\(\int \frac {A+B \log (\frac {e (c+d x)}{a+b x})}{(a g+b g x)^5} \, dx\) [181]

Optimal result

Integrand size = 30, antiderivative size = 206 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^5} \, dx=\frac {B}{16 b g^5 (a+b x)^4}-\frac {B d}{12 b (b c-a d) g^5 (a+b x)^3}+\frac {B d^2}{8 b (b c-a d)^2 g^5 (a+b x)^2}-\frac {B d^3}{4 b (b c-a d)^3 g^5 (a+b x)}-\frac {B d^4 \log (a+b x)}{4 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x)}{4 b (b c-a d)^4 g^5}-\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{4 b g^5 (a+b x)^4} \]

[Out]

1/16*B/b/g^5/(b*x+a)^4-1/12*B*d/b/(-a*d+b*c)/g^5/(b*x+a)^3+1/8*B*d^2/b/(-a*d+b*c)^2/g^5/(b*x+a)^2-1/4*B*d^3/b/
(-a*d+b*c)^3/g^5/(b*x+a)-1/4*B*d^4*ln(b*x+a)/b/(-a*d+b*c)^4/g^5+1/4*B*d^4*ln(d*x+c)/b/(-a*d+b*c)^4/g^5+1/4*(-A
-B*ln(e*(d*x+c)/(b*x+a)))/b/g^5/(b*x+a)^4

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2548, 21, 46} \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^5} \, dx=-\frac {B \log \left (\frac {e (c+d x)}{a+b x}\right )+A}{4 b g^5 (a+b x)^4}-\frac {B d^4 \log (a+b x)}{4 b g^5 (b c-a d)^4}+\frac {B d^4 \log (c+d x)}{4 b g^5 (b c-a d)^4}-\frac {B d^3}{4 b g^5 (a+b x) (b c-a d)^3}+\frac {B d^2}{8 b g^5 (a+b x)^2 (b c-a d)^2}-\frac {B d}{12 b g^5 (a+b x)^3 (b c-a d)}+\frac {B}{16 b g^5 (a+b x)^4} \]

[In]

Int[(A + B*Log[(e*(c + d*x))/(a + b*x)])/(a*g + b*g*x)^5,x]

[Out]

B/(16*b*g^5*(a + b*x)^4) - (B*d)/(12*b*(b*c - a*d)*g^5*(a + b*x)^3) + (B*d^2)/(8*b*(b*c - a*d)^2*g^5*(a + b*x)
^2) - (B*d^3)/(4*b*(b*c - a*d)^3*g^5*(a + b*x)) - (B*d^4*Log[a + b*x])/(4*b*(b*c - a*d)^4*g^5) + (B*d^4*Log[c
+ d*x])/(4*b*(b*c - a*d)^4*g^5) - (A + B*Log[(e*(c + d*x))/(a + b*x)])/(4*b*g^5*(a + b*x)^4)

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 46

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

Rule 2548

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))*((f_.) + (g_.)*(x_))^(m_.
), x_Symbol] :> Simp[(f + g*x)^(m + 1)*((A + B*Log[e*((a + b*x)^n/(c + d*x)^n)])/(g*(m + 1))), x] - Dist[B*n*(
(b*c - a*d)/(g*(m + 1))), Int[(f + g*x)^(m + 1)/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f, g, A
, B, m, n}, x] && EqQ[n + mn, 0] && NeQ[b*c - a*d, 0] && NeQ[m, -1] &&  !(EqQ[m, -2] && IntegerQ[n])

Rubi steps \begin{align*} \text {integral}& = -\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{4 b g^5 (a+b x)^4}-\frac {(B (b c-a d)) \int \frac {1}{(a+b x) (c+d x) (a g+b g x)^4} \, dx}{4 b g} \\ & = -\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{4 b g^5 (a+b x)^4}-\frac {(B (b c-a d)) \int \frac {1}{(a+b x)^5 (c+d x)} \, dx}{4 b g^5} \\ & = -\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{4 b g^5 (a+b x)^4}-\frac {(B (b c-a d)) \int \left (\frac {b}{(b c-a d) (a+b x)^5}-\frac {b d}{(b c-a d)^2 (a+b x)^4}+\frac {b d^2}{(b c-a d)^3 (a+b x)^3}-\frac {b d^3}{(b c-a d)^4 (a+b x)^2}+\frac {b d^4}{(b c-a d)^5 (a+b x)}-\frac {d^5}{(b c-a d)^5 (c+d x)}\right ) \, dx}{4 b g^5} \\ & = \frac {B}{16 b g^5 (a+b x)^4}-\frac {B d}{12 b (b c-a d) g^5 (a+b x)^3}+\frac {B d^2}{8 b (b c-a d)^2 g^5 (a+b x)^2}-\frac {B d^3}{4 b (b c-a d)^3 g^5 (a+b x)}-\frac {B d^4 \log (a+b x)}{4 b (b c-a d)^4 g^5}+\frac {B d^4 \log (c+d x)}{4 b (b c-a d)^4 g^5}-\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{4 b g^5 (a+b x)^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.81 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^5} \, dx=\frac {\frac {B (-b c+a d) \left (-\frac {3 (b c-a d)^4}{(a+b x)^4}+\frac {4 d (b c-a d)^3}{(a+b x)^3}-\frac {6 d^2 (b c-a d)^2}{(a+b x)^2}+\frac {12 d^3 (b c-a d)}{a+b x}+12 d^4 \log (a+b x)-12 d^4 \log (c+d x)\right )}{12 (b c-a d)^5}-\frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a+b x)^4}}{4 b g^5} \]

