Integrand size = 65, antiderivative size = 132 \[ \int \frac {x^2}{\left (1+x^2\right ) \sqrt [10]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx=\frac {\left (\left (3-13 x+4 x^2\right )^5\right )^{9/10} \left (-\frac {1}{2} \log \left (13-8 x+4 \sqrt {3-13 x+4 x^2}\right )+32 \text {RootSum}\left [14641-6292 \text {$\#$1}+1174 \text {$\#$1}^2-52 \text {$\#$1}^3+\text {$\#$1}^4\&,\frac {\log \left (13-8 x+4 \sqrt {3-13 x+4 x^2}-\text {$\#$1}\right ) \text {$\#$1}}{-1573+587 \text {$\#$1}-39 \text {$\#$1}^2+\text {$\#$1}^3}\&\right ]\right )}{\left (3-13 x+4 x^2\right )^{9/2}} \]
Time = 0.13 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.09 \[ \int \frac {x^2}{\left (1+x^2\right ) \sqrt [10]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx=\frac {\sqrt {3-13 x+4 x^2} \left (-\log \left (13-8 x+4 \sqrt {3-13 x+4 x^2}\right )+\text {RootSum}\left [178+104 \text {$\#$1}+10 \text {$\#$1}^2+\text {$\#$1}^4\&,\frac {13 \log \left (-2 x+\sqrt {3-13 x+4 x^2}-\text {$\#$1}\right )+4 \log \left (-2 x+\sqrt {3-13 x+4 x^2}-\text {$\#$1}\right ) \text {$\#$1}}{26+5 \text {$\#$1}+\text {$\#$1}^3}\&\right ]\right )}{2 \sqrt [10]{\left (3-13 x+4 x^2\right )^5}} \]
Integrate[x^2/((1 + x^2)*(243 - 5265*x + 47250*x^2 - 225810*x^3 + 615255*x ^4 - 954733*x^5 + 820340*x^6 - 401440*x^7 + 112000*x^8 - 16640*x^9 + 1024* x^10)^(1/10)),x]
(Sqrt[3 - 13*x + 4*x^2]*(-Log[13 - 8*x + 4*Sqrt[3 - 13*x + 4*x^2]] + RootS um[178 + 104*#1 + 10*#1^2 + #1^4 & , (13*Log[-2*x + Sqrt[3 - 13*x + 4*x^2] - #1] + 4*Log[-2*x + Sqrt[3 - 13*x + 4*x^2] - #1]*#1)/(26 + 5*#1 + #1^3) & ]))/(2*((3 - 13*x + 4*x^2)^5)^(1/10))
Time = 1.32 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.43, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.169, Rules used = {7239, 7271, 2144, 25, 1092, 219, 1318, 25, 1363, 217, 219}
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.
| \(\displaystyle \int \frac {x^2}{\left (x^2+1\right ) \sqrt [10]{1024 x^{10}-16640 x^9+112000 x^8-401440 x^7+820340 x^6-954733 x^5+615255 x^4-225810 x^3+47250 x^2-5265 x+243}} \, dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {x^2}{\left (x^2+1\right ) \sqrt [10]{\left (4 x^2-13 x+3\right )^5}}dx\) |
| \(\Big \downarrow \) 7271 |
| \(\displaystyle \frac {\sqrt {4 x^2-13 x+3} \int \frac {x^2}{\left (x^2+1\right ) \sqrt {4 x^2-13 x+3}}dx}{\sqrt [10]{\left (4 x^2-13 x+3\right )^5}}\) |
| \(\Big \downarrow \) 2144 |
| \(\displaystyle \frac {\sqrt {4 x^2-13 x+3} \left (\int \frac {1}{\sqrt {4 x^2-13 x+3}}dx+\int -\frac {1}{\left (x^2+1\right ) \sqrt {4 x^2-13 x+3}}dx\right )}{\sqrt [10]{\left (4 x^2-13 x+3\right )^5}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {4 x^2-13 x+3} \left (\int \frac {1}{\sqrt {4 x^2-13 x+3}}dx-\int \frac {1}{\left (x^2+1\right ) \sqrt {4 x^2-13 x+3}}dx\right )}{\sqrt [10]{\left (4 x^2-13 x+3\right )^5}}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {\sqrt {4 x^2-13 x+3} \left (2 \int \frac {1}{16-\frac {(13-8 x)^2}{4 x^2-13 x+3}}d\left (-\frac {13-8 x}{\sqrt {4 x^2-13 x+3}}\right )-\int \frac {1}{\left (x^2+1\right ) \sqrt {4 x^2-13 x+3}}dx\right )}{\sqrt [10]{\left (4 x^2-13 x+3\right )^5}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {4 x^2-13 x+3} \left (-\int \frac {1}{\left (x^2+1\right ) \sqrt {4 x^2-13 x+3}}dx-\frac {1}{2} \text {arctanh}\left (\frac {13-8 x}{4 \sqrt {4 x^2-13 x+3}}\right )\right )}{\sqrt [10]{\left (4 x^2-13 x+3\right )^5}}\) |
| \(\Big \downarrow \) 1318 |
| \(\displaystyle \frac {\sqrt {4 x^2-13 x+3} \left (-\frac {\int -\frac {13 x-\sqrt {170}+1}{\left (x^2+1\right ) \sqrt {4 x^2-13 x+3}}dx}{2 \sqrt {170}}+\frac {\int -\frac {13 x+\sqrt {170}+1}{\left (x^2+1\right ) \sqrt {4 x^2-13 x+3}}dx}{2 \sqrt {170}}-\frac {1}{2} \text {arctanh}\left (\frac {13-8 x}{4 \sqrt {4 x^2-13 x+3}}\right )\right )}{\sqrt [10]{\left (4 x^2-13 x+3\right )^5}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\sqrt {4 x^2-13 x+3} \left (\frac {\int \frac {13 x-\sqrt {170}+1}{\left (x^2+1\right ) \sqrt {4 x^2-13 x+3}}dx}{2 \sqrt {170}}-\frac {\int \frac {13 x+\sqrt {170}+1}{\left (x^2+1\right ) \sqrt {4 x^2-13 x+3}}dx}{2 \sqrt {170}}-\frac {1}{2} \text {arctanh}\left (\frac {13-8 x}{4 \sqrt {4 x^2-13 x+3}}\right )\right )}{\sqrt [10]{\left (4 x^2-13 x+3\right )^5}}\) |
| \(\Big \downarrow \) 1363 |
| \(\displaystyle \frac {\sqrt {4 x^2-13 x+3} \left (-\frac {13 \left (1-\sqrt {170}\right ) \int \frac {1}{26 \left (1-\sqrt {170}\right )-\frac {13 \left (13-\left (1-\sqrt {170}\right ) x\right )^2}{4 x^2-13 x+3}}d\frac {13-\left (1-\sqrt {170}\right ) x}{\sqrt {4 x^2-13 x+3}}}{\sqrt {170}}+\frac {13 \left (1+\sqrt {170}\right ) \int \frac {1}{26 \left (1+\sqrt {170}\right )-\frac {13 \left (13-\left (1+\sqrt {170}\right ) x\right )^2}{4 x^2-13 x+3}}d\frac {13-\left (1+\sqrt {170}\right ) x}{\sqrt {4 x^2-13 x+3}}}{\sqrt {170}}-\frac {1}{2} \text {arctanh}\left (\frac {13-8 x}{4 \sqrt {4 x^2-13 x+3}}\right )\right )}{\sqrt [10]{\left (4 x^2-13 x+3\right )^5}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\sqrt {4 x^2-13 x+3} \left (\frac {13 \left (1+\sqrt {170}\right ) \int \frac {1}{26 \left (1+\sqrt {170}\right )-\frac {13 \left (13-\left (1+\sqrt {170}\right ) x\right )^2}{4 x^2-13 x+3}}d\frac {13-\left (1+\sqrt {170}\right ) x}{\sqrt {4 x^2-13 x+3}}}{\sqrt {170}}+\frac {\left (1-\sqrt {170}\right ) \arctan \left (\frac {13-\left (1-\sqrt {170}\right ) x}{\sqrt {2 \left (\sqrt {170}-1\right )} \sqrt {4 x^2-13 x+3}}\right )}{2 \sqrt {85 \left (\sqrt {170}-1\right )}}-\frac {1}{2} \text {arctanh}\left (\frac {13-8 x}{4 \sqrt {4 x^2-13 x+3}}\right )\right )}{\sqrt [10]{\left (4 x^2-13 x+3\right )^5}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\sqrt {4 x^2-13 x+3} \left (\frac {\left (1-\sqrt {170}\right ) \arctan \left (\frac {13-\left (1-\sqrt {170}\right ) x}{\sqrt {2 \left (\sqrt {170}-1\right )} \sqrt {4 x^2-13 x+3}}\right )}{2 \sqrt {85 \left (\sqrt {170}-1\right )}}-\frac {1}{2} \text {arctanh}\left (\frac {13-8 x}{4 \sqrt {4 x^2-13 x+3}}\right )+\frac {1}{2} \sqrt {\frac {1}{85} \left (1+\sqrt {170}\right )} \text {arctanh}\left (\frac {13-\left (1+\sqrt {170}\right ) x}{\sqrt {2 \left (1+\sqrt {170}\right )} \sqrt {4 x^2-13 x+3}}\right )\right )}{\sqrt [10]{\left (4 x^2-13 x+3\right )^5}}\) |
Int[x^2/((1 + x^2)*(243 - 5265*x + 47250*x^2 - 225810*x^3 + 615255*x^4 - 9 54733*x^5 + 820340*x^6 - 401440*x^7 + 112000*x^8 - 16640*x^9 + 1024*x^10)^ (1/10)),x]
(Sqrt[3 - 13*x + 4*x^2]*(((1 - Sqrt[170])*ArcTan[(13 - (1 - Sqrt[170])*x)/ (Sqrt[2*(-1 + Sqrt[170])]*Sqrt[3 - 13*x + 4*x^2])])/(2*Sqrt[85*(-1 + Sqrt[ 170])]) - ArcTanh[(13 - 8*x)/(4*Sqrt[3 - 13*x + 4*x^2])]/2 + (Sqrt[(1 + Sq rt[170])/85]*ArcTanh[(13 - (1 + Sqrt[170])*x)/(Sqrt[2*(1 + Sqrt[170])]*Sqr t[3 - 13*x + 4*x^2])])/2))/((3 - 13*x + 4*x^2)^5)^(1/10)
3.20.3.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((a_.) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_S ymbol] :> With[{q = Rt[(c*d - a*f)^2 + a*c*e^2, 2]}, Simp[1/(2*q) Int[(c* d - a*f + q + c*e*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x] - Simp[1/( 2*q) Int[(c*d - a*f - q + c*e*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f}, x] && NeQ[e^2 - 4*d*f, 0] && NegQ[(-a)*c]
Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f _.)*(x_)^2]), x_Symbol] :> Simp[-2*a*g*h Subst[Int[1/Simp[2*a^2*g*h*c + a *e*x^2, x], x], x, Simp[a*h - g*c*x, x]/Sqrt[d + e*x + f*x^2]], x] /; FreeQ [{a, c, d, e, f, g, h}, x] && EqQ[a*h^2*e + 2*g*h*(c*d - a*f) - g^2*c*e, 0]
Int[(Px_)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{A = Coeff[Px, x, 0], B = Coeff[Px, x, 1], C = Coeff[Px, x, 2]}, Simp[C/c Int[1/Sqrt[d + e*x + f*x^2], x], x] + Simp[1/c Int[(A* c - a*C + B*c*x)/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x], x]] /; FreeQ[{a, c, d, e, f}, x] && PolyQ[Px, x, 2]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Simp[a^IntPart[p]*((a*v^m)^ FracPart[p]/v^(m*FracPart[p])) Int[u*v^(m*p), x], x] /; FreeQ[{a, m, p}, x] && !IntegerQ[p] && !FreeQ[v, x] && !(EqQ[a, 1] && EqQ[m, 1]) && !(Eq Q[v, x] && EqQ[m, 1])
Not integrable
Time = 0.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.48
\[\int \frac {x^{2}}{\left (x^{2}+1\right ) \left (1024 x^{10}-16640 x^{9}+112000 x^{8}-401440 x^{7}+820340 x^{6}-954733 x^{5}+615255 x^{4}-225810 x^{3}+47250 x^{2}-5265 x +243\right )^{\frac {1}{10}}}d x\]
int(x^2/(x^2+1)/(1024*x^10-16640*x^9+112000*x^8-401440*x^7+820340*x^6-9547 33*x^5+615255*x^4-225810*x^3+47250*x^2-5265*x+243)^(1/10),x)
