Integrand size = 15, antiderivative size = 117 \[ \int x^4 \sqrt [3]{-x+x^3} \, dx=\frac {1}{108} \sqrt [3]{-x+x^3} \left (-5 x-3 x^3+18 x^5\right )+\frac {5 \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{-x+x^3}}\right )}{54 \sqrt {3}}+\frac {5}{162} \log \left (-x+\sqrt [3]{-x+x^3}\right )-\frac {5}{324} \log \left (x^2+x \sqrt [3]{-x+x^3}+\left (-x+x^3\right )^{2/3}\right ) \]
1/108*(x^3-x)^(1/3)*(18*x^5-3*x^3-5*x)+5/162*arctan(3^(1/2)*x/(x+2*(x^3-x) ^(1/3)))*3^(1/2)+5/162*ln(-x+(x^3-x)^(1/3))-5/324*ln(x^2+x*(x^3-x)^(1/3)+( x^3-x)^(2/3))
Time = 1.14 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.44 \[ \int x^4 \sqrt [3]{-x+x^3} \, dx=\frac {15 x^2-6 x^4-63 x^6+54 x^8+10 \sqrt {3} x^{2/3} \left (-1+x^2\right )^{2/3} \arctan \left (\frac {\sqrt {3} x^{2/3}}{x^{2/3}+2 \sqrt [3]{-1+x^2}}\right )+10 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (-x^{2/3}+\sqrt [3]{-1+x^2}\right )-5 x^{2/3} \left (-1+x^2\right )^{2/3} \log \left (x^{4/3}+x^{2/3} \sqrt [3]{-1+x^2}+\left (-1+x^2\right )^{2/3}\right )}{324 \left (x \left (-1+x^2\right )\right )^{2/3}} \]
(15*x^2 - 6*x^4 - 63*x^6 + 54*x^8 + 10*Sqrt[3]*x^(2/3)*(-1 + x^2)^(2/3)*Ar cTan[(Sqrt[3]*x^(2/3))/(x^(2/3) + 2*(-1 + x^2)^(1/3))] + 10*x^(2/3)*(-1 + x^2)^(2/3)*Log[-x^(2/3) + (-1 + x^2)^(1/3)] - 5*x^(2/3)*(-1 + x^2)^(2/3)*L og[x^(4/3) + x^(2/3)*(-1 + x^2)^(1/3) + (-1 + x^2)^(2/3)])/(324*(x*(-1 + x ^2))^(2/3))
Time = 0.32 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.20, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {1927, 1930, 1930, 1938, 266, 807, 853}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int x^4 \sqrt [3]{x^3-x} \, dx\) |
| \(\Big \downarrow \) 1927 |
| \(\displaystyle \frac {1}{6} x^5 \sqrt [3]{x^3-x}-\frac {1}{9} \int \frac {x^5}{\left (x^3-x\right )^{2/3}}dx\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle \frac {1}{9} \left (-\frac {5}{6} \int \frac {x^3}{\left (x^3-x\right )^{2/3}}dx-\frac {1}{4} \sqrt [3]{x^3-x} x^3\right )+\frac {1}{6} \sqrt [3]{x^3-x} x^5\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle \frac {1}{9} \left (-\frac {5}{6} \left (\frac {2}{3} \int \frac {x}{\left (x^3-x\right )^{2/3}}dx+\frac {1}{2} \sqrt [3]{x^3-x} x\right )-\frac {1}{4} \sqrt [3]{x^3-x} x^3\right )+\frac {1}{6} \sqrt [3]{x^3-x} x^5\) |
| \(\Big \downarrow \) 1938 |
| \(\displaystyle \frac {1}{9} \left (-\frac {5}{6} \left (\frac {2 \left (x^2-1\right )^{2/3} x^{2/3} \int \frac {\sqrt [3]{x}}{\left (x^2-1\right )^{2/3}}dx}{3 \left (x^3-x\right )^{2/3}}+\frac {1}{2} \sqrt [3]{x^3-x} x\right )-\frac {1}{4} \sqrt [3]{x^3-x} x^3\right )+\frac {1}{6} \sqrt [3]{x^3-x} x^5\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {1}{9} \left (-\frac {5}{6} \left (\frac {2 \left (x^2-1\right )^{2/3} x^{2/3} \int \frac {x}{\left (x^2-1\right )^{2/3}}d\sqrt [3]{x}}{\left (x^3-x\right )^{2/3}}+\frac {1}{2} \sqrt [3]{x^3-x} x\right )-\frac {1}{4} \sqrt [3]{x^3-x} x^3\right )+\frac {1}{6} \sqrt [3]{x^3-x} x^5\) |
