3.20.3 \(\int \frac {x^2}{(1+x^2) \sqrt [10]{243-5265 x+47250 x^2-225810 x^3+615255 x^4-954733 x^5+820340 x^6-401440 x^7+112000 x^8-16640 x^9+1024 x^{10}}} \, dx\)
Optimal. Leaf size=132 \[ \frac {\left (\left (4 x^2-13 x+3\right )^5\right )^{9/10} \left (32 \text {RootSum}\left [\text {$\#$1}^4-52 \text {$\#$1}^3+1174 \text {$\#$1}^2-6292 \text {$\#$1}+14641\& ,\frac {\text {$\#$1} \log \left (-\text {$\#$1}+4 \sqrt {4 x^2-13 x+3}-8 x+13\right )}{\text {$\#$1}^3-39 \text {$\#$1}^2+587 \text {$\#$1}-1573}\& \right ]-\frac {1}{2} \log \left (4 \sqrt {4 x^2-13 x+3}-8 x+13\right )\right )}{\left (4 x^2-13 x+3\right )^{9/2}} \]
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Rubi [A] time = 1.24, antiderivative size = 241, normalized size of antiderivative = 1.83,
number of steps used = 10, number of rules used = 8, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.123, Rules used
= {6688, 6720, 1079, 621, 206, 987, 1030, 204} \begin {gather*} -\frac {\sqrt {\frac {1}{85} \left (\sqrt {170}-1\right )} \sqrt {4 x^2-13 x+3} \tan ^{-1}\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 [10]{\left (4 x^2-13 x+3\right )^5}}-\frac {\sqrt {4 x^2-13 x+3} \tanh ^{-1}\left (\frac {13-8 x}{4 \sqrt {4 x^2-13 x+3}}\right )}{2 \sqrt [10]{\left (4 x^2-13 x+3\right )^5}}+\frac {\sqrt {\frac {1}{85} \left (1+\sqrt {170}\right )} \sqrt {4 x^2-13 x+3} \tanh ^{-1}\left (\frac {13-\left (1+\sqrt {170}\right ) x}{\sqrt {2 \left (1+\sqrt {170}\right )} \sqrt {4 x^2-13 x+3}}\right )}{2 \sqrt [10]{\left (4 x^2-13 x+3\right )^5}} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[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]
[Out]
-1/2*(Sqrt[(-1 + Sqrt[170])/85]*Sqrt[3 - 13*x + 4*x^2]*ArcTan[(13 - (1 - Sqrt[170])*x)/(Sqrt[2*(-1 + Sqrt[170]
)]*Sqrt[3 - 13*x + 4*x^2])])/((3 - 13*x + 4*x^2)^5)^(1/10) - (Sqrt[3 - 13*x + 4*x^2]*ArcTanh[(13 - 8*x)/(4*Sqr
t[3 - 13*x + 4*x^2])])/(2*((3 - 13*x + 4*x^2)^5)^(1/10)) + (Sqrt[(1 + Sqrt[170])/85]*Sqrt[3 - 13*x + 4*x^2]*Ar
cTanh[(13 - (1 + Sqrt[170])*x)/(Sqrt[2*(1 + Sqrt[170])]*Sqrt[3 - 13*x + 4*x^2])])/(2*((3 - 13*x + 4*x^2)^5)^(1
/10))
Rule 204
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])
Rule 206
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])
Rule 621
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]
Rule 987
Int[1/(((a_.) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> With[{q = Rt[(c*d - a*f)^
2 + a*c*e^2, 2]}, Dist[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] - Dist
[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)]
Rule 1030
Int[((g_) + (h_.)*(x_))/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[-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] /; F
reeQ[{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]
Rule 1079
Int[((A_.) + (C_.)*(x_)^2)/(((a_) + (c_.)*(x_)^2)*Sqrt[(d_.) + (e_.)*(x_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[
C/c, Int[1/Sqrt[d + e*x + f*x^2], x], x] + Dist[(A*c - a*C)/c, Int[1/((a + c*x^2)*Sqrt[d + e*x + f*x^2]), x],
x] /; FreeQ[{a, c, d, e, f, A, C}, x] && NeQ[e^2 - 4*d*f, 0]
Rule 6688
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]
Rule 6720
Int[(u_.)*((a_.)*(v_)^(m_.))^(p_), x_Symbol] :> Dist[(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]) &&
