3.15.66 \(\int \frac {x^2 (-2+x^4)}{\sqrt [3]{-x+x^5} (-1+x^4+x^8)} \, dx\)

Optimal. Leaf size=103 \[ \frac {1}{4} \log \left (\sqrt [3]{x^5-x}+x^3\right )+\frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{x^5-x}}{\sqrt [3]{x^5-x}-2 x^3}\right )-\frac {1}{8} \log \left (x^6+\left (x^5-x\right )^{2/3}-\sqrt [3]{x^5-x} x^3\right ) \]

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Rubi [C]  time = 0.77, antiderivative size = 117, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2056, 6728, 466, 465, 511, 510} \begin {gather*} \frac {3 \sqrt [3]{1-x^4} x^3 F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^4,-\frac {2 x^4}{1-\sqrt {5}}\right )}{8 \sqrt [3]{x^5-x}}+\frac {3 \sqrt [3]{1-x^4} x^3 F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^4,-\frac {2 x^4}{1+\sqrt {5}}\right )}{8 \sqrt [3]{x^5-x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(x^2*(-2 + x^4))/((-x + x^5)^(1/3)*(-1 + x^4 + x^8)),x]

[Out]

(3*x^3*(1 - x^4)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^4, (-2*x^4)/(1 - Sqrt[5])])/(8*(-x + x^5)^(1/3)) + (3*x^3*
(1 - x^4)^(1/3)*AppellF1[2/3, 1/3, 1, 5/3, x^4, (-2*x^4)/(1 + Sqrt[5])])/(8*(-x + x^5)^(1/3))

Rule 465

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \frac {x^{5/3} \left (-2+x^4\right )}{\sqrt [3]{-1+x^4} \left (-1+x^4+x^8\right )} \, dx}{\sqrt [3]{-x+x^5}}\\ &=\frac {\left (\sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \left (\frac {\left (1-\sqrt {5}\right ) x^{5/3}}{\sqrt [3]{-1+x^4} \left (1-\sqrt {5}+2 x^4\right )}+\frac {\left (1+\sqrt {5}\right ) x^{5/3}}{\sqrt [3]{-1+x^4} \left (1+\sqrt {5}+2 x^4\right )}\right ) \, dx}{\sqrt [3]{-x+x^5}}\\ &=\frac {\left (\left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \frac {x^{5/3}}{\sqrt [3]{-1+x^4} \left (1-\sqrt {5}+2 x^4\right )} \, dx}{\sqrt [3]{-x+x^5}}+\frac {\left (\left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \int \frac {x^{5/3}}{\sqrt [3]{-1+x^4} \left (1+\sqrt {5}+2 x^4\right )} \, dx}{\sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{-1+x^{12}} \left (1-\sqrt {5}+2 x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x^7}{\sqrt [3]{-1+x^{12}} \left (1+\sqrt {5}+2 x^{12}\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^3} \left (1-\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{-1+x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{-1+x^3} \left (1+\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}\\ &=\frac {\left (3 \left (1-\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^3} \left (1-\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}+\frac {\left (3 \left (1+\sqrt {5}\right ) \sqrt [3]{x} \sqrt [3]{1-x^4}\right ) \operatorname {Subst}\left (\int \frac {x}{\sqrt [3]{1-x^3} \left (1+\sqrt {5}+2 x^3\right )} \, dx,x,x^{4/3}\right )}{4 \sqrt [3]{-x+x^5}}\\ &=\frac {3 x^3 \sqrt [3]{1-x^4} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^4,-\frac {2 x^4}{1-\sqrt {5}}\right )}{8 \sqrt [3]{-x+x^5}}+\frac {3 x^3 \sqrt [3]{1-x^4} F_1\left (\frac {2}{3};\frac {1}{3},1;\frac {5}{3};x^4,-\frac {2 x^4}{1+\sqrt {5}}\right )}{8 \sqrt [3]{-x+x^5}}\\ \end {align*}

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Mathematica [F]  time = 0.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^2 \left (-2+x^4\right )}{\sqrt [3]{-x+x^5} \left (-1+x^4+x^8\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(x^2*(-2 + x^4))/((-x + x^5)^(1/3)*(-1 + x^4 + x^8)),x]

[Out]

Integrate[(x^2*(-2 + x^4))/((-x + x^5)^(1/3)*(-1 + x^4 + x^8)), x]

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IntegrateAlgebraic [A]  time = 0.51, size = 103, normalized size = 1.00 \begin {gather*} \frac {1}{4} \sqrt {3} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{-x+x^5}}{-2 x^3+\sqrt [3]{-x+x^5}}\right )+\frac {1}{4} \log \left (x^3+\sqrt [3]{-x+x^5}\right )-\frac {1}{8} \log \left (x^6-x^3 \sqrt [3]{-x+x^5}+\left (-x+x^5\right )^{2/3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(x^2*(-2 + x^4))/((-x + x^5)^(1/3)*(-1 + x^4 + x^8)),x]

