\(\int \frac {\log (c (d+e x^2)^p)}{x (f+g x^2)} \, dx\) [341]

Optimal result

Integrand size = 25, antiderivative size = 119 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )} \, dx=\frac {\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 f}-\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 f}-\frac {p \operatorname {PolyLog}\left (2,-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 f}+\frac {p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right )}{2 f} \] Output:

1/2*ln(-e*x^2/d)*ln(c*(e*x^2+d)^p)/f-1/2*ln(c*(e*x^2+d)^p)*ln(e*(g*x^2+f)/ 
(-d*g+e*f))/f-1/2*p*polylog(2,-g*(e*x^2+d)/(-d*g+e*f))/f+1/2*p*polylog(2,1 
+e*x^2/d)/f
 

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.77 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )} \, dx=\frac {\log \left (c \left (d+e x^2\right )^p\right ) \left (\log \left (-\frac {e x^2}{d}\right )-\log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )\right )-p \operatorname {PolyLog}\left (2,\frac {g \left (d+e x^2\right )}{-e f+d g}\right )+p \operatorname {PolyLog}\left (2,1+\frac {e x^2}{d}\right )}{2 f} \] Input:

Integrate[Log[c*(d + e*x^2)^p]/(x*(f + g*x^2)),x]
 

Output:

(Log[c*(d + e*x^2)^p]*(Log[-((e*x^2)/d)] - Log[(e*(f + g*x^2))/(e*f - d*g) 
]) - p*PolyLog[2, (g*(d + e*x^2))/(-(e*f) + d*g)] + p*PolyLog[2, 1 + (e*x^ 
2)/d])/(2*f)
 

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.95, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2925, 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.

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

\(\Big \downarrow \) 2925

\(\displaystyle \frac {1}{2} \int \frac {\log \left (c \left (e x^2+d\right )^p\right )}{x^2 \left (g x^2+f\right )}dx^2\)

\(\Big \downarrow \) 2863

\(\displaystyle \frac {1}{2} \int \left (\frac {\log \left (c \left (e x^2+d\right )^p\right )}{f x^2}-\frac {g \log \left (c \left (e x^2+d\right )^p\right )}{f \left (g x^2+f\right )}\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (-\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{f}+\frac {\log \left (-\frac {e x^2}{d}\right ) \log \left (c \left (d+e x^2\right )^p\right )}{f}-\frac {p \operatorname {PolyLog}\left (2,-\frac {g \left (e x^2+d\right )}{e f-d g}\right )}{f}+\frac {p \operatorname {PolyLog}\left (2,\frac {e x^2}{d}+1\right )}{f}\right )\)

Input:

Int[Log[c*(d + e*x^2)^p]/(x*(f + g*x^2)),x]
 

Output:

((Log[-((e*x^2)/d)]*Log[c*(d + e*x^2)^p])/f - (Log[c*(d + e*x^2)^p]*Log[(e 
*(f + g*x^2))/(e*f - d*g)])/f - (p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g 
))])/f + (p*PolyLog[2, 1 + (e*x^2)/d])/f)/2
 

Defintions of rubi rules used

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 2925

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m 
_.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Simp[1/n   Subst[Int[x^(Si 
mplify[(m + 1)/n] - 1)*(f + g*x^(s/n))^r*(a + b*Log[c*(d + e*x)^p])^q, x], 
x, x^n], x] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, q, r, s}, x] && Integer 
Q[r] && IntegerQ[s/n] && IntegerQ[Simplify[(m + 1)/n]] && (GtQ[(m + 1)/n, 0 
] || IGtQ[q, 0])
 

Maple [C] (warning: unable to verify)

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

Time = 2.39 (sec) , antiderivative size = 420, normalized size of antiderivative = 3.53

method	result	size
parts	\(\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) \ln \left (x \right )}{f}-\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) \ln \left (g \,x^{2}+f \right )}{2 f}-e p \left (\frac {\frac {\ln \left (x \right ) \left (\ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )\right )}{e}+\frac {\operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )+\operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{e}}{f}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+d \right )}{\sum }\left (\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (g \,x^{2}+f \right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )}{2 f e}\right )\)	\(420\)
risch	\(\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) \ln \left (x \right )}{f}-\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) \ln \left (g \,x^{2}+f \right )}{2 f}-\frac {p \ln \left (x \right ) \ln \left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{f}-\frac {p \ln \left (x \right ) \ln \left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{f}-\frac {p \operatorname {dilog}\left (\frac {-e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{f}-\frac {p \operatorname {dilog}\left (\frac {e x +\sqrt {-d e}}{\sqrt {-d e}}\right )}{f}+\frac {p \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+d \right )}{\sum }\left (\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (g \,x^{2}+f \right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )\right )}{2 f}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (e \,x^{2}+d \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {\ln \left (x \right )}{f}-\frac {\ln \left (g \,x^{2}+f \right )}{2 f}\right )\)	\(555\)

