\(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{(a g+b g x) (c i+d i x)^3} \, dx\) [51]

Optimal result

Integrand size = 40, antiderivative size = 243 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x) (c i+d i x)^3} \, dx=-\frac {B \left (4 b-\frac {d (a+b x)}{c+d x}\right )^2}{4 (b c-a d)^3 g i^3}-\frac {b^2 B \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 (b c-a d)^3 g i^3}+\frac {d^2 (a+b x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{2 (b c-a d)^3 g i^3 (c+d x)^2}-\frac {2 b d (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g i^3 (c+d x)}+\frac {b^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )}{(b c-a d)^3 g i^3} \]

[Out]

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

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2562, 45, 2372, 2338} \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {b^2 \log \left (\frac {a+b x}{c+d x}\right ) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g i^3 (b c-a d)^3}+\frac {d^2 (a+b x)^2 \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{2 g i^3 (c+d x)^2 (b c-a d)^3}-\frac {2 b d (a+b x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g i^3 (c+d x) (b c-a d)^3}-\frac {b^2 B \log ^2\left (\frac {a+b x}{c+d x}\right )}{2 g i^3 (b c-a d)^3}-\frac {B \left (4 b-\frac {d (a+b x)}{c+d x}\right )^2}{4 g i^3 (b c-a d)^3} \]

[In]

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

[Out]

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

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 2338

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2372

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(x_)^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> With[{u = I
ntHide[x^m*(d + e*x^r)^q, x]}, Dist[a + b*Log[c*x^n], u, x] - Dist[b*n, Int[SimplifyIntegrand[u/x, x], x], x]]
/; FreeQ[{a, b, c, d, e, n, r}, x] && IGtQ[q, 0] && IntegerQ[m] &&  !(EqQ[q, 1] && EqQ[m, -1])

Rule 2562

Int[((A_.) + Log[(e_.)*((a_.) + (b_.)*(x_))^(n_.)*((c_.) + (d_.)*(x_))^(mn_)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_)
)^(m_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[x^m*(
(A + B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h,
i, A, B, n, p}, x] && EqQ[n + mn, 0] && IGtQ[n, 0] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i
, 0] && IntegersQ[m, q]

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

Mathematica [C] (verified)

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

Time = 0.27 (sec) , antiderivative size = 418, normalized size of antiderivative = 1.72 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {2 (b c-a d)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 b (b c-a d) (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+4 b^2 (c+d x)^2 \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-4 b^2 (c+d x)^2 \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-4 b B (c+d x) (b c-a d+b (c+d x) \log (a+b x)-b (c+d x) \log (c+d x))-B \left ((b c-a d)^2+2 b (b c-a d) (c+d x)+2 b^2 (c+d x)^2 \log (a+b x)-2 b^2 (c+d x)^2 \log (c+d x)\right )-2 b^2 B (c+d x)^2 \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+2 b^2 B (c+d x)^2 \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{4 (b c-a d)^3 g i^3 (c+d x)^2} \]

[In]

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

[Out]

(2*(b*c - a*d)^2*(A + B*Log[(e*(a + b*x))/(c + d*x)]) + 4*b*(b*c - a*d)*(c + d*x)*(A + B*Log[(e*(a + b*x))/(c
+ d*x)]) + 4*b^2*(c + d*x)^2*Log[a + b*x]*(A + B*Log[(e*(a + b*x))/(c + d*x)]) - 4*b^2*(c + d*x)^2*(A + B*Log[
(e*(a + b*x))/(c + d*x)])*Log[c + d*x] - 4*b*B*(c + d*x)*(b*c - a*d + b*(c + d*x)*Log[a + b*x] - b*(c + d*x)*L
og[c + d*x]) - B*((b*c - a*d)^2 + 2*b*(b*c - a*d)*(c + d*x) + 2*b^2*(c + d*x)^2*Log[a + b*x] - 2*b^2*(c + d*x)
^2*Log[c + d*x]) - 2*b^2*B*(c + d*x)^2*(Log[a + b*x]*(Log[a + b*x] - 2*Log[(b*(c + d*x))/(b*c - a*d)]) - 2*Pol
yLog[2, (d*(a + b*x))/(-(b*c) + a*d)]) + 2*b^2*B*(c + d*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*i^3*(c + d*x)^2)

