3.18.78 \(\int \frac {\sqrt [3]{-x+x^3} (8-10 x^2+x^4)}{x^4 (4-2 x^2+x^4)} \, dx\)
Optimal. Leaf size=120 \[ \frac {3 \left (4 x^2-1\right ) \sqrt [3]{x^3-x}}{4 x^3}-\frac {1}{8} \text {RootSum}\left [4 \text {$\#$1}^6-6 \text {$\#$1}^3+3\& ,\frac {-4 \text {$\#$1}^3 \log \left (\sqrt [3]{x^3-x}-\text {$\#$1} x\right )+4 \text {$\#$1}^3 \log (x)+9 \log \left (\sqrt [3]{x^3-x}-\text {$\#$1} x\right )-9 \log (x)}{4 \text {$\#$1}^5-3 \text {$\#$1}^2}\& \right ] \]
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Rubi [C] time = 1.90, antiderivative size = 427, normalized size of antiderivative = 3.56,
number of steps used = 14, number of rules used = 7, integrand size = 37, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used
= {2056, 6728, 264, 466, 465, 511, 510} \begin {gather*} \frac {\left (\sqrt {3}+6 i\right ) \sqrt [3]{x^3-x} \left (\left (3 \sqrt {3} x^2+\sqrt {3}+3 i\right ) x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {3} x^2}{\left (i-\sqrt {3}\right ) \left (1-x^2\right )}\right )-3 \left (-\sqrt {3} x^2+\sqrt {3}+3 i\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {\sqrt {3} x^2}{\left (i-\sqrt {3}\right ) \left (1-x^2\right )}\right )+2 \left (1-x^2\right ) \left (2 \left (\sqrt {3}+i\right )-3 \left (-\sqrt {3}+i\right ) x^2\right )\right )}{64 x^3 \left (1-x^2\right )}+\frac {\left (-\sqrt {3}+6 i\right ) \sqrt [3]{x^3-x} \left (\left (-3 \sqrt {3} x^2-\sqrt {3}+3 i\right ) x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {\sqrt {3} x^2}{\left (i+\sqrt {3}\right ) \left (1-x^2\right )}\right )-3 \left (\sqrt {3} x^2-\sqrt {3}+3 i\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {\sqrt {3} x^2}{\left (i+\sqrt {3}\right ) \left (1-x^2\right )}\right )+2 \left (1-x^2\right ) \left (2 \left (-\sqrt {3}+i\right )-3 \left (\sqrt {3}+i\right ) x^2\right )\right )}{64 x^3 \left (1-x^2\right )}-\frac {3 \sqrt [3]{x^3-x} \left (1-x^2\right )}{8 x^3} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Int[((-x + x^3)^(1/3)*(8 - 10*x^2 + x^4))/(x^4*(4 - 2*x^2 + x^4)),x]
[Out]
(-3*(1 - x^2)*(-x + x^3)^(1/3))/(8*x^3) + ((6*I + Sqrt[3])*(-x + x^3)^(1/3)*(2*(1 - x^2)*(2*(I + Sqrt[3]) - 3*
(I - Sqrt[3])*x^2) + x^2*(3*I + Sqrt[3] + 3*Sqrt[3]*x^2)*Hypergeometric2F1[2/3, 1, 5/3, (Sqrt[3]*x^2)/((I - Sq
rt[3])*(1 - x^2))] - 3*x^2*(3*I + Sqrt[3] - Sqrt[3]*x^2)*Hypergeometric2F1[2/3, 2, 5/3, (Sqrt[3]*x^2)/((I - Sq
rt[3])*(1 - x^2))]))/(64*x^3*(1 - x^2)) + ((6*I - Sqrt[3])*(-x + x^3)^(1/3)*(2*(1 - x^2)*(2*(I - Sqrt[3]) - 3*
(I + Sqrt[3])*x^2) + x^2*(3*I - Sqrt[3] - 3*Sqrt[3]*x^2)*Hypergeometric2F1[2/3, 1, 5/3, -((Sqrt[3]*x^2)/((I +
Sqrt[3])*(1 - x^2)))] - 3*x^2*(3*I - Sqrt[3] + Sqrt[3]*x^2)*Hypergeometric2F1[2/3, 2, 5/3, -((Sqrt[3]*x^2)/((I
+ Sqrt[3])*(1 - x^2)))]))/(64*x^3*(1 - x^2))
Rule 264
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
