\(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{(a g+b g x)^3 (c i+d i x)} \, dx\) [37]

Optimal result

Integrand size = 40, antiderivative size = 255 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=-\frac {B (c+d x)^2 \left (b-\frac {4 d (a+b x)}{c+d x}\right )^2}{4 (b c-a d)^3 g^3 i (a+b x)^2}-\frac {B d^2 \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^3 g^3 i}+\frac {2 b d (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g^3 i (a+b x)}-\frac {b^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g^3 i (a+b x)^2}+\frac {d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g^3 i} \]

[Out]

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

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.175, Rules used = {2562, 45, 2372, 12, 14, 37, 2338} \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=-\frac {b^2 (c+d x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g^3 i (a+b x)^2 (b c-a d)^3}+\frac {d^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i (b c-a d)^3}+\frac {2 b d (c+d x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g^3 i (a+b x) (b c-a d)^3}-\frac {B d^2 \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g^3 i (b c-a d)^3}-\frac {B (c+d x)^2 \left (b-\frac {4 d (a+b x)}{c+d x}\right )^2}{4 g^3 i (a+b x)^2 (b c-a d)^3} \]

[In]

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

[Out]

-1/4*(B*(c + d*x)^2*(b - (4*d*(a + b*x))/(c + d*x))^2)/((b*c - a*d)^3*g^3*i*(a + b*x)^2) - (B*d^2*Log[(a + b*x
)/(c + d*x)]^2)/(2*(b*c - a*d)^3*g^3*i) + (2*b*d*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^
3*g^3*i*(a + b*x)) - (b^2*(c + d*x)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/(2*(b*c - a*d)^3*g^3*i*(a + b*x)^2
) + (d^2*Log[(a + b*x)/(c + d*x)]*(A + B*Log[(e*(a + b*x))/(c + d*x)]))/((b*c - a*d)^3*g^3*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 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

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

Mathematica [C] (verified)

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

Time = 0.22 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.64 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=\frac {-2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d (b c-a d) (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 d^2 (a+b x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-4 d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)+4 B d (a+b x) (b c-a d+d (a+b x) \log (a+b x)-d (a+b x) \log (c+d x))-B \left ((b c-a d)^2+2 d (-b c+a d) (a+b x)-2 d^2 (a+b x)^2 \log (a+b x)+2 d^2 (a+b x)^2 \log (c+d x)\right )-2 B d^2 (a+b x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+2 B d^2 (a+b x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{4 (b c-a d)^3 g^3 i (a+b x)^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 365, normalized size of antiderivative = 1.43

