\(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{(a g+b g x)^5} \, dx\) [96]

Optimal result

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

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

Rubi [A] (verified)

Time = 0.11 (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} \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^5} \, dx=-\frac {B \log \left (\frac {e (a+b x)}{c+d x}\right )+A}{4 b g^5 (a+b x)^4}+\frac {B d^4 \log (a+b x)}{4 b g^5 (b c-a d)^4}-\frac {B d^4 \log (c+d x)}{4 b g^5 (b c-a d)^4}+\frac {B d^3}{4 b g^5 (a+b x) (b c-a d)^3}-\frac {B d^2}{8 b g^5 (a+b x)^2 (b c-a d)^2}+\frac {B d}{12 b g^5 (a+b x)^3 (b c-a d)}-\frac {B}{16 b g^5 (a+b x)^4} \]

[In]

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

[Out]

-1/16*B/(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) - (A + B*Log
[(e*(a + b*x))/(c + d*x)])/(4*b*g^5*(a + b*x)^4) - (B*d^4*Log[c + d*x])/(4*b*(b*c - a*d)^4*g^5)

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 (a+b x)}{c+d 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 (a+b x)}{c+d 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 (a+b x)}{c+d 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 {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{4 b g^5 (a+b x)^4}-\frac {B d^4 \log (c+d x)}{4 b (b c-a d)^4 g^5} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.77 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^5} \, dx=\frac {-\frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a+b x)^4}+\frac {B \left (-\frac {3 (b c-a d)^4}{(a+b x)^4}+\frac {4 d (b c-a d)^3}{(a+b x)^3}-\frac {6 d^2 (b c-a d)^2}{(a+b x)^2}+\frac {12 d^3 (b c-a d)}{a+b x}+12 d^4 \log (a+b x)-12 d^4 \log (c+d x)\right )}{12 (b c-a d)^4}}{4 b g^5} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(473\) vs. \(2(195)=390\).

Time = 2.16 (sec) , antiderivative size = 474, normalized size of antiderivative = 2.30

