3.1.53 \(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{(a g+b g x)^3 (c i+d i x)^3} \, dx\) [53]

Optimal. Leaf size=463 \[ -\frac {B d^4 (a+b x)^2}{4 (b c-a d)^5 g^3 i^3 (c+d x)^2}+\frac {4 b B d^3 (a+b x)}{(b c-a d)^5 g^3 i^3 (c+d x)}+\frac {4 b^3 B d (c+d x)}{(b c-a d)^5 g^3 i^3 (a+b x)}-\frac {b^4 B (c+d x)^2}{4 (b c-a d)^5 g^3 i^3 (a+b x)^2}-\frac {3 b^2 B d^2 \log ^2\left (\frac {a+b x}{c+d x}\right )}{(b c-a d)^5 g^3 i^3}+\frac {d^4 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^5 g^3 i^3 (c+d x)^2}-\frac {4 b d^3 (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^3 i^3 (c+d x)}+\frac {4 b^3 d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^3 i^3 (a+b x)}-\frac {b^4 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^5 g^3 i^3 (a+b x)^2}+\frac {6 b^2 d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^5 g^3 i^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.21, antiderivative size = 463, normalized size of antiderivative = 1.00, number of steps used = 5, 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)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 i^3 (a+b x)^2 (b c-a d)^5}+\frac {4 b^3 d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^3 (a+b x) (b c-a d)^5}+\frac {6 b^2 d^2 \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^3 i^3 (b c-a d)^5}+\frac {d^4 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 i^3 (c+d x)^2 (b c-a d)^5}-\frac {4 b d^3 (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i^3 (c+d x) (b c-a d)^5}-\frac {b^4 B (c+d x)^2}{4 g^3 i^3 (a+b x)^2 (b c-a d)^5}+\frac {4 b^3 B d (c+d x)}{g^3 i^3 (a+b x) (b c-a d)^5}-\frac {3 b^2 B d^2 \log ^2\left (\frac {a+b x}{c+d x}\right )}{g^3 i^3 (b c-a d)^5}-\frac {B d^4 (a+b x)^2}{4 g^3 i^3 (c+d x)^2 (b c-a d)^5}+\frac {4 b B d^3 (a+b x)}{g^3 i^3 (c+d x) (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)^3*(c*i + d*i*x)^3),x]

[Out]