[Out]

(Sqrt[3]*ArcTan[(Sqrt[3]*(-x + x^5)^(1/3))/(-2*x^3 + (-x + x^5)^(1/3))])/4 + Log[x^3 + (-x + x^5)^(1/3)]/4 - L
og[x^6 - x^3*(-x + x^5)^(1/3) + (-x + x^5)^(2/3)]/8

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fricas [A]  time = 4.01, size = 112, normalized size = 1.09 \begin {gather*} -\frac {1}{4} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{8} + 2 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {1}{3}} x^{5} + 4 \, \sqrt {3} {\left (x^{5} - x\right )}^{\frac {2}{3}} x^{2}}{x^{8} - 8 \, x^{4} + 8}\right ) + \frac {1}{8} \, \log \left (\frac {x^{8} + 3 \, {\left (x^{5} - x\right )}^{\frac {1}{3}} x^{5} + x^{4} + 3 \, {\left (x^{5} - x\right )}^{\frac {2}{3}} x^{2} - 1}{x^{8} + x^{4} - 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^4-2)/(x^5-x)^(1/3)/(x^8+x^4-1),x, algorithm="fricas")

[Out]

-1/4*sqrt(3)*arctan((sqrt(3)*x^8 + 2*sqrt(3)*(x^5 - x)^(1/3)*x^5 + 4*sqrt(3)*(x^5 - x)^(2/3)*x^2)/(x^8 - 8*x^4
+ 8)) + 1/8*log((x^8 + 3*(x^5 - x)^(1/3)*x^5 + x^4 + 3*(x^5 - x)^(2/3)*x^2 - 1)/(x^8 + x^4 - 1))

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^4-2)/(x^5-x)^(1/3)/(x^8+x^4-1),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Unab
le to convert to real 1/4 Error: Bad Argument ValueUnable to convert to real 1/4 Error: Bad Argument Value

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maple [C]  time = 53.16, size = 626, normalized size = 6.08

method	result	size
trager	\(-\frac {\ln \left (-\frac {1027936778409280454041232991842640 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2} x^{8}+4529039623257700492643576695162024 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x^{8}+610884417613243258599747967858176 x^{8}-6458608904506033300021952070673408 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {1}{3}} x^{5}-17543454351518386415637043060781056 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2} x^{4}-25262880761481365661024699935681 \left (x^{5}-x \right )^{\frac {1}{3}} x^{5}+6458608904506033300021952070673408 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {2}{3}} x^{2}-6572417063730249524797944388667308 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x^{4}+25262880761481365661024699935681 \left (x^{5}-x \right )^{\frac {2}{3}} x^{2}-575090408768717286416168985366486 x^{4}+17543454351518386415637043060781056 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2}+6572417063730249524797944388667308 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+575090408768717286416168985366486}{x^{8}+x^{4}-1}\right )}{4}-\ln \left (-\frac {1027936778409280454041232991842640 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2} x^{8}+4529039623257700492643576695162024 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x^{8}+610884417613243258599747967858176 x^{8}-6458608904506033300021952070673408 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {1}{3}} x^{5}-17543454351518386415637043060781056 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2} x^{4}-25262880761481365661024699935681 \left (x^{5}-x \right )^{\frac {1}{3}} x^{5}+6458608904506033300021952070673408 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {2}{3}} x^{2}-6572417063730249524797944388667308 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x^{4}+25262880761481365661024699935681 \left (x^{5}-x \right )^{\frac {2}{3}} x^{2}-575090408768717286416168985366486 x^{4}+17543454351518386415637043060781056 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2}+6572417063730249524797944388667308 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+575090408768717286416168985366486}{x^{8}+x^{4}-1}\right ) \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+\RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \ln \left (-\frac {1027936778409280454041232991842640 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2} x^{8}-4015071234053060265622960199240704 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x^{8}-457129439550601836183569143942165 x^{8}+6458608904506033300021952070673408 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {1}{3}} x^{5}-17543454351518386415637043060781056 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2} x^{4}+1589389345365026959344463317732671 \left (x^{5}-x \right )^{\frac {1}{3}} x^{5}-6458608904506033300021952070673408 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) \left (x^{5}-x \right )^{\frac {2}{3}} x^{2}-2199310112028943683020577141723220 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right ) x^{4}-1589389345365026959344463317732671 \left (x^{5}-x \right )^{\frac {2}{3}} x^{2}-28452039806054056193998079498475 x^{4}+17543454351518386415637043060781056 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )^{2}+2199310112028943683020577141723220 \RootOf \left (16 \textit {\_Z}^{2}+4 \textit {\_Z} +1\right )+28452039806054056193998079498475}{x^{8}+x^{4}-1}\right )\)	\(626\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x^4-2)/(x^5-x)^(1/3)/(x^8+x^4-1),x,method=_RETURNVERBOSE)

