3.19.93 \(\int \frac {(-2+x^6) \sqrt [3]{2+x^6}}{x^2 (2+2 x^3+x^6)} \, dx\)

Optimal. Leaf size=131 \[ \frac {\sqrt [3]{x^6+2}}{x}-\frac {1}{3} \sqrt [3]{2} \log \left (2^{2/3} \sqrt [3]{x^6+2}+2 x\right )+\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt {3} x}{2^{2/3} \sqrt [3]{x^6+2}-x}\right )}{\sqrt {3}}+\frac {\log \left (2^{2/3} \sqrt [3]{x^6+2} x-\sqrt [3]{2} \left (x^6+2\right )^{2/3}-2 x^2\right )}{3\ 2^{2/3}} \]

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Rubi [C]  time = 1.18, antiderivative size = 195, normalized size of antiderivative = 1.49, number of steps used = 29, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6728, 364, 1562, 465, 429, 510} \begin {gather*} \frac {i x^5 F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};-\frac {i x^6}{2},-\frac {x^6}{2}\right )}{5\ 2^{2/3}}-\frac {i x^5 F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {i x^6}{2},-\frac {x^6}{2}\right )}{5\ 2^{2/3}}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) x^2 F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};-\frac {i x^6}{2},-\frac {x^6}{2}\right )}{2^{2/3}}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) x^2 F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {i x^6}{2},-\frac {x^6}{2}\right )}{2^{2/3}}+\frac {\sqrt [3]{2} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};-\frac {x^6}{2}\right )}{x} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[((-2 + x^6)*(2 + x^6)^(1/3))/(x^2*(2 + 2*x^3 + x^6)),x]

[Out]

((1/2 - I/2)*x^2*AppellF1[1/3, 1, -1/3, 4/3, (-1/2*I)*x^6, -1/2*x^6])/2^(2/3) + ((1/2 + I/2)*x^2*AppellF1[1/3,
1, -1/3, 4/3, (I/2)*x^6, -1/2*x^6])/2^(2/3) + ((I/5)*x^5*AppellF1[5/6, 1, -1/3, 11/6, (-1/2*I)*x^6, -1/2*x^6]
)/2^(2/3) - ((I/5)*x^5*AppellF1[5/6, 1, -1/3, 11/6, (I/2)*x^6, -1/2*x^6])/2^(2/3) + (2^(1/3)*Hypergeometric2F1
[-1/3, -1/6, 5/6, -1/2*x^6])/x

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 429

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

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 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 1562

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_)*((a_) + (c_.)*(x_)^(n2_))^(p_), x_Symbol] :> Dist[(f*x)^m
/x^m, Int[ExpandIntegrand[x^m*(a + c*x^(2*n))^p, (d/(d^2 - e^2*x^(2*n)) - (e*x^n)/(d^2 - e^2*x^(2*n)))^(-q), x
], x], x] /; FreeQ[{a, c, d, e, f, m, n, p}, x] && EqQ[n2, 2*n] &&  !IntegerQ[p] && ILtQ[q, 0]