-1/4*(B*d^4*(a + b*x)^2)/((b*c - a*d)^5*g^3*i^3*(c + d*x)^2) + (4*b*B*d^3*(a + b*x))/((b*c - a*d)^5*g^3*i^3*(c
+ d*x)) + (4*b^3*B*d*(c + d*x))/((b*c - a*d)^5*g^3*i^3*(a + b*x)) - (b^4*B*(c + d*x)^2)/(4*(b*c - a*d)^5*g^3*
i^3*(a + b*x)^2) - (3*b^2*B*d^2*Log[(a + b*x)/(c + d*x)]^2)/((b*c - a*d)^5*g^3*i^3) + (d^4*(a + b*x)^2*(A + B*
Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^5*g^3*i^3*(c + d*x)^2) - (4*b*d^3*(a + b*x)*(A + B*Log[(e*(a + b
*x))/(c + d*x)]))/((b*c - a*d)^5*g^3*i^3*(c + d*x)) + (4*b^3*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))
/((b*c - a*d)^5*g^3*i^3*(a + b*x)) - (b^4*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^5*g
^3*i^3*(a + b*x)^2) + (6*b^2*d^2*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^5
*g^3*i^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 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 )}{(53 c+53 d x)^3 (a g+b g x)^3} \, dx &=\int \left (\frac {b^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^3 g^3 (a+b x)^3}-\frac {3 b^3 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (a+b x)^2}+\frac {6 b^3 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3 (a+b x)}-\frac {d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^3 g^3 (c+d x)^3}-\frac {3 b d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (c+d x)^2}-\frac {6 b^2 d^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3 (c+d x)}\right ) \, dx\\ &=\frac {\left (6 b^3 d^2\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x} \, dx}{148877 (b c-a d)^5 g^3}-\frac {\left (6 b^2 d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{c+d x} \, dx}{148877 (b c-a d)^5 g^3}-\frac {\left (3 b^3 d\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2} \, dx}{148877 (b c-a d)^4 g^3}-\frac {\left (3 b d^3\right ) \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^2} \, dx}{148877 (b c-a d)^4 g^3}+\frac {b^3 \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3} \, dx}{148877 (b c-a d)^3 g^3}-\frac {d^3 \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(c+d x)^3} \, dx}{148877 (b c-a d)^3 g^3}\\ &=-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2\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}{148877 (b c-a d)^5 g^3}+\frac {\left (6 b^2 B d^2\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}{148877 (b c-a d)^5 g^3}-\frac {\left (3 b^2 B d\right ) \int \frac {b c-a d}{(a+b x)^2 (c+d x)} \, dx}{148877 (b c-a d)^4 g^3}-\frac {\left (3 b B d^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^2} \, dx}{148877 (b c-a d)^4 g^3}+\frac {\left (b^2 B\right ) \int \frac {b c-a d}{(a+b x)^3 (c+d x)} \, dx}{297754 (b c-a d)^3 g^3}-\frac {\left (B d^2\right ) \int \frac {b c-a d}{(a+b x) (c+d x)^3} \, dx}{297754 (b c-a d)^3 g^3}\\ &=-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac {\left (3 b^2 B d\right ) \int \frac {1}{(a+b x)^2 (c+d x)} \, dx}{148877 (b c-a d)^3 g^3}-\frac {\left (3 b B d^2\right ) \int \frac {1}{(a+b x) (c+d x)^2} \, dx}{148877 (b c-a d)^3 g^3}+\frac {\left (b^2 B\right ) \int \frac {1}{(a+b x)^3 (c+d x)} \, dx}{297754 (b c-a d)^2 g^3}-\frac {\left (B d^2\right ) \int \frac {1}{(a+b x) (c+d x)^3} \, dx}{297754 (b c-a d)^2 g^3}-\frac {\left (6 b^2 B d^2\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}{148877 (b c-a d)^5 e g^3}+\frac {\left (6 b^2 B d^2\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}{148877 (b c-a d)^5 e g^3}\\ &=-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac {\left (3 b^2 B d\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}{148877 (b c-a d)^3 g^3}-\frac {\left (3 b B d^2\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}{148877 (b c-a d)^3 g^3}+\frac {\left (b^2 B\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}{297754 (b c-a d)^2 g^3}-\frac {\left (B d^2\right ) \int \left (\frac {b^3}{(b c-a d)^3 (a+b x)}-\frac {d}{(b c-a d) (c+d x)^3}-\frac {b d}{(b c-a d)^2 (c+d x)^2}-\frac {b^2 d}{(b c-a d)^3 (c+d x)}\right ) \, dx}{297754 (b c-a d)^2 g^3}-\frac {\left (6 b^2 B d^2\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}{148877 (b c-a d)^5 e g^3}+\frac {\left (6 b^2 B d^2\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}{148877 (b c-a d)^5 e g^3}\\ &=-\frac {b^2 B}{595508 (b c-a d)^3 g^3 (a+b x)^2}+\frac {7 b^2 B d}{297754 (b c-a d)^4 g^3 (a+b x)}-\frac {B d^2}{595508 (b c-a d)^3 g^3 (c+d x)^2}-\frac {7 b B d^2}{297754 (b c-a d)^4 g^3 (c+d x)}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac {\left (6 b^3 B d^2\right ) \int \frac {\log (a+b x)}{a+b x} \, dx}{148877 (b c-a d)^5 g^3}+\frac {\left (6 b^3 B d^2\right ) \int \frac {\log (c+d x)}{a+b x} \, dx}{148877 (b c-a d)^5 g^3}+\frac {\left (6 b^2 B d^3\right ) \int \frac {\log (a+b x)}{c+d x} \, dx}{148877 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^3\right ) \int \frac {\log (c+d x)}{c+d x} \, dx}{148877 (b c-a d)^5 g^3}\\ &=-\frac {b^2 B}{595508 (b c-a d)^3 g^3 (a+b x)^2}+\frac {7 b^2 B d}{297754 (b c-a d)^4 g^3 (a+b x)}-\frac {B d^2}{595508 (b c-a d)^3 g^3 (c+d x)^2}-\frac {7 b B d^2}{297754 (b c-a d)^4 g^3 (c+d x)}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{148877 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,a+b x\right )}{148877 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,c+d x\right )}{148877 (b c-a d)^5 g^3}-\frac {\left (6 b^3 B d^2\right ) \int \frac {\log \left (\frac {b (c+d x)}{b c-a d}\right )}{a+b x} \, dx}{148877 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^3\right ) \int \frac {\log \left (\frac {d (a+b x)}{-b c+a d}\right )}{c+d x} \, dx}{148877 (b c-a d)^5 g^3}\\ &=-\frac {b^2 B}{595508 (b c-a d)^3 g^3 (a+b x)^2}+\frac {7 b^2 B d}{297754 (b c-a d)^4 g^3 (a+b x)}-\frac {B d^2}{595508 (b c-a d)^3 g^3 (c+d x)^2}-\frac {7 b B d^2}{297754 (b c-a d)^4 g^3 (c+d x)}-\frac {3 b^2 B d^2 \log ^2(a+b x)}{148877 (b c-a d)^5 g^3}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac {3 b^2 B d^2 \log ^2(c+d x)}{148877 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{148877 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{b c-a d}\right )}{x} \, dx,x,a+b x\right )}{148877 (b c-a d)^5 g^3}-\frac {\left (6 b^2 B d^2\right ) \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b c+a d}\right )}{x} \, dx,x,c+d x\right )}{148877 (b c-a d)^5 g^3}\\ &=-\frac {b^2 B}{595508 (b c-a d)^3 g^3 (a+b x)^2}+\frac {7 b^2 B d}{297754 (b c-a d)^4 g^3 (a+b x)}-\frac {B d^2}{595508 (b c-a d)^3 g^3 (c+d x)^2}-\frac {7 b B d^2}{297754 (b c-a d)^4 g^3 (c+d x)}-\frac {3 b^2 B d^2 \log ^2(a+b x)}{148877 (b c-a d)^5 g^3}-\frac {b^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (a+b x)^2}+\frac {3 b^2 d \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (a+b x)}+\frac {d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{297754 (b c-a d)^3 g^3 (c+d x)^2}+\frac {3 b d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^4 g^3 (c+d x)}+\frac {6 b^2 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{148877 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 \log \left (-\frac {d (a+b x)}{b c-a d}\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac {6 b^2 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)}{148877 (b c-a d)^5 g^3}-\frac {3 b^2 B d^2 \log ^2(c+d x)}{148877 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )}{148877 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 \text {Li}_2\left (-\frac {d (a+b x)}{b c-a d}\right )}{148877 (b c-a d)^5 g^3}+\frac {6 b^2 B d^2 \text {Li}_2\left (\frac {b (c+d x)}{b c-a d}\right )}{148877 (b c-a d)^5 g^3}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
time = 0.76, size = 533, normalized size = 1.15 \begin {gather*} -\frac {\frac {b^2 B (b c-a d)^2}{(a+b x)^2}-\frac {12 b^3 B c d}{a+b x}+\frac {12 a b^2 B d^2}{a+b x}-\frac {2 b^2 B d (b c-a d)}{a+b x}+\frac {B d^2 (b c-a d)^2}{(c+d x)^2}+\frac {12 b^2 B c d^2}{c+d x}-\frac {12 a b B d^3}{c+d x}+\frac {2 b B d^2 (b c-a d)}{c+d x}+\frac {2 b^2 (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 {12 b^2 d (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{a+b x}-\frac {2 d^2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(c+d x)^2}-\frac {12 b d^2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{c+d x}-24 b^2 d^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+24 b^2 d^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+12 b^2 B d^2 \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 )-12 b^2 B d^2 \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 )}{4 (b c-a d)^5 g^3 i^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Log[(e*(a + b*x))/(c + d*x)])/((a*g + b*g*x)^3*(c*i + d*i*x)^3),x]

[Out]

-1/4*((b^2*B*(b*c - a*d)^2)/(a + b*x)^2 - (12*b^3*B*c*d)/(a + b*x) + (12*a*b^2*B*d^2)/(a + b*x) - (2*b^2*B*d*(
b*c - a*d))/(a + b*x) + (B*d^2*(b*c - a*d)^2)/(c + d*x)^2 + (12*b^2*B*c*d^2)/(c + d*x) - (12*a*b*B*d^3)/(c + d
*x) + (2*b*B*d^2*(b*c - a*d))/(c + d*x) + (2*b^2*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x)
^2 - (12*b^2*d*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(a + b*x) - (2*d^2*(b*c - a*d)^2*(A + B*Log[(
e*(a + b*x))/(c + d*x)]))/(c + d*x)^2 - (12*b*d^2*(b*c - a*d)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(c + d*x)
- 24*b^2*d^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 24*b^2*d^2*(A + B*Log[(e*(a + b*x))/(c + d*x)
])*Log[c + d*x] + 12*b^2*B*d^2*(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)]) - 12*b^2*B*d^2*((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^3*i^3)