method	result	size
parts	\(\frac {A \left (\frac {d^{2} \ln \left (d x +c \right )}{\left (a d -c b \right )^{3}}+\frac {1}{2 \left (a d -c b \right ) \left (b x +a \right )^{2}}+\frac {d}{\left (a d -c b \right )^{2} \left (b x +a \right )}-\frac {d^{2} \ln \left (b x +a \right )}{\left (a d -c b \right )^{3}}\right )}{g^{3} i}-\frac {B \left (\frac {d^{3} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )^{3}}-\frac {2 d^{2} b e \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{\left (a d -c b \right )^{3}}+\frac {d \,e^{2} b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{\left (a d -c b \right )^{3}}\right )}{g^{3} i d}\)	\(365\)
risch	\(\frac {A \,d^{2} \ln \left (d x +c \right )}{g^{3} i \left (a d -c b \right )^{3}}+\frac {A}{2 g^{3} i \left (a d -c b \right ) \left (b x +a \right )^{2}}+\frac {A d}{g^{3} i \left (a d -c b \right )^{2} \left (b x +a \right )}-\frac {A \,d^{2} \ln \left (b x +a \right )}{g^{3} i \left (a d -c b \right )^{3}}-\frac {B \,d^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 g^{3} i \left (a d -c b \right )^{3}}-\frac {2 B d b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{g^{3} i \left (a d -c b \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}-\frac {2 B d b e}{g^{3} i \left (a d -c b \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )}+\frac {B \,e^{2} b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 g^{3} i \left (a d -c b \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{2}}+\frac {B \,e^{2} b^{2}}{4 g^{3} i \left (a d -c b \right )^{3} \left (\frac {b e}{d}+\frac {e a}{d x +c}-\frac {e c b}{d \left (d x +c \right )}\right )^{2}}\)	\(447\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (-\frac {d^{2} e A \,b^{2}}{2 i \left (a d -c b \right )^{4} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {2 d^{3} A b}{i \left (a d -c b \right )^{4} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{4} A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (a d -c b \right )^{4} g^{3}}+\frac {d^{2} e B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i \left (a d -c b \right )^{4} g^{3}}-\frac {2 d^{3} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i \left (a d -c b \right )^{4} g^{3}}+\frac {d^{4} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e i \left (a d -c b \right )^{4} g^{3}}\right )}{d^{2}}\)	\(460\)
default	\(-\frac {e \left (a d -c b \right ) \left (-\frac {d^{2} e A \,b^{2}}{2 i \left (a d -c b \right )^{4} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}+\frac {2 d^{3} A b}{i \left (a d -c b \right )^{4} g^{3} \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}+\frac {d^{4} A \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e i \left (a d -c b \right )^{4} g^{3}}+\frac {d^{2} e B \,b^{2} \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}-\frac {1}{4 \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}\right )}{i \left (a d -c b \right )^{4} g^{3}}-\frac {2 d^{3} B b \left (-\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}-\frac {1}{\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}}\right )}{i \left (a d -c b \right )^{4} g^{3}}+\frac {d^{4} B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e i \left (a d -c b \right )^{4} g^{3}}\right )}{d^{2}}\)	\(460\)
parallelrisch	\(-\frac {2 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{4} b^{2} c^{2} d^{2}+4 A \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} b^{2} c^{2} d^{2}+6 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} b^{2} c^{2} d^{2}+4 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{5} b \,c^{2} d^{2}+8 A x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} b \,c^{2} d^{2}+8 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} b \,c^{2} d^{2}+4 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} b^{2} c^{3} d +6 A \,x^{2} a^{4} b^{2} c^{2} d^{2}-8 A \,x^{2} a^{3} b^{3} c^{3} d +7 B \,x^{2} a^{4} b^{2} c^{2} d^{2}-8 B \,x^{2} a^{3} b^{3} c^{3} d +8 A x \,a^{5} b \,c^{2} d^{2}-12 A x \,a^{4} b^{2} c^{3} d +8 B x \,a^{5} b \,c^{2} d^{2}-10 B x \,a^{4} b^{2} c^{3} d +8 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{5} b \,c^{3} d +2 A \,x^{2} a^{2} b^{4} c^{4}+B \,x^{2} a^{2} b^{4} c^{4}+2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{6} c^{2} d^{2}+4 A x \,a^{3} b^{3} c^{4}+4 A \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{6} c^{2} d^{2}+2 B x \,a^{3} b^{3} c^{4}-2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} b^{2} c^{4}}{4 i \,g^{3} \left (b x +a \right )^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) c^{2} \left (a d -c b \right ) a^{4}}\)	\(563\)
norman	\(\frac {\frac {6 A a \,b^{2} d -2 A \,b^{3} c +7 B a \,b^{2} d -B \,b^{3} c}{4 g i \left (a d -c b \right )^{2} b^{2}}-\frac {\left (2 A \,a^{2} d^{2}+4 B a b c d -B \,b^{2} c^{2}\right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (2 A \,b^{2} d +3 B \,b^{2} d \right ) x}{2 i g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b}-\frac {B \,a^{2} d^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b^{2} \left (2 A \,d^{2}+3 B \,d^{2}\right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b \left (2 A a \,d^{2}+2 B a \,d^{2}+B b c d \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b^{2} B \,d^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b B a \,d^{2} x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{\left (b x +a \right )^{2} g^{2}}\)	\(578\)

[In]

int((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3/(d*i*x+c*i),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.37 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=-\frac {{\left (2 \, A + B\right )} b^{2} c^{2} - 8 \, {\left (A + B\right )} a b c d + {\left (6 \, A + 7 \, B\right )} a^{2} d^{2} - 2 \, {\left (B b^{2} d^{2} x^{2} + 2 \, B a b d^{2} x + B a^{2} d^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} - 2 \, {\left ({\left (2 \, A + 3 \, B\right )} b^{2} c d - {\left (2 \, A + 3 \, B\right )} a b d^{2}\right )} x - 2 \, {\left ({\left (2 \, A + 3 \, B\right )} b^{2} d^{2} x^{2} - B b^{2} c^{2} + 4 \, B a b c d + 2 \, A a^{2} d^{2} + 2 \, {\left (B b^{2} c d + 2 \, {\left (A + B\right )} a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{4} c^{3} - 3 \, a^{2} b^{3} c^{2} d + 3 \, a^{3} b^{2} c d^{2} - a^{4} b d^{3}\right )} g^{3} i x + {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} g^{3} i\right )}} \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3/(d*i*x+c*i),x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 889 vs. \(2 (221) = 442\).

