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

Optimal result

Integrand size = 40, antiderivative size = 373 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=-\frac {3 b B d^2 (c+d x)}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 B d (c+d x)^2}{4 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {b^3 B (c+d x)^3}{9 (b c-a d)^4 g^4 i (a+b x)^3}+\frac {B d^3 \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^4 g^4 i}-\frac {3 b d^2 (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^4 i (a+b x)}+\frac {3 b^2 d (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^4 g^4 i (a+b x)^2}-\frac {b^3 (c+d x)^3 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{3 (b c-a d)^4 g^4 i (a+b x)^3}-\frac {d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^4 g^4 i} \]

[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] (verified)

Time = 0.19 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2562, 45, 2372, 12, 14, 2338} \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=-\frac {b^3 (c+d x)^3 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{3 g^4 i (a+b x)^3 (b c-a d)^4}+\frac {3 b^2 d (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^4 i (a+b x)^2 (b c-a d)^4}-\frac {d^3 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i (b c-a d)^4}-\frac {3 b d^2 (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^4 i (a+b x) (b c-a d)^4}-\frac {b^3 B (c+d x)^3}{9 g^4 i (a+b x)^3 (b c-a d)^4}+\frac {3 b^2 B d (c+d x)^2}{4 g^4 i (a+b x)^2 (b c-a d)^4}+\frac {B d^3 \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g^4 i (b c-a d)^4}-\frac {3 b B d^2 (c+d x)}{g^4 i (a+b x) (b c-a d)^4} \]

[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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(b-d x)^3 (A+B \log (e x))}{x^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{(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}-\frac {B \text {Subst}\left (\int \frac {-2 b^3+9 b^2 d x-18 b d^2 x^2-6 d^3 x^3 \log (x)}{6 x^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{(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}-\frac {B \text {Subst}\left (\int \frac {-2 b^3+9 b^2 d x-18 b d^2 x^2-6 d^3 x^3 \log (x)}{x^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{6 (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}-\frac {B \text {Subst}\left (\int \left (-\frac {b \left (2 b^2-9 b d x+18 d^2 x^2\right )}{x^4}-\frac {6 d^3 \log (x)}{x}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{6 (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}+\frac {(b B) \text {Subst}\left (\int \frac {2 b^2-9 b d x+18 d^2 x^2}{x^4} \, dx,x,\frac {a+b x}{c+d x}\right )}{6 (b c-a d)^4 g^4 i}+\frac {\left (B d^3\right ) \text {Subst}\left (\int \frac {\log (x)}{x} \, dx,x,\frac {a+b x}{c+d x}\right )}{(b c-a d)^4 g^4 i} \\ & = \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}+\frac {(b B) \text {Subst}\left (\int \left (\frac {2 b^2}{x^4}-\frac {9 b d}{x^3}+\frac {18 d^2}{x^2}\right ) \, dx,x,\frac {a+b x}{c+d x}\right )}{6 (b c-a d)^4 g^4 i} \\ & = -\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} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.

Time = 0.39 (sec) , antiderivative size = 492, normalized size of antiderivative = 1.32 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=\frac {-\frac {12 A (b c-a d)^3}{(a+b x)^3}-\frac {4 B (b c-a d)^3}{(a+b x)^3}+\frac {18 A d (b c-a d)^2}{(a+b x)^2}+\frac {15 B d (b c-a d)^2}{(a+b x)^2}+\frac {36 A d^2 (-b c+a d)}{a+b x}+\frac {66 B d^2 (-b c+a d)}{a+b x}-36 A d^3 \log (a+b x)-66 B d^3 \log (a+b x)+18 B d^3 \log ^2(a+b x)-\frac {12 B (b c-a d)^3 \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^3}+\frac {18 B d (b c-a d)^2 \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^2}+\frac {36 B d^2 (-b c+a d) \log \left (\frac {e (a+b x)}{c+d x}\right )}{a+b x}-36 B d^3 \log (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )+36 A d^3 \log (c+d x)+66 B d^3 \log (c+d x)-36 B d^3 \log \left (\frac {d (a+b x)}{-b c+a d}\right ) \log (c+d x)+36 B d^3 \log \left (\frac {e (a+b x)}{c+d x}\right ) \log (c+d x)+18 B d^3 \log ^2(c+d x)-36 B d^3 \log (a+b x) \log \left (\frac {b (c+d x)}{b c-a d}\right )-36 B d^3 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )-36 B d^3 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )}{36 (b c-a d)^4 g^4 i} \]

