\(\int \frac {x \log (e (\frac {a+b x}{c+d x})^n)}{f-g x^2} \, dx\) [78]

Optimal result

Integrand size = 30, antiderivative size = 403 \[ \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )}{2 g}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )}{2 g}-\frac {n \log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g}+\frac {n \log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g}+\frac {\left (n \log (a+b x)-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \log \left (f-g x^2\right )}{2 g}-\frac {n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g}-\frac {n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (a+b x)}{b \sqrt {f}+a \sqrt {g}}\right )}{2 g}+\frac {n \operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g}+\frac {n \operatorname {PolyLog}\left (2,\frac {\sqrt {g} (c+d x)}{d \sqrt {f}+c \sqrt {g}}\right )}{2 g} \] Output:

-1/2*n*ln(b*x+a)*ln(b*(f^(1/2)-g^(1/2)*x)/(b*f^(1/2)+a*g^(1/2)))/g+1/2*n*l 
n(d*x+c)*ln(d*(f^(1/2)-g^(1/2)*x)/(d*f^(1/2)+c*g^(1/2)))/g-1/2*n*ln(b*x+a) 
*ln(b*(f^(1/2)+g^(1/2)*x)/(b*f^(1/2)-a*g^(1/2)))/g+1/2*n*ln(d*x+c)*ln(d*(f 
^(1/2)+g^(1/2)*x)/(d*f^(1/2)-c*g^(1/2)))/g+1/2*(n*ln(b*x+a)-ln(e*((b*x+a)/ 
(d*x+c))^n)-n*ln(d*x+c))*ln(-g*x^2+f)/g-1/2*n*polylog(2,-g^(1/2)*(b*x+a)/( 
b*f^(1/2)-a*g^(1/2)))/g-1/2*n*polylog(2,g^(1/2)*(b*x+a)/(b*f^(1/2)+a*g^(1/ 
2)))/g+1/2*n*polylog(2,-g^(1/2)*(d*x+c)/(d*f^(1/2)-c*g^(1/2)))/g+1/2*n*pol 
ylog(2,g^(1/2)*(d*x+c)/(d*f^(1/2)+c*g^(1/2)))/g
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.02 \[ \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=-\frac {-n \log \left (\frac {\sqrt {g} (a+b x)}{b \sqrt {f}+a \sqrt {g}}\right ) \log \left (\sqrt {f}-\sqrt {g} x\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\sqrt {f}-\sqrt {g} x\right )+n \log \left (\frac {\sqrt {g} (c+d x)}{d \sqrt {f}+c \sqrt {g}}\right ) \log \left (\sqrt {f}-\sqrt {g} x\right )-n \log \left (-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right ) \log \left (\sqrt {f}+\sqrt {g} x\right )+\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right ) \log \left (\sqrt {f}+\sqrt {g} x\right )+n \log \left (-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right ) \log \left (\sqrt {f}+\sqrt {g} x\right )-n \operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{b \sqrt {f}+a \sqrt {g}}\right )+n \operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{d \sqrt {f}+c \sqrt {g}}\right )-n \operatorname {PolyLog}\left (2,\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )+n \operatorname {PolyLog}\left (2,\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g} \] Input:

Integrate[(x*Log[e*((a + b*x)/(c + d*x))^n])/(f - g*x^2),x]
 

Output:

-1/2*(-(n*Log[(Sqrt[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[g])]*Log[Sqrt[f] - S 
qrt[g]*x]) + Log[e*((a + b*x)/(c + d*x))^n]*Log[Sqrt[f] - Sqrt[g]*x] + n*L 
og[(Sqrt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])]*Log[Sqrt[f] - Sqrt[g]*x] - 
 n*Log[-((Sqrt[g]*(a + b*x))/(b*Sqrt[f] - a*Sqrt[g]))]*Log[Sqrt[f] + Sqrt[ 
g]*x] + Log[e*((a + b*x)/(c + d*x))^n]*Log[Sqrt[f] + Sqrt[g]*x] + n*Log[-( 
(Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g]))]*Log[Sqrt[f] + Sqrt[g]*x] - n 
*PolyLog[2, (b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt[g])] + n*PolyLog 
[2, (d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])] - n*PolyLog[2, (b*( 
Sqrt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])] + n*PolyLog[2, (d*(Sqrt[f] 
+ Sqrt[g]*x))/(d*Sqrt[f] - c*Sqrt[g])])/g
 

Rubi [A] (verified)

Time = 1.20 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2993, 240, 2863, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx\)

\(\Big \downarrow \) 2993

\(\displaystyle -\left (\left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right ) \int \frac {x}{f-g x^2}dx\right )+n \int \frac {x \log (a+b x)}{f-g x^2}dx-n \int \frac {x \log (c+d x)}{f-g x^2}dx\)

\(\Big \downarrow \) 240

\(\displaystyle n \int \frac {x \log (a+b x)}{f-g x^2}dx-n \int \frac {x \log (c+d x)}{f-g x^2}dx+\frac {\log \left (f-g x^2\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 g}\)