!(EqQ[v, x] && EqQ[m, 1])
Rubi steps
\begin {align*} \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 (1+x^2\right ) \sqrt [10]{\left (3-13 x+4 x^2\right )^5}} \, dx\\ &=\frac {\sqrt {3-13 x+4 x^2} \int \frac {x^2}{\left (1+x^2\right ) \sqrt {3-13 x+4 x^2}} \, dx}{\sqrt [10]{\left (3-13 x+4 x^2\right )^5}}\\ &=\frac {\sqrt {3-13 x+4 x^2} \int \frac {1}{\sqrt {3-13 x+4 x^2}} \, dx}{\sqrt [10]{\left (3-13 x+4 x^2\right )^5}}-\frac {\sqrt {3-13 x+4 x^2} \int \frac {1}{\left (1+x^2\right ) \sqrt {3-13 x+4 x^2}} \, dx}{\sqrt [10]{\left (3-13 x+4 x^2\right )^5}}\\ &=\frac {\left (2 \sqrt {3-13 x+4 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{16-x^2} \, dx,x,\frac {-13+8 x}{\sqrt {3-13 x+4 x^2}}\right )}{\sqrt [10]{\left (3-13 x+4 x^2\right )^5}}+\frac {\sqrt {3-13 x+4 x^2} \int \frac {-1-\sqrt {170}-13 x}{\left (1+x^2\right ) \sqrt {3-13 x+4 x^2}} \, dx}{2 \sqrt {170} \sqrt [10]{\left (3-13 x+4 x^2\right )^5}}-\frac {\sqrt {3-13 x+4 x^2} \int \frac {-1+\sqrt {170}-13 x}{\left (1+x^2\right ) \sqrt {3-13 x+4 x^2}} \, dx}{2 \sqrt {170} \sqrt [10]{\left (3-13 x+4 x^2\right )^5}}\\ &=-\frac {\sqrt {3-13 x+4 x^2} \tanh ^{-1}\left (\frac {13-8 x}{4 \sqrt {3-13 x+4 x^2}}\right )}{2 \sqrt [10]{\left (3-13 x+4 x^2\right )^5}}+\frac {\left (13 \left (1-\sqrt {170}\right ) \sqrt {3-13 x+4 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{26 \left (1-\sqrt {170}\right )-13 x^2} \, dx,x,\frac {-13+\left (1-\sqrt {170}\right ) x}{\sqrt {3-13 x+4 x^2}}\right )}{\sqrt {170} \sqrt [10]{\left (3-13 x+4 x^2\right )^5}}-\frac {\left (13 \left (1+\sqrt {170}\right ) \sqrt {3-13 x+4 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{26 \left (1+\sqrt {170}\right )-13 x^2} \, dx,x,\frac {-13+\left (1+\sqrt {170}\right ) x}{\sqrt {3-13 x+4 x^2}}\right )}{\sqrt {170} \sqrt [10]{\left (3-13 x+4 x^2\right )^5}}\\ &=-\frac {\sqrt {\frac {1}{85} \left (-1+\sqrt {170}\right )} \sqrt {3-13 x+4 x^2} \tan ^{-1}\left (\frac {13-\left (1-\sqrt {170}\right ) x}{\sqrt {2 \left (-1+\sqrt {170}\right )} \sqrt {3-13 x+4 x^2}}\right )}{2 \sqrt [10]{\left (3-13 x+4 x^2\right )^5}}-\frac {\sqrt {3-13 x+4 x^2} \tanh ^{-1}\left (\frac {13-8 x}{4 \sqrt {3-13 x+4 x^2}}\right )}{2 \sqrt [10]{\left (3-13 x+4 x^2\right )^5}}+\frac {\sqrt {\frac {1}{85} \left (1+\sqrt {170}\right )} \sqrt {3-13 x+4 x^2} \tanh ^{-1}\left (\frac {13-\left (1+\sqrt {170}\right ) x}{\sqrt {2 \left (1+\sqrt {170}\right )} \sqrt {3-13 x+4 x^2}}\right )}{2 \sqrt [10]{\left (3-13 x+4 x^2\right )^5}}\\ \end {align*}
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Mathematica [C] time = 0.14, size = 159, normalized size = 1.20 \begin {gather*} \frac {\left (\frac {1}{340}-\frac {i}{340}\right ) \sqrt {4 x^2-13 x+3} \left ((7+6 i) \sqrt {1-13 i} \tan ^{-1}\left (\frac {(6+13 i)-(13+8 i) x}{2 \sqrt {1-13 i} \sqrt {4 x^2-13 x+3}}\right )-(6+7 i) \sqrt {1+13 i} \tan ^{-1}\left (\frac {(13-8 i) x-(6-13 i)}{2 \sqrt {1+13 i} \sqrt {4 x^2-13 x+3}}\right )+(85+85 i) \tanh ^{-1}\left (\frac {8 x-13}{4 \sqrt {4 x^2-13 x+3}}\right )\right )}{\sqrt [10]{\left (4 x^2-13 x+3\right )^5}} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[x^2/((1 + x^2)*(243 - 5265*x + 47250*x^2 - 225810*x^3 + 615255*x^4 - 954733*x^5 + 820340*x^6 - 40144
0*x^7 + 112000*x^8 - 16640*x^9 + 1024*x^10)^(1/10)),x]