[In]

Integrate[(A + B*Log[(e*(c + d*x))/(a + b*x)])/(a*g + b*g*x)^5,x]

[Out]

((B*(-(b*c) + a*d)*((-3*(b*c - a*d)^4)/(a + b*x)^4 + (4*d*(b*c - a*d)^3)/(a + b*x)^3 - (6*d^2*(b*c - a*d)^2)/(
a + b*x)^2 + (12*d^3*(b*c - a*d))/(a + b*x) + 12*d^4*Log[a + b*x] - 12*d^4*Log[c + d*x]))/(12*(b*c - a*d)^5) -
(A + B*Log[(e*(c + d*x))/(a + b*x)])/(a + b*x)^4)/(4*b*g^5)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(436\) vs. \(2(195)=390\).

Time = 2.20 (sec) , antiderivative size = 437, normalized size of antiderivative = 2.12

method	result	size
parts	\(-\frac {A}{4 g^{5} \left (b x +a \right )^{4} b}-\frac {B \,b^{3} \left (\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{4} \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )}{4}-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{4}}{16}-\frac {3 d e \left (\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{3} \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )}{3}-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{3}}{9}\right )}{b}+\frac {3 d^{2} e^{2} \left (\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2} \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )}{2}-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2}}{4}\right )}{b^{2}}-\frac {d^{3} e^{3} \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )+\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}-\frac {d e}{b}\right )}{b^{3}}\right )}{g^{5} e^{4} \left (a d -c b \right )^{4}}\)	\(437\)
risch	\(-\frac {B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}{4 b \,g^{5} \left (b x +a \right )^{4}}-\frac {48 B a \,b^{3} c \,d^{3} x^{2}+72 B \,a^{2} b^{2} c \,d^{3} x -24 B a \,b^{3} c^{2} d^{2} x -48 A \,a^{3} b c \,d^{3}+72 A \,a^{2} b^{2} c^{2} d^{2}-48 A a \,b^{3} c^{3} d -12 B a \,b^{3} d^{4} x^{3}+12 B \,b^{4} c \,d^{3} x^{3}-42 B \,a^{2} b^{2} d^{4} x^{2}-6 B \,b^{4} c^{2} d^{2} x^{2}-52 B \,a^{3} b \,d^{4} x +4 B \,b^{4} c^{3} d x -3 B \,b^{4} c^{4}-36 B \,a^{2} b^{2} c^{2} d^{2}+12 A \,a^{4} d^{4}-25 B \,a^{4} d^{4}+48 B \ln \left (b x +a \right ) a \,b^{3} d^{4} x^{3}-48 B \ln \left (-d x -c \right ) a \,b^{3} d^{4} x^{3}+72 B \ln \left (b x +a \right ) a^{2} b^{2} d^{4} x^{2}+16 B a \,b^{3} c^{3} d +12 A \,b^{4} c^{4}+12 B \ln \left (b x +a \right ) a^{4} d^{4}-12 B \ln \left (-d x -c \right ) a^{4} d^{4}+48 B \,a^{3} b c \,d^{3}+12 B \ln \left (b x +a \right ) b^{4} d^{4} x^{4}-12 B \ln \left (-d x -c \right ) b^{4} d^{4} x^{4}-72 B \ln \left (-d x -c \right ) a^{2} b^{2} d^{4} x^{2}+48 B \ln \left (b x +a \right ) a^{3} b \,d^{4} x -48 B \ln \left (-d x -c \right ) a^{3} b \,d^{4} x}{48 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) g^{5} \left (b x +a \right )^{4} b}\)	\(525\)
derivativedivides	\(\frac {e \left (a d -c b \right ) \left (-\frac {b^{5} A \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{4}}{4 \left (a d -c b \right )^{5} e^{5} g^{5}}+\frac {b^{4} A d \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{3}}{\left (a d -c b \right )^{5} e^{4} g^{5}}-\frac {3 b^{3} A \,d^{2} \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2}}{2 \left (a d -c b \right )^{5} e^{3} g^{5}}+\frac {b^{2} A \,d^{3} \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )}{\left (a d -c b \right )^{5} e^{2} g^{5}}-\frac {b^{5} B \left (\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{4} \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )}{4}-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{4}}{16}\right )}{\left (a d -c b \right )^{5} e^{5} g^{5}}+\frac {3 b^{4} B d \left (\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{3} \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )}{3}-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{3}}{9}\right )}{\left (a d -c b \right )^{5} e^{4} g^{5}}-\frac {3 b^{3} B \,d^{2} \left (\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2} \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )}{2}-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{5} e^{3} g^{5}}+\frac {b^{2} B \,d^{3} \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )+\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}-\frac {d e}{b}\right )}{\left (a d -c b \right )^{5} e^{2} g^{5}}\right )}{b^{2}}\)	\(688\)
