {A\,b^2\,c^2}{3\,g^4\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^3}+\frac {85\,B\,a^2\,d^2}{36\,g^4\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^3}+\frac {B\,b^2\,c^2}{9\,g^4\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^3}+\frac {11\,B\,a^3\,d^3\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{6\,g^4\,i\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^3}-\frac {B\,b^3\,c^3\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{3\,g^4\,i\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^3}+\frac {A\,b^2\,d^2\,x^2}{g^4\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^3}+\frac {11\,B\,b^2\,d^2\,x^2}{6\,g^4\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^3}-\frac {7\,A\,a\,b\,c\,d}{6\,g^4\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^3}-\frac {23\,B\,a\,b\,c\,d}{36\,g^4\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^3}+\frac {5\,A\,a\,b\,d^2\,x}{2\,g^4\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^3}+\frac {49\,B\,a\,b\,d^2\,x}{12\,g^4\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^3}-\frac {A\,b^2\,c\,d\,x}{2\,g^4\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^3}-\frac {5\,B\,b^2\,c\,d\,x}{12\,g^4\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^3}+\frac {3\,B\,a\,b^2\,c^2\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^4\,i\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^3}-\frac {3\,B\,a^2\,b\,c\,d^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^4\,i\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^3}+\frac {5\,B\,a^2\,b\,d^3\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^4\,i\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^3}+\frac {B\,b^3\,c^2\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^4\,i\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^3}+\frac {B\,a\,b^2\,d^3\,x^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^4\,i\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^3}-\frac {B\,b^3\,c\,d^2\,x^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^4\,i\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^3}-\frac {3\,B\,a\,b^2\,c\,d^2\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^4\,i\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^3}+\frac {A\,d^3\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{g^4\,i\,{\left (a\,d-b\,c\right )}^4}+\frac {B\,d^3\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,11{}\mathrm {i}}{3\,g^4\,i\,{\left (a\,d-b\,c\right )}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(A*d^3*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*2i)/(g^4*i*(a*d - b*c)^4) + (B*d^3*atan((a*d*1i + b*c*1i
+ b*d*x*2i)/(a*d - b*c))*11i)/(3*g^4*i*(a*d - b*c)^4) - (B*d^3*log((e*(a + b*x))/(c + d*x))^2)/(2*g^4*i*(a*d
- b*c)^4) + (11*A*a^2*d^2)/(6*g^4*i*(a*d - b*c)^3*(a + b*x)^3) + (A*b^2*c^2)/(3*g^4*i*(a*d - b*c)^3*(a + b*x)^
3) + (85*B*a^2*d^2)/(36*g^4*i*(a*d - b*c)^3*(a + b*x)^3) + (B*b^2*c^2)/(9*g^4*i*(a*d - b*c)^3*(a + b*x)^3) + (
11*B*a^3*d^3*log((e*(a + b*x))/(c + d*x)))/(6*g^4*i*(a*d - b*c)^4*(a + b*x)^3) - (B*b^3*c^3*log((e*(a + b*x))/
(c + d*x)))/(3*g^4*i*(a*d - b*c)^4*(a + b*x)^3) + (A*b^2*d^2*x^2)/(g^4*i*(a*d - b*c)^3*(a + b*x)^3) + (11*B*b^
2*d^2*x^2)/(6*g^4*i*(a*d - b*c)^3*(a + b*x)^3) - (7*A*a*b*c*d)/(6*g^4*i*(a*d - b*c)^3*(a + b*x)^3) - (23*B*a*b
*c*d)/(36*g^4*i*(a*d - b*c)^3*(a + b*x)^3) + (5*A*a*b*d^2*x)/(2*g^4*i*(a*d - b*c)^3*(a + b*x)^3) + (49*B*a*b*d
^2*x)/(12*g^4*i*(a*d - b*c)^3*(a + b*x)^3) - (A*b^2*c*d*x)/(2*g^4*i*(a*d - b*c)^3*(a + b*x)^3) - (5*B*b^2*c*d*
x)/(12*g^4*i*(a*d - b*c)^3*(a + b*x)^3) + (3*B*a*b^2*c^2*d*log((e*(a + b*x))/(c + d*x)))/(2*g^4*i*(a*d - b*c)^
4*(a + b*x)^3) - (3*B*a^2*b*c*d^2*log((e*(a + b*x))/(c + d*x)))/(g^4*i*(a*d - b*c)^4*(a + b*x)^3) + (5*B*a^2*b
*d^3*x*log((e*(a + b*x))/(c + d*x)))/(2*g^4*i*(a*d - b*c)^4*(a + b*x)^3) + (B*b^3*c^2*d*x*log((e*(a + b*x))/(c
+ d*x)))/(2*g^4*i*(a*d - b*c)^4*(a + b*x)^3) + (B*a*b^2*d^3*x^2*log((e*(a + b*x))/(c + d*x)))/(g^4*i*(a*d - b
*c)^4*(a + b*x)^3) - (B*b^3*c*d^2*x^2*log((e*(a + b*x))/(c + d*x)))/(g^4*i*(a*d - b*c)^4*(a + b*x)^3) - (3*B*a
*b^2*c*d^2*x*log((e*(a + b*x))/(c + d*x)))/(g^4*i*(a*d - b*c)^4*(a + b*x)^3)

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