[Out]

-1/4*ln(-(1027936778409280454041232991842640*RootOf(16*_Z^2+4*_Z+1)^2*x^8+4529039623257700492643576695162024*R
ootOf(16*_Z^2+4*_Z+1)*x^8+610884417613243258599747967858176*x^8-6458608904506033300021952070673408*RootOf(16*_
Z^2+4*_Z+1)*(x^5-x)^(1/3)*x^5-17543454351518386415637043060781056*RootOf(16*_Z^2+4*_Z+1)^2*x^4-252628807614813
65661024699935681*(x^5-x)^(1/3)*x^5+6458608904506033300021952070673408*RootOf(16*_Z^2+4*_Z+1)*(x^5-x)^(2/3)*x^
2-6572417063730249524797944388667308*RootOf(16*_Z^2+4*_Z+1)*x^4+25262880761481365661024699935681*(x^5-x)^(2/3)
*x^2-575090408768717286416168985366486*x^4+17543454351518386415637043060781056*RootOf(16*_Z^2+4*_Z+1)^2+657241
7063730249524797944388667308*RootOf(16*_Z^2+4*_Z+1)+575090408768717286416168985366486)/(x^8+x^4-1))-ln(-(10279
36778409280454041232991842640*RootOf(16*_Z^2+4*_Z+1)^2*x^8+4529039623257700492643576695162024*RootOf(16*_Z^2+4
*_Z+1)*x^8+610884417613243258599747967858176*x^8-6458608904506033300021952070673408*RootOf(16*_Z^2+4*_Z+1)*(x^
5-x)^(1/3)*x^5-17543454351518386415637043060781056*RootOf(16*_Z^2+4*_Z+1)^2*x^4-252628807614813656610246999356
81*(x^5-x)^(1/3)*x^5+6458608904506033300021952070673408*RootOf(16*_Z^2+4*_Z+1)*(x^5-x)^(2/3)*x^2-6572417063730
249524797944388667308*RootOf(16*_Z^2+4*_Z+1)*x^4+25262880761481365661024699935681*(x^5-x)^(2/3)*x^2-5750904087
68717286416168985366486*x^4+17543454351518386415637043060781056*RootOf(16*_Z^2+4*_Z+1)^2+657241706373024952479
7944388667308*RootOf(16*_Z^2+4*_Z+1)+575090408768717286416168985366486)/(x^8+x^4-1))*RootOf(16*_Z^2+4*_Z+1)+Ro
otOf(16*_Z^2+4*_Z+1)*ln(-(1027936778409280454041232991842640*RootOf(16*_Z^2+4*_Z+1)^2*x^8-40150712340530602656
22960199240704*RootOf(16*_Z^2+4*_Z+1)*x^8-457129439550601836183569143942165*x^8+645860890450603330002195207067
3408*RootOf(16*_Z^2+4*_Z+1)*(x^5-x)^(1/3)*x^5-17543454351518386415637043060781056*RootOf(16*_Z^2+4*_Z+1)^2*x^4
+1589389345365026959344463317732671*(x^5-x)^(1/3)*x^5-6458608904506033300021952070673408*RootOf(16*_Z^2+4*_Z+1
)*(x^5-x)^(2/3)*x^2-2199310112028943683020577141723220*RootOf(16*_Z^2+4*_Z+1)*x^4-1589389345365026959344463317
732671*(x^5-x)^(2/3)*x^2-28452039806054056193998079498475*x^4+17543454351518386415637043060781056*RootOf(16*_Z
^2+4*_Z+1)^2+2199310112028943683020577141723220*RootOf(16*_Z^2+4*_Z+1)+28452039806054056193998079498475)/(x^8+
x^4-1))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - 2\right )} x^{2}}{{\left (x^{8} + x^{4} - 1\right )} {\left (x^{5} - x\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^4-2)/(x^5-x)^(1/3)/(x^8+x^4-1),x, algorithm="maxima")

[Out]

integrate((x^4 - 2)*x^2/((x^8 + x^4 - 1)*(x^5 - x)^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2\,\left (x^4-2\right )}{{\left (x^5-x\right )}^{1/3}\,\left (x^8+x^4-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*(x^4 - 2))/((x^5 - x)^(1/3)*(x^4 + x^8 - 1)),x)

[Out]

int((x^2*(x^4 - 2))/((x^5 - x)^(1/3)*(x^4 + x^8 - 1)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(x**4-2)/(x**5-x)**(1/3)/(x**8+x**4-1),x)

[Out]

Timed out

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