\(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{(a g+b g x) (c i+d i x)^2} \, dx\) [43]

Optimal result

Integrand size = 40, antiderivative size = 156 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x) (c i+d i x)^2} \, dx=-\frac {A d (a+b x)}{(b c-a d)^2 g i^2 (c+d x)}+\frac {B d (a+b x)}{(b c-a d)^2 g i^2 (c+d x)}-\frac {B d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{(b c-a d)^2 g i^2 (c+d x)}+\frac {b \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )^2}{2 B (b c-a d)^2 g i^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2562, 2388, 2338, 2332} \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {b \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )^2}{2 B g i^2 (b c-a d)^2}-\frac {A d (a+b x)}{g i^2 (c+d x) (b c-a d)^2}-\frac {B d (a+b x) \log \left (\frac {e (a+b x)}{c+d x}\right )}{g i^2 (c+d x) (b c-a d)^2}+\frac {B d (a+b x)}{g i^2 (c+d x) (b c-a d)^2} \]

[In]

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

[Out]

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

Rule 2332

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

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 2388

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[(d
+ e*x)^(q - 1)*((a + b*Log[c*x^n])^p/x), x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

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

Mathematica [C] (verified)

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

Time = 0.16 (sec) , antiderivative size = 292, normalized size of antiderivative = 1.87 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {2 (b c-a d) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )+2 b (c+d x) \log (a+b x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right )-2 b (c+d x) \left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (c+d x)-2 B (b c-a d+b (c+d x) \log (a+b x)-b (c+d x) \log (c+d x))-b B (c+d x) \left (\log (a+b x) \left (\log (a+b x)-2 \log \left (\frac {b (c+d x)}{b c-a d}\right )\right )-2 \operatorname {PolyLog}\left (2,\frac {d (a+b x)}{-b c+a d}\right )\right )+b B (c+d x) \left (\left (2 \log \left (\frac {d (a+b x)}{-b c+a d}\right )-\log (c+d x)\right ) \log (c+d x)+2 \operatorname {PolyLog}\left (2,\frac {b (c+d x)}{b c-a d}\right )\right )}{2 (b c-a d)^2 g i^2 (c+d x)} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.48

method	result	size
parts	\(\frac {A \left (-\frac {1}{\left (a d -c b \right ) \left (d x +c \right )}-\frac {b \ln \left (d x +c \right )}{\left (a d -c b \right )^{2}}+\frac {b \ln \left (b x +a \right )}{\left (a d -c b \right )^{2}}\right )}{g \,i^{2}}-\frac {B \left (\frac {d \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{a d -c b}-\frac {b e \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 \left (a d -c b \right )}\right )}{g \,i^{2} \left (a d -c b \right ) e}\)	\(231\)
parallelrisch	\(\frac {2 B a \,b^{2} d^{4}-2 B \,b^{3} c \,d^{3}-2 A a \,b^{2} d^{4}+2 A \,b^{3} c \,d^{3}+B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{3} d^{4}+2 A x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{4}-2 B x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} d^{4}+B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2} b^{3} c \,d^{3}+2 A \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) b^{3} c \,d^{3}-2 B \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right ) a \,b^{2} d^{4}}{2 i^{2} g \left (d x +c \right ) b^{2} d^{3} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\)	\(238\)
norman	\(\frac {\frac {\left (A b c -B a d \right ) \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {d \left (A b -B b \right ) x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )}{g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (-B +A \right ) d x}{g i c \left (a d -c b \right )}+\frac {B b c \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {b B d x \ln \left (\frac {e \left (b x +a \right )}{d x +c}\right )^{2}}{2 g i \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}}{i \left (d x +c \right )}\)	\(253\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (-\frac {d^{2} A b \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{2} \left (a d -c b \right )^{3} g}+\frac {d^{3} A \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2} \left (a d -c b \right )^{3} g}-\frac {d^{2} B b \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e \,i^{2} \left (a d -c b \right )^{3} g}+\frac {d^{3} B \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{2} \left (a d -c b \right )^{3} g}\right )}{d^{2}}\)	\(286\)
default	\(-\frac {e \left (a d -c b \right ) \left (-\frac {d^{2} A b \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e \,i^{2} \left (a d -c b \right )^{3} g}+\frac {d^{3} A \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )}{e^{2} i^{2} \left (a d -c b \right )^{3} g}-\frac {d^{2} B b \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 e \,i^{2} \left (a d -c b \right )^{3} g}+\frac {d^{3} B \left (\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}-\frac {b e}{d}\right )}{e^{2} i^{2} \left (a d -c b \right )^{3} g}\right )}{d^{2}}\)	\(286\)
risch	\(-\frac {A}{g \,i^{2} \left (a d -c b \right ) \left (d x +c \right )}-\frac {A b \ln \left (d x +c \right )}{g \,i^{2} \left (a d -c b \right )^{2}}+\frac {A b \ln \left (b x +a \right )}{g \,i^{2} \left (a d -c b \right )^{2}}-\frac {B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) b}{g \,i^{2} \left (a d -c b \right )^{2}}-\frac {B d \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) a}{g \,i^{2} \left (a d -c b \right )^{2} \left (d x +c \right )}+\frac {B \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) c b}{g \,i^{2} \left (a d -c b \right )^{2} \left (d x +c \right )}+\frac {B d a}{g \,i^{2} \left (a d -c b \right )^{2} \left (d x +c \right )}-\frac {B c b}{g \,i^{2} \left (a d -c b \right )^{2} \left (d x +c \right )}+\frac {B b}{g \,i^{2} \left (a d -c b \right )^{2}}+\frac {B b \ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )^{2}}{2 g \,i^{2} \left (a d -c b \right )^{2}}\)	\(362\)