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 {\left (-2+x^6\right ) \sqrt [3]{2+x^6}}{x^2 \left (2+2 x^3+x^6\right )} \, dx &=\int \left (-\frac {\sqrt [3]{2+x^6}}{x^2}+\frac {2 x \left (1+x^3\right ) \sqrt [3]{2+x^6}}{2+2 x^3+x^6}\right ) \, dx\\ &=2 \int \frac {x \left (1+x^3\right ) \sqrt [3]{2+x^6}}{2+2 x^3+x^6} \, dx-\int \frac {\sqrt [3]{2+x^6}}{x^2} \, dx\\ &=\frac {\sqrt [3]{2} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};-\frac {x^6}{2}\right )}{x}+2 \int \left (\frac {x \sqrt [3]{2+x^6}}{2+2 x^3+x^6}+\frac {x^4 \sqrt [3]{2+x^6}}{2+2 x^3+x^6}\right ) \, dx\\ &=\frac {\sqrt [3]{2} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};-\frac {x^6}{2}\right )}{x}+2 \int \frac {x \sqrt [3]{2+x^6}}{2+2 x^3+x^6} \, dx+2 \int \frac {x^4 \sqrt [3]{2+x^6}}{2+2 x^3+x^6} \, dx\\ &=\frac {\sqrt [3]{2} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};-\frac {x^6}{2}\right )}{x}+2 \int \left (\frac {i x \sqrt [3]{2+x^6}}{(-2+2 i)-2 x^3}+\frac {i x \sqrt [3]{2+x^6}}{(2+2 i)+2 x^3}\right ) \, dx+2 \int \left (-\frac {(1+i) x \sqrt [3]{2+x^6}}{(-2+2 i)-2 x^3}+\frac {(1-i) x \sqrt [3]{2+x^6}}{(2+2 i)+2 x^3}\right ) \, dx\\ &=\frac {\sqrt [3]{2} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};-\frac {x^6}{2}\right )}{x}+(-2-2 i) \int \frac {x \sqrt [3]{2+x^6}}{(-2+2 i)-2 x^3} \, dx+2 i \int \frac {x \sqrt [3]{2+x^6}}{(-2+2 i)-2 x^3} \, dx+2 i \int \frac {x \sqrt [3]{2+x^6}}{(2+2 i)+2 x^3} \, dx+(2-2 i) \int \frac {x \sqrt [3]{2+x^6}}{(2+2 i)+2 x^3} \, dx\\ &=\frac {\sqrt [3]{2} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};-\frac {x^6}{2}\right )}{x}+(-2-2 i) \int \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) x \sqrt [3]{2+x^6}}{2 i+x^6}-\frac {x^4 \sqrt [3]{2+x^6}}{2 \left (2 i+x^6\right )}\right ) \, dx+2 i \int \left (-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) x \sqrt [3]{2+x^6}}{-2 i+x^6}+\frac {x^4 \sqrt [3]{2+x^6}}{2 \left (-2 i+x^6\right )}\right ) \, dx+2 i \int \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) x \sqrt [3]{2+x^6}}{2 i+x^6}-\frac {x^4 \sqrt [3]{2+x^6}}{2 \left (2 i+x^6\right )}\right ) \, dx+(2-2 i) \int \left (-\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) x \sqrt [3]{2+x^6}}{-2 i+x^6}+\frac {x^4 \sqrt [3]{2+x^6}}{2 \left (-2 i+x^6\right )}\right ) \, dx\\ &=\frac {\sqrt [3]{2} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};-\frac {x^6}{2}\right )}{x}+i \int \frac {x^4 \sqrt [3]{2+x^6}}{-2 i+x^6} \, dx-i \int \frac {x^4 \sqrt [3]{2+x^6}}{2 i+x^6} \, dx+(1-i) \int \frac {x \sqrt [3]{2+x^6}}{-2 i+x^6} \, dx+(1-i) \int \frac {x^4 \sqrt [3]{2+x^6}}{-2 i+x^6} \, dx+(1+i) \int \frac {x \sqrt [3]{2+x^6}}{2 i+x^6} \, dx+(1+i) \int \frac {x^4 \sqrt [3]{2+x^6}}{2 i+x^6} \, dx-2 \int \frac {x \sqrt [3]{2+x^6}}{-2 i+x^6} \, dx-2 \int \frac {x \sqrt [3]{2+x^6}}{2 i+x^6} \, dx\\ &=\frac {i x^5 F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};-\frac {i x^6}{2},-\frac {x^6}{2}\right )}{5\ 2^{2/3}}-\frac {i x^5 F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {i x^6}{2},-\frac {x^6}{2}\right )}{5\ 2^{2/3}}+\frac {\sqrt [3]{2} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};-\frac {x^6}{2}\right )}{x}+\left (\frac {1}{2}-\frac {i}{2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{2+x^3}}{-2 i+x^3} \, dx,x,x^2\right )+\left (\frac {1}{2}+\frac {i}{2}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{2+x^3}}{2 i+x^3} \, dx,x,x^2\right )-\operatorname {Subst}\left (\int \frac {\sqrt [3]{2+x^3}}{-2 i+x^3} \, dx,x,x^2\right )-\operatorname {Subst}\left (\int \frac {\sqrt [3]{2+x^3}}{2 i+x^3} \, dx,x,x^2\right )\\ &=\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) x^2 F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};-\frac {i x^6}{2},-\frac {x^6}{2}\right )}{2^{2/3}}+\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) x^2 F_1\left (\frac {1}{3};1,-\frac {1}{3};\frac {4}{3};\frac {i x^6}{2},-\frac {x^6}{2}\right )}{2^{2/3}}+\frac {i x^5 F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};-\frac {i x^6}{2},-\frac {x^6}{2}\right )}{5\ 2^{2/3}}-\frac {i x^5 F_1\left (\frac {5}{6};1,-\frac {1}{3};\frac {11}{6};\frac {i x^6}{2},-\frac {x^6}{2}\right )}{5\ 2^{2/3}}+\frac {\sqrt [3]{2} \, _2F_1\left (-\frac {1}{3},-\frac {1}{6};\frac {5}{6};-\frac {x^6}{2}\right )}{x}\\ \end {align*}