method	result	size
parts	\(-\frac {A}{4 g^{5} \left (b x +a \right )^{4} b}-\frac {B \left (a d -c b \right ) e \left (\frac {d^{5} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{5}}-\frac {3 d^{4} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{5}}+\frac {3 d^{3} b^{2} e^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{3 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}-\frac {1}{9 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}\right )}{\left (a d -c b \right )^{5}}-\frac {d^{2} b^{3} e^{3} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (a d -c b \right )^{5}}\right )}{g^{5} d^{2}}\)	\(474\)
risch	\(-\frac {B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{4 b \,g^{5} \left (b x +a \right )^{4}}-\frac {-48 B a \,b^{3} c \,d^{3} x^{2}-72 B \,a^{2} b^{2} c \,d^{3} x +24 B a \,b^{3} c^{2} d^{2} x -48 A \,a^{3} b c \,d^{3}+72 A \,a^{2} b^{2} c^{2} d^{2}-48 A a \,b^{3} c^{3} d -48 B \ln \left (-b x -a \right ) a \,b^{3} d^{4} x^{3}+48 B \ln \left (d x +c \right ) a \,b^{3} d^{4} x^{3}-72 B \ln \left (-b x -a \right ) a^{2} b^{2} d^{4} x^{2}+72 B \ln \left (d x +c \right ) a^{2} b^{2} d^{4} x^{2}-48 B \ln \left (-b x -a \right ) a^{3} b \,d^{4} x +48 B \ln \left (d x +c \right ) a^{3} b \,d^{4} x +12 B \ln \left (d x +c \right ) a^{4} d^{4}+12 B a \,b^{3} d^{4} x^{3}-12 B \,b^{4} c \,d^{3} x^{3}+42 B \,a^{2} b^{2} d^{4} x^{2}+6 B \,b^{4} c^{2} d^{2} x^{2}+52 B \,a^{3} b \,d^{4} x -4 B \,b^{4} c^{3} d x -12 B \ln \left (-b x -a \right ) a^{4} d^{4}+3 B \,b^{4} c^{4}+36 B \,a^{2} b^{2} c^{2} d^{2}+12 A \,a^{4} d^{4}+25 B \,a^{4} d^{4}-16 B a \,b^{3} c^{3} d +12 A \,b^{4} c^{4}-48 B \,a^{3} b c \,d^{3}-12 B \ln \left (-b x -a \right ) b^{4} d^{4} x^{4}+12 B \ln \left (d x +c \right ) b^{4} d^{4} x^{4}}{48 \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right ) g^{5} \left (b x +a \right )^{4} b}\)	\(525\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A \,b^{3} e^{3}}{4 \left (a d -c b \right )^{5} g^{5} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {d^{3} A \,b^{2} e^{2}}{\left (a d -c b \right )^{5} g^{5} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}+\frac {3 d^{4} A b e}{2 \left (a d -c b \right )^{5} g^{5} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {d^{5} A}{\left (a d -c b \right )^{5} g^{5} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}-\frac {d^{2} B \,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 )}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (a d -c b \right )^{5} g^{5}}+\frac {3 d^{3} B \,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 )}{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 )^{5} g^{5}}-\frac {3 d^{4} B 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 )}{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 )^{5} g^{5}}+\frac {d^{5} 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 )}{\left (a d -c b \right )^{5} g^{5}}\right )}{d^{2}}\)	\(675\)
default	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A \,b^{3} e^{3}}{4 \left (a d -c b \right )^{5} g^{5} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {d^{3} A \,b^{2} e^{2}}{\left (a d -c b \right )^{5} g^{5} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{3}}+\frac {3 d^{4} A b e}{2 \left (a d -c b \right )^{5} g^{5} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {d^{5} A}{\left (a d -c b \right )^{5} g^{5} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}-\frac {d^{2} B \,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 )}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}-\frac {1}{16 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{4}}\right )}{\left (a d -c b \right )^{5} g^{5}}+\frac {3 d^{3} B \,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 )}{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 )^{5} g^{5}}-\frac {3 d^{4} B 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 )}{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 )^{5} g^{5}}+\frac {d^{5} 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 )}{\left (a d -c b \right )^{5} g^{5}}\right )}{d^{2}}\)	\(675\)
parallelrisch	\(\frac {12 B \,x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b^{3} c \,d^{4}+48 B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{7} b^{2} c \,d^{4}+72 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{8} b c \,d^{4}+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}+48 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{9} c^{2} d^{3}-12 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} b^{3} c^{5}-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}-72 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{8} b \,c^{3} d^{2}+48 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{7} b^{2} c^{4} d +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}+48 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{9} c \,d^{4}-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 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 (b x +a \right )}{d x +c}\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 \,b^{2} d^{4} B \,x^{3} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\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 (b x +a \right )}{d x +c}\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}+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 {\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}+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 {b^{3} d^{4} B \,x^{4} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\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 \,a^{2} b \,d^{4} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\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}}{\left (b x +a \right )^{4} g^{4}}\)	\(970\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 629 vs. \(2 (192) = 384\).

Time = 0.26 (sec) , antiderivative size = 629, normalized size of antiderivative = 3.05 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^5} \, dx=-\frac {3 \, {\left (4 \, A + B\right )} b^{4} c^{4} - 16 \, {\left (3 \, A + B\right )} a b^{3} c^{3} d + 36 \, {\left (2 \, A + B\right )} a^{2} b^{2} c^{2} d^{2} - 48 \, {\left (A + B\right )} a^{3} b c d^{3} + {\left (12 \, A + 25 \, B\right )} a^{4} d^{4} - 12 \, {\left (B b^{4} c d^{3} - B a b^{3} d^{4}\right )} x^{3} + 6 \, {\left (B b^{4} c^{2} d^{2} - 8 \, B a b^{3} c d^{3} + 7 \, B a^{2} b^{2} d^{4}\right )} x^{2} - 4 \, {\left (B b^{4} c^{3} d - 6 \, B a b^{3} c^{2} d^{2} + 18 \, B a^{2} b^{2} c d^{3} - 13 \, B a^{3} b d^{4}\right )} x - 12 \, {\left (B b^{4} d^{4} x^{4} + 4 \, B a b^{3} d^{4} x^{3} + 6 \, B a^{2} b^{2} d^{4} x^{2} + 4 \, B a^{3} b d^{4} x - B b^{4} c^{4} + 4 \, B a b^{3} c^{3} d - 6 \, B a^{2} b^{2} c^{2} d^{2} + 4 \, B a^{3} b c d^{3}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{48 \, {\left ({\left (b^{9} c^{4} - 4 \, a b^{8} c^{3} d + 6 \, a^{2} b^{7} c^{2} d^{2} - 4 \, a^{3} b^{6} c d^{3} + a^{4} b^{5} d^{4}\right )} g^{5} x^{4} + 4 \, {\left (a b^{8} c^{4} - 4 \, a^{2} b^{7} c^{3} d + 6 \, a^{3} b^{6} c^{2} d^{2} - 4 \, a^{4} b^{5} c d^{3} + a^{5} b^{4} d^{4}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{7} c^{4} - 4 \, a^{3} b^{6} c^{3} d + 6 \, a^{4} b^{5} c^{2} d^{2} - 4 \, a^{5} b^{4} c d^{3} + a^{6} b^{3} d^{4}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{6} c^{4} - 4 \, a^{4} b^{5} c^{3} d + 6 \, a^{5} b^{4} c^{2} d^{2} - 4 \, a^{6} b^{3} c d^{3} + a^{7} b^{2} d^{4}\right )} g^{5} x + {\left (a^{4} b^{5} c^{4} - 4 \, a^{5} b^{4} c^{3} d + 6 \, a^{6} b^{3} c^{2} d^{2} - 4 \, a^{7} b^{2} c d^{3} + a^{8} b d^{4}\right )} g^{5}\right )}} \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(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((b*e*x + a*e)/(d*x + c)))/((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 = 2.41 (sec) , antiderivative size = 944, normalized size of antiderivative = 4.58 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^5} \, dx=- \frac {B \log {\left (\frac {e \left (a + b x\right )}{c + d 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} \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*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**5,x)