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Maple [A]
time = 0.81, size = 804, normalized size = 1.74 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)^3/(d*i*x+c*i)^3,x,method=_RETURNVERBOSE)

[Out]

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

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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. 2298 vs. \(2 (430) = 860\).
time = 0.63, size = 2298, normalized size = 4.96 \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)^3/(d*i*x+c*i)^3,x, algorithm="maxima")

[Out]

1/2*B*(12*b^2*d^2*log(b*x + a)/((-I*b^5*c^5 + 5*I*a*b^4*c^4*d - 10*I*a^2*b^3*c^3*d^2 + 10*I*a^3*b^2*c^2*d^3 -
5*I*a^4*b*c*d^4 + I*a^5*d^5)*g^3) - 12*b^2*d^2*log(d*x + c)/((-I*b^5*c^5 + 5*I*a*b^4*c^4*d - 10*I*a^2*b^3*c^3*
d^2 + 10*I*a^3*b^2*c^2*d^3 - 5*I*a^4*b*c*d^4 + I*a^5*d^5)*g^3) + (12*b^3*d^3*x^3 - b^3*c^3 + 7*a*b^2*c^2*d + 7
*a^2*b*c*d^2 - a^3*d^3 + 18*(b^3*c*d^2 + a*b^2*d^3)*x^2 + 4*(b^3*c^2*d + 7*a*b^2*c*d^2 + a^2*b*d^3)*x)/((-I*b^
6*c^4*d^2 + 4*I*a*b^5*c^3*d^3 - 6*I*a^2*b^4*c^2*d^4 + 4*I*a^3*b^3*c*d^5 - I*a^4*b^2*d^6)*g^3*x^4 + 2*(-I*b^6*c
^5*d + 3*I*a*b^5*c^4*d^2 - 2*I*a^2*b^4*c^3*d^3 - 2*I*a^3*b^3*c^2*d^4 + 3*I*a^4*b^2*c*d^5 - I*a^5*b*d^6)*g^3*x^
3 + (-I*b^6*c^6 + 9*I*a^2*b^4*c^4*d^2 - 16*I*a^3*b^3*c^3*d^3 + 9*I*a^4*b^2*c^2*d^4 - I*a^6*d^6)*g^3*x^2 + 2*(-
I*a*b^5*c^6 + 3*I*a^2*b^4*c^5*d - 2*I*a^3*b^3*c^4*d^2 - 2*I*a^4*b^2*c^3*d^3 + 3*I*a^5*b*c^2*d^4 - I*a^6*c*d^5)
*g^3*x + (-I*a^2*b^4*c^6 + 4*I*a^3*b^3*c^5*d - 6*I*a^4*b^2*c^4*d^2 + 4*I*a^5*b*c^3*d^3 - I*a^6*c^2*d^4)*g^3))*
log(b*x*e/(d*x + c) + a*e/(d*x + c)) + 1/2*A*(12*b^2*d^2*log(b*x + a)/((-I*b^5*c^5 + 5*I*a*b^4*c^4*d - 10*I*a^
2*b^3*c^3*d^2 + 10*I*a^3*b^2*c^2*d^3 - 5*I*a^4*b*c*d^4 + I*a^5*d^5)*g^3) - 12*b^2*d^2*log(d*x + c)/((-I*b^5*c^
5 + 5*I*a*b^4*c^4*d - 10*I*a^2*b^3*c^3*d^2 + 10*I*a^3*b^2*c^2*d^3 - 5*I*a^4*b*c*d^4 + I*a^5*d^5)*g^3) + (12*b^
3*d^3*x^3 - b^3*c^3 + 7*a*b^2*c^2*d + 7*a^2*b*c*d^2 - a^3*d^3 + 18*(b^3*c*d^2 + a*b^2*d^3)*x^2 + 4*(b^3*c^2*d
+ 7*a*b^2*c*d^2 + a^2*b*d^3)*x)/((-I*b^6*c^4*d^2 + 4*I*a*b^5*c^3*d^3 - 6*I*a^2*b^4*c^2*d^4 + 4*I*a^3*b^3*c*d^5
- I*a^4*b^2*d^6)*g^3*x^4 + 2*(-I*b^6*c^5*d + 3*I*a*b^5*c^4*d^2 - 2*I*a^2*b^4*c^3*d^3 - 2*I*a^3*b^3*c^2*d^4 +
3*I*a^4*b^2*c*d^5 - I*a^5*b*d^6)*g^3*x^3 + (-I*b^6*c^6 + 9*I*a^2*b^4*c^4*d^2 - 16*I*a^3*b^3*c^3*d^3 + 9*I*a^4*
b^2*c^2*d^4 - I*a^6*d^6)*g^3*x^2 + 2*(-I*a*b^5*c^6 + 3*I*a^2*b^4*c^5*d - 2*I*a^3*b^3*c^4*d^2 - 2*I*a^4*b^2*c^3
*d^3 + 3*I*a^5*b*c^2*d^4 - I*a^6*c*d^5)*g^3*x + (-I*a^2*b^4*c^6 + 4*I*a^3*b^3*c^5*d - 6*I*a^4*b^2*c^4*d^2 + 4*
I*a^5*b*c^3*d^3 - I*a^6*c^2*d^4)*g^3)) - 1/4*(I*b^4*c^4 - 16*I*a*b^3*c^3*d + 30*I*a^2*b^2*c^2*d^2 - 16*I*a^3*b
*c*d^3 + I*a^4*d^4 - 12*(I*b^4*c^2*d^2 - 2*I*a*b^3*c*d^3 + I*a^2*b^2*d^4)*x^2 - 12*(-I*b^4*d^4*x^4 - I*a^2*b^2
*c^2*d^2 + 2*(-I*b^4*c*d^3 - I*a*b^3*d^4)*x^3 + (-I*b^4*c^2*d^2 - 4*I*a*b^3*c*d^3 - I*a^2*b^2*d^4)*x^2 + 2*(-I
*a*b^3*c^2*d^2 - I*a^2*b^2*c*d^3)*x)*log(b*x + a)^2 - 24*(I*b^4*d^4*x^4 + I*a^2*b^2*c^2*d^2 + 2*(I*b^4*c*d^3 +
I*a*b^3*d^4)*x^3 + (I*b^4*c^2*d^2 + 4*I*a*b^3*c*d^3 + I*a^2*b^2*d^4)*x^2 + 2*(I*a*b^3*c^2*d^2 + I*a^2*b^2*c*d
^3)*x)*log(b*x + a)*log(d*x + c) - 12*(-I*b^4*d^4*x^4 - I*a^2*b^2*c^2*d^2 + 2*(-I*b^4*c*d^3 - I*a*b^3*d^4)*x^3
+ (-I*b^4*c^2*d^2 - 4*I*a*b^3*c*d^3 - I*a^2*b^2*d^4)*x^2 + 2*(-I*a*b^3*c^2*d^2 - I*a^2*b^2*c*d^3)*x)*log(d*x
+ c)^2 - 12*(I*b^4*c^3*d - I*a*b^3*c^2*d^2 - I*a^2*b^2*c*d^3 + I*a^3*b*d^4)*x)*B/(a^2*b^5*c^7*g^3 - 5*a^3*b^4*
c^6*d*g^3 + 10*a^4*b^3*c^5*d^2*g^3 - 10*a^5*b^2*c^4*d^3*g^3 + 5*a^6*b*c^3*d^4*g^3 - a^7*c^2*d^5*g^3 + (b^7*c^5
*d^2*g^3 - 5*a*b^6*c^4*d^3*g^3 + 10*a^2*b^5*c^3*d^4*g^3 - 10*a^3*b^4*c^2*d^5*g^3 + 5*a^4*b^3*c*d^6*g^3 - a^5*b
^2*d^7*g^3)*x^4 + 2*(b^7*c^6*d*g^3 - 4*a*b^6*c^5*d^2*g^3 + 5*a^2*b^5*c^4*d^3*g^3 - 5*a^4*b^3*c^2*d^5*g^3 + 4*a
^5*b^2*c*d^6*g^3 - a^6*b*d^7*g^3)*x^3 + (b^7*c^7*g^3 - a*b^6*c^6*d*g^3 - 9*a^2*b^5*c^5*d^2*g^3 + 25*a^3*b^4*c^
4*d^3*g^3 - 25*a^4*b^3*c^3*d^4*g^3 + 9*a^5*b^2*c^2*d^5*g^3 + a^6*b*c*d^6*g^3 - a^7*d^7*g^3)*x^2 + 2*(a*b^6*c^7
*g^3 - 4*a^2*b^5*c^6*d*g^3 + 5*a^3*b^4*c^5*d^2*g^3 - 5*a^5*b^2*c^3*d^4*g^3 + 4*a^6*b*c^2*d^5*g^3 - a^7*c*d^6*g
^3)*x)