default	\(\frac {e \left (a d -c b \right ) \left (-\frac {b^{5} A \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{4}}{4 \left (a d -c b \right )^{5} e^{5} g^{5}}+\frac {b^{4} A d \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{3}}{\left (a d -c b \right )^{5} e^{4} g^{5}}-\frac {3 b^{3} A \,d^{2} \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2}}{2 \left (a d -c b \right )^{5} e^{3} g^{5}}+\frac {b^{2} A \,d^{3} \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )}{\left (a d -c b \right )^{5} e^{2} g^{5}}-\frac {b^{5} B \left (\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{4} \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )}{4}-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{4}}{16}\right )}{\left (a d -c b \right )^{5} e^{5} g^{5}}+\frac {3 b^{4} B d \left (\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{3} \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )}{3}-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{3}}{9}\right )}{\left (a d -c b \right )^{5} e^{4} g^{5}}-\frac {3 b^{3} B \,d^{2} \left (\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2} \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )}{2}-\frac {\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )^{2}}{4}\right )}{\left (a d -c b \right )^{5} e^{3} g^{5}}+\frac {b^{2} B \,d^{3} \left (\left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right ) \ln \left (\frac {d e}{b}-\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}\right )+\frac {e \left (a d -c b \right )}{b \left (b x +a \right )}-\frac {d e}{b}\right )}{\left (a d -c b \right )^{5} e^{2} g^{5}}\right )}{b^{2}}\)	\(688\)
parallelrisch	\(\frac {48 B x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{9} c \,d^{4}-72 B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{8} b \,c^{3} d^{2}+48 B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{7} b^{2} c^{4} d +12 A \,x^{4} a^{2} b^{7} c^{5}-3 B \,x^{4} a^{2} b^{7} c^{5}+48 A \,x^{3} a^{3} b^{6} c^{5}-12 B \,x^{3} a^{3} b^{6} c^{5}+72 A \,x^{2} a^{4} b^{5} c^{5}-18 B \,x^{2} a^{4} b^{5} c^{5}+48 A x \,a^{9} c \,d^{4}+48 A x \,a^{5} b^{4} c^{5}-48 B x \,a^{9} c \,d^{4}-12 B x \,a^{5} b^{4} c^{5}+12 B \,x^{4} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{6} b^{3} c \,d^{4}+48 B \,x^{3} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{7} b^{2} c \,d^{4}+72 B \,x^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{8} b c \,d^{4}+96 B \,x^{2} a^{5} b^{4} c^{4} d -48 A \,x^{4} a^{5} b^{4} c^{2} d^{3}+72 A \,x^{4} a^{4} b^{5} c^{3} d^{2}-48 A \,x^{4} a^{3} b^{6} c^{4} d -25 B \,x^{4} a^{6} b^{3} c \,d^{4}+48 B \,x^{4} a^{5} b^{4} c^{2} d^{3}+12 A \,x^{4} a^{6} b^{3} c \,d^{4}+180 B \,x^{3} a^{6} b^{3} c^{2} d^{3}-144 B \,x^{3} a^{5} b^{4} c^{3} d^{2}+64 B \,x^{3} a^{4} b^{5} c^{4} d +72 A \,x^{2} a^{8} b c \,d^{4}-288 A \,x^{2} a^{7} b^{2} c^{2} d^{3}+432 A \,x^{2} a^{6} b^{3} c^{3} d^{2}-288 A \,x^{2} a^{5} b^{4} c^{4} d -108 B \,x^{2} a^{8} b c \,d^{4}+240 B \,x^{2} a^{7} b^{2} c^{2} d^{3}-210 B \,x^{2} a^{6} b^{3} c^{3} d^{2}-192 A x \,a^{8} b \,c^{2} d^{3}+288 A x \,a^{7} b^{2} c^{3} d^{2}-192 A x \,a^{6} b^{3} c^{4} d +120 B x \,a^{8} b \,c^{2} d^{3}-120 B x \,a^{7} b^{2} c^{3} d^{2}+60 B x \,a^{6} b^{3} c^{4} d -192 A \,x^{3} a^{6} b^{3} c^{2} d^{3}+288 A \,x^{3} a^{5} b^{4} c^{3} d^{2}-192 A \,x^{3} a^{4} b^{5} c^{4} d -88 B \,x^{3} a^{7} b^{2} c \,d^{4}-36 B \,x^{4} a^{4} b^{5} c^{3} d^{2}+16 B \,x^{4} a^{3} b^{6} c^{4} d +48 A \,x^{3} a^{7} b^{2} c \,d^{4}+48 B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{9} c^{2} d^{3}-12 B \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right ) a^{6} b^{3} c^{5}}{48 g^{5} \left (b x +a \right )^{4} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) a^{6} c}\)	\(928\)
norman	\(\frac {\frac {B \,a^{3} d^{4} x \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}{\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) g}+\frac {a \,d^{4} B \,b^{2} x^{3} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}{\left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) g}+\frac {\left (4 A \,a^{3} d^{3}-12 A \,a^{2} b c \,d^{2}+12 A a \,b^{2} c^{2} d -4 A \,b^{3} c^{3}-4 B \,a^{3} d^{3}+6 B \,a^{2} b c \,d^{2}-4 B a \,b^{2} c^{2} d +B \,c^{3} b^{3}\right ) x}{4 g a \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {B c \left (4 a^{3} d^{3}-6 a^{2} b c \,d^{2}+4 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}{4 g \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}+\frac {\left (12 A \,a^{3} d^{3}-36 A \,a^{2} b c \,d^{2}+36 A a \,b^{2} c^{2} d -12 A \,b^{3} c^{3}-25 B \,a^{3} d^{3}+23 B \,a^{2} b c \,d^{2}-13 B a \,b^{2} c^{2} d +3 B \,c^{3} b^{3}\right ) b^{3} x^{4}}{48 g \,a^{4} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (12 A \,a^{3} d^{3}-36 A \,a^{2} b c \,d^{2}+36 A a \,b^{2} c^{2} d -12 A \,b^{3} c^{3}-22 B \,a^{3} d^{3}+23 B \,a^{2} b c \,d^{2}-13 B a \,b^{2} c^{2} d +3 B \,c^{3} b^{3}\right ) b^{2} x^{3}}{12 g \,a^{3} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (12 A \,a^{3} d^{3}-36 A \,a^{2} b c \,d^{2}+36 A a \,b^{2} c^{2} d -12 A \,b^{3} c^{3}-18 B \,a^{3} d^{3}+22 B \,a^{2} b c \,d^{2}-13 B a \,b^{2} c^{2} d +3 B \,c^{3} b^{3}\right ) b \,x^{2}}{8 g \,a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {b^{3} d^{4} B \,x^{4} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}{4 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) g}+\frac {3 B b \,a^{2} d^{4} x^{2} \ln \left (\frac {e \left (d x +c \right )}{b x +a}\right )}{2 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) g}}{g^{4} \left (b x +a \right )^{4}}\)	\(969\)

