3.30.53 \(\int \frac {x^3}{\sqrt [3]{x^2+x^4} (-1+x^6)} \, dx\)

Optimal. Leaf size=356 \[ -\frac {1}{6} \log \left (\sqrt [3]{x^4+x^2}-x\right )-\frac {1}{6} \log \left (\sqrt [3]{x^4+x^2}+x\right )+\frac {\log \left (2^{2/3} \sqrt [3]{x^4+x^2}-2 x\right )}{12 \sqrt [3]{2}}+\frac {\log \left (2^{2/3} \sqrt [3]{x^4+x^2}+2 x\right )}{12 \sqrt [3]{2}}+\frac {1}{12} \log \left (x^2-\sqrt [3]{x^4+x^2} x+\left (x^4+x^2\right )^{2/3}\right )+\frac {1}{12} \log \left (x^2+\sqrt [3]{x^4+x^2} x+\left (x^4+x^2\right )^{2/3}\right )-\frac {\log \left (-2 x^2+2^{2/3} \sqrt [3]{x^4+x^2} x-\sqrt [3]{2} \left (x^4+x^2\right )^{2/3}\right )}{24 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} \sqrt [3]{x^4+x^2} x+\sqrt [3]{2} \left (x^4+x^2\right )^{2/3}\right )}{24 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{x^2+2 \left (x^4+x^2\right )^{2/3}}\right )}{2 \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{x^2+\sqrt [3]{2} \left (x^4+x^2\right )^{2/3}}\right )}{4 \sqrt [3]{2} \sqrt {3}} \]

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Rubi [C]  time = 1.58, antiderivative size = 152, normalized size of antiderivative = 0.43, number of steps used = 43, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2056, 6725, 959, 466, 465, 510} \begin {gather*} -\frac {\sqrt [3]{x^2+1} x^4 F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};-x^2,-\sqrt [3]{-1} x^2\right )}{10 \sqrt [3]{x^4+x^2}}-\frac {\sqrt [3]{x^2+1} x^4 F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};-x^2,(-1)^{2/3} x^2\right )}{10 \sqrt [3]{x^4+x^2}}-\frac {\sqrt [3]{x^2+1} x^4 F_1\left (\frac {5}{3};1,\frac {1}{3};\frac {8}{3};x^2,-x^2\right )}{10 \sqrt [3]{x^4+x^2}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[x^3/((x^2 + x^4)^(1/3)*(-1 + x^6)),x]

[Out]

-1/10*(x^4*(1 + x^2)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -x^2, -((-1)^(1/3)*x^2)])/(x^2 + x^4)^(1/3) - (x^4*(1 +
x^2)^(1/3)*AppellF1[5/3, 1/3, 1, 8/3, -x^2, (-1)^(2/3)*x^2])/(10*(x^2 + x^4)^(1/3)) - (x^4*(1 + x^2)^(1/3)*App
ellF1[5/3, 1, 1/3, 8/3, x^2, -x^2])/(10*(x^2 + x^4)^(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 959

Int[(((g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_)^2)^(p_))/((d_) + (e_.)*(x_)), x_Symbol] :> Dist[(d*(g*x)^n)/x^n, In
t[(x^n*(a + c*x^2)^p)/(d^2 - e^2*x^2), x], x] - Dist[(e*(g*x)^n)/x^n, Int[(x^(n + 1)*(a + c*x^2)^p)/(d^2 - e^2
*x^2), x], x] /; FreeQ[{a, c, d, e, g, n, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IntegerQ[p] &&  !IntegersQ[n, 2
*p]