Maple [A] (verified)

Time = 1.22 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.51

method	result	size
parts	\(\frac {A \left (-\frac {1}{2 \left (a d -c b \right ) \left (d x +c \right )^{2}}+\frac {b^{2} \ln \left (d x +c \right )}{\left (a d -c b \right )^{3}}+\frac {b}{\left (a d -c b \right )^{2} \left (d x +c \right )}-\frac {b^{2} \ln \left (b x +a \right )}{\left (a d -c b \right )^{3}}\right )}{g \,i^{3}}-\frac {B d \left (\frac {d \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{a d -c b}-\frac {2 b e \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{a d -c b}+\frac {e^{2} b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 d \left (a d -c b \right )}\right )}{g \,i^{3} \left (a d -c b \right )^{2} e^{2}}\)	\(367\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{3} \left (a d -c b \right )^{4} g}-\frac {2 d^{3} A b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{3} \left (a d -c b \right )^{4} g}+\frac {d^{4} A \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e^{3} i^{3} \left (a d -c b \right )^{4} g}+\frac {d^{2} B \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e \,i^{3} \left (a d -c b \right )^{4} g}-\frac {2 d^{3} B b \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{3} \left (a d -c b \right )^{4} g}+\frac {d^{4} B \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{e^{3} i^{3} \left (a d -c b \right )^{4} g}\right )}{d^{2}}\)	\(462\)
default	\(-\frac {e \left (a d -c b \right ) \left (\frac {d^{2} A \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{3} \left (a d -c b \right )^{4} g}-\frac {2 d^{3} A b \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{3} \left (a d -c b \right )^{4} g}+\frac {d^{4} A \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e^{3} i^{3} \left (a d -c b \right )^{4} g}+\frac {d^{2} B \,b^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e \,i^{3} \left (a d -c b \right )^{4} g}-\frac {2 d^{3} B b \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{3} \left (a d -c b \right )^{4} g}+\frac {d^{4} B \left (\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2} \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{2}-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{4}\right )}{e^{3} i^{3} \left (a d -c b \right )^{4} g}\right )}{d^{2}}\)	\(462\)
parallelrisch	\(-\frac {-4 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{4} d^{2}-8 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{2} c^{5} d +12 A x \,a^{3} b \,c^{4} d^{2}-8 A x \,a^{2} b^{2} c^{5} d -10 B x \,a^{3} b \,c^{4} d^{2}+8 B x \,a^{2} b^{2} c^{5} d -8 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{3} b \,c^{5} d +8 A \,x^{2} a^{3} b \,c^{3} d^{3}-6 A \,x^{2} a^{2} b^{2} c^{4} d^{2}-8 B \,x^{2} a^{3} b \,c^{3} d^{3}+7 B \,x^{2} a^{2} b^{2} c^{4} d^{2}-2 A \,x^{2} a^{4} c^{2} d^{4}+B \,x^{2} a^{4} c^{2} d^{4}+2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{2} b^{2} c^{6}-4 A x \,a^{4} c^{3} d^{3}+4 A \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{2} c^{6}+2 B x \,a^{4} c^{3} d^{3}+2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{4} c^{4} d^{2}+2 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{2} b^{2} c^{4} d^{2}+4 A \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{2} c^{4} d^{2}-6 B \,x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{2} c^{4} d^{2}+4 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} a^{2} b^{2} c^{5} d +8 A x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a^{2} b^{2} c^{5} d}{4 i^{3} g \left (d x +c \right )^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) a^{2} \left (a d -c b \right ) c^{4}}\)	\(563\)
norman	\(\frac {-\frac {2 A a \,d^{3}-6 A b c \,d^{2}-B a \,d^{3}+7 B b c \,d^{2}}{4 g i \left (a d -c b \right )^{2} d^{2}}-\frac {\left (2 A \,b^{2} c^{2}+B \,a^{2} d^{2}-4 B a b c d \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {\left (2 A b \,d^{2}-3 B b \,d^{2}\right ) x}{2 i g \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) d}-\frac {B \,b^{2} c^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d^{2} \left (2 A \,b^{2}-3 B \,b^{2}\right ) x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{2 g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d \left (2 A \,b^{2} c -B a b d -2 B \,b^{2} c \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {b^{2} B \,d^{2} x^{2} \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 i g \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}-\frac {d B \,b^{2} c x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{g i \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}}{i^{2} \left (d x +c \right )^{2}}\)	\(578\)
risch	\(-\frac {A}{2 g \,i^{3} \left (a d -c b \right ) \left (d x +c \right )^{2}}+\frac {A \,b^{2} \ln \left (d x +c \right )}{g \,i^{3} \left (a d -c b \right )^{3}}+\frac {A b}{g \,i^{3} \left (a d -c b \right )^{2} \left (d x +c \right )}-\frac {A \,b^{2} \ln \left (b x +a \right )}{g \,i^{3} \left (a d -c b \right )^{3}}+\frac {3 B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) b^{2}}{2 g \,i^{3} \left (a d -c b \right )^{3}}+\frac {B d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) b a}{g \,i^{3} \left (a d -c b \right )^{3} \left (d x +c \right )}-\frac {B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) b^{2} c}{g \,i^{3} \left (a d -c b \right )^{3} \left (d x +c \right )}-\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 ) a^{2}}{2 g \,i^{3} \left (a d -c b \right )^{3} \left (d x +c \right )^{2}}+\frac {B d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a c b}{g \,i^{3} \left (a d -c b \right )^{3} \left (d x +c \right )^{2}}-\frac {B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) c^{2} b^{2}}{2 g \,i^{3} \left (a d -c b \right )^{3} \left (d x +c \right )^{2}}-\frac {7 B \,b^{2}}{4 g \,i^{3} \left (a d -c b \right )^{3}}-\frac {3 B d b a}{2 g \,i^{3} \left (a d -c b \right )^{3} \left (d x +c \right )}+\frac {3 B \,b^{2} c}{2 g \,i^{3} \left (a d -c b \right )^{3} \left (d x +c \right )}+\frac {B \,d^{2} a^{2}}{4 g \,i^{3} \left (a d -c b \right )^{3} \left (d x +c \right )^{2}}-\frac {B d a c b}{2 g \,i^{3} \left (a d -c b \right )^{3} \left (d x +c \right )^{2}}+\frac {B \,c^{2} b^{2}}{4 g \,i^{3} \left (a d -c b \right )^{3} \left (d x +c \right )^{2}}-\frac {B \,b^{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 \,i^{3} \left (a d -c b \right )^{3}}\)	\(677\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 355, normalized size of antiderivative = 1.46 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {{\left (6 \, A - 7 \, B\right )} b^{2} c^{2} - 8 \, {\left (A - B\right )} a b c d + {\left (2 \, A - B\right )} a^{2} d^{2} + 2 \, {\left (B b^{2} d^{2} x^{2} + 2 \, B b^{2} c d x + B b^{2} c^{2}\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left ({\left (2 \, A - 3 \, B\right )} b^{2} c d - {\left (2 \, A - 3 \, B\right )} a b d^{2}\right )} x + 2 \, {\left ({\left (2 \, A - 3 \, B\right )} b^{2} d^{2} x^{2} + 2 \, A b^{2} c^{2} - 4 \, B a b c d + B a^{2} d^{2} + 2 \, {\left (2 \, {\left (A - B\right )} b^{2} c d - B a b d^{2}\right )} x\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{4 \, {\left ({\left (b^{3} c^{3} d^{2} - 3 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} g i^{3} x^{2} + 2 \, {\left (b^{3} c^{4} d - 3 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\right )} g i^{3} x + {\left (b^{3} c^{5} - 3 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3}\right )} g i^{3}\right )}} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 889 vs. \(2 (207) = 414\).

