\(\int \frac {A+B \log (\frac {e (a+b x)}{c+d x})}{f+g x} \, dx\) [235]

Optimal result

Integrand size = 27, antiderivative size = 140 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{f+g x} \, dx=-\frac {B \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{g}+\frac {B \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {B \operatorname {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {B \operatorname {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )}{g} \]

[Out]

-B*ln(-g*(b*x+a)/(-a*g+b*f))*ln(g*x+f)/g+(A+B*ln(e*(b*x+a)/(d*x+c)))*ln(g*x+f)/g+B*ln(-g*(d*x+c)/(-c*g+d*f))*l
n(g*x+f)/g-B*polylog(2,b*(g*x+f)/(-a*g+b*f))/g+B*polylog(2,d*(g*x+f)/(-c*g+d*f))/g

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2546, 2441, 2440, 2438} \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{f+g x} \, dx=\frac {\log (f+g x) \left (B \log \left (\frac {e (a+b x)}{c+d x}\right )+A\right )}{g}-\frac {B \operatorname {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )}{g}-\frac {B \log (f+g x) \log \left (-\frac {g (a+b x)}{b f-a g}\right )}{g}+\frac {B \operatorname {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )}{g}+\frac {B \log (f+g x) \log \left (-\frac {g (c+d x)}{d f-c g}\right )}{g} \]

[In]

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

[Out]

-((B*Log[-((g*(a + b*x))/(b*f - a*g))]*Log[f + g*x])/g) + ((A + B*Log[(e*(a + b*x))/(c + d*x)])*Log[f + g*x])/
g + (B*Log[-((g*(c + d*x))/(d*f - c*g))]*Log[f + g*x])/g - (B*PolyLog[2, (b*(f + g*x))/(b*f - a*g)])/g + (B*Po
lyLog[2, (d*(f + g*x))/(d*f - c*g)])/g

Rule 2438

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2, (-c)*e*x^n]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 2440

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + c*e*(x/g)])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2441

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[Log[e*((f + g
*x)/(e*f - d*g))]*((a + b*Log[c*(d + e*x)^n])/g), x] - Dist[b*e*(n/g), Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2546

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

Rubi steps \begin{align*} \text {integral}& = \frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{g}-\frac {(b B) \int \frac {\log (f+g x)}{a+b x} \, dx}{g}+\frac {(B d) \int \frac {\log (f+g x)}{c+d x} \, dx}{g} \\ & = -\frac {B \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{g}+\frac {B \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}+B \int \frac {\log \left (\frac {g (a+b x)}{-b f+a g}\right )}{f+g x} \, dx-B \int \frac {\log \left (\frac {g (c+d x)}{-d f+c g}\right )}{f+g x} \, dx \\ & = -\frac {B \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{g}+\frac {B \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}+\frac {B \text {Subst}\left (\int \frac {\log \left (1+\frac {b x}{-b f+a g}\right )}{x} \, dx,x,f+g x\right )}{g}-\frac {B \text {Subst}\left (\int \frac {\log \left (1+\frac {d x}{-d f+c g}\right )}{x} \, dx,x,f+g x\right )}{g} \\ & = -\frac {B \log \left (-\frac {g (a+b x)}{b f-a g}\right ) \log (f+g x)}{g}+\frac {\left (A+B \log \left (\frac {e (a+b x)}{c+d x}\right )\right ) \log (f+g x)}{g}+\frac {B \log \left (-\frac {g (c+d x)}{d f-c g}\right ) \log (f+g x)}{g}-\frac {B \text {Li}_2\left (\frac {b (f+g x)}{b f-a g}\right )}{g}+\frac {B \text {Li}_2\left (\frac {d (f+g x)}{d f-c g}\right )}{g} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.82 \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{f+g x} \, dx=\frac {\left (A-B \log \left (\frac {g (a+b x)}{-b f+a g}\right )+B \log \left (\frac {e (a+b x)}{c+d x}\right )+B \log \left (\frac {g (c+d x)}{-d f+c g}\right )\right ) \log (f+g x)-B \operatorname {PolyLog}\left (2,\frac {b (f+g x)}{b f-a g}\right )+B \operatorname {PolyLog}\left (2,\frac {d (f+g x)}{d f-c g}\right )}{g} \]

[In]

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

[Out]

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(371\) vs. \(2(140)=280\).

Time = 3.98 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.66

method	result	size
parts	\(\frac {A \ln \left (g x +f \right )}{g}-\frac {B \left (a d -c b \right ) e \left (-\frac {d^{2} \left (c g -d f \right ) \left (\frac {\operatorname {dilog}\left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}\right )}{e g \left (a d -c b \right )}+\frac {d^{3} \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{e g \left (a d -c b \right )}\right )}{d^{2}}\)	\(372\)
derivativedivides	\(-\frac {e \left (a d -c b \right ) \left (-d^{2} A \left (-\frac {\left (c g -d f \right ) \ln \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )}{e g \left (a d -c b \right ) \left (-c g +d f \right )}-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e g \left (a d -c b \right )}\right )-d^{2} B \left (\frac {\left (c g -d f \right ) \left (\frac {\operatorname {dilog}\left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}\right )}{e g \left (a d -c b \right )}-\frac {d \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{e g \left (a d -c b \right )}\right )\right )}{d^{2}}\)	\(529\)
default	\(-\frac {e \left (a d -c b \right ) \left (-d^{2} A \left (-\frac {\left (c g -d f \right ) \ln \left (a e g -b e f -c g \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )+d f \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )\right )}{e g \left (a d -c b \right ) \left (-c g +d f \right )}-\frac {\ln \left (b e -\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d \right )}{e g \left (a d -c b \right )}\right )-d^{2} B \left (\frac {\left (c g -d f \right ) \left (\frac {\operatorname {dilog}\left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (\frac {\left (c g -d f \right ) \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right )-a e g +b e f}{-a e g +b e f}\right )}{c g -d f}\right )}{e g \left (a d -c b \right )}-\frac {d \left (\frac {\operatorname {dilog}\left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}+\frac {\ln \left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) \ln \left (-\frac {\left (\frac {b e}{d}+\frac {\left (a d -c b \right ) e}{d \left (d x +c \right )}\right ) d -b e}{b e}\right )}{d}\right )}{e g \left (a d -c b \right )}\right )\right )}{d^{2}}\)	\(529\)
risch	\(\text {Expression too large to display}\)	\(1127\)

[In]

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

[Out]

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

Fricas [F]

\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{f+g x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A}{g x + f} \,d x } \]

[In]

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

[Out]

integral((B*log((b*e*x + a*e)/(d*x + c)) + A)/(g*x + f), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{f+g x} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{f+g x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A}{g x + f} \,d x } \]

[In]

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

[Out]

-B*integrate(-(log(b*x + a) - log(d*x + c) + log(e))/(g*x + f), x) + A*log(g*x + f)/g

Giac [F]

\[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{f+g x} \, dx=\int { \frac {B \log \left (\frac {{\left (b x + a\right )} e}{d x + c}\right ) + A}{g x + f} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B \log \left (\frac {e (a+b x)}{c+d x}\right )}{f+g x} \, dx=\int \frac {A+B\,\ln \left (\frac {e\,\left (a+b\,x\right )}{c+d\,x}\right )}{f+g\,x} \,d x \]

[In]

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

[Out]

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