[In]

int((A+B*ln(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g)^5,x,method=_RETURNVERBOSE)

[Out]

-1/4*A/g^5/(b*x+a)^4/b-B/g^5*b^3/e^4/(a*d-b*c)^4*(1/4*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^4*ln(d*e/b-e*(a*d-b*c)/b/(
b*x+a))-1/16*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^4-3*d*e/b*(1/3*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^3*ln(d*e/b-e*(a*d-b*c)
/b/(b*x+a))-1/9*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^3)+3*d^2*e^2/b^2*(1/2*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^2*ln(d*e/b-e
*(a*d-b*c)/b/(b*x+a))-1/4*(d*e/b-e*(a*d-b*c)/b/(b*x+a))^2)-d^3*e^3/b^3*((d*e/b-e*(a*d-b*c)/b/(b*x+a))*ln(d*e/b
-e*(a*d-b*c)/b/(b*x+a))+e*(a*d-b*c)/b/(b*x+a)-d*e/b))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 637 vs. \(2 (192) = 384\).

Time = 0.27 (sec) , antiderivative size = 637, normalized size of antiderivative = 3.09 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^5} \, dx=-\frac {3 \, {\left (4 \, A - B\right )} b^{4} c^{4} - 16 \, {\left (3 \, A - B\right )} a b^{3} c^{3} d + 36 \, {\left (2 \, A - B\right )} a^{2} b^{2} c^{2} d^{2} - 48 \, {\left (A - B\right )} a^{3} b c d^{3} + {\left (12 \, A - 25 \, B\right )} a^{4} d^{4} + 12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} x^{3} - 6 \, {\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} x^{2} + 4 \, {\left (B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 18 \, B a^{2} b^{2} c d^{3} - 13 \, B a^{3} b d^{4}\right )} x - 12 \, {\left (B b^{4} d^{4} x^{4} + 4 \, B a b^{3} d^{4} x^{3} + 6 \, B a^{2} b^{2} d^{4} x^{2} + 4 \, B a^{3} b d^{4} x - B b^{4} c^{4} + 4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3}\right )} \log \left (\frac {d e x + c e}{b x + a}\right )}{48 \, {\left ({\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} g^{5} x + {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} g^{5}\right )}} \]

[In]

integrate((A+B*log(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g)^5,x, algorithm="fricas")

[Out]

-1/48*(3*(4*A - B)*b^4*c^4 - 16*(3*A - B)*a*b^3*c^3*d + 36*(2*A - B)*a^2*b^2*c^2*d^2 - 48*(A - B)*a^3*b*c*d^3
+ (12*A - 25*B)*a^4*d^4 + 12*(B*b^4*c*d^3 - B*a*b^3*d^4)*x^3 - 6*(B*b^4*c^2*d^2 - 8*B*a*b^3*c*d^3 + 7*B*a^2*b^
2*d^4)*x^2 + 4*(B*b^4*c^3*d - 6*B*a*b^3*c^2*d^2 + 18*B*a^2*b^2*c*d^3 - 13*B*a^3*b*d^4)*x - 12*(B*b^4*d^4*x^4 +
4*B*a*b^3*d^4*x^3 + 6*B*a^2*b^2*d^4*x^2 + 4*B*a^3*b*d^4*x - B*b^4*c^4 + 4*B*a*b^3*c^3*d - 6*B*a^2*b^2*c^2*d^2
+ 4*B*a^3*b*c*d^3)*log((d*e*x + c*e)/(b*x + a)))/((b^9*c^4 - 4*a*b^8*c^3*d + 6*a^2*b^7*c^2*d^2 - 4*a^3*b^6*c*
d^3 + a^4*b^5*d^4)*g^5*x^4 + 4*(a*b^8*c^4 - 4*a^2*b^7*c^3*d + 6*a^3*b^6*c^2*d^2 - 4*a^4*b^5*c*d^3 + a^5*b^4*d^
4)*g^5*x^3 + 6*(a^2*b^7*c^4 - 4*a^3*b^6*c^3*d + 6*a^4*b^5*c^2*d^2 - 4*a^5*b^4*c*d^3 + a^6*b^3*d^4)*g^5*x^2 + 4
*(a^3*b^6*c^4 - 4*a^4*b^5*c^3*d + 6*a^5*b^4*c^2*d^2 - 4*a^6*b^3*c*d^3 + a^7*b^2*d^4)*g^5*x + (a^4*b^5*c^4 - 4*
a^5*b^4*c^3*d + 6*a^6*b^3*c^2*d^2 - 4*a^7*b^2*c*d^3 + a^8*b*d^4)*g^5)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 944 vs. \(2 (178) = 356\).