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Mathematica [F]  time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-2+x^6\right ) \sqrt [3]{2+x^6}}{x^2 \left (2+2 x^3+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((-2 + x^6)*(2 + x^6)^(1/3))/(x^2*(2 + 2*x^3 + x^6)),x]

[Out]

Integrate[((-2 + x^6)*(2 + x^6)^(1/3))/(x^2*(2 + 2*x^3 + x^6)), x]

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IntegrateAlgebraic [A]  time = 1.09, size = 131, normalized size = 1.00 \begin {gather*} \frac {\sqrt [3]{2+x^6}}{x}+\frac {\sqrt [3]{2} \tan ^{-1}\left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{2+x^6}}\right )}{\sqrt {3}}-\frac {1}{3} \sqrt [3]{2} \log \left (2 x+2^{2/3} \sqrt [3]{2+x^6}\right )+\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{2+x^6}-\sqrt [3]{2} \left (2+x^6\right )^{2/3}\right )}{3\ 2^{2/3}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[((-2 + x^6)*(2 + x^6)^(1/3))/(x^2*(2 + 2*x^3 + x^6)),x]

[Out]

(2 + x^6)^(1/3)/x + (2^(1/3)*ArcTan[(Sqrt[3]*x)/(-x + 2^(2/3)*(2 + x^6)^(1/3))])/Sqrt[3] - (2^(1/3)*Log[2*x +
2^(2/3)*(2 + x^6)^(1/3)])/3 + Log[-2*x^2 + 2^(2/3)*x*(2 + x^6)^(1/3) - 2^(1/3)*(2 + x^6)^(2/3)]/(3*2^(2/3))

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fricas [B]  time = 175.69, size = 334, normalized size = 2.55 \begin {gather*} \frac {2 \, \sqrt {3} \left (-2\right )^{\frac {1}{3}} x \arctan \left (\frac {6 \, \sqrt {3} \left (-2\right )^{\frac {2}{3}} {\left (x^{14} - 14 \, x^{11} + 8 \, x^{8} - 28 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{6} + 2\right )}^{\frac {1}{3}} + 6 \, \sqrt {3} \left (-2\right )^{\frac {1}{3}} {\left (x^{13} - 2 \, x^{10} - 4 \, x^{7} - 4 \, x^{4} + 4 \, x\right )} {\left (x^{6} + 2\right )}^{\frac {2}{3}} + \sqrt {3} {\left (x^{18} - 30 \, x^{15} + 54 \, x^{12} - 112 \, x^{9} + 108 \, x^{6} - 120 \, x^{3} + 8\right )}}{3 \, {\left (x^{18} + 6 \, x^{15} - 90 \, x^{12} + 32 \, x^{9} - 180 \, x^{6} + 24 \, x^{3} + 8\right )}}\right ) + 2 \, \left (-2\right )^{\frac {1}{3}} x \log \left (\frac {6 \, \left (-2\right )^{\frac {1}{3}} {\left (x^{6} + 2\right )}^{\frac {1}{3}} x^{2} - \left (-2\right )^{\frac {2}{3}} {\left (x^{6} + 2 \, x^{3} + 2\right )} - 6 \, {\left (x^{6} + 2\right )}^{\frac {2}{3}} x}{x^{6} + 2 \, x^{3} + 2}\right ) - \left (-2\right )^{\frac {1}{3}} x \log \left (-\frac {3 \, \left (-2\right )^{\frac {2}{3}} {\left (x^{7} - 4 \, x^{4} + 2 \, x\right )} {\left (x^{6} + 2\right )}^{\frac {2}{3}} + \left (-2\right )^{\frac {1}{3}} {\left (x^{12} - 14 \, x^{9} + 8 \, x^{6} - 28 \, x^{3} + 4\right )} - 12 \, {\left (x^{8} - x^{5} + 2 \, x^{2}\right )} {\left (x^{6} + 2\right )}^{\frac {1}{3}}}{x^{12} + 4 \, x^{9} + 8 \, x^{6} + 8 \, x^{3} + 4}\right ) + 18 \, {\left (x^{6} + 2\right )}^{\frac {1}{3}}}{18 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-2)*(x^6+2)^(1/3)/x^2/(x^6+2*x^3+2),x, algorithm="fricas")