Time = 2.51 (sec) , antiderivative size = 889, normalized size of antiderivative = 3.66 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x) (c i+d i x)^3} \, dx=- \frac {B b^{2} \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{3} d^{3} g i^{3} - 6 a^{2} b c d^{2} g i^{3} + 6 a b^{2} c^{2} d g i^{3} - 2 b^{3} c^{3} g i^{3}} + \frac {b^{2} \cdot \left (2 A - 3 B\right ) \log {\left (x + \frac {2 A a b^{2} d + 2 A b^{3} c - 3 B a b^{2} d - 3 B b^{3} c - \frac {a^{4} b^{2} d^{4} \cdot \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}} + \frac {4 a^{3} b^{3} c d^{3} \cdot \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}} - \frac {6 a^{2} b^{4} c^{2} d^{2} \cdot \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}} + \frac {4 a b^{5} c^{3} d \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}} - \frac {b^{6} c^{4} \cdot \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}}}{4 A b^{3} d - 6 B b^{3} d} \right )}}{2 g i^{3} \left (a d - b c\right )^{3}} - \frac {b^{2} \cdot \left (2 A - 3 B\right ) \log {\left (x + \frac {2 A a b^{2} d + 2 A b^{3} c - 3 B a b^{2} d - 3 B b^{3} c + \frac {a^{4} b^{2} d^{4} \cdot \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}} - \frac {4 a^{3} b^{3} c d^{3} \cdot \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}} + \frac {6 a^{2} b^{4} c^{2} d^{2} \cdot \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}} - \frac {4 a b^{5} c^{3} d \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}} + \frac {b^{6} c^{4} \cdot \left (2 A - 3 B\right )}{\left (a d - b c\right )^{3}}}{4 A b^{3} d - 6 B b^{3} d} \right )}}{2 g i^{3} \left (a d - b c\right )^{3}} + \frac {\left (- B a d + 3 B b c + 2 B b d x\right ) \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{2 a^{2} c^{2} d^{2} g i^{3} + 4 a^{2} c d^{3} g i^{3} x + 2 a^{2} d^{4} g i^{3} x^{2} - 4 a b c^{3} d g i^{3} - 8 a b c^{2} d^{2} g i^{3} x - 4 a b c d^{3} g i^{3} x^{2} + 2 b^{2} c^{4} g i^{3} + 4 b^{2} c^{3} d g i^{3} x + 2 b^{2} c^{2} d^{2} g i^{3} x^{2}} + \frac {- 2 A a d + 6 A b c + B a d - 7 B b c + x \left (4 A b d - 6 B b d\right )}{4 a^{2} c^{2} d^{2} g i^{3} - 8 a b c^{3} d g i^{3} + 4 b^{2} c^{4} g i^{3} + x^{2} \cdot \left (4 a^{2} d^{4} g i^{3} - 8 a b c d^{3} g i^{3} + 4 b^{2} c^{2} d^{2} g i^{3}\right ) + x \left (8 a^{2} c d^{3} g i^{3} - 16 a b c^{2} d^{2} g i^{3} + 8 b^{2} c^{3} d g i^{3}\right )} \]