Time = 2.67 (sec) , antiderivative size = 944, normalized size of antiderivative = 4.58 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^5} \, dx=- \frac {B \log {\left (\frac {e \left (c + d x\right )}{a + b x} \right )}}{4 a^{4} b g^{5} + 16 a^{3} b^{2} g^{5} x + 24 a^{2} b^{3} g^{5} x^{2} + 16 a b^{4} g^{5} x^{3} + 4 b^{5} g^{5} x^{4}} + \frac {B d^{4} \log {\left (x + \frac {- \frac {B a^{5} d^{9}}{\left (a d - b c\right )^{4}} + \frac {5 B a^{4} b c d^{8}}{\left (a d - b c\right )^{4}} - \frac {10 B a^{3} b^{2} c^{2} d^{7}}{\left (a d - b c\right )^{4}} + \frac {10 B a^{2} b^{3} c^{3} d^{6}}{\left (a d - b c\right )^{4}} - \frac {5 B a b^{4} c^{4} d^{5}}{\left (a d - b c\right )^{4}} + B a d^{5} + \frac {B b^{5} c^{5} d^{4}}{\left (a d - b c\right )^{4}} + B b c d^{4}}{2 B b d^{5}} \right )}}{4 b g^{5} \left (a d - b c\right )^{4}} - \frac {B d^{4} \log {\left (x + \frac {\frac {B a^{5} d^{9}}{\left (a d - b c\right )^{4}} - \frac {5 B a^{4} b c d^{8}}{\left (a d - b c\right )^{4}} + \frac {10 B a^{3} b^{2} c^{2} d^{7}}{\left (a d - b c\right )^{4}} - \frac {10 B a^{2} b^{3} c^{3} d^{6}}{\left (a d - b c\right )^{4}} + \frac {5 B a b^{4} c^{4} d^{5}}{\left (a d - b c\right )^{4}} + B a d^{5} - \frac {B b^{5} c^{5} d^{4}}{\left (a d - b c\right )^{4}} + B b c d^{4}}{2 B b d^{5}} \right )}}{4 b g^{5} \left (a d - b c\right )^{4}} + \frac {- 12 A a^{3} d^{3} + 36 A a^{2} b c d^{2} - 36 A a b^{2} c^{2} d + 12 A b^{3} c^{3} + 25 B a^{3} d^{3} - 23 B a^{2} b c d^{2} + 13 B a b^{2} c^{2} d - 3 B b^{3} c^{3} + 12 B b^{3} d^{3} x^{3} + x^{2} \cdot \left (42 B a b^{2} d^{3} - 6 B b^{3} c d^{2}\right ) + x \left (52 B a^{2} b d^{3} - 20 B a b^{2} c d^{2} + 4 B b^{3} c^{2} d\right )}{48 a^{7} b d^{3} g^{5} - 144 a^{6} b^{2} c d^{2} g^{5} + 144 a^{5} b^{3} c^{2} d g^{5} - 48 a^{4} b^{4} c^{3} g^{5} + x^{4} \cdot \left (48 a^{3} b^{5} d^{3} g^{5} - 144 a^{2} b^{6} c d^{2} g^{5} + 144 a b^{7} c^{2} d g^{5} - 48 b^{8} c^{3} g^{5}\right ) + x^{3} \cdot \left (192 a^{4} b^{4} d^{3} g^{5} - 576 a^{3} b^{5} c d^{2} g^{5} + 576 a^{2} b^{6} c^{2} d g^{5} - 192 a b^{7} c^{3} g^{5}\right ) + x^{2} \cdot \left (288 a^{5} b^{3} d^{3} g^{5} - 864 a^{4} b^{4} c d^{2} g^{5} + 864 a^{3} b^{5} c^{2} d g^{5} - 288 a^{2} b^{6} c^{3} g^{5}\right ) + x \left (192 a^{6} b^{2} d^{3} g^{5} - 576 a^{5} b^{3} c d^{2} g^{5} + 576 a^{4} b^{4} c^{2} d g^{5} - 192 a^{3} b^{5} c^{3} g^{5}\right )} \]