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 6725

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

Rubi steps

\begin {align*} \int \frac {x^3}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\sqrt [3]{1+x^2} \left (-1+x^6\right )} \, dx}{\sqrt [3]{x^2+x^4}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \left (-\frac {x^{7/3}}{2 \sqrt [3]{1+x^2} \left (1-x^3\right )}-\frac {x^{7/3}}{2 \sqrt [3]{1+x^2} \left (1+x^3\right )}\right ) \, dx}{\sqrt [3]{x^2+x^4}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\sqrt [3]{1+x^2} \left (1-x^3\right )} \, dx}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\sqrt [3]{1+x^2} \left (1+x^3\right )} \, dx}{2 \sqrt [3]{x^2+x^4}}\\ &=-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \left (-\frac {x^{7/3}}{3 (-1-x) \sqrt [3]{1+x^2}}-\frac {x^{7/3}}{3 \left (-1+\sqrt [3]{-1} x\right ) \sqrt [3]{1+x^2}}-\frac {x^{7/3}}{3 \left (-1-(-1)^{2/3} x\right ) \sqrt [3]{1+x^2}}\right ) \, dx}{2 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \left (\frac {x^{7/3}}{3 (1-x) \sqrt [3]{1+x^2}}+\frac {x^{7/3}}{3 \left (1+\sqrt [3]{-1} x\right ) \sqrt [3]{1+x^2}}+\frac {x^{7/3}}{3 \left (1-(-1)^{2/3} x\right ) \sqrt [3]{1+x^2}}\right ) \, dx}{2 \sqrt [3]{x^2+x^4}}\\ &=\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{(-1-x) \sqrt [3]{1+x^2}} \, dx}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{(1-x) \sqrt [3]{1+x^2}} \, dx}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\left (-1+\sqrt [3]{-1} x\right ) \sqrt [3]{1+x^2}} \, dx}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\left (1+\sqrt [3]{-1} x\right ) \sqrt [3]{1+x^2}} \, dx}{6 \sqrt [3]{x^2+x^4}}+\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\left (-1-(-1)^{2/3} x\right ) \sqrt [3]{1+x^2}} \, dx}{6 \sqrt [3]{x^2+x^4}}-\frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\left (1-(-1)^{2/3} x\right ) \sqrt [3]{1+x^2}} \, dx}{6 \sqrt [3]{x^2+x^4}}\\ &=-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\left (1-x^2\right ) \sqrt [3]{1+x^2}} \, dx}{6 \sqrt [3]{x^2+x^4}}-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\sqrt [3]{1+x^2} \left (1+\sqrt [3]{-1} x^2\right )} \, dx}{6 \sqrt [3]{x^2+x^4}}-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \int \frac {x^{7/3}}{\sqrt [3]{1+x^2} \left (1-(-1)^{2/3} x^2\right )} \, dx}{6 \sqrt [3]{x^2+x^4}}\\ &=-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^9}{\left (1-x^6\right ) \sqrt [3]{1+x^6}} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^9}{\sqrt [3]{1+x^6} \left (1+\sqrt [3]{-1} x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^9}{\sqrt [3]{1+x^6} \left (1-(-1)^{2/3} x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{2 \sqrt [3]{x^2+x^4}}\\ &=-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\left (1-x^3\right ) \sqrt [3]{1+x^3}} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{x^2+x^4}}-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{1+x^3} \left (1+\sqrt [3]{-1} x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{x^2+x^4}}-2 \frac {\left (x^{2/3} \sqrt [3]{1+x^2}\right ) \operatorname {Subst}\left (\int \frac {x^4}{\sqrt [3]{1+x^3} \left (1-(-1)^{2/3} x^3\right )} \, dx,x,x^{2/3}\right )}{4 \sqrt [3]{x^2+x^4}}\\ &=-\frac {x^4 \sqrt [3]{1+x^2} F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};-x^2,-\sqrt [3]{-1} x^2\right )}{10 \sqrt [3]{x^2+x^4}}-\frac {x^4 \sqrt [3]{1+x^2} F_1\left (\frac {5}{3};\frac {1}{3},1;\frac {8}{3};-x^2,(-1)^{2/3} x^2\right )}{10 \sqrt [3]{x^2+x^4}}-\frac {x^4 \sqrt [3]{1+x^2} F_1\left (\frac {5}{3};1,\frac {1}{3};\frac {8}{3};x^2,-x^2\right )}{10 \sqrt [3]{x^2+x^4}}\\ \end {align*}

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Mathematica [F]  time = 1.34, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{\sqrt [3]{x^2+x^4} \left (-1+x^6\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x^3/((x^2 + x^4)^(1/3)*(-1 + x^6)),x]

[Out]

Integrate[x^3/((x^2 + x^4)^(1/3)*(-1 + x^6)), x]