[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)

Maple [A] (verified)

Time = 1.63 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.34

method	result	size
parts	\(\frac {A \left (\frac {d^{3} \ln \left (d x +c \right )}{\left (a d -c b \right )^{4}}+\frac {1}{3 \left (a d -c b \right ) \left (b x +a \right )^{3}}+\frac {d}{2 \left (a d -c b \right )^{2} \left (b x +a \right )^{2}}+\frac {d^{2}}{\left (a d -c b \right )^{3} \left (b x +a \right )}-\frac {d^{3} \ln \left (b x +a \right )}{\left (a d -c b \right )^{4}}\right )}{g^{4} i}-\frac {B \left (\frac {d^{4} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )^{4}}-\frac {3 d^{3} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{4}}+\frac {3 d^{2} b^{2} e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{4}}-\frac {d \,b^{3} e^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{4}}\right )}{g^{4} i d}\)	\(500\)
risch	\(\frac {A \,d^{3} \ln \left (d x +c \right )}{g^{4} i \left (a d -c b \right )^{4}}+\frac {A}{3 g^{4} i \left (a d -c b \right ) \left (b x +a \right )^{3}}+\frac {A d}{2 g^{4} i \left (a d -c b \right )^{2} \left (b x +a \right )^{2}}+\frac {A \,d^{2}}{g^{4} i \left (a d -c b \right )^{3} \left (b x +a \right )}-\frac {A \,d^{3} \ln \left (b x +a \right )}{g^{4} i \left (a d -c b \right )^{4}}-\frac {B \,d^{3} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 g^{4} i \left (a d -c b \right )^{4}}-\frac {3 B \,d^{2} b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{4} i \left (a d -c b \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {3 B \,d^{2} b e}{g^{4} i \left (a d -c b \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}+\frac {3 B d \,b^{2} e^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 g^{4} i \left (a d -c b \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{2}}+\frac {3 B d \,b^{2} e^{2}}{4 g^{4} i \left (a d -c b \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{2}}-\frac {B \,b^{3} e^{3} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 g^{4} i \left (a d -c b \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{3}}-\frac {B \,b^{3} e^{3}}{9 g^{4} i \left (a d -c b \right )^{4} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{3}}\)	\(628\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} e^{2} A \,b^{3}}{3 i \left (a d -c b \right )^{5} g^{4} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {3 d^{3} e A \,b^{2}}{2 i \left (a d -c b \right )^{5} g^{4} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {3 d^{4} A b}{i \left (a d -c b \right )^{5} g^{4} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{5} A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (a d -c b \right )^{5} g^{4}}-\frac {d^{2} e^{2} B \,b^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{i \left (a d -c b \right )^{5} g^{4}}+\frac {3 d^{3} e B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i \left (a d -c b \right )^{5} g^{4}}-\frac {3 d^{4} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i \left (a d -c b \right )^{5} g^{4}}+\frac {d^{5} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e i \left (a d -c b \right )^{5} g^{4}}\right )}{d^{2}}\)	\(637\)
default	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} e^{2} A \,b^{3}}{3 i \left (a d -c b \right )^{5} g^{4} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {3 d^{3} e A \,b^{2}}{2 i \left (a d -c b \right )^{5} g^{4} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {3 d^{4} A b}{i \left (a d -c b \right )^{5} g^{4} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{5} A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (a d -c b \right )^{5} g^{4}}-\frac {d^{2} e^{2} B \,b^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{i \left (a d -c b \right )^{5} g^{4}}+\frac {3 d^{3} e B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i \left (a d -c b \right )^{5} g^{4}}-\frac {3 d^{4} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i \left (a d -c b \right )^{5} g^{4}}+\frac {d^{5} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e i \left (a d -c b \right )^{5} g^{4}}\right )}{d^{2}}\)	\(637\)
parallelrisch	\(-\frac {-288 A \,x^{2} a^{5} b^{3} c^{3} d^{2}+162 A \,x^{2} a^{4} b^{4} c^{4} d -12 A \,x^{3} a^{2} b^{6} c^{5}-4 B \,x^{3} a^{2} b^{6} c^{5}-36 A \,x^{2} a^{3} b^{5} c^{5}+189 B \,x^{2} a^{6} b^{2} c^{2} d^{3}-258 B \,x^{2} a^{5} b^{3} c^{3} d^{2}+81 B \,x^{2} a^{4} b^{4} c^{4} d +108 A x \,a^{7} b \,c^{2} d^{3}-216 A x \,a^{6} b^{2} c^{3} d^{2}+144 A x \,a^{5} b^{3} c^{4} d +108 B x \,a^{7} b \,c^{2} d^{3}-162 B x \,a^{6} b^{2} c^{3} d^{2}+66 B x \,a^{5} b^{3} c^{4} d +108 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{7} b \,c^{3} d^{2}-54 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b^{2} c^{4} d +66 A \,x^{3} a^{5} b^{3} c^{2} d^{3}-108 A \,x^{3} a^{4} b^{4} c^{3} d^{2}+54 A \,x^{3} a^{3} b^{5} c^{4} d +85 B \,x^{3} a^{5} b^{3} c^{2} d^{3}-108 B \,x^{3} a^{4} b^{4} c^{3} d^{2}+27 B \,x^{3} a^{3} b^{5} c^{4} d +162 A \,x^{2} a^{6} b^{2} c^{2} d^{3}-12 B \,x^{2} a^{3} b^{5} c^{5}+18 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{8} c^{2} d^{3}-36 A x \,a^{4} b^{4} c^{5}+36 A \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{8} c^{2} d^{3}-12 B x \,a^{4} b^{4} c^{5}+12 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} b^{3} c^{5}+108 A \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b^{2} c^{2} d^{3}+162 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b^{2} c^{2} d^{3}+36 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} b^{3} c^{3} d^{2}+54 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{7} b \,c^{2} d^{3}+108 A x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{7} b \,c^{2} d^{3}+108 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{7} b \,c^{2} d^{3}+108 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b^{2} c^{3} d^{2}-18 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} b^{3} c^{4} d +18 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{5} b^{3} c^{2} d^{3}+36 A \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} b^{3} c^{2} d^{3}+66 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} b^{3} c^{2} d^{3}+54 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{6} b^{2} c^{2} d^{3}}{36 i \,g^{4} \left (b x +a \right )^{3} \left (a d -c b \right )^{2} c^{2} a^{5} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\)	\(961\)
norman	\(\text {Expression too large to display}\)	\(1038\)