[In]

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

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 885 vs. \(2 (237) = 474\).

Time = 0.26 (sec) , antiderivative size = 885, normalized size of antiderivative = 3.64 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {1}{2} \, B {\left (\frac {2 \, b d x + 3 \, b c - a d}{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} g i^{3} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} g i^{3} x + {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} g i^{3}} + \frac {2 \, b^{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 i^{3}} - \frac {2 \, b^{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 i^{3}}\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 + 3 \, b c - a d}{{\left (b^{2} c^{2} d^{2} - 2 \, a b c d^{3} + a^{2} d^{4}\right )} g i^{3} x^{2} + 2 \, {\left (b^{2} c^{3} d - 2 \, a b c^{2} d^{2} + a^{2} c d^{3}\right )} g i^{3} x + {\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} g i^{3}} + \frac {2 \, b^{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 i^{3}} - \frac {2 \, b^{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 i^{3}}\right )} - \frac {{\left (7 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (b x + a\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{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 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (b x + a\right ) - 2 \, {\left (3 \, b^{2} d^{2} x^{2} + 6 \, b^{2} c d x + 3 \, b^{2} c^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2}\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B}{4 \, {\left (b^{3} c^{5} g i^{3} - 3 \, a b^{2} c^{4} d g i^{3} + 3 \, a^{2} b c^{3} d^{2} g i^{3} - a^{3} c^{2} d^{3} g i^{3} + {\left (b^{3} c^{3} d^{2} g i^{3} - 3 \, a b^{2} c^{2} d^{3} g i^{3} + 3 \, a^{2} b c d^{4} g i^{3} - a^{3} d^{5} g i^{3}\right )} x^{2} + 2 \, {\left (b^{3} c^{4} d g i^{3} - 3 \, a b^{2} c^{3} d^{2} g i^{3} + 3 \, a^{2} b c^{2} d^{3} g i^{3} - a^{3} c d^{4} g i^{3}\right )} x\right )}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.41 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.82 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {1}{4} \, {\left (\frac {2 \, B b^{2} e \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}} + \frac {4 \, A b^{2} e \log \left (\frac {b e x + a e}{d x + c}\right )}{b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}} - 2 \, {\left (\frac {4 \, {\left (b e x + a e\right )} B b d}{{\left (b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}\right )} {\left (d x + c\right )}} - \frac {{\left (b e x + a e\right )}^{2} B d^{2}}{{\left (b^{2} c^{2} e g i^{3} - 2 \, a b c d e g i^{3} + a^{2} d^{2} e g i^{3}\right )} {\left (d x + c\right )}^{2}}\right )} \log \left (\frac {b e x + a e}{d x + c}\right ) + \frac {{\left (2 \, A d^{2} - B d^{2}\right )} {\left (b e x + a e\right )}^{2}}{{\left (b^{2} c^{2} e g i^{3} - 2 \, a b c d e g i^{3} + a^{2} d^{2} e g i^{3}\right )} {\left (d x + c\right )}^{2}} - \frac {8 \, {\left (A b d - B b d\right )} {\left (b e x + a e\right )}}{{\left (b^{2} c^{2} g i^{3} - 2 \, a b c d g i^{3} + a^{2} d^{2} g i^{3}\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]