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IntegrateAlgebraic [A]  time = 0.93, size = 360, normalized size = 1.01 \begin {gather*} \frac {\tan ^{-1}\left (\frac {\sqrt {3} x^2}{x^2+\sqrt [3]{2} \left (x^2+x^4\right )^{2/3}}\right )}{4 \sqrt [3]{2} \sqrt {3}}+\frac {\tan ^{-1}\left (\frac {\frac {x^2}{\sqrt {3}}+\frac {2 \left (x^2+x^4\right )^{2/3}}{\sqrt {3}}}{x^2}\right )}{2 \sqrt {3}}-\frac {1}{6} \log \left (-x+\sqrt [3]{x^2+x^4}\right )-\frac {1}{6} \log \left (x+\sqrt [3]{x^2+x^4}\right )+\frac {\log \left (-2 x+2^{2/3} \sqrt [3]{x^2+x^4}\right )}{12 \sqrt [3]{2}}+\frac {\log \left (2 x+2^{2/3} \sqrt [3]{x^2+x^4}\right )}{12 \sqrt [3]{2}}+\frac {1}{12} \log \left (x^2-x \sqrt [3]{x^2+x^4}+\left (x^2+x^4\right )^{2/3}\right )+\frac {1}{12} \log \left (x^2+x \sqrt [3]{x^2+x^4}+\left (x^2+x^4\right )^{2/3}\right )-\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{x^2+x^4}-\sqrt [3]{2} \left (x^2+x^4\right )^{2/3}\right )}{24 \sqrt [3]{2}}-\frac {\log \left (2 x^2+2^{2/3} x \sqrt [3]{x^2+x^4}+\sqrt [3]{2} \left (x^2+x^4\right )^{2/3}\right )}{24 \sqrt [3]{2}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^3/((x^2 + x^4)^(1/3)*(-1 + x^6)),x]

[Out]

ArcTan[(Sqrt[3]*x^2)/(x^2 + 2^(1/3)*(x^2 + x^4)^(2/3))]/(4*2^(1/3)*Sqrt[3]) + ArcTan[(x^2/Sqrt[3] + (2*(x^2 +
x^4)^(2/3))/Sqrt[3])/x^2]/(2*Sqrt[3]) - Log[-x + (x^2 + x^4)^(1/3)]/6 - Log[x + (x^2 + x^4)^(1/3)]/6 + Log[-2*
x + 2^(2/3)*(x^2 + x^4)^(1/3)]/(12*2^(1/3)) + Log[2*x + 2^(2/3)*(x^2 + x^4)^(1/3)]/(12*2^(1/3)) + Log[x^2 - x*
(x^2 + x^4)^(1/3) + (x^2 + x^4)^(2/3)]/12 + Log[x^2 + x*(x^2 + x^4)^(1/3) + (x^2 + x^4)^(2/3)]/12 - Log[-2*x^2
+ 2^(2/3)*x*(x^2 + x^4)^(1/3) - 2^(1/3)*(x^2 + x^4)^(2/3)]/(24*2^(1/3)) - Log[2*x^2 + 2^(2/3)*x*(x^2 + x^4)^(
1/3) + 2^(1/3)*(x^2 + x^4)^(2/3)]/(24*2^(1/3))

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fricas [A]  time = 3.23, size = 418, normalized size = 1.17 \begin {gather*} -\frac {1}{72} \, \sqrt {6} 2^{\frac {1}{6}} \arctan \left (\frac {2^{\frac {1}{6}} {\left (24 \, \sqrt {6} 2^{\frac {2}{3}} {\left (x^{8} + 2 \, x^{6} - 6 \, x^{4} + 2 \, x^{2} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + \sqrt {6} 2^{\frac {1}{3}} {\left (x^{12} - 42 \, x^{10} - 417 \, x^{8} - 812 \, x^{6} - 417 \, x^{4} - 42 \, x^{2} + 1\right )} - 12 \, \sqrt {6} {\left (x^{10} + 33 \, x^{8} + 110 \, x^{6} + 110 \, x^{4} + 33 \, x^{2} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (x^{12} + 102 \, x^{10} + 447 \, x^{8} + 628 \, x^{6} + 447 \, x^{4} + 102 \, x^{2} + 1\right )}}\right ) - \frac {1}{144} \cdot 2^{\frac {2}{3}} \log \left (\frac {12 \cdot 2^{\frac {2}{3}} {\left (x^{4} + 4 \, x^{2} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (x^{8} + 32 \, x^{6} + 78 \, x^{4} + 32 \, x^{2} + 1\right )} + 6 \, {\left (x^{6} + 11 \, x^{4} + 11 \, x^{2} + 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}}{x^{8} - 4 \, x^{6} + 6 \, x^{4} - 4 \, x^{2} + 1}\right ) + \frac {1}{72} \cdot 2^{\frac {2}{3}} \log \left (-\frac {2^{\frac {2}{3}} {\left (x^{4} - 2 \, x^{2} + 1\right )} - 6 \cdot 2^{\frac {1}{3}} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )} + 12 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}}}{x^{4} - 2 \, x^{2} + 1}\right ) - \frac {1}{6} \, \sqrt {3} \arctan \left (\frac {\sqrt {3} {\left (x^{2} + 1\right )} + 2 \, \sqrt {3} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}}{3 \, {\left (x^{2} + 1\right )}}\right ) - \frac {1}{12} \, \log \left (\frac {x^{4} + x^{2} - 3 \, {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}} {\left (x^{2} + 1\right )} + 3 \, {\left (x^{4} + x^{2}\right )}^{\frac {2}{3}} + 1}{x^{4} + x^{2} + 1}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^4+x^2)^(1/3)/(x^6-1),x, algorithm="fricas")