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 {\sqrt [3]{-x+x^3} \left (8-10 x^2+x^4\right )}{x^4 \left (4-2 x^2+x^4\right )} \, dx &=\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{-1+x^2} \left (8-10 x^2+x^4\right )}{x^{11/3} \left (4-2 x^2+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=\frac {\sqrt [3]{-x+x^3} \int \left (\frac {\sqrt [3]{-1+x^2}}{x^{11/3}}+\frac {4 \left (1-2 x^2\right ) \sqrt [3]{-1+x^2}}{x^{11/3} \left (4-2 x^2+x^4\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=\frac {\sqrt [3]{-x+x^3} \int \frac {\sqrt [3]{-1+x^2}}{x^{11/3}} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}+\frac {\left (4 \sqrt [3]{-x+x^3}\right ) \int \frac {\left (1-2 x^2\right ) \sqrt [3]{-1+x^2}}{x^{11/3} \left (4-2 x^2+x^4\right )} \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (4 \sqrt [3]{-x+x^3}\right ) \int \left (\frac {\left (-2+\frac {i}{\sqrt {3}}\right ) \sqrt [3]{-1+x^2}}{x^{11/3} \left (-2-2 i \sqrt {3}+2 x^2\right )}+\frac {\left (-2-\frac {i}{\sqrt {3}}\right ) \sqrt [3]{-1+x^2}}{x^{11/3} \left (-2+2 i \sqrt {3}+2 x^2\right )}\right ) \, dx}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (4 \left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \int \frac {\sqrt [3]{-1+x^2}}{x^{11/3} \left (-2-2 i \sqrt {3}+2 x^2\right )} \, dx}{3 \sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (4 \left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \int \frac {\sqrt [3]{-1+x^2}}{x^{11/3} \left (-2+2 i \sqrt {3}+2 x^2\right )} \, dx}{3 \sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (4 \left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x^6}}{x^9 \left (-2-2 i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (4 \left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x^6}}{x^9 \left (-2+2 i \sqrt {3}+2 x^6\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (2 \left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{x^5 \left (-2-2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}-\frac {\left (2 \left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{-1+x^3}}{x^5 \left (-2+2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{-1+x^2}}\\ &=-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (2 \left (-6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-x^3}}{x^5 \left (-2-2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^2}}-\frac {\left (2 \left (6+i \sqrt {3}\right ) \sqrt [3]{-x+x^3}\right ) \operatorname {Subst}\left (\int \frac {\sqrt [3]{1-x^3}}{x^5 \left (-2+2 i \sqrt {3}+2 x^3\right )} \, dx,x,x^{2/3}\right )}{\sqrt [3]{x} \sqrt [3]{1-x^2}}\\ &=-\frac {3 \left (1-x^2\right ) \sqrt [3]{-x+x^3}}{8 x^3}+\frac {\left (6 i+\sqrt {3}\right ) \sqrt [3]{-x+x^3} \left (2 \left (1-x^2\right ) \left (2 \left (i+\sqrt {3}\right )-3 \left (i-\sqrt {3}\right ) x^2\right )+x^2 \left (3 i+\sqrt {3}+3 \sqrt {3} x^2\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {3} x^2}{\left (i-\sqrt {3}\right ) \left (1-x^2\right )}\right )-3 x^2 \left (3 i+\sqrt {3}-\sqrt {3} x^2\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {\sqrt {3} x^2}{\left (i-\sqrt {3}\right ) \left (1-x^2\right )}\right )\right )}{64 x^3 \left (1-x^2\right )}+\frac {\left (6 i-\sqrt {3}\right ) \sqrt [3]{-x+x^3} \left (2 \left (1-x^2\right ) \left (2 \left (i-\sqrt {3}\right )-3 \left (i+\sqrt {3}\right ) x^2\right )+x^2 \left (3 i-\sqrt {3}-3 \sqrt {3} x^2\right ) \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};-\frac {\sqrt {3} x^2}{\left (i+\sqrt {3}\right ) \left (1-x^2\right )}\right )-3 x^2 \left (3 i-\sqrt {3}+\sqrt {3} x^2\right ) \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};-\frac {\sqrt {3} x^2}{\left (i+\sqrt {3}\right ) \left (1-x^2\right )}\right )\right )}{64 x^3 \left (1-x^2\right )}\\ \end {align*}
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Mathematica [C] time = 0.57, size = 356, normalized size = 2.97 \begin {gather*} \frac {-\left (\sqrt {3}+6 i\right ) \left (\left (3 \sqrt {3} x^2+\sqrt {3}+3 i\right ) x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {3} x^2}{\left (-i+\sqrt {3}\right ) \left (x^2-1\right )}\right )-3 \left (-\sqrt {3} x^2+\sqrt {3}+3 i\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {\sqrt {3} x^2}{\left (-i+\sqrt {3}\right ) \left (x^2-1\right )}\right )-2 \left (x^2-1\right ) \left (3 \left (\sqrt {3}-i\right ) x^2+2 \left (\sqrt {3}+i\right )\right )\right )-\left (-\sqrt {3}+6 i\right ) \left (-\left (\left (3 \sqrt {3} x^2+\sqrt {3}-3 i\right ) x^2 \, _2F_1\left (\frac {2}{3},1;\frac {5}{3};\frac {\sqrt {3} x^2}{\left (i+\sqrt {3}\right ) \left (x^2-1\right )}\right )\right )-3 \left (\sqrt {3} x^2-\sqrt {3}+3 i\right ) x^2 \, _2F_1\left (\frac {2}{3},2;\frac {5}{3};\frac {\sqrt {3} x^2}{\left (i+\sqrt {3}\right ) \left (x^2-1\right )}\right )+2 \left (x^2-1\right ) \left (3 \left (\sqrt {3}+i\right ) x^2+2 \left (\sqrt {3}-i\right )\right )\right )+24 \left (x^2-1\right )^2}{64 x^2 \left (x \left (x^2-1\right )\right )^{2/3}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
Integrate[((-x + x^3)^(1/3)*(8 - 10*x^2 + x^4))/(x^4*(4 - 2*x^2 + x^4)),x]
[Out]
(24*(-1 + x^2)^2 - (6*I + Sqrt[3])*(-2*(-1 + x^2)*(2*(I + Sqrt[3]) + 3*(-I + Sqrt[3])*x^2) + x^2*(3*I + Sqrt[3
] + 3*Sqrt[3]*x^2)*Hypergeometric2F1[2/3, 1, 5/3, (Sqrt[3]*x^2)/((-I + Sqrt[3])*(-1 + x^2))] - 3*x^2*(3*I + Sq
rt[3] - Sqrt[3]*x^2)*Hypergeometric2F1[2/3, 2, 5/3, (Sqrt[3]*x^2)/((-I + Sqrt[3])*(-1 + x^2))]) - (6*I - Sqrt[
3])*(2*(-1 + x^2)*(2*(-I + Sqrt[3]) + 3*(I + Sqrt[3])*x^2) - x^2*(-3*I + Sqrt[3] + 3*Sqrt[3]*x^2)*Hypergeometr
ic2F1[2/3, 1, 5/3, (Sqrt[3]*x^2)/((I + Sqrt[3])*(-1 + x^2))] - 3*x^2*(3*I - Sqrt[3] + Sqrt[3]*x^2)*Hypergeomet
ric2F1[2/3, 2, 5/3, (Sqrt[3]*x^2)/((I + Sqrt[3])*(-1 + x^2))]))/(64*x^2*(x*(-1 + x^2))^(2/3))
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IntegrateAlgebraic [A] time = 0.31, size = 120, normalized size = 1.00 \begin {gather*} \frac {3 \left (-1+4 x^2\right ) \sqrt [3]{-x+x^3}}{4 x^3}-\frac {1}{8} \text {RootSum}\left [3-6 \text {$\#$1}^3+4 \text {$\#$1}^6\&,\frac {-9 \log (x)+9 \log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right )+4 \log (x) \text {$\#$1}^3-4 \log \left (\sqrt [3]{-x+x^3}-x \text {$\#$1}\right ) \text {$\#$1}^3}{-3 \text {$\#$1}^2+4 \text {$\#$1}^5}\&\right ] \end {gather*}
Antiderivative was successfully verified.