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1022 vs. \(2 (430) = 860\).
time = 0.43, size = 1022, normalized size = 2.21 \begin {gather*} -\frac {{\left (2 i \, A + i \, B\right )} b^{4} c^{4} - 16 \, {\left (i \, A + i \, B\right )} a b^{3} c^{3} d + 30 i \, B a^{2} b^{2} c^{2} d^{2} - 16 \, {\left (-i \, A + i \, B\right )} a^{3} b c d^{3} + {\left (-2 i \, A + i \, B\right )} a^{4} d^{4} - 24 \, {\left (i \, A b^{4} c d^{3} - i \, A a b^{3} d^{4}\right )} x^{3} - 12 \, {\left ({\left (3 i \, A + i \, B\right )} b^{4} c^{2} d^{2} - 2 i \, B a b^{3} c d^{3} + {\left (-3 i \, A + i \, B\right )} a^{2} b^{2} d^{4}\right )} x^{2} - 12 \, {\left (i \, B b^{4} d^{4} x^{4} + i \, B a^{2} b^{2} c^{2} d^{2} + 2 \, {\left (i \, B b^{4} c d^{3} + i \, B a b^{3} d^{4}\right )} x^{3} + {\left (i \, B b^{4} c^{2} d^{2} + 4 i \, B a b^{3} c d^{3} + i \, B a^{2} b^{2} d^{4}\right )} x^{2} + 2 \, {\left (i \, B a b^{3} c^{2} d^{2} + i \, B a^{2} b^{2} c d^{3}\right )} x\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )^{2} - 4 \, {\left ({\left (2 i \, A + 3 i \, B\right )} b^{4} c^{3} d + 3 \, {\left (4 i \, A - i \, B\right )} a b^{3} c^{2} d^{2} + 3 \, {\left (-4 i \, A - i \, B\right )} a^{2} b^{2} c d^{3} + {\left (-2 i \, A + 3 i \, B\right )} a^{3} b d^{4}\right )} x - 2 \, {\left (12 i \, A b^{4} d^{4} x^{4} - i \, B b^{4} c^{4} + 8 i \, B a b^{3} c^{3} d + 12 i \, A a^{2} b^{2} c^{2} d^{2} - 8 i \, B a^{3} b c d^{3} + i \, B a^{4} d^{4} + 12 \, {\left ({\left (2 i \, A + i \, B\right )} b^{4} c d^{3} + {\left (2 i \, A - i \, B\right )} a b^{3} d^{4}\right )} x^{3} + 6 \, {\left ({\left (2 i \, A + 3 i \, B\right )} b^{4} c^{2} d^{2} + 8 i \, A a b^{3} c d^{3} + {\left (2 i \, A - 3 i \, B\right )} a^{2} b^{2} d^{4}\right )} x^{2} + 4 \, {\left (i \, B b^{4} c^{3} d + 6 \, {\left (i \, A + i \, B\right )} a b^{3} c^{2} d^{2} + 6 \, {\left (i \, A - i \, B\right )} a^{2} b^{2} c d^{3} - i \, B a^{3} b d^{4}\right )} x\right )} \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right )}{4 \, {\left ({\left (b^{7} c^{5} d^{2} - 5 \, a b^{6} c^{4} d^{3} + 10 \, a^{2} b^{5} c^{3} d^{4} - 10 \, a^{3} b^{4} c^{2} d^{5} + 5 \, a^{4} b^{3} c d^{6} - a^{5} b^{2} d^{7}\right )} g^{3} x^{4} + 2 \, {\left (b^{7} c^{6} d - 4 \, a b^{6} c^{5} d^{2} + 5 \, a^{2} b^{5} c^{4} d^{3} - 5 \, a^{4} b^{3} c^{2} d^{5} + 4 \, a^{5} b^{2} c d^{6} - a^{6} b d^{7}\right )} g^{3} x^{3} + {\left (b^{7} c^{7} - a b^{6} c^{6} d - 9 \, a^{2} b^{5} c^{5} d^{2} + 25 \, a^{3} b^{4} c^{4} d^{3} - 25 \, a^{4} b^{3} c^{3} d^{4} + 9 \, a^{5} b^{2} c^{2} d^{5} + a^{6} b c d^{6} - a^{7} d^{7}\right )} g^{3} x^{2} + 2 \, {\left (a b^{6} c^{7} - 4 \, a^{2} b^{5} c^{6} d + 5 \, a^{3} b^{4} c^{5} d^{2} - 5 \, a^{5} b^{2} c^{3} d^{4} + 4 \, a^{6} b c^{2} d^{5} - a^{7} c d^{6}\right )} g^{3} x + {\left (a^{2} b^{5} c^{7} - 5 \, a^{3} b^{4} c^{6} d + 10 \, a^{4} b^{3} c^{5} d^{2} - 10 \, a^{5} b^{2} c^{4} d^{3} + 5 \, a^{6} b c^{3} d^{4} - a^{7} c^{2} d^{5}\right )} g^{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)^3/(d*i*x+c*i)^3,x, algorithm="fricas")