[In]

integrate((A+B*ln(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g)**5,x)

[Out]

-B*log(e*(c + d*x)/(a + b*x))/(4*a**4*b*g**5 + 16*a**3*b**2*g**5*x + 24*a**2*b**3*g**5*x**2 + 16*a*b**4*g**5*x
**3 + 4*b**5*g**5*x**4) + B*d**4*log(x + (-B*a**5*d**9/(a*d - b*c)**4 + 5*B*a**4*b*c*d**8/(a*d - b*c)**4 - 10*
B*a**3*b**2*c**2*d**7/(a*d - b*c)**4 + 10*B*a**2*b**3*c**3*d**6/(a*d - b*c)**4 - 5*B*a*b**4*c**4*d**5/(a*d - b
*c)**4 + B*a*d**5 + B*b**5*c**5*d**4/(a*d - b*c)**4 + B*b*c*d**4)/(2*B*b*d**5))/(4*b*g**5*(a*d - b*c)**4) - B*
d**4*log(x + (B*a**5*d**9/(a*d - b*c)**4 - 5*B*a**4*b*c*d**8/(a*d - b*c)**4 + 10*B*a**3*b**2*c**2*d**7/(a*d -
b*c)**4 - 10*B*a**2*b**3*c**3*d**6/(a*d - b*c)**4 + 5*B*a*b**4*c**4*d**5/(a*d - b*c)**4 + B*a*d**5 - B*b**5*c*
*5*d**4/(a*d - b*c)**4 + B*b*c*d**4)/(2*B*b*d**5))/(4*b*g**5*(a*d - b*c)**4) + (-12*A*a**3*d**3 + 36*A*a**2*b*
c*d**2 - 36*A*a*b**2*c**2*d + 12*A*b**3*c**3 + 25*B*a**3*d**3 - 23*B*a**2*b*c*d**2 + 13*B*a*b**2*c**2*d - 3*B*
b**3*c**3 + 12*B*b**3*d**3*x**3 + x**2*(42*B*a*b**2*d**3 - 6*B*b**3*c*d**2) + x*(52*B*a**2*b*d**3 - 20*B*a*b**
2*c*d**2 + 4*B*b**3*c**2*d))/(48*a**7*b*d**3*g**5 - 144*a**6*b**2*c*d**2*g**5 + 144*a**5*b**3*c**2*d*g**5 - 48
*a**4*b**4*c**3*g**5 + x**4*(48*a**3*b**5*d**3*g**5 - 144*a**2*b**6*c*d**2*g**5 + 144*a*b**7*c**2*d*g**5 - 48*
b**8*c**3*g**5) + x**3*(192*a**4*b**4*d**3*g**5 - 576*a**3*b**5*c*d**2*g**5 + 576*a**2*b**6*c**2*d*g**5 - 192*
a*b**7*c**3*g**5) + x**2*(288*a**5*b**3*d**3*g**5 - 864*a**4*b**4*c*d**2*g**5 + 864*a**3*b**5*c**2*d*g**5 - 28
8*a**2*b**6*c**3*g**5) + x*(192*a**6*b**2*d**3*g**5 - 576*a**5*b**3*c*d**2*g**5 + 576*a**4*b**4*c**2*d*g**5 -
192*a**3*b**5*c**3*g**5))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 647 vs. \(2 (192) = 384\).

Time = 0.22 (sec) , antiderivative size = 647, normalized size of antiderivative = 3.14 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^5} \, dx=-\frac {1}{48} \, B {\left (\frac {12 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 13 \, a b^{2} c^{2} d - 23 \, a^{2} b c d^{2} + 25 \, a^{3} d^{3} - 6 \, {\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + 13 \, a^{2} b d^{3}\right )} x}{{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} g^{5} x^{4} + 4 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} g^{5} x + {\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} g^{5}} + \frac {12 \, \log \left (\frac {d e x}{b x + a} + \frac {c e}{b x + a}\right )}{b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}} + \frac {12 \, d^{4} \log \left (b x + a\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}} - \frac {12 \, d^{4} \log \left (d x + c\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}}\right )} - \frac {A}{4 \, {\left (b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}\right )}} \]

[In]

integrate((A+B*log(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g)^5,x, algorithm="maxima")

[Out]

-1/48*B*((12*b^3*d^3*x^3 - 3*b^3*c^3 + 13*a*b^2*c^2*d - 23*a^2*b*c*d^2 + 25*a^3*d^3 - 6*(b^3*c*d^2 - 7*a*b^2*d
^3)*x^2 + 4*(b^3*c^2*d - 5*a*b^2*c*d^2 + 13*a^2*b*d^3)*x)/((b^8*c^3 - 3*a*b^7*c^2*d + 3*a^2*b^6*c*d^2 - a^3*b^
5*d^3)*g^5*x^4 + 4*(a*b^7*c^3 - 3*a^2*b^6*c^2*d + 3*a^3*b^5*c*d^2 - a^4*b^4*d^3)*g^5*x^3 + 6*(a^2*b^6*c^3 - 3*
a^3*b^5*c^2*d + 3*a^4*b^4*c*d^2 - a^5*b^3*d^3)*g^5*x^2 + 4*(a^3*b^5*c^3 - 3*a^4*b^4*c^2*d + 3*a^5*b^3*c*d^2 -
a^6*b^2*d^3)*g^5*x + (a^4*b^4*c^3 - 3*a^5*b^3*c^2*d + 3*a^6*b^2*c*d^2 - a^7*b*d^3)*g^5) + 12*log(d*e*x/(b*x +
a) + c*e/(b*x + a))/(b^5*g^5*x^4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5) + 12*d^4
*log(b*x + a)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5) - 12*d^4*log(d
*x + c)/((b^5*c^4 - 4*a*b^4*c^3*d + 6*a^2*b^3*c^2*d^2 - 4*a^3*b^2*c*d^3 + a^4*b*d^4)*g^5)) - 1/4*A/(b^5*g^5*x^
4 + 4*a*b^4*g^5*x^3 + 6*a^2*b^3*g^5*x^2 + 4*a^3*b^2*g^5*x + a^4*b*g^5)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 751 vs. \(2 (192) = 384\).