[Out]

1/18*(2*sqrt(3)*(-2)^(1/3)*x*arctan(1/3*(6*sqrt(3)*(-2)^(2/3)*(x^14 - 14*x^11 + 8*x^8 - 28*x^5 + 4*x^2)*(x^6 +
2)^(1/3) + 6*sqrt(3)*(-2)^(1/3)*(x^13 - 2*x^10 - 4*x^7 - 4*x^4 + 4*x)*(x^6 + 2)^(2/3) + sqrt(3)*(x^18 - 30*x^
15 + 54*x^12 - 112*x^9 + 108*x^6 - 120*x^3 + 8))/(x^18 + 6*x^15 - 90*x^12 + 32*x^9 - 180*x^6 + 24*x^3 + 8)) +
2*(-2)^(1/3)*x*log((6*(-2)^(1/3)*(x^6 + 2)^(1/3)*x^2 - (-2)^(2/3)*(x^6 + 2*x^3 + 2) - 6*(x^6 + 2)^(2/3)*x)/(x^
6 + 2*x^3 + 2)) - (-2)^(1/3)*x*log(-(3*(-2)^(2/3)*(x^7 - 4*x^4 + 2*x)*(x^6 + 2)^(2/3) + (-2)^(1/3)*(x^12 - 14*
x^9 + 8*x^6 - 28*x^3 + 4) - 12*(x^8 - x^5 + 2*x^2)*(x^6 + 2)^(1/3))/(x^12 + 4*x^9 + 8*x^6 + 8*x^3 + 4)) + 18*(
x^6 + 2)^(1/3))/x

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 2\right )}^{\frac {1}{3}} {\left (x^{6} - 2\right )}}{{\left (x^{6} + 2 \, x^{3} + 2\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-2)*(x^6+2)^(1/3)/x^2/(x^6+2*x^3+2),x, algorithm="giac")

[Out]

integrate((x^6 + 2)^(1/3)*(x^6 - 2)/((x^6 + 2*x^3 + 2)*x^2), x)

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maple [C]  time = 145.59, size = 1516, normalized size = 11.57

method	result	size
trager	\(\text {Expression too large to display}\)	\(1516\)
risch	\(\text {Expression too large to display}\)	\(1600\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^6-2)*(x^6+2)^(1/3)/x^2/(x^6+2*x^3+2),x,method=_RETURNVERBOSE)

[Out]