[Out]

-1/4*((2*I*A + I*B)*b^4*c^4 - 16*(I*A + I*B)*a*b^3*c^3*d + 30*I*B*a^2*b^2*c^2*d^2 - 16*(-I*A + I*B)*a^3*b*c*d^
3 + (-2*I*A + I*B)*a^4*d^4 - 24*(I*A*b^4*c*d^3 - I*A*a*b^3*d^4)*x^3 - 12*((3*I*A + I*B)*b^4*c^2*d^2 - 2*I*B*a*
b^3*c*d^3 + (-3*I*A + I*B)*a^2*b^2*d^4)*x^2 - 12*(I*B*b^4*d^4*x^4 + I*B*a^2*b^2*c^2*d^2 + 2*(I*B*b^4*c*d^3 + I
*B*a*b^3*d^4)*x^3 + (I*B*b^4*c^2*d^2 + 4*I*B*a*b^3*c*d^3 + I*B*a^2*b^2*d^4)*x^2 + 2*(I*B*a*b^3*c^2*d^2 + I*B*a
^2*b^2*c*d^3)*x)*log((b*x + a)*e/(d*x + c))^2 - 4*((2*I*A + 3*I*B)*b^4*c^3*d + 3*(4*I*A - I*B)*a*b^3*c^2*d^2 +
3*(-4*I*A - I*B)*a^2*b^2*c*d^3 + (-2*I*A + 3*I*B)*a^3*b*d^4)*x - 2*(12*I*A*b^4*d^4*x^4 - I*B*b^4*c^4 + 8*I*B*
a*b^3*c^3*d + 12*I*A*a^2*b^2*c^2*d^2 - 8*I*B*a^3*b*c*d^3 + I*B*a^4*d^4 + 12*((2*I*A + I*B)*b^4*c*d^3 + (2*I*A
- I*B)*a*b^3*d^4)*x^3 + 6*((2*I*A + 3*I*B)*b^4*c^2*d^2 + 8*I*A*a*b^3*c*d^3 + (2*I*A - 3*I*B)*a^2*b^2*d^4)*x^2
+ 4*(I*B*b^4*c^3*d + 6*(I*A + I*B)*a*b^3*c^2*d^2 + 6*(I*A - I*B)*a^2*b^2*c*d^3 - I*B*a^3*b*d^4)*x)*log((b*x +
a)*e/(d*x + c)))/((b^7*c^5*d^2 - 5*a*b^6*c^4*d^3 + 10*a^2*b^5*c^3*d^4 - 10*a^3*b^4*c^2*d^5 + 5*a^4*b^3*c*d^6 -
a^5*b^2*d^7)*g^3*x^4 + 2*(b^7*c^6*d - 4*a*b^6*c^5*d^2 + 5*a^2*b^5*c^4*d^3 - 5*a^4*b^3*c^2*d^5 + 4*a^5*b^2*c*d
^6 - a^6*b*d^7)*g^3*x^3 + (b^7*c^7 - a*b^6*c^6*d - 9*a^2*b^5*c^5*d^2 + 25*a^3*b^4*c^4*d^3 - 25*a^4*b^3*c^3*d^4
+ 9*a^5*b^2*c^2*d^5 + a^6*b*c*d^6 - a^7*d^7)*g^3*x^2 + 2*(a*b^6*c^7 - 4*a^2*b^5*c^6*d + 5*a^3*b^4*c^5*d^2 - 5
*a^5*b^2*c^3*d^4 + 4*a^6*b*c^2*d^5 - a^7*c*d^6)*g^3*x + (a^2*b^5*c^7 - 5*a^3*b^4*c^6*d + 10*a^4*b^3*c^5*d^2 -
10*a^5*b^2*c^4*d^3 + 5*a^6*b*c^3*d^4 - a^7*c^2*d^5)*g^3)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 2106 vs. \(2 (430) = 860\).
time = 144.96, size = 2106, normalized size = 4.55 \begin {gather*} \frac {6 A b^{2} d^{2} \log {\left (x + \frac {- \frac {6 A a^{6} b^{2} d^{8}}{\left (a d - b c\right )^{5}} + \frac {36 A a^{5} b^{3} c d^{7}}{\left (a d - b c\right )^{5}} - \frac {90 A a^{4} b^{4} c^{2} d^{6}}{\left (a d - b c\right )^{5}} + \frac {120 A a^{3} b^{5} c^{3} d^{5}}{\left (a d - b c\right )^{5}} - \frac {90 A a^{2} b^{6} c^{4} d^{4}}{\left (a d - b c\right )^{5}} + \frac {36 A a b^{7} c^{5} d^{3}}{\left (a d - b c\right )^{5}} + 6 A a b^{2} d^{3} - \frac {6 A b^{8} c^{6} d^{2}}{\left (a d - b c\right )^{5}} + 6 A b^{3} c d^{2}}{12 A b^{3} d^{3}} \right )}}{g^{3} i^{3} \left (a d - b c\right )^{5}} - \frac {6 A b^{2} d^{2} \log {\left (x + \frac {\frac {6 A a^{6} b^{2} d^{8}}{\left (a d - b c\right )^{5}} - \frac {36 A a^{5} b^{3} c d^{7}}{\left (a d - b c\right )^{5}} + \frac {90 A a^{4} b^{4} c^{2} d^{6}}{\left (a d - b c\right )^{5}} - \frac {120 A a^{3} b^{5} c^{3} d^{5}}{\left (a d - b c\right )^{5}} + \frac {90 A a^{2} b^{6} c^{4} d^{4}}{\left (a d - b c\right )^{5}} - \frac {36 A a b^{7} c^{5} d^{3}}{\left (a d - b c\right )^{5}} + 6 A a b^{2} d^{3} + \frac {6 A b^{8} c^{6} d^{2}}{\left (a d - b c\right )^{5}} + 6 A b^{3} c d^{2}}{12 A b^{3} d^{3}} \right )}}{g^{3} i^{3} \left (a d - b c\right )^{5}} - \frac {3 B b^{2} d^{2} \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{a^{5} d^{5} g^{3} i^{3} - 