int(x^2/(x^2+1)/(1024*x^10-16640*x^9+112000*x^8-401440*x^7+820340*x^6-9547 33*x^5+615255*x^4-225810*x^3+47250*x^2-5265*x+243)^(1/10),x)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 0.30 (sec) , antiderivative size = 367, normalized size of antiderivative = 2.78 \[ \int \frac {x^2}{\left (1+x^2\right ) \sqrt [10]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx=\frac {1}{340} \, \sqrt {170} \sqrt {13 i + 1} \log \left (\left (i + 13\right ) \, \sqrt {170} \sqrt {13 i + 1} - 340 \, x + 170 \, {\left (1024 \, x^{10} - 16640 \, x^{9} + 112000 \, x^{8} - 401440 \, x^{7} + 820340 \, x^{6} - 954733 \, x^{5} + 615255 \, x^{4} - 225810 \, x^{3} + 47250 \, x^{2} - 5265 \, x + 243\right )}^{\frac {1}{10}} - 340 i\right ) - \frac {1}{340} \, \sqrt {170} \sqrt {13 i + 1} \log \left (-\left (i + 13\right ) \, \sqrt {170} \sqrt {13 i + 1} - 340 \, x + 170 \, {\left (1024 \, x^{10} - 16640 \, x^{9} + 112000 \, x^{8} - 401440 \, x^{7} + 820340 \, x^{6} - 954733 \, x^{5} + 615255 \, x^{4} - 225810 \, x^{3} + 47250 \, x^{2} - 5265 \, x + 243\right )}^{\frac {1}{10}} - 340 i\right ) + \frac {1}{340} \, \sqrt {170} \sqrt {-13 i + 1} \log \left (-\left (i - 13\right ) \, \sqrt {170} \sqrt {-13 i + 1} - 340 \, x + 170 \, {\left (1024 \, x^{10} - 16640 \, x^{9} + 112000 \, x^{8} - 401440 \, x^{7} + 820340 \, x^{6} - 954733 \, x^{5} + 615255 \, x^{4} - 225810 \, x^{3} + 47250 \, x^{2} - 5265 \, x + 243\right )}^{\frac {1}{10}} + 340 i\right ) - \frac {1}{340} \, \sqrt {170} \sqrt {-13 i + 1} \log \left (\left (i - 13\right ) \, \sqrt {170} \sqrt {-13 i + 1} - 340 \, x + 170 \, {\left (1024 \, x^{10} - 16640 \, x^{9} + 112000 \, x^{8} - 401440 \, x^{7} + 820340 \, x^{6} - 954733 \, x^{5} + 615255 \, x^{4} - 225810 \, x^{3} + 47250 \, x^{2} - 5265 \, x + 243\right )}^{\frac {1}{10}} + 340 i\right ) - \frac {1}{2} \, \log \left (-8 \, x + 4 \, {\left (1024 \, x^{10} - 16640 \, x^{9} + 112000 \, x^{8} - 401440 \, x^{7} + 820340 \, x^{6} - 954733 \, x^{5} + 615255 \, x^{4} - 225810 \, x^{3} + 47250 \, x^{2} - 5265 \, x + 243\right )}^{\frac {1}{10}} + 13\right ) \]
integrate(x^2/(x^2+1)/(1024*x^10-16640*x^9+112000*x^8-401440*x^7+820340*x^ 6-954733*x^5+615255*x^4-225810*x^3+47250*x^2-5265*x+243)^(1/10),x, algorit hm="fricas")
1/340*sqrt(170)*sqrt(13*I + 1)*log((I + 13)*sqrt(170)*sqrt(13*I + 1) - 340 *x + 170*(1024*x^10 - 16640*x^9 + 112000*x^8 - 401440*x^7 + 820340*x^6 - 9 54733*x^5 + 615255*x^4 - 225810*x^3 + 47250*x^2 - 5265*x + 243)^(1/10) - 3 40*I) - 1/340*sqrt(170)*sqrt(13*I + 1)*log(-(I + 13)*sqrt(170)*sqrt(13*I + 1) - 340*x + 170*(1024*x^10 - 16640*x^9 + 112000*x^8 - 401440*x^7 + 82034 0*x^6 - 954733*x^5 + 615255*x^4 - 225810*x^3 + 47250*x^2 - 5265*x + 243)^( 1/10) - 340*I) + 1/340*sqrt(170)*sqrt(-13*I + 1)*log(-(I - 13)*sqrt(170)*s qrt(-13*I + 1) - 340*x + 170*(1024*x^10 - 16640*x^9 + 112000*x^8 - 401440* x^7 + 820340*x^6 - 954733*x^5 + 615255*x^4 - 225810*x^3 + 47250*x^2 - 5265 *x + 243)^(1/10) + 340*I) - 1/340*sqrt(170)*sqrt(-13*I + 1)*log((I - 13)*s qrt(170)*sqrt(-13*I + 1) - 340*x + 170*(1024*x^10 - 16640*x^9 + 112000*x^8 - 401440*x^7 + 820340*x^6 - 954733*x^5 + 615255*x^4 - 225810*x^3 + 47250* x^2 - 5265*x + 243)^(1/10) + 340*I) - 1/2*log(-8*x + 4*(1024*x^10 - 16640* x^9 + 112000*x^8 - 401440*x^7 + 820340*x^6 - 954733*x^5 + 615255*x^4 - 225 810*x^3 + 47250*x^2 - 5265*x + 243)^(1/10) + 13)