Time = 0.83 (sec) , antiderivative size = 751, normalized size of antiderivative = 3.65 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^5} \, dx=-\frac {1}{48} \, {\left (12 \, {\left (\frac {{\left (d e x + c e\right )}^{4} B b^{3}}{{\left (b^{3} c^{3} e^{3} g^{5} - 3 \, a b^{2} c^{2} d e^{3} g^{5} + 3 \, a^{2} b c d^{2} e^{3} g^{5} - a^{3} d^{3} e^{3} g^{5}\right )} {\left (b x + a\right )}^{4}} - \frac {4 \, {\left (d e x + c e\right )}^{3} B b^{2} d}{{\left (b^{3} c^{3} e^{2} g^{5} - 3 \, a b^{2} c^{2} d e^{2} g^{5} + 3 \, a^{2} b c d^{2} e^{2} g^{5} - a^{3} d^{3} e^{2} g^{5}\right )} {\left (b x + a\right )}^{3}} + \frac {6 \, {\left (d e x + c e\right )}^{2} B b d^{2}}{{\left (b^{3} c^{3} e g^{5} - 3 \, a b^{2} c^{2} d e g^{5} + 3 \, a^{2} b c d^{2} e g^{5} - a^{3} d^{3} e g^{5}\right )} {\left (b x + a\right )}^{2}} - \frac {4 \, {\left (d e x + c e\right )} B d^{3}}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (b x + a\right )}}\right )} \log \left (\frac {d e x + c e}{b x + a}\right ) + \frac {3 \, {\left (4 \, A b^{3} - B b^{3}\right )} {\left (d e x + c e\right )}^{4}}{{\left (b^{3} c^{3} e^{3} g^{5} - 3 \, a b^{2} c^{2} d e^{3} g^{5} + 3 \, a^{2} b c d^{2} e^{3} g^{5} - a^{3} d^{3} e^{3} g^{5}\right )} {\left (b x + a\right )}^{4}} - \frac {16 \, {\left (3 \, A b^{2} d - B b^{2} d\right )} {\left (d e x + c e\right )}^{3}}{{\left (b^{3} c^{3} e^{2} g^{5} - 3 \, a b^{2} c^{2} d e^{2} g^{5} + 3 \, a^{2} b c d^{2} e^{2} g^{5} - a^{3} d^{3} e^{2} g^{5}\right )} {\left (b x + a\right )}^{3}} + \frac {36 \, {\left (2 \, A b d^{2} - B b d^{2}\right )} {\left (d e x + c e\right )}^{2}}{{\left (b^{3} c^{3} e g^{5} - 3 \, a b^{2} c^{2} d e g^{5} + 3 \, a^{2} b c d^{2} e g^{5} - a^{3} d^{3} e g^{5}\right )} {\left (b x + a\right )}^{2}} - \frac {48 \, {\left (A d^{3} - B d^{3}\right )} {\left (d e x + c e\right )}}{{\left (b^{3} c^{3} g^{5} - 3 \, a b^{2} c^{2} d g^{5} + 3 \, a^{2} b c d^{2} g^{5} - a^{3} d^{3} g^{5}\right )} {\left (b x + a\right )}}\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 )} \]

[In]

integrate((A+B*log(e*(d*x+c)/(b*x+a)))/(b*g*x+a*g)^5,x, algorithm="giac")

[Out]