[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]

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

Fricas [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 611, normalized size of antiderivative = 1.64 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=-\frac {4 \, {\left (3 \, A + B\right )} b^{3} c^{3} - 27 \, {\left (2 \, A + B\right )} a b^{2} c^{2} d + 108 \, {\left (A + B\right )} a^{2} b c d^{2} - {\left (66 \, A + 85 \, B\right )} a^{3} d^{3} + 6 \, {\left ({\left (6 \, A + 11 \, B\right )} b^{3} c d^{2} - {\left (6 \, A + 11 \, B\right )} a b^{2} d^{3}\right )} x^{2} + 18 \, {\left (B b^{3} d^{3} x^{3} + 3 \, B a b^{2} d^{3} x^{2} + 3 \, B a^{2} b d^{3} x + B a^{3} d^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 3 \, {\left ({\left (6 \, A + 5 \, B\right )} b^{3} c^{2} d - 18 \, {\left (2 \, A + 3 \, B\right )} a b^{2} c d^{2} + {\left (30 \, A + 49 \, B\right )} a^{2} b d^{3}\right )} x + 6 \, {\left ({\left (6 \, A + 11 \, B\right )} b^{3} d^{3} x^{3} + 2 \, B b^{3} c^{3} - 9 \, B a b^{2} c^{2} d + 18 \, B a^{2} b c d^{2} + 6 \, A a^{3} d^{3} + 3 \, {\left (2 \, B b^{3} c d^{2} + 3 \, {\left (2 \, A + 3 \, B\right )} a b^{2} d^{3}\right )} x^{2} - 3 \, {\left (B b^{3} c^{2} d - 6 \, B a b^{2} c d^{2} - 6 \, {\left (A + B\right )} a^{2} b d^{3}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{36 \, {\left ({\left (b^{7} c^{4} - 4 \, a b^{6} c^{3} d + 6 \, a^{2} b^{5} c^{2} d^{2} - 4 \, a^{3} b^{4} c d^{3} + a^{4} b^{3} d^{4}\right )} g^{4} i x^{3} + 3 \, {\left (a b^{6} c^{4} - 4 \, a^{2} b^{5} c^{3} d + 6 \, a^{3} b^{4} c^{2} d^{2} - 4 \, a^{4} b^{3} c d^{3} + a^{5} b^{2} d^{4}\right )} g^{4} i x^{2} + 3 \, {\left (a^{2} b^{5} c^{4} - 4 \, a^{3} b^{4} c^{3} d + 6 \, a^{4} b^{3} c^{2} d^{2} - 4 \, a^{5} b^{2} c d^{3} + a^{6} b d^{4}\right )} g^{4} i x + {\left (a^{3} b^{4} c^{4} - 4 \, a^{4} b^{3} c^{3} d + 6 \, a^{5} b^{2} c^{2} d^{2} - 4 \, a^{6} b c d^{3} + a^{7} d^{4}\right )} g^{4} i\right )}} \]