\(\Big \downarrow \) 2863

\(\displaystyle n \int \left (\frac {\log (a+b x)}{2 \sqrt {g} \left (\sqrt {f}-\sqrt {g} x\right )}-\frac {\log (a+b x)}{2 \sqrt {g} \left (\sqrt {g} x+\sqrt {f}\right )}\right )dx-n \int \left (\frac {\log (c+d x)}{2 \sqrt {g} \left (\sqrt {f}-\sqrt {g} x\right )}-\frac {\log (c+d x)}{2 \sqrt {g} \left (\sqrt {g} x+\sqrt {f}\right )}\right )dx+\frac {\log \left (f-g x^2\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 g}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\log \left (f-g x^2\right ) \left (-\log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{2 g}+n \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (a+b x)}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (a+b x)}{\sqrt {g} a+b \sqrt {f}}\right )}{2 g}-\frac {\log (a+b x) \log \left (\frac {b \left (\sqrt {f}-\sqrt {g} x\right )}{a \sqrt {g}+b \sqrt {f}}\right )}{2 g}-\frac {\log (a+b x) \log \left (\frac {b \left (\sqrt {f}+\sqrt {g} x\right )}{b \sqrt {f}-a \sqrt {g}}\right )}{2 g}\right )-n \left (-\frac {\operatorname {PolyLog}\left (2,-\frac {\sqrt {g} (c+d x)}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g}-\frac {\operatorname {PolyLog}\left (2,\frac {\sqrt {g} (c+d x)}{\sqrt {g} c+d \sqrt {f}}\right )}{2 g}-\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt {f}-\sqrt {g} x\right )}{c \sqrt {g}+d \sqrt {f}}\right )}{2 g}-\frac {\log (c+d x) \log \left (\frac {d \left (\sqrt {f}+\sqrt {g} x\right )}{d \sqrt {f}-c \sqrt {g}}\right )}{2 g}\right )\)

Input:

Int[(x*Log[e*((a + b*x)/(c + d*x))^n])/(f - g*x^2),x]
 

Output:

((n*Log[a + b*x] - Log[e*((a + b*x)/(c + d*x))^n] - n*Log[c + d*x])*Log[f 
- g*x^2])/(2*g) + n*(-1/2*(Log[a + b*x]*Log[(b*(Sqrt[f] - Sqrt[g]*x))/(b*S 
qrt[f] + a*Sqrt[g])])/g - (Log[a + b*x]*Log[(b*(Sqrt[f] + Sqrt[g]*x))/(b*S 
qrt[f] - a*Sqrt[g])])/(2*g) - PolyLog[2, -((Sqrt[g]*(a + b*x))/(b*Sqrt[f] 
- a*Sqrt[g]))]/(2*g) - PolyLog[2, (Sqrt[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[ 
g])]/(2*g)) - n*(-1/2*(Log[c + d*x]*Log[(d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[ 
f] + c*Sqrt[g])])/g - (Log[c + d*x]*Log[(d*(Sqrt[f] + Sqrt[g]*x))/(d*Sqrt[ 
f] - c*Sqrt[g])])/(2*g) - PolyLog[2, -((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c* 
Sqrt[g]))]/(2*g) - PolyLog[2, (Sqrt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])] 
/(2*g))
 

Defintions of rubi rules used

rule 240

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

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2863

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) 
^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a 
 + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
 

rule 2993

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.)) 
^(r_.)]*(RFx_.), x_Symbol] :> Simp[p*r   Int[RFx*Log[a + b*x], x], x] + (Si 
mp[q*r   Int[RFx*Log[c + d*x], x], x] - Simp[(p*r*Log[a + b*x] + q*r*Log[c 
+ d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])   Int[RFx, x], x]) /; FreeQ[ 
{a, b, c, d, e, f, p, q, r}, x] && RationalFunctionQ[RFx, x] && NeQ[b*c - a 
*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; IntegersQ[ 
m, n]]
 

Maple [A] (verified)

