\(\int x^2 \sqrt [3]{-x+x^3} \, dx\) [1651]

Optimal result

Integrand size = 15, antiderivative size = 112 \[ \int x^2 \sqrt [3]{-x+x^3} \, dx=\frac {1}{12} \sqrt [3]{-x+x^3} \left (-x+3 x^3\right )+\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )}{6 \sqrt {3}}+\frac {1}{18} \log \left (-x+\sqrt [3]{-x+x^3}\right )-\frac {1}{36} \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right ) \]

[Out]

1/12*(x^3-x)^(1/3)*(3*x^3-x)+1/18*arctan(3^(1/2)*x/(x+2*(x^3-x)^(1/3)))*3^(1/2)+1/18*ln(-x+(x^3-x)^(1/3))-1/36
*ln(x^2+x*(x^3-x)^(1/3)+(x^3-x)^(2/3))

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.26, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {2046, 2049, 2057, 335, 281, 337} \[ \int x^2 \sqrt [3]{-x+x^3} \, dx=\frac {\left (x^2-1\right )^{2/3} x^{2/3} \arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x^2-1}}+1}{\sqrt {3}}\right )}{6 \sqrt {3} \left (x^3-x\right )^{2/3}}+\frac {1}{4} \sqrt [3]{x^3-x} x^3-\frac {1}{12} \sqrt [3]{x^3-x} x+\frac {\left (x^2-1\right )^{2/3} x^{2/3} \log \left (x^{2/3}-\sqrt [3]{x^2-1}\right )}{12 \left (x^3-x\right )^{2/3}} \]

[In]

Int[x^2*(-x + x^3)^(1/3),x]

[Out]

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

Rule 281

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

Rule 335

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Dist[k/c, Subst[I
nt[x^(k*(m + 1) - 1)*(a + b*(x^(k*n)/c^n))^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0]
&& FractionQ[m] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 337

Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Simp[-ArcTan[(1 + 2*q*(x/(a + b*x^3)^(
1/3)))/Sqrt[3]]/(Sqrt[3]*q^2), x] - Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*q^2), x]] /; FreeQ[{a, b}, x]

Rule 2046

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a*x^j + b
*x^n)^p/(c*(m + n*p + 1))), x] + Dist[a*(n - j)*(p/(c^j*(m + n*p + 1))), Int[(c*x)^(m + j)*(a*x^j + b*x^n)^(p
- 1), x], x] /; FreeQ[{a, b, c, m}, x] &&  !IntegerQ[p] && LtQ[0, j, n] && (IntegersQ[j, n] || GtQ[c, 0]) && G
tQ[p, 0] && NeQ[m + n*p + 1, 0]

Rule 2049

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

Rule 2057

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Dist[c^IntPart[m]*(c*x)^FracPa
rt[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x^(n - j))^FracPart[p])), Int[x^(m
+ j*p)*(a + b*x^(n - j))^p, x], x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[n, j] && PosQ[n
- j]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {1}{6} \int \frac {x^3}{\left (-x+x^3\right )^{2/3}} \, dx \\ & = -\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {1}{9} \int \frac {x}{\left (-x+x^3\right )^{2/3}} \, dx \\ & = -\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \int \frac {\sqrt [3]{x}}{\left (-1+x^2\right )^{2/3}} \, dx}{9 \left (-x+x^3\right )^{2/3}} \\ & = -\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x^3}{\left (-1+x^6\right )^{2/3}} \, dx,x,\sqrt [3]{x}\right )}{3 \left (-x+x^3\right )^{2/3}} \\ & = -\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}-\frac {\left (x^{2/3} \left (-1+x^2\right )^{2/3}\right ) \text {Subst}\left (\int \frac {x}{\left (-1+x^3\right )^{2/3}} \, dx,x,x^{2/3}\right )}{6 \left (-x+x^3\right )^{2/3}} \\ & = -\frac {1}{12} x \sqrt [3]{-x+x^3}+\frac {1}{4} x^3 \sqrt [3]{-x+x^3}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \arctan \left (\frac {1+\frac {2 x^{2/3}}{\sqrt [3]{-1+x^2}}}{\sqrt {3}}\right )}{6 \sqrt {3} \left (-x+x^3\right )^{2/3}}+\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (x^{2/3}-\sqrt [3]{-1+x^2}\right )}{12 \left (-x+x^3\right )^{2/3}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.70 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.37 \[ \int x^2 \sqrt [3]{-x+x^3} \, dx=\frac {x^{2/3} \left (-1+x^2\right )^{2/3} \left (-3 x^{4/3} \sqrt [3]{-1+x^2}+9 x^{10/3} \sqrt [3]{-1+x^2}+2 \sqrt {3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )+2 \log \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )-\log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )\right )}{36 \left (x \left (-1+x^2\right )\right )^{2/3}} \]

[In]

Integrate[x^2*(-x + x^3)^(1/3),x]

[Out]

(x^(2/3)*(-1 + x^2)^(2/3)*(-3*x^(4/3)*(-1 + x^2)^(1/3) + 9*x^(10/3)*(-1 + x^2)^(1/3) + 2*Sqrt[3]*ArcTan[(Sqrt[
3]*x^(2/3))/(x^(2/3) + 2*(-1 + x^2)^(1/3))] + 2*Log[-x^(2/3) + (-1 + x^2)^(1/3)] - Log[x^(4/3) + x^(2/3)*(-1 +
x^2)^(1/3) + (-1 + x^2)^(2/3)]))/(36*(x*(-1 + x^2))^(2/3))