| \(\Big \downarrow \) 807 |
| \(\displaystyle \frac {1}{9} \left (-\frac {5}{6} \left (\frac {\left (x^2-1\right )^{2/3} x^{2/3} \int \frac {x^{2/3}}{(x-1)^{2/3}}dx^{2/3}}{\left (x^3-x\right )^{2/3}}+\frac {1}{2} \sqrt [3]{x^3-x} x\right )-\frac {1}{4} \sqrt [3]{x^3-x} x^3\right )+\frac {1}{6} \sqrt [3]{x^3-x} x^5\) |
| \(\Big \downarrow \) 853 |
| \(\displaystyle \frac {1}{9} \left (-\frac {5}{6} \left (\frac {\left (x^2-1\right )^{2/3} x^{2/3} \left (-\frac {\arctan \left (\frac {\frac {2 x^{2/3}}{\sqrt [3]{x-1}}+1}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{2} \log \left (x^{2/3}-\sqrt [3]{x-1}\right )\right )}{\left (x^3-x\right )^{2/3}}+\frac {1}{2} \sqrt [3]{x^3-x} x\right )-\frac {1}{4} \sqrt [3]{x^3-x} x^3\right )+\frac {1}{6} \sqrt [3]{x^3-x} x^5\) |
(x^5*(-x + x^3)^(1/3))/6 + (-1/4*(x^3*(-x + x^3)^(1/3)) - (5*((x*(-x + x^3 )^(1/3))/2 + (x^(2/3)*(-1 + x^2)^(2/3)*(-(ArcTan[(1 + (2*x^(2/3))/(-1 + x) ^(1/3))/Sqrt[3]]/Sqrt[3]) - Log[-(-1 + x)^(1/3) + x^(2/3)]/2))/(-x + x^3)^ (2/3)))/6)/9
3.18.29.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[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]
Int[(x_)/((a_) + (b_.)*(x_)^3)^(2/3), x_Symbol] :> With[{q = Rt[b, 3]}, Sim p[-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]
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] + Simp[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] && (Int egersQ[j, n] || GtQ[c, 0]) && GtQ[p, 0] && NeQ[m + n*p + 1, 0]
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] - Simp[a*c^(n - j)*((m + j*p - n + j + 1)/(b*(m + n*p + 1))) I nt[(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]) && Gt Q[m + j*p - n + j + 1, 0] && NeQ[m + n*p + 1, 0]
Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol ] :> Simp[c^IntPart[m]*(c*x)^FracPart[m]*((a*x^j + b*x^n)^FracPart[p]/(x^(F racPart[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] && !Inte gerQ[p] && NeQ[n, j] && PosQ[n - j]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.88 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.28
| method | result | size |
| meijerg | \(\frac {3 \operatorname {signum}\left (x^{2}-1\right )^{\frac {1}{3}} x^{\frac {16}{3}} \operatorname {hypergeom}\left (\left [-\frac {1}{3}, \frac {8}{3}\right ], \left [\frac {11}{3}\right ], x^{2}\right )}{16 {\left (-\operatorname {signum}\left (x^{2}-1\right )\right )}^{\frac {1}{3}}}\) | \(33\) |
| pseudoelliptic | \(-\frac {x^{3} \left (\left (-54 x^{5}+9 x^{3}+15 x \right ) \left (x^{3}-x \right )^{\frac {1}{3}}+10 \sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (x +2 \left (x^{3}-x \right )^{\frac {1}{3}}\right )}{3 x}\right )+5 \ln \left (\frac {x^{2}+x \left (x^{3}-x \right )^{\frac {1}{3}}+\left (x^{3}-x \right )^{\frac {2}{3}}}{x^{2}}\right )-10 \ln \left (\frac {-x +\left (x^{3}-x \right )^{\frac {1}{3}}}{x}\right )\right )}{324 {\left (\left (x^{3}-x \right )^{\frac {2}{3}}+x \left (x +\left (x^{3}-x \right )^{\frac {1}{3}}\right )\right )}^{3} {\left (x -\left (x^{3}-x \right )^{\frac {1}{3}}\right )}^{3}}\) | \(149\) |
