Integrand size = 25, antiderivative size = 188 \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\frac {f p x^2}{2 g^2}+\frac {d p x^2}{4 e g}-\frac {p x^4}{8 g}-\frac {d^2 p \log \left (d+e x^2\right )}{4 e^2 g}+\frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{4 g}-\frac {f \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{2 e g^2}+\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{2 g^3}+\frac {f^2 p \operatorname {PolyLog}\left (2,-\frac {g \left (d+e x^2\right )}{e f-d g}\right )}{2 g^3} \] Output:
1/2*f*p*x^2/g^2+1/4*d*p*x^2/e/g-1/8*p*x^4/g-1/4*d^2*p*ln(e*x^2+d)/e^2/g+1/ 4*x^4*ln(c*(e*x^2+d)^p)/g-1/2*f*(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)*ln(e*(g*x^2+f)/(-d*g+e*f))/g^3+1/2*f^2*p*polylog(2,-g*(e *x^2+d)/(-d*g+e*f))/g^3
Time = 0.12 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.76 \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\frac {e g p x^2 \left (4 e f+2 d g-e g x^2\right )-2 d^2 g^2 p \log \left (d+e x^2\right )+e \log \left (c \left (d+e x^2\right )^p\right ) \left (2 g \left (-2 d f-2 e f x^2+e g x^4\right )+4 e f^2 \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )\right )+4 e^2 f^2 p \operatorname {PolyLog}\left (2,\frac {g \left (d+e x^2\right )}{-e f+d g}\right )}{8 e^2 g^3} \] Input:
Integrate[(x^5*Log[c*(d + e*x^2)^p])/(f + g*x^2),x]
Output:
(e*g*p*x^2*(4*e*f + 2*d*g - e*g*x^2) - 2*d^2*g^2*p*Log[d + e*x^2] + e*Log[ c*(d + e*x^2)^p]*(2*g*(-2*d*f - 2*e*f*x^2 + e*g*x^4) + 4*e*f^2*Log[(e*(f + g*x^2))/(e*f - d*g)]) + 4*e^2*f^2*p*PolyLog[2, (g*(d + e*x^2))/(-(e*f) + d*g)])/(8*e^2*g^3)
Time = 0.77 (sec) , antiderivative size = 181, 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.
| \(\displaystyle \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx\) |
| \(\Big \downarrow \) 2925 |
| \(\displaystyle \frac {1}{2} \int \frac {x^4 \log \left (c \left (e x^2+d\right )^p\right )}{g x^2+f}dx^2\) |
| \(\Big \downarrow \) 2863 |
| \(\displaystyle \frac {1}{2} \int \left (\frac {\log \left (c \left (e x^2+d\right )^p\right ) f^2}{g^2 \left (g x^2+f\right )}-\frac {\log \left (c \left (e x^2+d\right )^p\right ) f}{g^2}+\frac {x^2 \log \left (c \left (e x^2+d\right )^p\right )}{g}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1}{2} \left (\frac {f^2 \log \left (c \left (d+e x^2\right )^p\right ) \log \left (\frac {e \left (f+g x^2\right )}{e f-d g}\right )}{g^3}-\frac {f \left (d+e x^2\right ) \log \left (c \left (d+e x^2\right )^p\right )}{e g^2}+\frac {x^4 \log \left (c \left (d+e x^2\right )^p\right )}{2 g}-\frac {d^2 p \log \left (d+e x^2\right )}{2 e^2 g}+\frac {f^2 p \operatorname {PolyLog}\left (2,-\frac {g \left (e x^2+d\right )}{e f-d g}\right )}{g^3}+\frac {d p x^2}{2 e g}+\frac {f p x^2}{g^2}-\frac {p x^4}{4 g}\right )\) |
Input:
Int[(x^5*Log[c*(d + e*x^2)^p])/(f + g*x^2),x]
Output:
((f*p*x^2)/g^2 + (d*p*x^2)/(2*e*g) - (p*x^4)/(4*g) - (d^2*p*Log[d + e*x^2] )/(2*e^2*g) + (x^4*Log[c*(d + e*x^2)^p])/(2*g) - (f*(d + e*x^2)*Log[c*(d + e*x^2)^p])/(e*g^2) + (f^2*Log[c*(d + e*x^2)^p]*Log[(e*(f + g*x^2))/(e*f - d*g)])/g^3 + (f^2*p*PolyLog[2, -((g*(d + e*x^2))/(e*f - d*g))])/g^3)/2
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 = 4.70 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.18
| method | result | size |
| parts | \(\frac {x^{4} \ln \left (c \left (e \,x^{2}+d \right )^{p}\right )}{4 g}-\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f \,x^{2}}{2 g^{2}}+\frac {\ln \left (c \left (e \,x^{2}+d \right )^{p}\right ) f^{2} \ln \left (g \,x^{2}+f \right )}{2 g^{3}}-e p \left (\frac {f^{2} \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^{3} e}-\frac {-\frac {\frac {1}{2} e g \,x^{4}-d g \,x^{2}-2 f \,x^{2} e}{2 e^{2}}-\frac {d \left (d g +2 e f \right ) \ln \left (e \,x^{2}+d \right )}{2 e^{3}}}{2 g^{2}}\right )\) | \(410\) |