[Out]
((1/340 - I/340)*Sqrt[3 - 13*x + 4*x^2]*((7 + 6*I)*Sqrt[1 - 13*I]*ArcTan[((6 + 13*I) - (13 + 8*I)*x)/(2*Sqrt[1
- 13*I]*Sqrt[3 - 13*x + 4*x^2])] - (6 + 7*I)*Sqrt[1 + 13*I]*ArcTan[((-6 + 13*I) + (13 - 8*I)*x)/(2*Sqrt[1 + 1
3*I]*Sqrt[3 - 13*x + 4*x^2])] + (85 + 85*I)*ArcTanh[(-13 + 8*x)/(4*Sqrt[3 - 13*x + 4*x^2])]))/((3 - 13*x + 4*x
^2)^5)^(1/10)
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IntegrateAlgebraic [A] time = 19.97, size = 147, normalized size = 1.11 \begin {gather*} \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 )+\frac {1}{2} \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 )}{\left (3-13 x+4 x^2\right )^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[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]
[Out]
(((3 - 13*x + 4*x^2)^5)^(9/10)*(-1/2*Log[13 - 8*x + 4*Sqrt[3 - 13*x + 4*x^2]] + RootSum[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)^(9/2)
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fricas [B] time = 0.68, size = 1333, normalized size = 10.10
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
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+4
7250*x^2-5265*x+243)^(1/10),x, algorithm="fricas")
[Out]
-1/8840*170^(1/4)*(sqrt(170) + 1)*sqrt(-2*sqrt(170) + 340)*log(115600*x^2 + 340/13*170^(1/4)*(sqrt(170)*(13*x
+ 1) + 170)*sqrt(-2*sqrt(170) + 340) - 170*(1024*x^10 - 16640*x^9 + 112000*x^8 - 401440*x^7 + 820340*x^6 - 954
733*x^5 + 615255*x^4 - 225810*x^3 + 47250*x^2 - 5265*x + 243)^(1/10)*(170^(3/4)*sqrt(-2*sqrt(170) + 340) + 680
*x) + 28900*sqrt(170) + 28900*(1024*x^10 - 16640*x^9 + 112000*x^8 - 401440*x^7 + 820340*x^6 - 954733*x^5 + 615
255*x^4 - 225810*x^3 + 47250*x^2 - 5265*x + 243)^(1/5) + 115600) + 1/8840*170^(1/4)*(sqrt(170) + 1)*sqrt(-2*sq
rt(170) + 340)*log(115600*x^2 - 340/13*170^(1/4)*(sqrt(170)*(13*x + 1) + 170)*sqrt(-2*sqrt(170) + 340) + 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)*(170^(3/4)*sqrt(-2*sqrt(170) + 340) - 680*x) + 28900*sqrt(170) + 28900*(1024*x^10 - 1
6640*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)^(1/5) + 115600) + 1/170*170^(1/4)*sqrt(-2*sqrt(170) + 340)*arctan(1/52596872900*sqrt(1502800*x^2 - 340*17
0^(1/4)*(sqrt(170)*(13*x + 1) + 170)*sqrt(-2*sqrt(170) + 340) + 2210*(1024*x^10 - 16640*x^9 + 112000*x^8 - 401
440*x^7 + 820340*x^6 - 954733*x^5 + 615255*x^4 - 225810*x^3 + 47250*x^2 - 5265*x + 243)^(1/10)*(170^(3/4)*sqrt
(-2*sqrt(170) + 340) - 680*x) + 375700*sqrt(170) + 375700*(1024*x^10 - 16640*x^9 + 112000*x^8 - 401440*x^7 + 8
20340*x^6 - 954733*x^5 + 615255*x^4 - 225810*x^3 + 47250*x^2 - 5265*x + 243)^(1/5) + 1502800)*(52*sqrt(170)*(2
33*sqrt(170)*sqrt(13) - 17850*sqrt(13)) + (170^(3/4)*(128*sqrt(170)*sqrt(13) - 17617*sqrt(13)) + 8*170^(1/4)*(
233*sqrt(170)*sqrt(13) - 17850*sqrt(13)))*sqrt(-2*sqrt(170) + 340) + 141440*sqrt(170)*sqrt(13) - 7567040*sqrt(
13)) + 4/915365*sqrt(170)*(sqrt(170)*(233*x + 1391) - 17850*x - 4420) + 1/10769*sqrt(170)*(128*x + 1365) + 1/1
1899745*(170^(3/4)*(sqrt(170)*(128*x + 1365) - 17617*x - 3029) + 8*170^(1/4)*(sqrt(170)*(233*x + 1391) - 17850
*x - 4420))*sqrt(-2*sqrt(170) + 340) - 1/23799490*(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)*(52*sqrt(170)*(233*sqrt(170) - 178
