\(\int \frac {x \log (c (d+e x^2)^p)}{f+g x^2} \, dx\) [340]

Optimal result

Integrand size = 23, antiderivative size = 70 \[ \int \frac {x \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g}+\frac {p \operatorname {PolyLog}\left (2,-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 g} \] Output:

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.91 \[ \int \frac {x \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\frac {\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )+p \operatorname {PolyLog}\left (2,\frac {g \left (d+e x^2\right )}{-e f+d g}\right )}{2 g} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.97, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2925, 2841, 2840, 2838}

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.

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

\(\Big \downarrow \) 2925

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

\(\Big \downarrow \) 2841

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

\(\Big \downarrow \) 2840

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 2838

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 2840

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_ 
Symbol] :> Simp[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 2841

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] - Simp[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 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 = 3.60 (sec) , antiderivative size = 301, normalized size of antiderivative = 4.30

method	result	size
parts	\(\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) \ln \left (g \,x^{2}+f \right )}{2 g}-\frac {p \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (e \,\textit {\_Z}^{2}+d \right )}{\sum }\left (\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \ln \left (g \,x^{2}+f \right )-\ln \left (x -\underline {\hspace {1.25 ex}}\alpha \right ) \left (\ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )+\ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =1\right )}\right )-\operatorname {dilog}\left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )-x +\underline {\hspace {1.25 ex}}\alpha }{\operatorname {RootOf}\left (\textit {\_Z}^{2} e g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )\right )}{2 g}\)	\(301\)
risch	\(\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) \ln \left (g \,x^{2}+f \right )}{2 g}-\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 g}+\frac {\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 ) \ln \left (g \,x^{2}+f \right )}{2 g}\)	\(429\)

Input:

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

Output:

1/2*ln(c*(e*x^2+d)^p)/g*ln(g*x^2+f)-1/2/g*p*sum(ln(x-_alpha)*ln(g*x^2+f)-l 
n(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,index=2)))-dilog((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))-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]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 138 vs. \(2 (65) = 130\).

Time = 0.03 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.97 \[ \int \frac {x \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\frac {e p {\left (\frac {\log \left (e x^{2} + d\right ) \log \left (g x^{2} + f\right )}{e} - \frac {\log \left (g x^{2} + f\right ) \log \left (-\frac {e g x^{2} + e f}{e f - d g} + 1\right ) + {\rm Li}_2\left (\frac {e g x^{2} + e f}{e f - d g}\right )}{e}\right )}}{2 \, g} - \frac {p \log \left (e x^{2} + d\right ) \log \left (g x^{2} + f\right )}{2 \, g} + \frac {\log \left (g x^{2} + f\right ) \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{2 \, g} \] Input:

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\frac {4 \left (\int \frac {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) x}{e g \,x^{4}+d g \,x^{2}+e f \,x^{2}+d f}d x \right ) d g p -4 \left (\int \frac {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) x}{e g \,x^{4}+d g \,x^{2}+e f \,x^{2}+d f}d x \right ) e f p +{\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}^{2}}{4 g p} \] Input:

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

Output:

(4*int((log((d + e*x**2)**p*c)*x)/(d*f + d*g*x**2 + e*f*x**2 + e*g*x**4),x 
)*d*g*p - 4*int((log((d + e*x**2)**p*c)*x)/(d*f + d*g*x**2 + e*f*x**2 + e* 
g*x**4),x)*e*f*p + log((d + e*x**2)**p*c)**2)/(4*g*p)