[In]

int((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)/(d*i*x+c*i)^2,x,method=_RETURNVERBOSE)

[Out]

A/g/i^2*(-1/(a*d-b*c)/(d*x+c)-b/(a*d-b*c)^2*ln(d*x+c)+b/(a*d-b*c)^2*ln(b*x+a))-B/g/i^2/(a*d-b*c)/e*(d/(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*b*e/(a*d-b*c
)*ln(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 = 151, normalized size of antiderivative = 0.97 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {2 \, {\left (A - B\right )} b c - 2 \, {\left (A - B\right )} a d + {\left (B b d x + B b c\right )} \log \left (\frac {b e x + a e}{d x + c}\right )^{2} + 2 \, {\left ({\left (A - B\right )} b d x + A b c - B a d\right )} \log \left (\frac {b e x + a e}{d x + c}\right )}{2 \, {\left ({\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} g i^{2} x + {\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} g i^{2}\right )}} \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)/(d*i*x+c*i)^2,x, algorithm="fricas")

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (131) = 262\).

Time = 0.60 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.47 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {B b \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}^{2}}{2 a^{2} d^{2} g i^{2} - 4 a b c d g i^{2} + 2 b^{2} c^{2} g i^{2}} - \frac {B \log {\left (\frac {e \left (a + b x\right )}{c + d x} \right )}}{a c d g i^{2} + a d^{2} g i^{2} x - b c^{2} g i^{2} - b c d g i^{2} x} + \left (A - B\right ) \left (- \frac {b \log {\left (x + \frac {- \frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} + \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} - \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d + \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{g i^{2} \left (a d - b c\right )^{2}} + \frac {b \log {\left (x + \frac {\frac {a^{3} b d^{3}}{\left (a d - b c\right )^{2}} - \frac {3 a^{2} b^{2} c d^{2}}{\left (a d - b c\right )^{2}} + \frac {3 a b^{3} c^{2} d}{\left (a d - b c\right )^{2}} + a b d - \frac {b^{4} c^{3}}{\left (a d - b c\right )^{2}} + b^{2} c}{2 b^{2} d} \right )}}{g i^{2} \left (a d - b c\right )^{2}} - \frac {1}{a c d g i^{2} - b c^{2} g i^{2} + x \left (a d^{2} g i^{2} - b c d g i^{2}\right )}\right ) \]

[In]

integrate((A+B*ln(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)/(d*i*x+c*i)**2,x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 421 vs. \(2 (154) = 308\).

