Integrand size = 25, antiderivative size = 199 \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {p x^2}{2 g^2}+\frac {e f^2 p \log \left (d+e x^2\right )}{2 g^3 (e f-d g)}+\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g^2}-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{2 g^3 \left (f+g x^2\right )}-\frac {e f^2 p \log \left (f+g x^2\right )}{2 g^3 (e f-d g)}-\frac {f \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{g^3}-\frac {f p \operatorname {PolyLog}\left (2,-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{g^3} \]
-1/2*p*x^2/g^2+1/2*e*f^2*p*ln(e*x^2+d)/g^3/(-d*g+e*f)+1/2*(e*x^2+d)*ln(c*( e*x^2+d)^p)/e/g^2-1/2*f^2*ln(c*(e*x^2+d)^p)/g^3/(g*x^2+f)-1/2*e*f^2*p*ln(g *x^2+f)/g^3/(-d*g+e*f)-f*ln(c*(e*x^2+d)^p)*ln(e*(g*x^2+f)/(-d*g+e*f))/g^3- f*p*polylog(2,-g*(e*x^2+d)/(-d*g+e*f))/g^3
Time = 0.18 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.83 \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=-\frac {p x^2-\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e}+\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{g \left (f+g x^2\right )}+\frac {e f^2 p \left (\log \left (d+e x^2\right )-\log \left (f+g x^2\right )\right )}{g (-e f+d g)}+\frac {2 f \left (\log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )+p \operatorname {PolyLog}\left (2,\frac {g \left (d+e x^2\right )}{-e f+d g}\right )\right )}{g}}{2 g^2} \]
-1/2*(p*x^2 - ((d + e*x^2)*Log[c*(d + e*x^2)^p])/e + (f^2*Log[c*(d + e*x^2 )^p])/(g*(f + g*x^2)) + (e*f^2*p*(Log[d + e*x^2] - Log[f + g*x^2]))/(g*(-( e*f) + d*g)) + (2*f*(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)]))/g)/g^2
Time = 0.46 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.96, 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.
| \(\displaystyle \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx\) |
| \(\Big \downarrow \) 2925 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4 \log \left (c \left (e x^2+d\right )^p\right )}{\left (g x^2+f\right )^2}dx^2\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\log \left (c \left (e x^2+d\right )^p\right ) f^2}{g^2 \left (g x^2+f\right )^2}-\frac {2 \log \left (c \left (e x^2+d\right )^p\right ) f}{g^2 \left (g x^2+f\right )}+\frac {\log \left (c \left (e x^2+d\right )^p\right )}{g^2}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right )}{g^3 \left (f+g x^2\right )}-\frac {2 f \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{g^3}+\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e g^2}+\frac {e f^2 p \log \left (d+e x^2\right )}{g^3 (e f-d g)}-\frac {e f^2 p \log \left (f+g x^2\right )}{g^3 (e f-d g)}-\frac {2 f p \operatorname {PolyLog}\left (2,-\frac {g \left (e x^2+d\right )}{e f-d g}\right )}{g^3}-\frac {p x^2}{g^2}\right )\) |
(-((p*x^2)/g^2) + (e*f^2*p*Log[d + e*x^2])/(g^3*(e*f - d*g)) + ((d + e*x^2 )*Log[c*(d + e*x^2)^p])/(e*g^2) - (f^2*Log[c*(d + e*x^2)^p])/(g^3*(f + g*x ^2)) - (e*f^2*p*Log[f + g*x^2])/(g^3*(e*f - d*g)) - (2*f*Log[c*(d + e*x^2) ^p]*Log[(e*(f + g*x^2))/(e*f - d*g)])/g^3 - (2*f*p*PolyLog[2, -((g*(d + e* x^2))/(e*f - d*g))])/g^3)/2
3.4.48.3.1 Defintions of rubi rules used
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]
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])
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 2.09 (sec) , antiderivative size = 442, normalized size of antiderivative = 2.22
| method | result | size |
| parts | \(\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) x^{2}}{2 g^{2}}-\frac {f^{2} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{2 g^{3} \left (g \,x^{2}+f \right )}-\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f \ln \left (g \,x^{2}+f \right )}{g^{3}}-p e \left (-\frac {f \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 (e \,\textit {\_Z}^{2} 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 (e \,\textit {\_Z}^{2} 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 (e \,\textit {\_Z}^{2} 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 (e \,\textit {\_Z}^{2} 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 (e \,\textit {\_Z}^{2} 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 (e \,\textit {\_Z}^{2} 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 (e \,\textit {\_Z}^{2} 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 (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )\right )}{g^{3} e}-\frac {-\frac {g \,x^{2}}{2 e}+\frac {\left (g^{2} d^{2}-d e f g -e^{2} f^{2}\right ) \ln \left (e \,x^{2}+d \right )}{2 \left (d g -e f \right ) e^{2}}+\frac {f^{2} \ln \left (g \,x^{2}+f \right )}{2 d g -2 e f}}{g^{3}}\right )\) | \(442\) |
| risch | \(\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) x^{2}}{2 g^{2}}-\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f^{2}}{2 g^{3} \left (g \,x^{2}+f \right )}-\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f \ln \left (g \,x^{2}+f \right )}{g^{3}}-\frac {p \,x^{2}}{2 g^{2}}+\frac {p \ln \left (e \,x^{2}+d \right ) d^{2}}{2 e g \left (d g -e f \right )}-\frac {p \ln \left (e \,x^{2}+d \right ) d f}{2 g^{2} \left (d g -e f \right )}-\frac {p e \ln \left (e \,x^{2}+d \right ) f^{2}}{2 g^{3} \left (d g -e f \right )}+\frac {p e \,f^{2} \ln \left (g \,x^{2}+f \right )}{2 g^{3} \left (d g -e f \right )}+\frac {p f \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 (e \,\textit {\_Z}^{2} 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 (e \,\textit {\_Z}^{2} 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 (e \,\textit {\_Z}^{2} 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 (e \,\textit {\_Z}^{2} 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 (e \,\textit {\_Z}^{2} 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 (e \,\textit {\_Z}^{2} 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 (e \,\textit {\_Z}^{2} 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 (e \,\textit {\_Z}^{2} g +2 \underline {\hspace {1.25 ex}}\alpha \textit {\_Z} g e -d g +e f , \operatorname {index} =2\right )}\right )\right )\right )}{g^{3}}+\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 {x^{2}}{2 g^{2}}-\frac {f \left (\frac {f}{g \left (g \,x^{2}+f \right )}+\frac {2 \ln \left (g \,x^{2}+f \right )}{g}\right )}{2 g^{2}}\right )\) | \(627\) |
1/2*ln(c*(e*x^2+d)^p)*x^2/g^2-1/2*f^2*ln(c*(e*x^2+d)^p)/g^3/(g*x^2+f)-ln(c *(e*x^2+d)^p)*f/g^3*ln(g*x^2+f)-p*e*(-f/g^3/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+_alp ha)/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((RootO f(_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))-1/g^3*(-1/2*g*x^2/e +1/2*(d^2*g^2-d*e*f*g-e^2*f^2)/(d*g-e*f)/e^2*ln(e*x^2+d)+1/2*f^2/(d*g-e*f) *ln(g*x^2+f)))
\[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
\[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int { \frac {x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{{\left (g x^{2} + f\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{\left (f+g x^2\right )^2} \, dx=\int \frac {x^5\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{{\left (g\,x^2+f\right )}^2} \,d x \]