[Out]

-1/72*sqrt(6)*2^(1/6)*arctan(1/6*2^(1/6)*(24*sqrt(6)*2^(2/3)*(x^8 + 2*x^6 - 6*x^4 + 2*x^2 + 1)*(x^4 + x^2)^(2/
3) + sqrt(6)*2^(1/3)*(x^12 - 42*x^10 - 417*x^8 - 812*x^6 - 417*x^4 - 42*x^2 + 1) - 12*sqrt(6)*(x^10 + 33*x^8 +
110*x^6 + 110*x^4 + 33*x^2 + 1)*(x^4 + x^2)^(1/3))/(x^12 + 102*x^10 + 447*x^8 + 628*x^6 + 447*x^4 + 102*x^2 +
1)) - 1/144*2^(2/3)*log((12*2^(2/3)*(x^4 + 4*x^2 + 1)*(x^4 + x^2)^(2/3) + 2^(1/3)*(x^8 + 32*x^6 + 78*x^4 + 32
*x^2 + 1) + 6*(x^6 + 11*x^4 + 11*x^2 + 1)*(x^4 + x^2)^(1/3))/(x^8 - 4*x^6 + 6*x^4 - 4*x^2 + 1)) + 1/72*2^(2/3)
*log(-(2^(2/3)*(x^4 - 2*x^2 + 1) - 6*2^(1/3)*(x^4 + x^2)^(1/3)*(x^2 + 1) + 12*(x^4 + x^2)^(2/3))/(x^4 - 2*x^2
+ 1)) - 1/6*sqrt(3)*arctan(1/3*(sqrt(3)*(x^2 + 1) + 2*sqrt(3)*(x^4 + x^2)^(1/3))/(x^2 + 1)) - 1/12*log((x^4 +
x^2 - 3*(x^4 + x^2)^(1/3)*(x^2 + 1) + 3*(x^4 + x^2)^(2/3) + 1)/(x^4 + x^2 + 1))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{{\left (x^{6} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^4+x^2)^(1/3)/(x^6-1),x, algorithm="giac")

[Out]

integrate(x^3/((x^6 - 1)*(x^4 + x^2)^(1/3)), x)

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maple [C]  time = 59.64, size = 2401, normalized size = 6.74

method	result	size
trager	\(\text {Expression too large to display}\)	\(2401\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(x^4+x^2)^(1/3)/(x^6-1),x,method=_RETURNVERBOSE)

[Out]

