Integrand size = 25, antiderivative size = 112 \[ \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=-\frac {p x^2}{2 g}+\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g}-\frac {f \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^2}-\frac {f p \operatorname {PolyLog}\left (2,-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 g^2} \]
-1/2*p*x^2/g+1/2*(e*x^2+d)*ln(c*(e*x^2+d)^p)/e/g-1/2*f*ln(c*(e*x^2+d)^p)*l n(e*(g*x^2+f)/(-d*g+e*f))/g^2-1/2*f*p*polylog(2,-g*(e*x^2+d)/(-d*g+e*f))/g ^2
Time = 0.03 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.81 \[ \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=-\frac {e g p x^2-\log \left (c \left (d+e x^2\right )^p\right ) \left (d g+e g x^2-e f \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )\right )+e f p \operatorname {PolyLog}\left (2,\frac {g \left (d+e x^2\right )}{-e f+d g}\right )}{2 e g^2} \]
-1/2*(e*g*p*x^2 - Log[c*(d + e*x^2)^p]*(d*g + e*g*x^2 - e*f*Log[(e*(f + g* x^2))/(e*f - d*g)]) + e*f*p*PolyLog[2, (g*(d + e*x^2))/(-(e*f) + d*g)])/(e *g^2)
Time = 0.35 (sec) , antiderivative size = 107, 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.
| \(\displaystyle \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx\) |
| \(\Big \downarrow \) 2925 |
| \(\displaystyle \frac {1}{2} \int \frac {x^2 \log \left (c \left (e x^2+d\right )^p\right )}{g x^2+f}dx^2\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\log \left (c \left (e x^2+d\right )^p\right )}{g}-\frac {f \log \left (c \left (e x^2+d\right )^p\right )}{g \left (g x^2+f\right )}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (-\frac {f \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{g^2}+\frac {\left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e g}-\frac {f p \operatorname {PolyLog}\left (2,-\frac {g \left (e x^2+d\right )}{e f-d g}\right )}{g^2}-\frac {p x^2}{g}\right )\) |
(-((p*x^2)/g) + ((d + e*x^2)*Log[c*(d + e*x^2)^p])/(e*g) - (f*Log[c*(d + e *x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)])/g^2 - (f*p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g))])/g^2)/2
3.4.39.3.1 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 = 1.50 (sec) , antiderivative size = 358, normalized size of antiderivative = 3.20
| method | result | size |
| parts | \(\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) x^{2}}{2 g}-\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f \ln \left (g \,x^{2}+f \right )}{2 g^{2}}-p e \left (\frac {x^{2}}{2 g e}-\frac {d \ln \left (e \,x^{2}+d \right )}{2 g \,e^{2}}-\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 )}{2 g^{2} e}\right )\) | \(358\) |
| risch | \(\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) x^{2}}{2 g}-\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f \ln \left (g \,x^{2}+f \right )}{2 g^{2}}-\frac {p \,x^{2}}{2 g}+\frac {p d \ln \left (e \,x^{2}+d \right )}{2 e g}+\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 )}{2 g^{2}}+\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}-\frac {f \ln \left (g \,x^{2}+f \right )}{2 g^{2}}\right )\) | \(487\) |
1/2*ln(c*(e*x^2+d)^p)*x^2/g-1/2*ln(c*(e*x^2+d)^p)*f/g^2*ln(g*x^2+f)-p*e*(1 /2/g/e*x^2-1/2/g*d/e^2*ln(e*x^2+d)-1/2*f/g^2/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+_al pha)/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((Root Of(_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)))
\[ \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int { \frac {x^{3} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
\[ \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int \frac {x^{3} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{f + g x^{2}}\, dx \]
\[ \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int { \frac {x^{3} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
\[ \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int { \frac {x^{3} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \]
Timed out. \[ \int \frac {x^3 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int \frac {x^3\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{g\,x^2+f} \,d x \]