Time = 2.94 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.09

method	result	size
parts	\(-\frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right ) \ln \left (-g \,x^{2}+f \right )}{2 g}+\frac {n \left (-d a +b c \right ) \left (\frac {\left (\frac {\ln \left (d x +c \right ) \ln \left (-g \,x^{2}+f \right )}{d}+\frac {2 g \left (-\frac {\ln \left (d x +c \right ) \left (\ln \left (\frac {d \sqrt {g f}-g \left (d x +c \right )+c g}{d \sqrt {g f}+c g}\right )+\ln \left (\frac {d \sqrt {g f}+g \left (d x +c \right )-c g}{d \sqrt {g f}-c g}\right )\right )}{2 g}-\frac {\operatorname {dilog}\left (\frac {d \sqrt {g f}-g \left (d x +c \right )+c g}{d \sqrt {g f}+c g}\right )+\operatorname {dilog}\left (\frac {d \sqrt {g f}+g \left (d x +c \right )-c g}{d \sqrt {g f}-c g}\right )}{2 g}\right )}{d}\right ) d}{d a -b c}-\frac {\left (\frac {\ln \left (b x +a \right ) \ln \left (-g \,x^{2}+f \right )}{b}+\frac {2 g \left (-\frac {\ln \left (b x +a \right ) \left (\ln \left (\frac {b \sqrt {g f}-g \left (b x +a \right )+a g}{b \sqrt {g f}+a g}\right )+\ln \left (\frac {b \sqrt {g f}+g \left (b x +a \right )-a g}{b \sqrt {g f}-a g}\right )\right )}{2 g}-\frac {\operatorname {dilog}\left (\frac {b \sqrt {g f}-g \left (b x +a \right )+a g}{b \sqrt {g f}+a g}\right )+\operatorname {dilog}\left (\frac {b \sqrt {g f}+g \left (b x +a \right )-a g}{b \sqrt {g f}-a g}\right )}{2 g}\right )}{b}\right ) b}{d a -b c}\right )}{2 g}\)	\(440\)

Input:

int(x*ln(e*((b*x+a)/(d*x+c))^n)/(-g*x^2+f),x,method=_RETURNVERBOSE)
 

Output:

-1/2*ln(e*((b*x+a)/(d*x+c))^n)/g*ln(-g*x^2+f)+1/2/g*n*(-a*d+b*c)*((ln(d*x+ 
c)/d*ln(-g*x^2+f)+2/d*g*(-1/2*ln(d*x+c)*(ln((d*(g*f)^(1/2)-g*(d*x+c)+c*g)/ 
(d*(g*f)^(1/2)+c*g))+ln((d*(g*f)^(1/2)+g*(d*x+c)-c*g)/(d*(g*f)^(1/2)-c*g)) 
)/g-1/2*(dilog((d*(g*f)^(1/2)-g*(d*x+c)+c*g)/(d*(g*f)^(1/2)+c*g))+dilog((d 
*(g*f)^(1/2)+g*(d*x+c)-c*g)/(d*(g*f)^(1/2)-c*g)))/g))*d/(a*d-b*c)-(ln(b*x+ 
a)/b*ln(-g*x^2+f)+2/b*g*(-1/2*ln(b*x+a)*(ln((b*(g*f)^(1/2)-g*(b*x+a)+a*g)/ 
(b*(g*f)^(1/2)+a*g))+ln((b*(g*f)^(1/2)+g*(b*x+a)-a*g)/(b*(g*f)^(1/2)-a*g)) 
)/g-1/2*(dilog((b*(g*f)^(1/2)-g*(b*x+a)+a*g)/(b*(g*f)^(1/2)+a*g))+dilog((b 
*(g*f)^(1/2)+g*(b*x+a)-a*g)/(b*(g*f)^(1/2)-a*g)))/g))*b/(a*d-b*c))
 

Fricas [F]

\[ \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\int { -\frac {x \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{g x^{2} - f} \,d x } \] Input:

integrate(x*log(e*((b*x+a)/(d*x+c))^n)/(-g*x^2+f),x, algorithm="fricas")
 

Output:

integral(-x*log(e*((b*x + a)/(d*x + c))^n)/(g*x^2 - f), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\text {Timed out} \] Input:

integrate(x*ln(e*((b*x+a)/(d*x+c))**n)/(-g*x**2+f),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\int { -\frac {x \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{g x^{2} - f} \,d x } \] Input:

integrate(x*log(e*((b*x+a)/(d*x+c))^n)/(-g*x^2+f),x, algorithm="maxima")
 

Output:

-integrate(x*log(e*((b*x + a)/(d*x + c))^n)/(g*x^2 - f), x)
 

Giac [F]

\[ \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\int { -\frac {x \log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )}{g x^{2} - f} \,d x } \] Input:

integrate(x*log(e*((b*x+a)/(d*x+c))^n)/(-g*x^2+f),x, algorithm="giac")
 

Output:

integrate(-x*log(e*((b*x + a)/(d*x + c))^n)/(g*x^2 - f), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\int \frac {x\,\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}{f-g\,x^2} \,d x \] Input:

int((x*log(e*((a + b*x)/(c + d*x))^n))/(f - g*x^2),x)
 

Output:

int((x*log(e*((a + b*x)/(c + d*x))^n))/(f - g*x^2), x)
 

Reduce [F]

\[ \int \frac {x \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx=\int \frac {\mathrm {log}\left (\frac {\left (b x +a \right )^{n} e}{\left (d x +c \right )^{n}}\right ) x}{-g \,x^{2}+f}d x \] Input:

int(x*log(e*((b*x+a)/(d*x+c))^n)/(-g*x^2+f),x)
 

Output:

int((log(((a + b*x)**n*e)/(c + d*x)**n)*x)/(f - g*x**2),x)