50) + (170^(3/4)*(128*sqrt(170) - 17617) + 8*170^(1/4)*(233*sqrt(170) - 17850))*sqrt(-2*sqrt(170) + 340) + 141
440*sqrt(170) - 7567040) - 6848/10769*x - 3029/10769) + 1/170*170^(1/4)*sqrt(-2*sqrt(170) + 340)*arctan(-1/525
96872900*sqrt(1502800*x^2 + 340*170^(1/4)*(sqrt(170)*(13*x + 1) + 170)*sqrt(-2*sqrt(170) + 340) - 2210*(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 - 52
65*x + 243)^(1/10)*(170^(3/4)*sqrt(-2*sqrt(170) + 340) + 680*x) + 375700*sqrt(170) + 375700*(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)
^(1/5) + 1502800)*(52*sqrt(170)*(233*sqrt(170)*sqrt(13) - 17850*sqrt(13)) - (170^(3/4)*(128*sqrt(170)*sqrt(13)
- 17617*sqrt(13)) + 8*170^(1/4)*(233*sqrt(170)*sqrt(13) - 17850*sqrt(13)))*sqrt(-2*sqrt(170) + 340) + 141440*
sqrt(170)*sqrt(13) - 7567040*sqrt(13)) - 4/915365*sqrt(170)*(sqrt(170)*(233*x + 1391) - 17850*x - 4420) - 1/10
769*sqrt(170)*(128*x + 1365) + 1/11899745*(170^(3/4)*(sqrt(170)*(128*x + 1365) - 17617*x - 3029) + 8*170^(1/4)
*(sqrt(170)*(233*x + 1391) - 17850*x - 4420))*sqrt(-2*sqrt(170) + 340) + 1/23799490*(1024*x^10 - 16640*x^9 + 1
12000*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)*
(52*sqrt(170)*(233*sqrt(170) - 17850) - (170^(3/4)*(128*sqrt(170) - 17617) + 8*170^(1/4)*(233*sqrt(170) - 1785
0))*sqrt(-2*sqrt(170) + 340) + 141440*sqrt(170) - 7567040) + 6848/10769*x + 3029/10769) - 1/2*log(-8*x + 4*(10
24*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) + 13)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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+4
7250*x^2-5265*x+243)^(1/10),x, algorithm="giac")
[Out]
integrate(x^2/((1024*x^10 - 16640*x^9 + 112000*x^8 - 401440*x^7 + 820340*x^6 - 954733*x^5 + 615255*x^4 - 22581
0*x^3 + 47250*x^2 - 5265*x + 243)^(1/10)*(x^2 + 1)), x)
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2}}{\left (x^{2}+1\right ) \left (1024 x^{10}-16640 x^{9}+112000 x^{8}-401440 x^{7}+820340 x^{6}-954733 x^{5}+615255 x^{4}-225810 x^{3}+47250 x^{2}-5265 x +243\right )^{\frac {1}{10}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(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)
[Out]
int(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)
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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+4
7250*x^2-5265*x+243)^(1/10),x, algorithm="maxima")
[Out]
integrate(x^2/((1024*x^10 - 16640*x^9 + 112000*x^8 - 401440*x^7 + 820340*x^6 - 954733*x^5 + 615255*x^4 - 22581
0*x^3 + 47250*x^2 - 5265*x + 243)^(1/10)*(x^2 + 1)), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \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 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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 + 1120
00*x^8 - 16640*x^9 + 1024*x^10 + 243)^(1/10)),x)
[Out]
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 + 1120
00*x^8 - 16640*x^9 + 1024*x^10 + 243)^(1/10)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2}}{\sqrt [10]{\left (x - 3\right )^{5} \left (4 x - 1\right )^{5}} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
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**4-225
810*x**3+47250*x**2-5265*x+243)**(1/10),x)
[Out]
Integral(x**2/(((x - 3)**5*(4*x - 1)**5)**(1/10)*(x**2 + 1)), x)
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