Not integrable
Time = 2.83 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.18 \[ \int \frac {x^2}{\left (1+x^2\right ) \sqrt [10]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx=\int \frac {x^{2}}{\sqrt [10]{\left (x - 3\right )^{5} \left (4 x - 1\right )^{5}} \left (x^{2} + 1\right )}\, dx \]
integrate(x**2/(x**2+1)/(1024*x**10-16640*x**9+112000*x**8-401440*x**7+820 340*x**6-954733*x**5+615255*x**4-225810*x**3+47250*x**2-5265*x+243)**(1/10 ),x)
Not integrable
Time = 0.31 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.49 \[ \int \frac {x^2}{\left (1+x^2\right ) \sqrt [10]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx=\int { \frac {x^{2}}{{\left (1024 \, x^{10} - 16640 \, x^{9} + 112000 \, x^{8} - 401440 \, x^{7} + 820340 \, x^{6} - 954733 \, x^{5} + 615255 \, x^{4} - 225810 \, x^{3} + 47250 \, x^{2} - 5265 \, x + 243\right )}^{\frac {1}{10}} {\left (x^{2} + 1\right )}} \,d x } \]
integrate(x^2/(x^2+1)/(1024*x^10-16640*x^9+112000*x^8-401440*x^7+820340*x^ 6-954733*x^5+615255*x^4-225810*x^3+47250*x^2-5265*x+243)^(1/10),x, algorit hm="maxima")
integrate(x^2/((1024*x^10 - 16640*x^9 + 112000*x^8 - 401440*x^7 + 820340*x ^6 - 954733*x^5 + 615255*x^4 - 225810*x^3 + 47250*x^2 - 5265*x + 243)^(1/1 0)*(x^2 + 1)), x)
Not integrable
Time = 0.45 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.49 \[ \int \frac {x^2}{\left (1+x^2\right ) \sqrt [10]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx=\int { \frac {x^{2}}{{\left (1024 \, x^{10} - 16640 \, x^{9} + 112000 \, x^{8} - 401440 \, x^{7} + 820340 \, x^{6} - 954733 \, x^{5} + 615255 \, x^{4} - 225810 \, x^{3} + 47250 \, x^{2} - 5265 \, x + 243\right )}^{\frac {1}{10}} {\left (x^{2} + 1\right )}} \,d x } \]
integrate(x^2/(x^2+1)/(1024*x^10-16640*x^9+112000*x^8-401440*x^7+820340*x^ 6-954733*x^5+615255*x^4-225810*x^3+47250*x^2-5265*x+243)^(1/10),x, algorit hm="giac")
integrate(x^2/((1024*x^10 - 16640*x^9 + 112000*x^8 - 401440*x^7 + 820340*x ^6 - 954733*x^5 + 615255*x^4 - 225810*x^3 + 47250*x^2 - 5265*x + 243)^(1/1 0)*(x^2 + 1)), x)
Not integrable
Time = 5.77 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.49 \[ \int \frac {x^2}{\left (1+x^2\right ) \sqrt [10]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx=\int \frac {x^2}{\left (x^2+1\right )\,{\left (1024\,x^{10}-16640\,x^9+112000\,x^8-401440\,x^7+820340\,x^6-954733\,x^5+615255\,x^4-225810\,x^3+47250\,x^2-5265\,x+243\right )}^{1/10}} \,d x \]
int(x^2/((x^2 + 1)*(47250*x^2 - 5265*x - 225810*x^3 + 615255*x^4 - 954733* x^5 + 820340*x^6 - 401440*x^7 + 112000*x^8 - 16640*x^9 + 1024*x^10 + 243)^ (1/10)),x)