| trager | \(\frac {x \left (18 x^{4}-3 x^{2}-5\right ) \left (x^{3}-x \right )^{\frac {1}{3}}}{108}-\frac {5 \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 )}{162}-\frac {5 \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 )}{54}+\frac {5 \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 )}{54}\) | \(468\) |
| risch | \(\text {Expression too large to display}\) | \(798\) |
Time = 0.43 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.96 \[ \int x^4 \sqrt [3]{-x+x^3} \, dx=\frac {5}{162} \, \sqrt {3} \arctan \left (-\frac {44032959556 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {1}{3}} x + \sqrt {3} {\left (16754327161 \, x^{2} - 2707204793\right )} - 10524305234 \, \sqrt {3} {\left (x^{3} - x\right )}^{\frac {2}{3}}}{81835897185 \, x^{2} - 1102302937}\right ) + \frac {1}{108} \, {\left (18 \, x^{5} - 3 \, x^{3} - 5 \, x\right )} {\left (x^{3} - x\right )}^{\frac {1}{3}} + \frac {5}{324} \, \log \left (-3 \, {\left (x^{3} - x\right )}^{\frac {1}{3}} x + 3 \, {\left (x^{3} - x\right )}^{\frac {2}{3}} + 1\right ) \]
5/162*sqrt(3)*arctan(-(44032959556*sqrt(3)*(x^3 - x)^(1/3)*x + sqrt(3)*(16 754327161*x^2 - 2707204793) - 10524305234*sqrt(3)*(x^3 - x)^(2/3))/(818358 97185*x^2 - 1102302937)) + 1/108*(18*x^5 - 3*x^3 - 5*x)*(x^3 - x)^(1/3) + 5/324*log(-3*(x^3 - x)^(1/3)*x + 3*(x^3 - x)^(2/3) + 1)
\[ \int x^4 \sqrt [3]{-x+x^3} \, dx=\int x^{4} \sqrt [3]{x \left (x - 1\right ) \left (x + 1\right )}\, dx \]
\[ \int x^4 \sqrt [3]{-x+x^3} \, dx=\int { {\left (x^{3} - x\right )}^{\frac {1}{3}} x^{4} \,d x } \]
Time = 0.29 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.93 \[ \int x^4 \sqrt [3]{-x+x^3} \, dx=-\frac {1}{108} \, {\left (5 \, {\left (\frac {1}{x^{2}} - 1\right )}^{2} {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 13 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {4}{3}} - 10 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}}\right )} x^{6} - \frac {5}{162} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - \frac {5}{324} \, \log \left ({\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {2}{3}} + {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {5}{162} \, \log \left ({\left | {\left (-\frac {1}{x^{2}} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) \]
-1/108*(5*(1/x^2 - 1)^2*(-1/x^2 + 1)^(1/3) - 13*(-1/x^2 + 1)^(4/3) - 10*(- 1/x^2 + 1)^(1/3))*x^6 - 5/162*sqrt(3)*arctan(1/3*sqrt(3)*(2*(-1/x^2 + 1)^( 1/3) + 1)) - 5/324*log((-1/x^2 + 1)^(2/3) + (-1/x^2 + 1)^(1/3) + 1) + 5/16 2*log(abs((-1/x^2 + 1)^(1/3) - 1))
Timed out. \[ \int x^4 \sqrt [3]{-x+x^3} \, dx=\int x^4\,{\left (x^3-x\right )}^{1/3} \,d x \]