[In]
IntegrateAlgebraic[((-x + x^3)^(1/3)*(8 - 10*x^2 + x^4))/(x^4*(4 - 2*x^2 + x^4)),x]
[Out]
(3*(-1 + 4*x^2)*(-x + x^3)^(1/3))/(4*x^3) - RootSum[3 - 6*#1^3 + 4*#1^6 & , (-9*Log[x] + 9*Log[(-x + x^3)^(1/3
) - x*#1] + 4*Log[x]*#1^3 - 4*Log[(-x + x^3)^(1/3) - x*#1]*#1^3)/(-3*#1^2 + 4*#1^5) & ]/8
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fricas [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^3-x)^(1/3)*(x^4-10*x^2+8)/x^4/(x^4-2*x^2+4),x, algorithm="fricas")
[Out]
Exception raised: TypeError >> Error detected within library code: integrate: implementation incomplete (tr
ace 0)
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - 10 \, x^{2} + 8\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}}{{\left (x^{4} - 2 \, x^{2} + 4\right )} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^3-x)^(1/3)*(x^4-10*x^2+8)/x^4/(x^4-2*x^2+4),x, algorithm="giac")
[Out]
integrate((x^4 - 10*x^2 + 8)*(x^3 - x)^(1/3)/((x^4 - 2*x^2 + 4)*x^4), x)
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maple [B] time = 43.11, size = 3326, normalized size = 27.72 \[\text {output too large to display}\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((x^3-x)^(1/3)*(x^4-10*x^2+8)/x^4/(x^4-2*x^2+4),x)
[Out]
3/4*(4*x^4-5*x^2+1)/x^3*(x*(x^2-1))^(1/3)/(x^2-1)+(-53/408*ln((-1976*RootOf(4*_Z^6+1602*_Z^3+177957)^6*RootOf(
_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)*x^4+9880*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)*
RootOf(4*_Z^6+1602*_Z^3+177957)^6*x^2+99042*(x^6-2*x^4+x^2)^(1/3)*RootOf(4*_Z^6+1602*_Z^3+177957)^3*RootOf(_Z^
3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)^2*x^2-1779258*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204
)*RootOf(4*_Z^6+1602*_Z^3+177957)^3*x^4-7904*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)*RootOf(4*_Z
^6+1602*_Z^3+177957)^6-99042*(x^6-2*x^4+x^2)^(1/3)*RootOf(4*_Z^6+1602*_Z^3+177957)^3*RootOf(_Z^3+8*RootOf(4*_Z
^6+1602*_Z^3+177957)^3+3204)^2+5721846*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)*RootOf(4*_Z^6+160
2*_Z^3+177957)^3*x^2+13652496*(x^6-2*x^4+x^2)^(2/3)*RootOf(4*_Z^6+1602*_Z^3+177957)^3+23426442*(x^6-2*x^4+x^2)
^(1/3)*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)^2*x^2-314185365*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*
_Z^3+177957)^3+3204)*x^4-3942588*RootOf(4*_Z^6+1602*_Z^3+177957)^3*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+17795
7)^3+3204)-23426442*(x^6-2*x^4+x^2)^(1/3)*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)^2+805301055*Ro
otOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)*x^2+3184910118*(x^6-2*x^4+x^2)^(2/3)-491115690*RootOf(_Z^3
+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204))/(-1+x)/(1+x)/(2*RootOf(4*_Z^6+1602*_Z^3+177957)^3*x^2-8*RootOf(4*_
Z^6+1602*_Z^3+177957)^3+171*x^2-1602))*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)-1/1836*ln((-1976*
RootOf(4*_Z^6+1602*_Z^3+177957)^6*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)*x^4+9880*RootOf(_Z^3+8
*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)*RootOf(4*_Z^6+1602*_Z^3+177957)^6*x^2+99042*(x^6-2*x^4+x^2)^(1/3)*Roo
tOf(4*_Z^6+1602*_Z^3+177957)^3*RootOf(_Z^3+8*RootOf(4*_Z^6+1602*_Z^3+177957)^3+3204)^2*x^2-1779258*RootOf(_Z^3
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (x^{4} - 10 \, x^{2} + 8\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}}}{{\left (x^{4} - 2 \, x^{2} + 4\right )} x^{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x^3-x)^(1/3)*(x^4-10*x^2+8)/x^4/(x^4-2*x^2+4),x, algorithm="maxima")
[Out]
integrate((x^4 - 10*x^2 + 8)*(x^3 - x)^(1/3)/((x^4 - 2*x^2 + 4)*x^4), x)
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x^3-x\right )}^{1/3}\,\left (x^4-10\,x^2+8\right )}{x^4\,\left (x^4-2\,x^2+4\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((x^3 - x)^(1/3)*(x^4 - 10*x^2 + 8))/(x^4*(x^4 - 2*x^2 + 4)),x)
[Out]
int(((x^3 - x)^(1/3)*(x^4 - 10*x^2 + 8))/(x^4*(x^4 - 2*x^2 + 4)), x)
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )} \left (x^{4} - 10 x^{2} + 8\right )}{x^{4} \left (x^{4} - 2 x^{2} + 4\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((x**3-x)**(1/3)*(x**4-10*x**2+8)/x**4/(x**4-2*x**2+4),x)
[Out]
Integral((x*(x - 1)*(x + 1))**(1/3)*(x**4 - 10*x**2 + 8)/(x**4*(x**4 - 2*x**2 + 4)), x)
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