5 a^{4} b c d^{4} g^{3} i^{3} + 10 a^{3} b^{2} c^{2} d^{3} g^{3} i^{3} - 10 a^{2} b^{3} c^{3} d^{2} g^{3} i^{3} + 5 a b^{4} c^{4} d g^{3} i^{3} - b^{5} c^{5} g^{3} i^{3}} + \frac {\left (- B a^{3} d^{3} + 7 B a^{2} b c d^{2} + 4 B a^{2} b d^{3} x + 7 B a b^{2} c^{2} d + 28 B a b^{2} c d^{2} x + 18 B a b^{2} d^{3} x^{2} - B b^{3} c^{3} + 4 B b^{3} c^{2} d x + 18 B b^{3} c d^{2} x^{2} + 12 B b^{3} d^{3} x^{3}\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 a^{6} c^{2} d^{4} g^{3} i^{3} + 4 a^{6} c d^{5} g^{3} i^{3} x + 2 a^{6} d^{6} g^{3} i^{3} x^{2} - 8 a^{5} b c^{3} d^{3} g^{3} i^{3} - 12 a^{5} b c^{2} d^{4} g^{3} i^{3} x + 4 a^{5} b d^{6} g^{3} i^{3} x^{3} + 12 a^{4} b^{2} c^{4} d^{2} g^{3} i^{3} + 8 a^{4} b^{2} c^{3} d^{3} g^{3} i^{3} x - 18 a^{4} b^{2} c^{2} d^{4} g^{3} i^{3} x^{2} - 12 a^{4} b^{2} c d^{5} g^{3} i^{3} x^{3} + 2 a^{4} b^{2} d^{6} g^{3} i^{3} x^{4} - 8 a^{3} b^{3} c^{5} d g^{3} i^{3} + 8 a^{3} b^{3} c^{4} d^{2} g^{3} i^{3} x + 32 a^{3} b^{3} c^{3} d^{3} g^{3} i^{3} x^{2} + 8 a^{3} b^{3} c^{2} d^{4} g^{3} i^{3} x^{3} - 8 a^{3} b^{3} c d^{5} g^{3} i^{3} x^{4} + 2 a^{2} b^{4} c^{6} g^{3} i^{3} - 12 a^{2} b^{4} c^{5} d g^{3} i^{3} x - 18 a^{2} b^{4} c^{4} d^{2} g^{3} i^{3} x^{2} + 8 a^{2} b^{4} c^{3} d^{3} g^{3} i^{3} x^{3} + 12 a^{2} b^{4} c^{2} d^{4} g^{3} i^{3} x^{4} + 4 a b^{5} c^{6} g^{3} i^{3} x - 12 a b^{5} c^{4} d^{2} g^{3} i^{3} x^{3} - 8 a b^{5} c^{3} d^{3} g^{3} i^{3} x^{4} + 2 b^{6} c^{6} g^{3} i^{3} x^{2} + 4 b^{6} c^{5} d g^{3} i^{3} x^{3} + 2 b^{6} c^{4} d^{2} g^{3} i^{3} x^{4}} + \frac {- 2 A a^{3} d^{3} + 14 A a^{2} b c d^{2} + 14 A a b^{2} c^{2} d - 2 A b^{3} c^{3} + 24 A b^{3} d^{3} x^{3} + B a^{3} d^{3} - 15 B a^{2} b c d^{2} + 15 B a b^{2} c^{2} d - B b^{3} c^{3} + x^{2} \cdot \left (36 A a b^{2} d^{3} + 36 A b^{3} c d^{2} - 12 B a b^{2} d^{3} + 12 B b^{3} c d^{2}\right ) + x \left (8 A a^{2} b d^{3} + 56 A a b^{2} c d^{2} + 8 A b^{3} c^{2} d - 12 B a^{2} b d^{3} + 12 B b^{3} c^{2} d\right )}{4 a^{6} c^{2} d^{4} g^{3} i^{3} - 16 a^{5} b c^{3} d^{3} g^{3} i^{3} + 24 a^{4} b^{2} c^{4} d^{2} g^{3} i^{3} - 16 a^{3} b^{3} c^{5} d g^{3} i^{3} + 4 a^{2} b^{4} c^{6} g^{3} i^{3} + x^{4} \cdot \left (4 a^{4} b^{2} d^{6} g^{3} i^{3} - 16 a^{3} b^{3} c d^{5} g^{3} i^{3} + 24 a^{2} b^{4} c^{2} d^{4} g^{3} i^{3} - 16 a b^{5} c^{3} d^{3} g^{3} i^{3} + 4 b^{6} c^{4} d^{2} g^{3} i^{3}\right ) + x^{3} \cdot \left (8 a^{5} b d^{6} g^{3} i^{3} - 24 a^{4} b^{2} c d^{5} g^{3} i^{3} + 16 a^{3} b^{3} c^{2} d^{4} g^{3} i^{3} + 16 a^{2} b^{4} c^{3} d^{3} g^{3} i^{3} - 24 a b^{5} c^{4} d^{2} g^{3} i^{3} + 8 b^{6} c^{5} d g^{3} i^{3}\right ) + x^{2} \cdot \left (4 a^{6} d^{6} g^{3} i^{3} - 36 a^{4} b^{2} c^{2} d^{4} g^{3} i^{3} + 64 a^{3} b^{3} c^{3} d^{3} g^{3} i^{3} - 36 a^{2} b^{4} c^{4} d^{2} g^{3} i^{3} + 4 b^{6} c^{6} g^{3} i^{3}\right ) + x \left (8 a^{6} c d^{5} g^{3} i^{3} - 24 a^{5} b c^{2} d^{4} g^{3} i^{3} + 16 a^{4} b^{2} c^{3} d^{3} g^{3} i^{3} + 16 a^{3} b^{3} c^{4} d^{2} g^{3} i^{3} - 24 a^{2} b^{4} c^{5} d g^{3} i^{3} + 8 a b^{5} c^{6} g^{3} i^{3}\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)**3/(d*i*x+c*i)**3,x)