Input:

int(ln(c*(e*x^2+d)^p)/x/(g*x^2+f),x,method=_RETURNVERBOSE)
 

Output:

ln(c*(e*x^2+d)^p)/f*ln(x)-1/2*ln(c*(e*x^2+d)^p)/f*ln(g*x^2+f)-e*p*(2/f*(1/ 
2*ln(x)*(ln((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+ln((e*x+(-d*e)^(1/2))/(-d*e) 
^(1/2)))/e+1/2*(dilog((-e*x+(-d*e)^(1/2))/(-d*e)^(1/2))+dilog((e*x+(-d*e)^ 
(1/2))/(-d*e)^(1/2)))/e)-1/2/f/e*sum(ln(x-_alpha)*ln(g*x^2+f)-ln(x-_alpha) 
*(ln((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=1)-x+_alpha)/RootOf(_Z 
^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=1))+ln((RootOf(_Z^2*e*g+2*_Z*_alpha*e 
*g-d*g+e*f,index=2)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,inde 
x=2)))-dilog((RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=1)-x+_alpha)/R 
ootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d*g+e*f,index=1))-dilog((RootOf(_Z^2*e*g+2* 
_Z*_alpha*e*g-d*g+e*f,index=2)-x+_alpha)/RootOf(_Z^2*e*g+2*_Z*_alpha*e*g-d 
*g+e*f,index=2)),_alpha=RootOf(_Z^2*e+d)))
 

Fricas [F]

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

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

Output:

integral(log((e*x^2 + d)^p*c)/(g*x^3 + f*x), x)
 

Sympy [F(-1)]

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

integrate(ln(c*(e*x**2+d)**p)/x/(g*x**2+f),x)
 

Output:

Timed out
 

Maxima [A] (verification not implemented)

Time = 0.16 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.18 \[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )} \, dx=-\frac {1}{2} \, e p {\left (\frac {2 \, \log \left (\frac {e x^{2}}{d} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-\frac {e x^{2}}{d}\right )}{e f} - \frac {\log \left (g x^{2} + f\right ) \log \left (-\frac {e g x^{2} + e f}{e f - d g} + 1\right ) + {\rm Li}_2\left (\frac {e g x^{2} + e f}{e f - d g}\right )}{e f}\right )} - \frac {1}{2} \, {\left (\frac {\log \left (g x^{2} + f\right )}{f} - \frac {\log \left (x^{2}\right )}{f}\right )} \log \left ({\left (e x^{2} + d\right )}^{p} c\right ) \] Input:

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

Output:

-1/2*e*p*((2*log(e*x^2/d + 1)*log(x) + dilog(-e*x^2/d))/(e*f) - (log(g*x^2 
 + f)*log(-(e*g*x^2 + e*f)/(e*f - d*g) + 1) + dilog((e*g*x^2 + e*f)/(e*f - 
 d*g)))/(e*f)) - 1/2*(log(g*x^2 + f)/f - log(x^2)/f)*log((e*x^2 + d)^p*c)
 

Giac [F]

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

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

Output:

integrate(log((e*x^2 + d)^p*c)/((g*x^2 + f)*x), x)
 

Mupad [F(-1)]

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

int(log(c*(d + e*x^2)^p)/(x*(f + g*x^2)),x)
 

Output:

int(log(c*(d + e*x^2)^p)/(x*(f + g*x^2)), x)
 

Reduce [F]

\[ \int \frac {\log \left (c \left (d+e x^2\right )^p\right )}{x \left (f+g x^2\right )} \, dx=\int \frac {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}{g \,x^{3}+f x}d x \] Input:

int(log(c*(e*x^2+d)^p)/x/(g*x^2+f),x)
                                                                                    
                                                                                    
 

Output:

int(log((d + e*x**2)**p*c)/(f*x + g*x**3),x)