[In]

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

[Out]

1/4*(2*B*b^2*e*log((b*e*x + a*e)/(d*x + c))^2/(b^2*c^2*g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3) + 4*A*b^2*e*lo
g((b*e*x + a*e)/(d*x + c))/(b^2*c^2*g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3) - 2*(4*(b*e*x + a*e)*B*b*d/((b^2*
c^2*g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3)*(d*x + c)) - (b*e*x + a*e)^2*B*d^2/((b^2*c^2*e*g*i^3 - 2*a*b*c*d*
e*g*i^3 + a^2*d^2*e*g*i^3)*(d*x + c)^2))*log((b*e*x + a*e)/(d*x + c)) + (2*A*d^2 - B*d^2)*(b*e*x + a*e)^2/((b^
2*c^2*e*g*i^3 - 2*a*b*c*d*e*g*i^3 + a^2*d^2*e*g*i^3)*(d*x + c)^2) - 8*(A*b*d - B*b*d)*(b*e*x + a*e)/((b^2*c^2*
g*i^3 - 2*a*b*c*d*g*i^3 + a^2*d^2*g*i^3)*(d*x + c)))*(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 = 3.61 (sec) , antiderivative size = 545, normalized size of antiderivative = 2.24 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x) (c i+d i x)^3} \, dx=\frac {3\,A\,b\,c}{2\,g\,i^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {A\,a\,d}{2\,g\,i^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {B\,b^2\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{2\,g\,i^3\,{\left (a\,d-b\,c\right )}^3}+\frac {B\,a\,d}{4\,g\,i^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {7\,B\,b\,c}{4\,g\,i^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {B\,a^2\,d^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g\,i^3\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b^2\,c^2\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{2\,g\,i^3\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^2}+\frac {A\,b\,d\,x}{g\,i^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^2}-\frac {3\,B\,b\,d\,x}{2\,g\,i^3\,{\left (a\,d-b\,c\right )}^2\,{\left (c+d\,x\right )}^2}+\frac {B\,a\,b\,d^2\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g\,i^3\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^2}-\frac {B\,b^2\,c\,d\,x\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g\,i^3\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^2}+\frac {2\,B\,a\,b\,c\,d\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{g\,i^3\,{\left (a\,d-b\,c\right )}^3\,{\left (c+d\,x\right )}^2}+\frac {A\,b^2\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,2{}\mathrm {i}}{g\,i^3\,{\left (a\,d-b\,c\right )}^3}-\frac {B\,b^2\,\mathrm {atan}\left (\frac {a\,d\,1{}\mathrm {i}+b\,c\,1{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}\right )\,3{}\mathrm {i}}{g\,i^3\,{\left (a\,d-b\,c\right )}^3} \]

[In]

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

[Out]

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