| risch | \(\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) x^{4}}{4 g}-\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f \,x^{2}}{2 g^{2}}+\frac {\ln \left (\left (e \,x^{2}+d \right )^{p}\right ) f^{2} \ln \left (g \,x^{2}+f \right )}{2 g^{3}}-\frac {p \,f^{2} \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^{3}}-\frac {p \,x^{4}}{8 g}+\frac {d p \,x^{2}}{4 e g}+\frac {f p \,x^{2}}{2 g^{2}}-\frac {d^{2} p \ln \left (e \,x^{2}+d \right )}{4 e^{2} g}-\frac {p d \ln \left (e \,x^{2}+d \right ) f}{2 e \,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 {\frac {1}{2} g \,x^{4}-f \,x^{2}}{2 g^{2}}+\frac {f^{2} \ln \left (g \,x^{2}+f \right )}{2 g^{3}}\right )\) | \(566\) |
Input:
int(x^5*ln(c*(e*x^2+d)^p)/(g*x^2+f),x,method=_RETURNVERBOSE)
Output:
1/4*x^4*ln(c*(e*x^2+d)^p)/g-1/2*ln(c*(e*x^2+d)^p)/g^2*f*x^2+1/2*ln(c*(e*x^ 2+d)^p)*f^2/g^3*ln(g*x^2+f)-e*p*(1/2*f^2/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+_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))-1/2/g^2*(-1/2/e^2* (1/2*e*g*x^4-d*g*x^2-2*f*x^2*e)-1/2*d*(d*g+2*e*f)/e^3*ln(e*x^2+d)))
\[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int { \frac {x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \] Input:
integrate(x^5*log(c*(e*x^2+d)^p)/(g*x^2+f),x, algorithm="fricas")
Output:
integral(x^5*log((e*x^2 + d)^p*c)/(g*x^2 + f), x)
\[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int \frac {x^{5} \log {\left (c \left (d + e x^{2}\right )^{p} \right )}}{f + g x^{2}}\, dx \] Input:
integrate(x**5*ln(c*(e*x**2+d)**p)/(g*x**2+f),x)
Output:
Integral(x**5*log(c*(d + e*x**2)**p)/(f + g*x**2), x)
\[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int { \frac {x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \] Input:
integrate(x^5*log(c*(e*x^2+d)^p)/(g*x^2+f),x, algorithm="maxima")
Output:
integrate(x^5*log((e*x^2 + d)^p*c)/(g*x^2 + f), x)
\[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int { \frac {x^{5} \log \left ({\left (e x^{2} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \] Input:
integrate(x^5*log(c*(e*x^2+d)^p)/(g*x^2+f),x, algorithm="giac")
Output:
integrate(x^5*log((e*x^2 + d)^p*c)/(g*x^2 + f), x)
Timed out. \[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\int \frac {x^5\,\ln \left (c\,{\left (e\,x^2+d\right )}^p\right )}{g\,x^2+f} \,d x \] Input:
int((x^5*log(c*(d + e*x^2)^p))/(f + g*x^2),x)
Output:
int((x^5*log(c*(d + e*x^2)^p))/(f + g*x^2), x)
\[ \int \frac {x^5 \log \left (c \left (d+e x^2\right )^p\right )}{f+g x^2} \, dx=\frac {8 \left (\int \frac {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) x}{e g \,x^{4}+d g \,x^{2}+e f \,x^{2}+d f}d x \right ) d \,e^{2} f^{2} g p -8 \left (\int \frac {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) x}{e g \,x^{4}+d g \,x^{2}+e f \,x^{2}+d f}d x \right ) e^{3} f^{3} p +2 {\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right )}^{2} e^{2} f^{2}-2 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d^{2} g^{2} p -4 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) d e f g p -4 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) e^{2} f g p \,x^{2}+2 \,\mathrm {log}\left (\left (e \,x^{2}+d \right )^{p} c \right ) e^{2} g^{2} p \,x^{4}+2 d e \,g^{2} p^{2} x^{2}+4 e^{2} f g \,p^{2} x^{2}-e^{2} g^{2} p^{2} x^{4}}{8 e^{2} g^{3} p} \] Input:
int(x^5*log(c*(e*x^2+d)^p)/(g*x^2+f),x)
Output:
(8*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*e**2*f**2*g*p - 8*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**3*f**3*p + 2*log((d + e*x**2)**p*c)**2*e**2*f**2 - 2*log((d + e*x**2)**p*c)*d**2*g**2*p - 4*log((d + e*x**2)**p*c)*d*e*f*g*p - 4*log((d + e*x**2)**p*c)*e**2*f*g*p*x**2 + 2*log((d + e*x**2)**p*c)*e** 2*g**2*p*x**4 + 2*d*e*g**2*p**2*x**2 + 4*e**2*f*g*p**2*x**2 - e**2*g**2*p* *2*x**4)/(8*e**2*g**3*p)