[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*A + B)*b^3*c^3 - 27*(2*A + B)*a*b^2*c^2*d + 108*(A + B)*a^2*b*c*d^2 - (66*A + 85*B)*a^3*d^3 + 6*((
6*A + 11*B)*b^3*c*d^2 - (6*A + 11*B)*a*b^2*d^3)*x^2 + 18*(B*b^3*d^3*x^3 + 3*B*a*b^2*d^3*x^2 + 3*B*a^2*b*d^3*x
+ B*a^3*d^3)*log((b*e*x + a*e)/(d*x + c))^2 - 3*((6*A + 5*B)*b^3*c^2*d - 18*(2*A + 3*B)*a*b^2*c*d^2 + (30*A +
49*B)*a^2*b*d^3)*x + 6*((6*A + 11*B)*b^3*d^3*x^3 + 2*B*b^3*c^3 - 9*B*a*b^2*c^2*d + 18*B*a^2*b*c*d^2 + 6*A*a^3*
d^3 + 3*(2*B*b^3*c*d^2 + 3*(2*A + 3*B)*a*b^2*d^3)*x^2 - 3*(B*b^3*c^2*d - 6*B*a*b^2*c*d^2 - 6*(A + B)*a^2*b*d^3
)*x)*log((b*e*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*i*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*i*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*i*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*i)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1392 vs. \(2 (332) = 664\).

Time = 9.88 (sec) , antiderivative size = 1392, normalized size of antiderivative = 3.73 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=\text {Too large to display} \]

[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))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1469 vs. \(2 (363) = 726\).

Time = 0.31 (sec) , antiderivative size = 1469, normalized size of antiderivative = 3.94 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=\text {Too large to display} \]

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

Giac [A] (verification not implemented)

none

Time = 55.04 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.72 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=-\frac {1}{36} \, {\left (\frac {6 \, {\left (2 \, B b e^{4} - \frac {3 \, {\left (b e x + a e\right )} B d e^{3}}{d x + c}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{3} b c g^{4} i}{{\left (d x + c\right )}^{3}} - \frac {{\left (b e x + a e\right )}^{3} a d g^{4} i}{{\left (d x + c\right )}^{3}}} + \frac {12 \, A b e^{4} + 4 \, B b e^{4} - \frac {18 \, {\left (b e x + a e\right )} A d e^{3}}{d x + c} - \frac {9 \, {\left (b e x + a e\right )} B d e^{3}}{d x + c}}{\frac {{\left (b e x + a e\right )}^{3} b c g^{4} i}{{\left (d x + c\right )}^{3}} - \frac {{\left (b e x + a e\right )}^{3} a d g^{4} i}{{\left (d x + c\right )}^{3}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2} \]

[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*(6*(2*B*b*e^4 - 3*(b*e*x + a*e)*B*d*e^3/(d*x + c))*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^3*b*c*g^4
*i/(d*x + c)^3 - (b*e*x + a*e)^3*a*d*g^4*i/(d*x + c)^3) + (12*A*b*e^4 + 4*B*b*e^4 - 18*(b*e*x + a*e)*A*d*e^3/(
d*x + c) - 9*(b*e*x + a*e)*B*d*e^3/(d*x + c))/((b*e*x + a*e)^3*b*c*g^4*i/(d*x + c)^3 - (b*e*x + a*e)^3*a*d*g^4
*i/(d*x + c)^3))*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c - a*d)))^2

Mupad [B] (verification not implemented)

Time = 5.99 (sec) , antiderivative size = 970, normalized size of antiderivative = 2.60 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^4 (c i+d i x)} \, dx=\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} \]

[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)