-1/48*(12*((d*e*x + c*e)^4*B*b^3/((b^3*c^3*e^3*g^5 - 3*a*b^2*c^2*d*e^3*g^5 + 3*a^2*b*c*d^2*e^3*g^5 - a^3*d^3*e
^3*g^5)*(b*x + a)^4) - 4*(d*e*x + c*e)^3*B*b^2*d/((b^3*c^3*e^2*g^5 - 3*a*b^2*c^2*d*e^2*g^5 + 3*a^2*b*c*d^2*e^2
*g^5 - a^3*d^3*e^2*g^5)*(b*x + a)^3) + 6*(d*e*x + c*e)^2*B*b*d^2/((b^3*c^3*e*g^5 - 3*a*b^2*c^2*d*e*g^5 + 3*a^2
*b*c*d^2*e*g^5 - a^3*d^3*e*g^5)*(b*x + a)^2) - 4*(d*e*x + c*e)*B*d^3/((b^3*c^3*g^5 - 3*a*b^2*c^2*d*g^5 + 3*a^2
*b*c*d^2*g^5 - a^3*d^3*g^5)*(b*x + a)))*log((d*e*x + c*e)/(b*x + a)) + 3*(4*A*b^3 - B*b^3)*(d*e*x + c*e)^4/((b
^3*c^3*e^3*g^5 - 3*a*b^2*c^2*d*e^3*g^5 + 3*a^2*b*c*d^2*e^3*g^5 - a^3*d^3*e^3*g^5)*(b*x + a)^4) - 16*(3*A*b^2*d
- B*b^2*d)*(d*e*x + c*e)^3/((b^3*c^3*e^2*g^5 - 3*a*b^2*c^2*d*e^2*g^5 + 3*a^2*b*c*d^2*e^2*g^5 - a^3*d^3*e^2*g^
5)*(b*x + a)^3) + 36*(2*A*b*d^2 - B*b*d^2)*(d*e*x + c*e)^2/((b^3*c^3*e*g^5 - 3*a*b^2*c^2*d*e*g^5 + 3*a^2*b*c*d
^2*e*g^5 - a^3*d^3*e*g^5)*(b*x + a)^2) - 48*(A*d^3 - B*d^3)*(d*e*x + c*e)/((b^3*c^3*g^5 - 3*a*b^2*c^2*d*g^5 +
3*a^2*b*c*d^2*g^5 - a^3*d^3*g^5)*(b*x + a)))*(b*c/((b*c*e - a*d*e)*(b*c - a*d)) - a*d/((b*c*e - a*d*e)*(b*c -
a*d)))

Mupad [B] (verification not implemented)

Time = 3.25 (sec) , antiderivative size = 578, normalized size of antiderivative = 2.81 \[ \int \frac {A+B \log \left (\frac {e (c+d x)}{a+b x}\right )}{(a g+b g x)^5} \, dx=\frac {B\,d^4\,\mathrm {atanh}\left (\frac {-4\,a^4\,b\,d^4\,g^5+8\,a^3\,b^2\,c\,d^3\,g^5-8\,a\,b^4\,c^3\,d\,g^5+4\,b^5\,c^4\,g^5}{4\,b\,g^5\,{\left (a\,d-b\,c\right )}^4}-\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{2\,b\,g^5\,{\left (a\,d-b\,c\right )}^4}-\frac {B\,\ln \left (\frac {e\,\left (c+d\,x\right )}{a+b\,x}\right )}{4\,b^2\,g^5\,\left (4\,a^3\,x+\frac {a^4}{b}+b^3\,x^4+6\,a^2\,b\,x^2+4\,a\,b^2\,x^3\right )}-\frac {\frac {12\,A\,a^3\,d^3-12\,A\,b^3\,c^3-25\,B\,a^3\,d^3+3\,B\,b^3\,c^3+36\,A\,a\,b^2\,c^2\,d-36\,A\,a^2\,b\,c\,d^2-13\,B\,a\,b^2\,c^2\,d+23\,B\,a^2\,b\,c\,d^2}{12\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {d^2\,x^2\,\left (B\,b^3\,c-7\,B\,a\,b^2\,d\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {d\,x\,\left (13\,B\,a^2\,b\,d^2-5\,B\,a\,b^2\,c\,d+B\,b^3\,c^2\right )}{3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {B\,b^3\,d^3\,x^3}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{4\,a^4\,b\,g^5+16\,a^3\,b^2\,g^5\,x+24\,a^2\,b^3\,g^5\,x^2+16\,a\,b^4\,g^5\,x^3+4\,b^5\,g^5\,x^4} \]

[In]

int((A + B*log((e*(c + d*x))/(a + b*x)))/(a*g + b*g*x)^5,x)

[Out]

(B*d^4*atanh((4*b^5*c^4*g^5 - 4*a^4*b*d^4*g^5 - 8*a*b^4*c^3*d*g^5 + 8*a^3*b^2*c*d^3*g^5)/(4*b*g^5*(a*d - b*c)^
4) - (2*b*d*x*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a*d - b*c)^4))/(2*b*g^5*(a*d - b*c)^4) - (
B*log((e*(c + d*x))/(a + b*x)))/(4*b^2*g^5*(4*a^3*x + a^4/b + b^3*x^4 + 6*a^2*b*x^2 + 4*a*b^2*x^3)) - ((12*A*a
^3*d^3 - 12*A*b^3*c^3 - 25*B*a^3*d^3 + 3*B*b^3*c^3 + 36*A*a*b^2*c^2*d - 36*A*a^2*b*c*d^2 - 13*B*a*b^2*c^2*d +
23*B*a^2*b*c*d^2)/(12*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) + (d^2*x^2*(B*b^3*c - 7*B*a*b^2*d))
/(2*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (d*x*(B*b^3*c^2 + 13*B*a^2*b*d^2 - 5*B*a*b^2*c*d))/
(3*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) - (B*b^3*d^3*x^3)/(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d -
3*a^2*b*c*d^2))/(4*a^4*b*g^5 + 4*b^5*g^5*x^4 + 16*a^3*b^2*g^5*x + 16*a*b^4*g^5*x^3 + 24*a^2*b^3*g^5*x^2)