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.71 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.29

method	result	size
meijerg	\(\frac {3 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}} x^{\frac {10}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {5}{3}\right ], \left [\frac {8}{3}\right ], x^{2}\right )}{10 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}}}\)	\(33\)
pseudoelliptic	\(-\frac {x^{2} \left (\left (-9 x^{3}+3 x \right ) \left (x^{3}-x \right )^{\frac {1}{3}}+2 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right )+\ln \left (\frac {x^{2}+x \left (x^{3}-x \right )^{\frac {1}{3}}+\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )-2 \ln \left (\frac {-x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )\right )}{36 {\left (\left (x^{3}-x \right )^{\frac {2}{3}}+x \left (x +\left (x^{3}-x \right )^{\frac {1}{3}}\right )\right )}^{2} {\left (x -\left (x^{3}-x \right )^{\frac {1}{3}}\right )}^{2}}\)	\(142\)
trager	\(\frac {x \left (3 x^{2}-1\right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{12}+\frac {\operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (1395 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}+6768 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}+6768 \left (x^{3}-x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x +7233 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-5580 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+7611 \left (x^{3}-x \right )^{\frac {2}{3}}+7611 x \left (x^{3}-x \right )^{\frac {1}{3}}+7766 x^{2}-11727 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-4942\right )}{6}-\frac {\ln \left (1395 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-6768 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-6768 \left (x^{3}-x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -6303 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-5580 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}+5355 x \left (x^{3}-x \right )^{\frac {1}{3}}+5510 x^{2}+8007 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1653\right )}{18}-\frac {\ln \left (1395 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{2}-6768 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{3}-x \right )^{\frac {2}{3}}-6768 \left (x^{3}-x \right )^{\frac {1}{3}} \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x -6303 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{2}-5580 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+5355 \left (x^{3}-x \right )^{\frac {2}{3}}+5355 x \left (x^{3}-x \right )^{\frac {1}{3}}+5510 x^{2}+8007 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-1653\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{6}\)	\(463\)
risch	\(\frac {x \left (3 x^{2}-1\right ) {\left (x \left (x^{2}-1\right )\right )}^{\frac {1}{3}}}{12}+\frac {\left (\frac {\ln \left (\frac {35 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}+1956 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-175 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}+4104 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}-23364 x^{4}-5850 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-2010 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}+35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+140 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}-4104 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-10476 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+38232 x^{2}+54 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )-35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}-14868}{\left (-1+x \right ) \left (1+x \right )}\right )}{18}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \ln \left (\frac {59 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{4}-3750 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{4}-295 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2} x^{2}-1746 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+12600 x^{4}+5850 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}+5652 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) x^{2}-35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}} x^{2}+236 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )^{2}+1746 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right ) \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}+24624 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {2}{3}}-16380 x^{2}-1902 \operatorname {RootOf}\left (\textit {\_Z}^{2}+6 \textit {\_Z} +36\right )+35100 \left (x^{6}-2 x^{4}+x^{2}\right )^{\frac {1}{3}}+3780}{\left (-1+x \right ) \left (1+x \right )}\right )}{108}\right ) {\left (x \left (x^{2}-1\right )\right )}^{\frac {1}{3}} \left (x^{2} \left (x^{2}-1\right )^{2}\right )^{\frac {1}{3}}}{x \left (x^{2}-1\right )}\)	\(542\)

[In]

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

[Out]

3/10*signum(x^2-1)^(1/3)/(-signum(x^2-1))^(1/3)*x^(10/3)*hypergeom([-1/3,5/3],[8/3],x^2)

Fricas [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.96 \[ \int x^2 \sqrt [3]{-x+x^3} \, dx=\frac {1}{18} \, \sqrt {3} \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + \frac {1}{12} \, {\left (3 \, x^{3} - x\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}} + \frac {1}{36} \, \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) \]

[In]

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

[Out]

1/18*sqrt(3)*arctan(-(44032959556*sqrt(3)*(x^3 - x)^(1/3)*x + sqrt(3)*(16754327161*x^2 - 2707204793) - 1052430
5234*sqrt(3)*(x^3 - x)^(2/3))/(81835897185*x^2 - 1102302937)) + 1/12*(3*x^3 - x)*(x^3 - x)^(1/3) + 1/36*log(-3
*(x^3 - x)^(1/3)*x + 3*(x^3 - x)^(2/3) + 1)

Sympy [F]

\[ \int x^2 \sqrt [3]{-x+x^3} \, dx=\int x^{2} \sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int x^2 \sqrt [3]{-x+x^3} \, dx=\int { {\left (x^{3} - x\right )}^{\frac {1}{3}} x^{2} \,d x } \]

[In]

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

[Out]

integrate((x^3 - x)^(1/3)*x^2, x)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.79 \[ \int x^2 \sqrt [3]{-x+x^3} \, dx=\frac {1}{12} \, {\left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} + 2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )} x^{4} - \frac {1}{18} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {1}{36} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{18} \, \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]

[In]

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

[Out]

1/12*((-1/x^2 + 1)^(4/3) + 2*(-1/x^2 + 1)^(1/3))*x^4 - 1/18*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-1/x^2 + 1)^(1/3) +
1)) - 1/36*log((-1/x^2 + 1)^(2/3) + (-1/x^2 + 1)^(1/3) + 1) + 1/18*log(abs((-1/x^2 + 1)^(1/3) - 1))

Mupad [F(-1)]

Timed out. \[ \int x^2 \sqrt [3]{-x+x^3} \, dx=\int x^2\,{\left (x^3-x\right )}^{1/3} \,d x \]

[In]

int(x^2*(x^3 - x)^(1/3),x)

[Out]

int(x^2*(x^3 - x)^(1/3), x)