[Out]

-B*log(e*(a + b*x)/(c + d*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 - 4
8*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 - 19
2*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 -
288*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 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^5} \, dx=\frac {1}{48} \, B {\left (\frac {12 \, b^{3} d^{3} x^{3} - 3 \, b^{3} c^{3} + 13 \, a b^{2} c^{2} d - 23 \, a^{2} b c d^{2} + 25 \, a^{3} d^{3} - 6 \, {\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 4 \, {\left (b^{3} c^{2} d - 5 \, a b^{2} c d^{2} + 13 \, a^{2} b d^{3}\right )} x}{{\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} g^{5} x^{4} + 4 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} g^{5} x^{3} + 6 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} g^{5} x^{2} + 4 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} g^{5} x + {\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} g^{5}} - \frac {12 \, \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right )}{b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}} + \frac {12 \, d^{4} \log \left (b x + a\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}} - \frac {12 \, d^{4} \log \left (d x + c\right )}{{\left (b^{5} c^{4} - 4 \, a b^{4} c^{3} d + 6 \, a^{2} b^{3} c^{2} d^{2} - 4 \, a^{3} b^{2} c d^{3} + a^{4} b d^{4}\right )} g^{5}}\right )} - \frac {A}{4 \, {\left (b^{5} g^{5} x^{4} + 4 \, a b^{4} g^{5} x^{3} + 6 \, a^{2} b^{3} g^{5} x^{2} + 4 \, a^{3} b^{2} g^{5} x + a^{4} b g^{5}\right )}} \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(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(b*e*x/(d*x + c
) + a*e/(d*x + c))/(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. 573 vs. \(2 (192) = 384\).

Time = 0.49 (sec) , antiderivative size = 573, normalized size of antiderivative = 2.78 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^5} \, dx=-\frac {1}{48} \, {\left (\frac {12 \, {\left (B b^{3} e^{5} - \frac {4 \, {\left (b e x + a e\right )} B b^{2} d e^{4}}{d x + c} + \frac {6 \, {\left (b e x + a e\right )}^{2} B b d^{2} e^{3}}{{\left (d x + c\right )}^{2}} - \frac {4 \, {\left (b e x + a e\right )}^{3} B d^{3} e^{2}}{{\left (d x + c\right )}^{3}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{\frac {{\left (b e x + a e\right )}^{4} b^{3} c^{3} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {3 \, {\left (b e x + a e\right )}^{4} a b^{2} c^{2} d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {3 \, {\left (b e x + a e\right )}^{4} a^{2} b c d^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b e x + a e\right )}^{4} a^{3} d^{3} g^{5}}{{\left (d x + c\right )}^{4}}} + \frac {12 \, A b^{3} e^{5} + 3 \, B b^{3} e^{5} - \frac {48 \, {\left (b e x + a e\right )} A b^{2} d e^{4}}{d x + c} - \frac {16 \, {\left (b e x + a e\right )} B b^{2} d e^{4}}{d x + c} + \frac {72 \, {\left (b e x + a e\right )}^{2} A b d^{2} e^{3}}{{\left (d x + c\right )}^{2}} + \frac {36 \, {\left (b e x + a e\right )}^{2} B b d^{2} e^{3}}{{\left (d x + c\right )}^{2}} - \frac {48 \, {\left (b e x + a e\right )}^{3} A d^{3} e^{2}}{{\left (d x + c\right )}^{3}} - \frac {48 \, {\left (b e x + a e\right )}^{3} B d^{3} e^{2}}{{\left (d x + c\right )}^{3}}}{\frac {{\left (b e x + a e\right )}^{4} b^{3} c^{3} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {3 \, {\left (b e x + a e\right )}^{4} a b^{2} c^{2} d g^{5}}{{\left (d x + c\right )}^{4}} + \frac {3 \, {\left (b e x + a e\right )}^{4} a^{2} b c d^{2} g^{5}}{{\left (d x + c\right )}^{4}} - \frac {{\left (b e x + a e\right )}^{4} a^{3} d^{3} g^{5}}{{\left (d x + c\right )}^{4}}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]