[Out]

6*A*b**2*d**2*log(x + (-6*A*a**6*b**2*d**8/(a*d - b*c)**5 + 36*A*a**5*b**3*c*d**7/(a*d - b*c)**5 - 90*A*a**4*b
**4*c**2*d**6/(a*d - b*c)**5 + 120*A*a**3*b**5*c**3*d**5/(a*d - b*c)**5 - 90*A*a**2*b**6*c**4*d**4/(a*d - b*c)
**5 + 36*A*a*b**7*c**5*d**3/(a*d - b*c)**5 + 6*A*a*b**2*d**3 - 6*A*b**8*c**6*d**2/(a*d - b*c)**5 + 6*A*b**3*c*
d**2)/(12*A*b**3*d**3))/(g**3*i**3*(a*d - b*c)**5) - 6*A*b**2*d**2*log(x + (6*A*a**6*b**2*d**8/(a*d - b*c)**5
- 36*A*a**5*b**3*c*d**7/(a*d - b*c)**5 + 90*A*a**4*b**4*c**2*d**6/(a*d - b*c)**5 - 120*A*a**3*b**5*c**3*d**5/(
a*d - b*c)**5 + 90*A*a**2*b**6*c**4*d**4/(a*d - b*c)**5 - 36*A*a*b**7*c**5*d**3/(a*d - b*c)**5 + 6*A*a*b**2*d*
*3 + 6*A*b**8*c**6*d**2/(a*d - b*c)**5 + 6*A*b**3*c*d**2)/(12*A*b**3*d**3))/(g**3*i**3*(a*d - b*c)**5) - 3*B*b
**2*d**2*log(e*(a + b*x)/(c + d*x))**2/(a**5*d**5*g**3*i**3 - 5*a**4*b*c*d**4*g**3*i**3 + 10*a**3*b**2*c**2*d*
*3*g**3*i**3 - 10*a**2*b**3*c**3*d**2*g**3*i**3 + 5*a*b**4*c**4*d*g**3*i**3 - b**5*c**5*g**3*i**3) + (-B*a**3*
d**3 + 7*B*a**2*b*c*d**2 + 4*B*a**2*b*d**3*x + 7*B*a*b**2*c**2*d + 28*B*a*b**2*c*d**2*x + 18*B*a*b**2*d**3*x**
2 - B*b**3*c**3 + 4*B*b**3*c**2*d*x + 18*B*b**3*c*d**2*x**2 + 12*B*b**3*d**3*x**3)*log(e*(a + b*x)/(c + d*x))/
(2*a**6*c**2*d**4*g**3*i**3 + 4*a**6*c*d**5*g**3*i**3*x + 2*a**6*d**6*g**3*i**3*x**2 - 8*a**5*b*c**3*d**3*g**3
*i**3 - 12*a**5*b*c**2*d**4*g**3*i**3*x + 4*a**5*b*d**6*g**3*i**3*x**3 + 12*a**4*b**2*c**4*d**2*g**3*i**3 + 8*
a**4*b**2*c**3*d**3*g**3*i**3*x - 18*a**4*b**2*c**2*d**4*g**3*i**3*x**2 - 12*a**4*b**2*c*d**5*g**3*i**3*x**3 +
2*a**4*b**2*d**6*g**3*i**3*x**4 - 8*a**3*b**3*c**5*d*g**3*i**3 + 8*a**3*b**3*c**4*d**2*g**3*i**3*x + 32*a**3*
b**3*c**3*d**3*g**3*i**3*x**2 + 8*a**3*b**3*c**2*d**4*g**3*i**3*x**3 - 8*a**3*b**3*c*d**5*g**3*i**3*x**4 + 2*a
**2*b**4*c**6*g**3*i**3 - 12*a**2*b**4*c**5*d*g**3*i**3*x - 18*a**2*b**4*c**4*d**2*g**3*i**3*x**2 + 8*a**2*b**
4*c**3*d**3*g**3*i**3*x**3 + 12*a**2*b**4*c**2*d**4*g**3*i**3*x**4 + 4*a*b**5*c**6*g**3*i**3*x - 12*a*b**5*c**
4*d**2*g**3*i**3*x**3 - 8*a*b**5*c**3*d**3*g**3*i**3*x**4 + 2*b**6*c**6*g**3*i**3*x**2 + 4*b**6*c**5*d*g**3*i*
*3*x**3 + 2*b**6*c**4*d**2*g**3*i**3*x**4) + (-2*A*a**3*d**3 + 14*A*a**2*b*c*d**2 + 14*A*a*b**2*c**2*d - 2*A*b
**3*c**3 + 24*A*b**3*d**3*x**3 + B*a**3*d**3 - 15*B*a**2*b*c*d**2 + 15*B*a*b**2*c**2*d - B*b**3*c**3 + x**2*(3
6*A*a*b**2*d**3 + 36*A*b**3*c*d**2 - 12*B*a*b**2*d**3 + 12*B*b**3*c*d**2) + x*(8*A*a**2*b*d**3 + 56*A*a*b**2*c
*d**2 + 8*A*b**3*c**2*d - 12*B*a**2*b*d**3 + 12*B*b**3*c**2*d))/(4*a**6*c**2*d**4*g**3*i**3 - 16*a**5*b*c**3*d
**3*g**3*i**3 + 24*a**4*b**2*c**4*d**2*g**3*i**3 - 16*a**3*b**3*c**5*d*g**3*i**3 + 4*a**2*b**4*c**6*g**3*i**3
+ x**4*(4*a**4*b**2*d**6*g**3*i**3 - 16*a**3*b**3*c*d**5*g**3*i**3 + 24*a**2*b**4*c**2*d**4*g**3*i**3 - 16*a*b
**5*c**3*d**3*g**3*i**3 + 4*b**6*c**4*d**2*g**3*i**3) + x**3*(8*a**5*b*d**6*g**3*i**3 - 24*a**4*b**2*c*d**5*g*
*3*i**3 + 16*a**3*b**3*c**2*d**4*g**3*i**3 + 16*a**2*b**4*c**3*d**3*g**3*i**3 - 24*a*b**5*c**4*d**2*g**3*i**3
+ 8*b**6*c**5*d*g**3*i**3) + x**2*(4*a**6*d**6*g**3*i**3 - 36*a**4*b**2*c**2*d**4*g**3*i**3 + 64*a**3*b**3*c**
3*d**3*g**3*i**3 - 36*a**2*b**4*c**4*d**2*g**3*i**3 + 4*b**6*c**6*g**3*i**3) + x*(8*a**6*c*d**5*g**3*i**3 - 24
*a**5*b*c**2*d**4*g**3*i**3 + 16*a**4*b**2*c**3*d**3*g**3*i**3 + 16*a**3*b**3*c**4*d**2*g**3*i**3 - 24*a**2*b*
*4*c**5*d*g**3*i**3 + 8*a*b**5*c**6*g**3*i**3))