1/4*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*ln((-1051752*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3
-4)+36*_Z^2)*RootOf(_Z^3-4)^3*x^4+7172976*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4
)^2*x^4+4469946*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^3*x^2-30485148*RootOf(Root
Of(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+34713150*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(
_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*(x^4+x^2)^(2/3)-5785525*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^2*x^2-20043774*(x^4+x
^2)^(1/3)*RootOf(_Z^3-4)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^2+1095575*RootOf(_Z^3-4)*x^4-7
471850*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^4-1051752*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z
^3-4)+36*_Z^2)*RootOf(_Z^3-4)^3+7172976*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^
2-5785525*RootOf(_Z^3-4)^2*(x^4+x^2)^(1/3)-20043774*(x^4+x^2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4
)+36*_Z^2)*RootOf(_Z^3-4)+9378122*RootOf(_Z^3-4)*x^2-63959036*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_
Z^2)*x^2+9779584*(x^4+x^2)^(2/3)+1095575*RootOf(_Z^3-4)-7471850*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36
*_Z^2))/(-1+x)^2/(1+x)^2)+1/24*RootOf(_Z^3-4)*ln(-(3586488*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2
)*RootOf(_Z^3-4)^3*x^4-18931536*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^2*x^4-15
242574*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^3*x^2+80459028*RootOf(RootOf(_Z^3-4
)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^2*x^2+104139450*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+
36*_Z^2)*RootOf(_Z^3-4)^2*(x^4+x^2)^(2/3)-17356575*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^2*x^2-44008128*(x^4+x^2)^(1/
3)*RootOf(_Z^3-4)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^2+6126917*RootOf(_Z^3-4)*x^4-32341374
*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^4+3586488*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+
36*_Z^2)*RootOf(_Z^3-4)^3-18931536*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^2-173
56575*RootOf(_Z^3-4)^2*(x^4+x^2)^(1/3)-44008128*(x^4+x^2)^(1/3)*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36
*_Z^2)*RootOf(_Z^3-4)[phone]*RootOf(_Z^3-4)*x^2-115166844*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^
2)*x^2+40087548*(x^4+x^2)^(2/3)+6126917*RootOf(_Z^3-4)-32341374*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36
*_Z^2))/(-1+x)^2/(1+x)^2)-1/6*ln((545052*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)
^4*x^4-2316471*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^4*x^2-1675418*RootOf(Root
Of(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*x^4-8496*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^2*RootOf(Ro
otOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^2+545052*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*
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-9707954*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*x^2-8496*RootOf(RootOf(_Z^3-4)^
2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*(x^4+x^2)^(1/3)-2481800*x^4+4083236*(x^4+x^2)^(1/3)*x^2-167541
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36*(x^4+x^2)^(1/3)-2481800)/(x^2+x+1)/(x^2-x+1))+1/6*ln(-(-348396*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+
36*_Z^2)^2*RootOf(_Z^3-4)^4*x^4+1480683*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^
4*x^2-2279546*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*x^4-6124854*(x^4+x^2)^(1/3
)*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^2-348396*RootOf(RootOf(_Z^3-4)^2+6*_
Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^4+6133350*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*Root
Of(_Z^3-4)^2*(x^4+x^2)^(2/3)-1068656*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*x^2
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4*(x^4+x^2)^(1/3)*x^2-2279546*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2+4083236*(x
^4+x^2)^(2/3)-6154864*x^2+5664*(x^4+x^2)^(1/3)-2878888)/(x^2+x+1)/(x^2-x+1))+1/4*ln(-(-348396*RootOf(RootOf(_Z
^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^4*x^4+1480683*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4
)+36*_Z^2)^2*RootOf(_Z^3-4)^4*x^2-2279546*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^
2*x^4-6124854*(x^4+x^2)^(1/3)*RootOf(_Z^3-4)^2*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*x^2-348396
*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)^2*RootOf(_Z^3-4)^4+6133350*RootOf(RootOf(_Z^3-4)^2+6*_Z*
RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*(x^4+x^2)^(2/3)-1068656*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+3
6*_Z^2)*RootOf(_Z^3-4)^2*x^2-6124854*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2*(x^
4+x^2)^(1/3)-2878888*x^4+5664*(x^4+x^2)^(1/3)*x^2-2279546*RootOf(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)
*RootOf(_Z^3-4)^2+4083236*(x^4+x^2)^(2/3)-6154864*x^2+5664*(x^4+x^2)^(1/3)-2878888)/(x^2+x+1)/(x^2-x+1))*RootO
f(RootOf(_Z^3-4)^2+6*_Z*RootOf(_Z^3-4)+36*_Z^2)*RootOf(_Z^3-4)^2

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{{\left (x^{6} - 1\right )} {\left (x^{4} + x^{2}\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(x^4+x^2)^(1/3)/(x^6-1),x, algorithm="maxima")

[Out]

integrate(x^3/((x^6 - 1)*(x^4 + x^2)^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3}{{\left (x^4+x^2\right )}^{1/3}\,\left (x^6-1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((x^2 + x^4)^(1/3)*(x^6 - 1)),x)

[Out]

int(x^3/((x^2 + x^4)^(1/3)*(x^6 - 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(x**4+x**2)**(1/3)/(x**6-1),x)

[Out]

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

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