\(\int \frac {\log (g (a+b x+c x^2)^n)}{d+e x^2} \, dx\) [94]

Optimal result

Integrand size = 25, antiderivative size = 762 \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x^2} \, dx=-\frac {n \log \left (\frac {\sqrt {e} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \log \left (\frac {\sqrt {e} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (-\frac {\sqrt {e} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}-\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (-\frac {\sqrt {e} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}-\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 c \sqrt {-d}+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 c \sqrt {-d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{2 c \sqrt {-d}-\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{2 c \sqrt {-d}-\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}} \] Output:

-1/2*n*ln(e^(1/2)*(b-(-4*a*c+b^2)^(1/2)+2*c*x)/(2*c*(-d)^(1/2)+(b-(-4*a*c+ 
b^2)^(1/2))*e^(1/2)))*ln((-d)^(1/2)-e^(1/2)*x)/(-d)^(1/2)/e^(1/2)-1/2*n*ln 
(e^(1/2)*(b+(-4*a*c+b^2)^(1/2)+2*c*x)/(2*c*(-d)^(1/2)+(b+(-4*a*c+b^2)^(1/2 
))*e^(1/2)))*ln((-d)^(1/2)-e^(1/2)*x)/(-d)^(1/2)/e^(1/2)+1/2*n*ln(-e^(1/2) 
*(b-(-4*a*c+b^2)^(1/2)+2*c*x)/(2*c*(-d)^(1/2)-(b-(-4*a*c+b^2)^(1/2))*e^(1/ 
2)))*ln((-d)^(1/2)+e^(1/2)*x)/(-d)^(1/2)/e^(1/2)+1/2*n*ln(-e^(1/2)*(b+(-4* 
a*c+b^2)^(1/2)+2*c*x)/(2*c*(-d)^(1/2)-(b+(-4*a*c+b^2)^(1/2))*e^(1/2)))*ln( 
(-d)^(1/2)+e^(1/2)*x)/(-d)^(1/2)/e^(1/2)+1/2*ln((-d)^(1/2)-e^(1/2)*x)*ln(g 
*(c*x^2+b*x+a)^n)/(-d)^(1/2)/e^(1/2)-1/2*ln((-d)^(1/2)+e^(1/2)*x)*ln(g*(c* 
x^2+b*x+a)^n)/(-d)^(1/2)/e^(1/2)-1/2*n*polylog(2,2*c*((-d)^(1/2)-e^(1/2)*x 
)/(2*c*(-d)^(1/2)+(b-(-4*a*c+b^2)^(1/2))*e^(1/2)))/(-d)^(1/2)/e^(1/2)-1/2* 
n*polylog(2,2*c*((-d)^(1/2)-e^(1/2)*x)/(2*c*(-d)^(1/2)+(b+(-4*a*c+b^2)^(1/ 
2))*e^(1/2)))/(-d)^(1/2)/e^(1/2)+1/2*n*polylog(2,2*c*((-d)^(1/2)+e^(1/2)*x 
)/(2*c*(-d)^(1/2)-(b-(-4*a*c+b^2)^(1/2))*e^(1/2)))/(-d)^(1/2)/e^(1/2)+1/2* 
n*polylog(2,2*c*((-d)^(1/2)+e^(1/2)*x)/(2*c*(-d)^(1/2)-(b+(-4*a*c+b^2)^(1/ 
2))*e^(1/2)))/(-d)^(1/2)/e^(1/2)
 

Mathematica [A] (verified)

Time = 1.23 (sec) , antiderivative size = 626, normalized size of antiderivative = 0.82 \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x^2} \, dx=\frac {-n \log \left (\frac {\sqrt {e} \left (b-\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )-n \log \left (\frac {\sqrt {e} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{2 c \sqrt {-d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}-\sqrt {e} x\right )+n \log \left (\frac {\sqrt {e} \left (-b+\sqrt {b^2-4 a c}-2 c x\right )}{2 c \sqrt {-d}+\left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )+n \log \left (\frac {\sqrt {e} \left (b+\sqrt {b^2-4 a c}+2 c x\right )}{-2 c \sqrt {-d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right ) \log \left (\sqrt {-d}+\sqrt {e} x\right )+\log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (g (a+x (b+c x))^n\right )-\log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (g (a+x (b+c x))^n\right )-n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 c \sqrt {-d}+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )-n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 c \sqrt {-d}+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )+n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{2 c \sqrt {-d}+\left (-b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )+n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}+\sqrt {e} x\right )}{2 c \sqrt {-d}-\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}} \] Input:

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

Output:

(-(n*Log[(Sqrt[e]*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] + (b - Sq 
rt[b^2 - 4*a*c])*Sqrt[e])]*Log[Sqrt[-d] - Sqrt[e]*x]) - n*Log[(Sqrt[e]*(b 
+ Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] + (b + Sqrt[b^2 - 4*a*c])*Sqrt 
[e])]*Log[Sqrt[-d] - Sqrt[e]*x] + n*Log[(Sqrt[e]*(-b + Sqrt[b^2 - 4*a*c] - 
 2*c*x))/(2*c*Sqrt[-d] + (-b + Sqrt[b^2 - 4*a*c])*Sqrt[e])]*Log[Sqrt[-d] + 
 Sqrt[e]*x] + n*Log[(Sqrt[e]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(-2*c*Sqrt[- 
d] + (b + Sqrt[b^2 - 4*a*c])*Sqrt[e])]*Log[Sqrt[-d] + Sqrt[e]*x] + Log[Sqr 
t[-d] - Sqrt[e]*x]*Log[g*(a + x*(b + c*x))^n] - Log[Sqrt[-d] + Sqrt[e]*x]* 
Log[g*(a + x*(b + c*x))^n] - n*PolyLog[2, (2*c*(Sqrt[-d] - Sqrt[e]*x))/(2* 
c*Sqrt[-d] + (b - Sqrt[b^2 - 4*a*c])*Sqrt[e])] - n*PolyLog[2, (2*c*(Sqrt[- 
d] - Sqrt[e]*x))/(2*c*Sqrt[-d] + (b + Sqrt[b^2 - 4*a*c])*Sqrt[e])] + n*Pol 
yLog[2, (2*c*(Sqrt[-d] + Sqrt[e]*x))/(2*c*Sqrt[-d] + (-b + Sqrt[b^2 - 4*a* 
c])*Sqrt[e])] + n*PolyLog[2, (2*c*(Sqrt[-d] + Sqrt[e]*x))/(2*c*Sqrt[-d] - 
(b + Sqrt[b^2 - 4*a*c])*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e])
 

Rubi [A] (verified)

Time = 1.80 (sec) , antiderivative size = 762, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {3008, 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.

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

\(\Big \downarrow \) 3008

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} c+\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {-d}-\sqrt {e} x\right )}{2 \sqrt {-d} c+\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{2 c \sqrt {-d}-\left (b-\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \operatorname {PolyLog}\left (2,\frac {2 c \left (\sqrt {e} x+\sqrt {-d}\right )}{2 c \sqrt {-d}-\left (b+\sqrt {b^2-4 a c}\right ) \sqrt {e}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {\sqrt {e} \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{\sqrt {e} \left (b-\sqrt {b^2-4 a c}\right )+2 c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {n \log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (\frac {\sqrt {e} \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{\sqrt {e} \left (\sqrt {b^2-4 a c}+b\right )+2 c \sqrt {-d}}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (-\frac {\sqrt {e} \left (-\sqrt {b^2-4 a c}+b+2 c x\right )}{2 c \sqrt {-d}-\sqrt {e} \left (b-\sqrt {b^2-4 a c}\right )}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {n \log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (-\frac {\sqrt {e} \left (\sqrt {b^2-4 a c}+b+2 c x\right )}{2 c \sqrt {-d}-\sqrt {e} \left (\sqrt {b^2-4 a c}+b\right )}\right )}{2 \sqrt {-d} \sqrt {e}}+\frac {\log \left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}-\frac {\log \left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (g \left (a+b x+c x^2\right )^n\right )}{2 \sqrt {-d} \sqrt {e}}\)

Input:

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

Output:

-1/2*(n*Log[(Sqrt[e]*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] + (b - 
 Sqrt[b^2 - 4*a*c])*Sqrt[e])]*Log[Sqrt[-d] - Sqrt[e]*x])/(Sqrt[-d]*Sqrt[e] 
) - (n*Log[(Sqrt[e]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] + (b + 
Sqrt[b^2 - 4*a*c])*Sqrt[e])]*Log[Sqrt[-d] - Sqrt[e]*x])/(2*Sqrt[-d]*Sqrt[e 
]) + (n*Log[-((Sqrt[e]*(b - Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] - (b 
 - Sqrt[b^2 - 4*a*c])*Sqrt[e]))]*Log[Sqrt[-d] + Sqrt[e]*x])/(2*Sqrt[-d]*Sq 
rt[e]) + (n*Log[-((Sqrt[e]*(b + Sqrt[b^2 - 4*a*c] + 2*c*x))/(2*c*Sqrt[-d] 
- (b + Sqrt[b^2 - 4*a*c])*Sqrt[e]))]*Log[Sqrt[-d] + Sqrt[e]*x])/(2*Sqrt[-d 
]*Sqrt[e]) + (Log[Sqrt[-d] - Sqrt[e]*x]*Log[g*(a + b*x + c*x^2)^n])/(2*Sqr 
t[-d]*Sqrt[e]) - (Log[Sqrt[-d] + Sqrt[e]*x]*Log[g*(a + b*x + c*x^2)^n])/(2 
*Sqrt[-d]*Sqrt[e]) - (n*PolyLog[2, (2*c*(Sqrt[-d] - Sqrt[e]*x))/(2*c*Sqrt[ 
-d] + (b - Sqrt[b^2 - 4*a*c])*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) - (n*PolyLog 
[2, (2*c*(Sqrt[-d] - Sqrt[e]*x))/(2*c*Sqrt[-d] + (b + Sqrt[b^2 - 4*a*c])*S 
qrt[e])])/(2*Sqrt[-d]*Sqrt[e]) + (n*PolyLog[2, (2*c*(Sqrt[-d] + Sqrt[e]*x) 
)/(2*c*Sqrt[-d] - (b - Sqrt[b^2 - 4*a*c])*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e]) 
+ (n*PolyLog[2, (2*c*(Sqrt[-d] + Sqrt[e]*x))/(2*c*Sqrt[-d] - (b + Sqrt[b^2 
 - 4*a*c])*Sqrt[e])])/(2*Sqrt[-d]*Sqrt[e])
 

Defintions of rubi rules used

rule 2009

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

rule 3008

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With 
[{u = ExpandIntegrand[(a + b*Log[c*RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u 
]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalFuncti 
onQ[RGx, x] && IGtQ[n, 0]
 

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.88 (sec) , antiderivative size = 555, normalized size of antiderivative = 0.73

method	result	size
risch	\(\text {Expression too large to display}\)	\(555\)

Input:

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

Output:

(ln((c*x^2+b*x+a)^n)-n*ln(c*x^2+b*x+a))/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2) 
)+1/2*n/e*sum(1/_alpha*(ln(x-_alpha)*ln(c*x^2+b*x+a)-ln(x-_alpha)*ln((Root 
Of(_Z^2*c*e+(2*_alpha*c*e+b*e)*_Z+b*_alpha*e+e*a-c*d,index=1)-x+_alpha)/Ro 
otOf(_Z^2*c*e+(2*_alpha*c*e+b*e)*_Z+b*_alpha*e+e*a-c*d,index=1))-ln(x-_alp 
ha)*ln((RootOf(_Z^2*c*e+(2*_alpha*c*e+b*e)*_Z+b*_alpha*e+e*a-c*d,index=2)- 
x+_alpha)/RootOf(_Z^2*c*e+(2*_alpha*c*e+b*e)*_Z+b*_alpha*e+e*a-c*d,index=2 
))-dilog((RootOf(_Z^2*c*e+(2*_alpha*c*e+b*e)*_Z+b*_alpha*e+e*a-c*d,index=1 
)-x+_alpha)/RootOf(_Z^2*c*e+(2*_alpha*c*e+b*e)*_Z+b*_alpha*e+e*a-c*d,index 
=1))-dilog((RootOf(_Z^2*c*e+(2*_alpha*c*e+b*e)*_Z+b*_alpha*e+e*a-c*d,index 
=2)-x+_alpha)/RootOf(_Z^2*c*e+(2*_alpha*c*e+b*e)*_Z+b*_alpha*e+e*a-c*d,ind 
ex=2))),_alpha=RootOf(_Z^2*e+d))+(-1/2*I*Pi*csgn(I*(c*x^2+b*x+a)^n)*csgn(I 
*g)*csgn(I*g*(c*x^2+b*x+a)^n)+1/2*I*Pi*csgn(I*(c*x^2+b*x+a)^n)*csgn(I*g*(c 
*x^2+b*x+a)^n)^2+1/2*I*Pi*csgn(I*g)*csgn(I*g*(c*x^2+b*x+a)^n)^2-1/2*I*Pi*c 
sgn(I*g*(c*x^2+b*x+a)^n)^3+ln(g))/(d*e)^(1/2)*arctan(x*e/(d*e)^(1/2))
 

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\log \left (g \left (a+b x+c x^2\right )^n\right )}{d+e x^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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