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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)^3/(d*i*x+c*i)^3,x, algorithm="giac")

[Out]

integrate((B*log((b*x + a)*e/(d*x + c)) + A)/((b*g*x + a*g)^3*(I*d*x + I*c)^3), x)

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Mupad [B]
time = 12.78, size = 1443, normalized size = 3.12 \begin {gather*} \frac {B\,a^3\,d^3}{4\,g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {A\,a^3\,d^3}{2\,g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {A\,b^3\,c^3}{2\,g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b^2\,d^2\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^5}-\frac {B\,b^3\,c^3}{4\,g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {B\,a\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^3\,i^3\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {B\,b\,c\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^3\,i^3\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {6\,A\,b^3\,d^3\,x^3}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {7\,A\,a\,b^2\,c^2\,d}{2\,g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {7\,A\,a^2\,b\,c\,d^2}{2\,g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {15\,B\,a\,b^2\,c^2\,d}{4\,g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {15\,B\,a^2\,b\,c\,d^2}{4\,g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {2\,A\,a^2\,b\,d^3\,x}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,a^2\,b\,d^3\,x}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {2\,A\,b^3\,c^2\,d\,x}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,b^3\,c^2\,d\,x}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {9\,A\,a\,b^2\,d^3\,x^2}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,a\,b^2\,d^3\,x^2}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {9\,A\,b^3\,c\,d^2\,x^2}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,b^3\,c\,d^2\,x^2}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {B\,b\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {6\,B\,b^3\,d^3\,x^3\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {9\,B\,a\,b^2\,d^3\,x^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {9\,B\,b^3\,c\,d^2\,x^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {14\,A\,a\,b^2\,c\,d^2\,x}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,a\,b^2\,c^2\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,a^2\,b\,c\,d^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,a^2\,b\,d^3\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {3\,B\,b^3\,c^2\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {12\,B\,a\,b^2\,c\,d^2\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i^3\,{\left (a\,d-b\,c\right )}^4\,{\left (a+b\,x\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {A\,b^2\,d^2\,\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 )\,12{}\mathrm {i}}{g^3\,i^3\,{\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)^3*(c*i + d*i*x)^3),x)

[Out]

(A*b^2*d^2*atan((a*d*1i + b*c*1i + b*d*x*2i)/(a*d - b*c))*12i)/(g^3*i^3*(a*d - b*c)^5) - (3*B*b^2*d^2*log((e*(
a + b*x))/(c + d*x))^2)/(g^3*i^3*(a*d - b*c)^5) - (A*a^3*d^3)/(2*g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2
) - (A*b^3*c^3)/(2*g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (B*a^3*d^3)/(4*g^3*i^3*(a*d - b*c)^4*(a +
b*x)^2*(c + d*x)^2) - (B*b^3*c^3)/(4*g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) - (B*a*d*log((e*(a + b*x))
/(c + d*x)))/(2*g^3*i^3*(a*d - b*c)^2*(a + b*x)^2*(c + d*x)^2) - (B*b*c*log((e*(a + b*x))/(c + d*x)))/(2*g^3*i
^3*(a*d - b*c)^2*(a + b*x)^2*(c + d*x)^2) + (6*A*b^3*d^3*x^3)/(g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2)
+ (7*A*a*b^2*c^2*d)/(2*g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (7*A*a^2*b*c*d^2)/(2*g^3*i^3*(a*d - b*
c)^4*(a + b*x)^2*(c + d*x)^2) + (15*B*a*b^2*c^2*d)/(4*g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) - (15*B*a
^2*b*c*d^2)/(4*g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (2*A*a^2*b*d^3*x)/(g^3*i^3*(a*d - b*c)^4*(a +
b*x)^2*(c + d*x)^2) - (3*B*a^2*b*d^3*x)/(g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (2*A*b^3*c^2*d*x)/(g
^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (3*B*b^3*c^2*d*x)/(g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)
^2) + (9*A*a*b^2*d^3*x^2)/(g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) - (3*B*a*b^2*d^3*x^2)/(g^3*i^3*(a*d
- b*c)^4*(a + b*x)^2*(c + d*x)^2) + (9*A*b^3*c*d^2*x^2)/(g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (3*B
*b^3*c*d^2*x^2)/(g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) - (B*b*d*x*log((e*(a + b*x))/(c + d*x)))/(g^3*
i^3*(a*d - b*c)^2*(a + b*x)^2*(c + d*x)^2) + (6*B*b^3*d^3*x^3*log((e*(a + b*x))/(c + d*x)))/(g^3*i^3*(a*d - b*
c)^4*(a + b*x)^2*(c + d*x)^2) + (9*B*a*b^2*d^3*x^2*log((e*(a + b*x))/(c + d*x)))/(g^3*i^3*(a*d - b*c)^4*(a + b
*x)^2*(c + d*x)^2) + (9*B*b^3*c*d^2*x^2*log((e*(a + b*x))/(c + d*x)))/(g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c +
d*x)^2) + (14*A*a*b^2*c*d^2*x)/(g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (3*B*a*b^2*c^2*d*log((e*(a +
b*x))/(c + d*x)))/(g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (3*B*a^2*b*c*d^2*log((e*(a + b*x))/(c + d*
x)))/(g^3*i^3*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (3*B*a^2*b*d^3*x*log((e*(a + b*x))/(c + d*x)))/(g^3*i^3
*(a*d - b*c)^4*(a + b*x)^2*(c + d*x)^2) + (3*B*b^3*c^2*d*x*log((e*(a + b*x))/(c + d*x)))/(g^3*i^3*(a*d - b*c)^
4*(a + b*x)^2*(c + d*x)^2) + (12*B*a*b^2*c*d^2*x*log((e*(a + b*x))/(c + d*x)))/(g^3*i^3*(a*d - b*c)^4*(a + b*x
)^2*(c + d*x)^2)

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