[In]

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

[Out]

-1/48*(12*(B*b^3*e^5 - 4*(b*e*x + a*e)*B*b^2*d*e^4/(d*x + c) + 6*(b*e*x + a*e)^2*B*b*d^2*e^3/(d*x + c)^2 - 4*(
b*e*x + a*e)^3*B*d^3*e^2/(d*x + c)^3)*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^4*b^3*c^3*g^5/(d*x + c)^4 -
3*(b*e*x + a*e)^4*a*b^2*c^2*d*g^5/(d*x + c)^4 + 3*(b*e*x + a*e)^4*a^2*b*c*d^2*g^5/(d*x + c)^4 - (b*e*x + a*e)^
4*a^3*d^3*g^5/(d*x + c)^4) + (12*A*b^3*e^5 + 3*B*b^3*e^5 - 48*(b*e*x + a*e)*A*b^2*d*e^4/(d*x + c) - 16*(b*e*x
+ a*e)*B*b^2*d*e^4/(d*x + c) + 72*(b*e*x + a*e)^2*A*b*d^2*e^3/(d*x + c)^2 + 36*(b*e*x + a*e)^2*B*b*d^2*e^3/(d*
x + c)^2 - 48*(b*e*x + a*e)^3*A*d^3*e^2/(d*x + c)^3 - 48*(b*e*x + a*e)^3*B*d^3*e^2/(d*x + c)^3)/((b*e*x + a*e)
^4*b^3*c^3*g^5/(d*x + c)^4 - 3*(b*e*x + a*e)^4*a*b^2*c^2*d*g^5/(d*x + c)^4 + 3*(b*e*x + a*e)^4*a^2*b*c*d^2*g^5
/(d*x + c)^4 - (b*e*x + a*e)^4*a^3*d^3*g^5/(d*x + c)^4))*(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 = 2.97 (sec) , antiderivative size = 577, normalized size of antiderivative = 2.80 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^5} \, dx=-\frac {\frac {12\,A\,a^3\,d^3-12\,A\,b^3\,c^3+25\,B\,a^3\,d^3-3\,B\,b^3\,c^3+36\,A\,a\,b^2\,c^2\,d-36\,A\,a^2\,b\,c\,d^2+13\,B\,a\,b^2\,c^2\,d-23\,B\,a^2\,b\,c\,d^2}{12\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}-\frac {d^2\,x^2\,\left (B\,b^3\,c-7\,B\,a\,b^2\,d\right )}{2\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {d\,x\,\left (13\,B\,a^2\,b\,d^2-5\,B\,a\,b^2\,c\,d+B\,b^3\,c^2\right )}{3\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}+\frac {B\,b^3\,d^3\,x^3}{a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3}}{4\,a^4\,b\,g^5+16\,a^3\,b^2\,g^5\,x+24\,a^2\,b^3\,g^5\,x^2+16\,a\,b^4\,g^5\,x^3+4\,b^5\,g^5\,x^4}-\frac {B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{4\,b^2\,g^5\,\left (4\,a^3\,x+\frac {a^4}{b}+b^3\,x^4+6\,a^2\,b\,x^2+4\,a\,b^2\,x^3\right )}-\frac {B\,d^4\,\mathrm {atanh}\left (\frac {-4\,a^4\,b\,d^4\,g^5+8\,a^3\,b^2\,c\,d^3\,g^5-8\,a\,b^4\,c^3\,d\,g^5+4\,b^5\,c^4\,g^5}{4\,b\,g^5\,{\left (a\,d-b\,c\right )}^4}-\frac {2\,b\,d\,x\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{{\left (a\,d-b\,c\right )}^4}\right )}{2\,b\,g^5\,{\left (a\,d-b\,c\right )}^4} \]

[In]

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

[Out]

- ((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) - (B*log((e*(a + b*x))/(c + d*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)) -
(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)