Time = 0.22 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.70 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x) (c i+d i x)^2} \, dx=B {\left (\frac {1}{{\left (b c d - a d^{2}\right )} g i^{2} x + {\left (b c^{2} - a c d\right )} g i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} \log \left (\frac {b e x}{d x + c} + \frac {a e}{d x + c}\right ) + A {\left (\frac {1}{{\left (b c d - a d^{2}\right )} g i^{2} x + {\left (b c^{2} - a c d\right )} g i^{2}} + \frac {b \log \left (b x + a\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}} - \frac {b \log \left (d x + c\right )}{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} g i^{2}}\right )} - \frac {{\left ({\left (b d x + b c\right )} \log \left (b x + a\right )^{2} + {\left (b d x + b c\right )} \log \left (d x + c\right )^{2} + 2 \, b c - 2 \, a d + 2 \, {\left (b d x + b c\right )} \log \left (b x + a\right ) - 2 \, {\left (b d x + b c + {\left (b d x + b c\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )\right )} B}{2 \, {\left (b^{2} c^{3} g i^{2} - 2 \, a b c^{2} d g i^{2} + a^{2} c d^{2} g i^{2} + {\left (b^{2} c^{2} d g i^{2} - 2 \, a b c d^{2} g i^{2} + a^{2} d^{3} g i^{2}\right )} x\right )}} \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)/(d*i*x+c*i)^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.49 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {1}{2} \, {\left (\frac {B b e \log \left (\frac {b e x + a e}{d x + c}\right )^{2}}{b c g i^{2} - a d g i^{2}} + \frac {2 \, A b e \log \left (\frac {b e x + a e}{d x + c}\right )}{b c g i^{2} - a d g i^{2}} - \frac {2 \, {\left (b e x + a e\right )} B d \log \left (\frac {b e x + a e}{d x + c}\right )}{{\left (b c g i^{2} - a d g i^{2}\right )} {\left (d x + c\right )}} - \frac {2 \, {\left (b e x + a e\right )} {\left (A d - B d\right )}}{{\left (b c g i^{2} - a d g i^{2}\right )} {\left (d x + c\right )}}\right )} {\left (\frac {b c}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}} - \frac {a d}{{\left (b c e - a d e\right )} {\left (b c - a d\right )}}\right )} \]

[In]

integrate((A+B*log(e*(b*x+a)/(d*x+c)))/(b*g*x+a*g)/(d*i*x+c*i)^2,x, algorithm="giac")

[Out]

1/2*(B*b*e*log((b*e*x + a*e)/(d*x + c))^2/(b*c*g*i^2 - a*d*g*i^2) + 2*A*b*e*log((b*e*x + a*e)/(d*x + c))/(b*c*
g*i^2 - a*d*g*i^2) - 2*(b*e*x + a*e)*B*d*log((b*e*x + a*e)/(d*x + c))/((b*c*g*i^2 - a*d*g*i^2)*(d*x + c)) - 2*
(b*e*x + a*e)*(A*d - B*d)/((b*c*g*i^2 - a*d*g*i^2)*(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 = 2.50 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.58 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{(a g+b g x) (c i+d i x)^2} \, dx=\frac {B\,b\,{\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}^2}{2\,g\,i^2\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {A-B}{\left (a\,d-b\,c\right )\,\left (c\,g\,i^2+d\,g\,i^2\,x\right )}-\frac {B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )\,\left (a\,d-b\,c\right )}{b\,d\,g\,i^2\,\left (\frac {x}{b}+\frac {c}{b\,d}\right )\,\left (a^2\,d^2-2\,a\,b\,c\,d+b^2\,c^2\right )}-\frac {b\,\mathrm {atan}\left (\frac {\left (2\,b\,d\,x+\frac {a^2\,d^2\,g\,i^2-b^2\,c^2\,g\,i^2}{g\,i^2\,\left (a\,d-b\,c\right )}\right )\,1{}\mathrm {i}}{a\,d-b\,c}\right )\,\left (A-B\right )\,2{}\mathrm {i}}{g\,i^2\,{\left (a\,d-b\,c\right )}^2} \]

[In]

int((A + B*log((e*(a + b*x))/(c + d*x)))/((a*g + b*g*x)*(c*i + d*i*x)^2),x)

[Out]

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