(x^6+2)^(1/3)/x+2*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*ln((4164520586334*RootOf(_Z^3+2)^4*Root
Of(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^6+37852911926640*RootOf(_Z^3+2)^3*RootOf(RootOf(_Z^3+2)^2+6
*_Z*RootOf(_Z^3+2)+36*_Z^2)^2*x^6-12493561759002*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*RootOf(_
Z^3+2)^4*x^3-113558735779920*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)^2*RootOf(_Z^3+2)^3*x^3+69408
6764389*RootOf(_Z^3+2)^2*x^6+6308818654440*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*RootOf(_Z^3+2)
*x^6+4412671686090*RootOf(_Z^3+2)^2*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*(x^6+2)^(2/3)*x+83290
41172668*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*RootOf(_Z^3+2)^4+75705823853280*RootOf(RootOf(_Z
^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)^2*RootOf(_Z^3+2)^3+2776347057556*x^3*RootOf(_Z^3+2)^2+25235274617760*Root
Of(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*RootOf(_Z^3+2)*x^3-9126524228622*RootOf(_Z^3+2)*(x^6+2)^(1/3)
*x^2-8825343372180*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*(x^6+2)^(1/3)*x^2-9126524228622*(x^6+2
)^(2/3)*x+1388173528778*RootOf(_Z^3+2)^2+12617637308880*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*R
ootOf(_Z^3+2))/(x^6+2*x^3+2))-1/3*ln((2144298068106*RootOf(_Z^3+2)^4*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+
2)+36*_Z^2)*x^6+37852911926640*RootOf(_Z^3+2)^3*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)^2*x^6-643
2894204318*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*RootOf(_Z^3+2)^4*x^3-113558735779920*RootOf(Ro
otOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)^2*RootOf(_Z^3+2)^3*x^3-1072149034053*RootOf(_Z^3+2)^2*x^6-18926455
963320*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*RootOf(_Z^3+2)*x^6-4412671686090*RootOf(_Z^3+2)^2*
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Z*RootOf(_Z^3+2)+36*_Z^2)*RootOf(_Z^3+2)^4+75705823853280*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)
^2*RootOf(_Z^3+2)^3+714766022702*x^3*RootOf(_Z^3+2)^2+12617637308880*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+
2)+36*_Z^2)*RootOf(_Z^3+2)*x^3-7655633666592*RootOf(_Z^3+2)*(x^6+2)^(1/3)*x^2+8825343372180*RootOf(RootOf(_Z^3
+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*(x^6+2)^(1/3)*x^2-7655633666592*(x^6+2)^(2/3)*x-2144298068106*RootOf(_Z^3+2
)^2-37852911926640*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*RootOf(_Z^3+2))/(x^6+2*x^3+2))*RootOf(
_Z^3+2)-2*ln((2144298068106*RootOf(_Z^3+2)^4*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*x^6+37852911
926640*RootOf(_Z^3+2)^3*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)^2*x^6-6432894204318*RootOf(RootOf
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Of(_Z^3+2)+36*_Z^2)^2*RootOf(_Z^3+2)^3*x^3-1072149034053*RootOf(_Z^3+2)^2*x^6-18926455963320*RootOf(RootOf(_Z^
3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*RootOf(_Z^3+2)*x^6-4412671686090*RootOf(_Z^3+2)^2*RootOf(RootOf(_Z^3+2)^2+
6*_Z*RootOf(_Z^3+2)+36*_Z^2)*(x^6+2)^(2/3)*x+4288596136212*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2
)*RootOf(_Z^3+2)^4+75705823853280*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)^2*RootOf(_Z^3+2)^3+7147
66022702*x^3*RootOf(_Z^3+2)^2+12617637308880*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*RootOf(_Z^3+
2)*x^3-7655633666592*RootOf(_Z^3+2)*(x^6+2)^(1/3)*x^2+8825343372180*RootOf(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2
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f(RootOf(_Z^3+2)^2+6*_Z*RootOf(_Z^3+2)+36*_Z^2)*RootOf(_Z^3+2))/(x^6+2*x^3+2))*RootOf(RootOf(_Z^3+2)^2+6*_Z*Ro
otOf(_Z^3+2)+36*_Z^2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{6} + 2\right )}^{\frac {1}{3}} {\left (x^{6} - 2\right )}}{{\left (x^{6} + 2 \, x^{3} + 2\right )} x^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^6-2)*(x^6+2)^(1/3)/x^2/(x^6+2*x^3+2),x, algorithm="maxima")

[Out]

integrate((x^6 + 2)^(1/3)*(x^6 - 2)/((x^6 + 2*x^3 + 2)*x^2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^6-2\right )\,{\left (x^6+2\right )}^{1/3}}{x^2\,\left (x^6+2\,x^3+2\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^6 - 2)*(x^6 + 2)^(1/3))/(x^2*(2*x^3 + x^6 + 2)),x)

[Out]

int(((x^6 - 2)*(x^6 + 2)^(1/3))/(x^2*(2*x^3 + x^6 + 2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{6} - 2\right ) \sqrt [3]{x^{6} + 2}}{x^{2} \left (x^{6} + 2 x^{3} + 2\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**6-2)*(x**6+2)**(1/3)/x**2/(x**6+2*x**3+2),x)

[Out]

Integral((x**6 - 2)*(x**6 + 2)**(1/3)/(x**2*(x**6 + 2*x**3 + 2)), x)

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