Time = 2.59 (sec) , antiderivative size = 889, normalized size of antiderivative = 3.49 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=- \frac {B d^{2} \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{3} d^{3} g^{3} i - 6 a^{2} b c d^{2} g^{3} i + 6 a b^{2} c^{2} d g^{3} i - 2 b^{3} c^{3} g^{3} i} + \frac {d^{2} \cdot \left (2 A + 3 B\right ) \log {\left (x + \frac {2 A a d^{3} + 2 A b c d^{2} + 3 B a d^{3} + 3 B b c d^{2} - \frac {a^{4} d^{6} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b c d^{5} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{2} c^{2} d^{4} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}} + \frac {4 a b^{3} c^{3} d^{3} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}} - \frac {b^{4} c^{4} d^{2} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}}}{4 A b d^{3} + 6 B b d^{3}} \right )}}{2 g^{3} i \left (a d - b c\right )^{3}} - \frac {d^{2} \cdot \left (2 A + 3 B\right ) \log {\left (x + \frac {2 A a d^{3} + 2 A b c d^{2} + 3 B a d^{3} + 3 B b c d^{2} + \frac {a^{4} d^{6} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b c d^{5} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{2} c^{2} d^{4} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}} - \frac {4 a b^{3} c^{3} d^{3} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}} + \frac {b^{4} c^{4} d^{2} \cdot \left (2 A + 3 B\right )}{\left (a d - b c\right )^{3}}}{4 A b d^{3} + 6 B b d^{3}} \right )}}{2 g^{3} i \left (a d - b c\right )^{3}} + \frac {\left (3 B a d - B b c + 2 B b d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 a^{4} d^{2} g^{3} i - 4 a^{3} b c d g^{3} i + 4 a^{3} b d^{2} g^{3} i x + 2 a^{2} b^{2} c^{2} g^{3} i - 8 a^{2} b^{2} c d g^{3} i x + 2 a^{2} b^{2} d^{2} g^{3} i x^{2} + 4 a b^{3} c^{2} g^{3} i x - 4 a b^{3} c d g^{3} i x^{2} + 2 b^{4} c^{2} g^{3} i x^{2}} + \frac {6 A a d - 2 A b c + 7 B a d - B b c + x \left (4 A b d + 6 B b d\right )}{4 a^{4} d^{2} g^{3} i - 8 a^{3} b c d g^{3} i + 4 a^{2} b^{2} c^{2} g^{3} i + x^{2} \cdot \left (4 a^{2} b^{2} d^{2} g^{3} i - 8 a b^{3} c d g^{3} i + 4 b^{4} c^{2} g^{3} i\right ) + x \left (8 a^{3} b d^{2} g^{3} i - 16 a^{2} b^{2} c d g^{3} i + 8 a b^{3} c^{2} g^{3} i\right )} \]

[In]

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)**3/(d*i*x+c*i),x)

[Out]

-B*d**2*log(e*(a + b*x)/(c + d*x))**2/(2*a**3*d**3*g**3*i - 6*a**2*b*c*d**2*g**3*i + 6*a*b**2*c**2*d*g**3*i -
2*b**3*c**3*g**3*i) + d**2*(2*A + 3*B)*log(x + (2*A*a*d**3 + 2*A*b*c*d**2 + 3*B*a*d**3 + 3*B*b*c*d**2 - a**4*d
**6*(2*A + 3*B)/(a*d - b*c)**3 + 4*a**3*b*c*d**5*(2*A + 3*B)/(a*d - b*c)**3 - 6*a**2*b**2*c**2*d**4*(2*A + 3*B
)/(a*d - b*c)**3 + 4*a*b**3*c**3*d**3*(2*A + 3*B)/(a*d - b*c)**3 - b**4*c**4*d**2*(2*A + 3*B)/(a*d - b*c)**3)/
(4*A*b*d**3 + 6*B*b*d**3))/(2*g**3*i*(a*d - b*c)**3) - d**2*(2*A + 3*B)*log(x + (2*A*a*d**3 + 2*A*b*c*d**2 + 3
*B*a*d**3 + 3*B*b*c*d**2 + a**4*d**6*(2*A + 3*B)/(a*d - b*c)**3 - 4*a**3*b*c*d**5*(2*A + 3*B)/(a*d - b*c)**3 +
6*a**2*b**2*c**2*d**4*(2*A + 3*B)/(a*d - b*c)**3 - 4*a*b**3*c**3*d**3*(2*A + 3*B)/(a*d - b*c)**3 + b**4*c**4*
d**2*(2*A + 3*B)/(a*d - b*c)**3)/(4*A*b*d**3 + 6*B*b*d**3))/(2*g**3*i*(a*d - b*c)**3) + (3*B*a*d - B*b*c + 2*B
*b*d*x)*log(e*(a + b*x)/(c + d*x))/(2*a**4*d**2*g**3*i - 4*a**3*b*c*d*g**3*i + 4*a**3*b*d**2*g**3*i*x + 2*a**2
*b**2*c**2*g**3*i - 8*a**2*b**2*c*d*g**3*i*x + 2*a**2*b**2*d**2*g**3*i*x**2 + 4*a*b**3*c**2*g**3*i*x - 4*a*b**
3*c*d*g**3*i*x**2 + 2*b**4*c**2*g**3*i*x**2) + (6*A*a*d - 2*A*b*c + 7*B*a*d - B*b*c + x*(4*A*b*d + 6*B*b*d))/(
4*a**4*d**2*g**3*i - 8*a**3*b*c*d*g**3*i + 4*a**2*b**2*c**2*g**3*i + x**2*(4*a**2*b**2*d**2*g**3*i - 8*a*b**3*
c*d*g**3*i + 4*b**4*c**2*g**3*i) + x*(8*a**3*b*d**2*g**3*i - 16*a**2*b**2*c*d*g**3*i + 8*a*b**3*c**2*g**3*i))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 885 vs. \(2 (249) = 498\).

Time = 0.24 (sec) , antiderivative size = 885, normalized size of antiderivative = 3.47 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=\frac {1}{2} \, B {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} g^{3} i x + {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} g^{3} i} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{3} i} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{3} i}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + \frac {1}{2} \, A {\left (\frac {2 \, b d x - b c + 3 \, a d}{{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} g^{3} i x^{2} + 2 \, {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} g^{3} i x + {\left (a^{2} b^{2} c^{2} - 2 \, a^{3} b c d + a^{4} d^{2}\right )} g^{3} i} + \frac {2 \, d^{2} \log \left (b x + a\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{3} i} - \frac {2 \, d^{2} \log \left (d x + c\right )}{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} g^{3} i}\right )} - \frac {{\left (b^{2} c^{2} - 8 \, a b c d + 7 \, a^{2} d^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (d x + c\right )^{2} - 6 \, {\left (b^{2} c d - a b d^{2}\right )} x - 6 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 6 \, a b d^{2} x + 3 \, a^{2} d^{2} - 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, a b d^{2} x + a^{2} d^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B}{4 \, {\left (a^{2} b^{3} c^{3} g^{3} i - 3 \, a^{3} b^{2} c^{2} d g^{3} i + 3 \, a^{4} b c d^{2} g^{3} i - a^{5} d^{3} g^{3} i + {\left (b^{5} c^{3} g^{3} i - 3 \, a b^{4} c^{2} d g^{3} i + 3 \, a^{2} b^{3} c d^{2} g^{3} i - a^{3} b^{2} d^{3} g^{3} i\right )} x^{2} + 2 \, {\left (a b^{4} c^{3} g^{3} i - 3 \, a^{2} b^{3} c^{2} d g^{3} i + 3 \, a^{3} b^{2} c d^{2} g^{3} i - a^{4} b d^{3} g^{3} i\right )} x\right )}} \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3/(d*i*x+c*i),x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 42.82 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.55 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=-\frac {1}{4} \, {\left (\frac {2 \, {\left (d x + c\right )}^{2} B e^{3} \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b e x + a e\right )}^{2} g^{3} i} + \frac {{\left (2 \, A e^{3} + B e^{3}\right )} {\left (d x + c\right )}^{2}}{{\left (b e x + a e\right )}^{2} g^{3} i}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )}^{2} \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)^3/(d*i*x+c*i),x, algorithm="giac")

[Out]

-1/4*(2*(d*x + c)^2*B*e^3*log((b*e*x + a*e)/(d*x + c))/((b*e*x + a*e)^2*g^3*i) + (2*A*e^3 + B*e^3)*(d*x + c)^2
/((b*e*x + a*e)^2*g^3*i))*(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 = 3.50 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.14 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x)^3 (c i+d i x)} \, dx=\frac {3\,A\,a\,d}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}-\frac {B\,d^2\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^3}-\frac {A\,b\,c}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}+\frac {7\,B\,a\,d}{4\,g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}-\frac {B\,b\,c}{4\,g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}+\frac {3\,B\,a^2\,d^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2}+\frac {B\,b^2\,c^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2}+\frac {A\,b\,d\,x}{g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}+\frac {3\,B\,b\,d\,x}{2\,g^3\,i\,{\left (a\,d-b\,c\right )}^2\,{\left (a+b\,x\right )}^2}+\frac {B\,a\,b\,d^2\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2}-\frac {B\,b^2\,c\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2}-\frac {2\,B\,a\,b\,c\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g^3\,i\,{\left (a\,d-b\,c\right )}^3\,{\left (a+b\,x\right )}^2}+\frac {A\,d^2\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{g^3\,i\,{\left (a\,d-b\,c\right )}^3}+\frac {B\,d^2\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,3{}\mathrm {i}}{g^3\,i\,{\left (a\,d-b\,c\right )}^3} \]

[In]

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

[Out]

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