Integrand size = 15, antiderivative size = 63 \[ \int x^4 \sqrt [4]{-x+x^4} \, dx=\frac {1}{24} \sqrt [4]{-x+x^4} \left (-x^2+4 x^5\right )+\frac {1}{16} \arctan \left (\frac {x}{\sqrt [4]{-x+x^4}}\right )-\frac {1}{16} \text {arctanh}\left (\frac {x}{\sqrt [4]{-x+x^4}}\right ) \]
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 10.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.89 \[ \int x^4 \sqrt [4]{-x+x^4} \, dx=\frac {x^2 \sqrt [4]{x \left (-1+x^3\right )} \left (-\left (1-x^3\right )^{5/4}+\operatorname {Hypergeometric2F1}\left (-\frac {1}{4},\frac {3}{4},\frac {7}{4},x^3\right )\right )}{6 \sqrt [4]{1-x^3}} \]
(x^2*(x*(-1 + x^3))^(1/4)*(-(1 - x^3)^(5/4) + Hypergeometric2F1[-1/4, 3/4, 7/4, x^3]))/(6*(1 - x^3)^(1/4))
Time = 0.30 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.68, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {1927, 1930, 1938, 851, 807, 854, 827, 216, 219}
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.
\(\displaystyle \int x^4 \sqrt [4]{x^4-x} \, dx\) |
\(\Big \downarrow \) 1927 |
\(\displaystyle \frac {1}{6} x^5 \sqrt [4]{x^4-x}-\frac {1}{8} \int \frac {x^5}{\left (x^4-x\right )^{3/4}}dx\) |
\(\Big \downarrow \) 1930 |
\(\displaystyle \frac {1}{8} \left (-\frac {3}{4} \int \frac {x^2}{\left (x^4-x\right )^{3/4}}dx-\frac {1}{3} \sqrt [4]{x^4-x} x^2\right )+\frac {1}{6} \sqrt [4]{x^4-x} x^5\) |
\(\Big \downarrow \) 1938 |
\(\displaystyle \frac {1}{8} \left (-\frac {3 \left (x^3-1\right )^{3/4} x^{3/4} \int \frac {x^{5/4}}{\left (x^3-1\right )^{3/4}}dx}{4 \left (x^4-x\right )^{3/4}}-\frac {1}{3} \sqrt [4]{x^4-x} x^2\right )+\frac {1}{6} \sqrt [4]{x^4-x} x^5\) |
\(\Big \downarrow \) 851 |
\(\displaystyle \frac {1}{8} \left (-\frac {3 \left (x^3-1\right )^{3/4} x^{3/4} \int \frac {x^2}{\left (x^3-1\right )^{3/4}}d\sqrt [4]{x}}{\left (x^4-x\right )^{3/4}}-\frac {1}{3} \sqrt [4]{x^4-x} x^2\right )+\frac {1}{6} \sqrt [4]{x^4-x} x^5\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{8} \left (-\frac {\left (x^3-1\right )^{3/4} x^{3/4} \int \frac {\sqrt {x}}{(x-1)^{3/4}}dx^{3/4}}{\left (x^4-x\right )^{3/4}}-\frac {1}{3} \sqrt [4]{x^4-x} x^2\right )+\frac {1}{6} \sqrt [4]{x^4-x} x^5\) |
\(\Big \downarrow \) 854 |
\(\displaystyle \frac {1}{8} \left (-\frac {\left (x^3-1\right )^{3/4} x^{3/4} \int \frac {\sqrt {x}}{1-x}d\frac {x^{3/4}}{\sqrt [4]{x-1}}}{\left (x^4-x\right )^{3/4}}-\frac {1}{3} \sqrt [4]{x^4-x} x^2\right )+\frac {1}{6} \sqrt [4]{x^4-x} x^5\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {1}{8} \left (-\frac {\left (x^3-1\right )^{3/4} x^{3/4} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x}}d\frac {x^{3/4}}{\sqrt [4]{x-1}}-\frac {1}{2} \int \frac {1}{\sqrt {x}+1}d\frac {x^{3/4}}{\sqrt [4]{x-1}}\right )}{\left (x^4-x\right )^{3/4}}-\frac {1}{3} \sqrt [4]{x^4-x} x^2\right )+\frac {1}{6} \sqrt [4]{x^4-x} x^5\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{8} \left (-\frac {\left (x^3-1\right )^{3/4} x^{3/4} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x}}d\frac {x^{3/4}}{\sqrt [4]{x-1}}-\frac {1}{2} \arctan \left (\frac {x^{3/4}}{\sqrt [4]{x-1}}\right )\right )}{\left (x^4-x\right )^{3/4}}-\frac {1}{3} \sqrt [4]{x^4-x} x^2\right )+\frac {1}{6} \sqrt [4]{x^4-x} x^5\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{8} \left (-\frac {\left (x^3-1\right )^{3/4} x^{3/4} \left (\frac {1}{2} \text {arctanh}\left (\frac {x^{3/4}}{\sqrt [4]{x-1}}\right )-\frac {1}{2} \arctan \left (\frac {x^{3/4}}{\sqrt [4]{x-1}}\right )\right )}{\left (x^4-x\right )^{3/4}}-\frac {1}{3} \sqrt [4]{x^4-x} x^2\right )+\frac {1}{6} \sqrt [4]{x^4-x} x^5\) |
(x^5*(-x + x^4)^(1/4))/6 + (-1/3*(x^2*(-x + x^4)^(1/4)) - (x^(3/4)*(-1 + x ^3)^(3/4)*(-1/2*ArcTan[x^(3/4)/(-1 + x)^(1/4)] + ArcTanh[x^(3/4)/(-1 + x)^ (1/4)]/2))/(-x + x^4)^(3/4))/8
3.9.32.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
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_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k/c Subst[Int[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]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
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 = 4.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.52
method | result | size |
meijerg | \(\frac {4 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {21}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], x^{3}\right )}{21 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}}}\) | \(33\) |
pseudoelliptic | \(\frac {x^{2} \left (16 \left (x^{4}-x \right )^{\frac {1}{4}} x^{5}-4 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}-6 \arctan \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}}{x}\right )+3 \ln \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}-x}{x}\right )-3 \ln \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}+x}{x}\right )\right )}{96 {\left (\left (x^{4}-x \right )^{\frac {1}{4}}-x \right )}^{2} \left (x^{2}+\sqrt {x^{4}-x}\right )^{2} {\left (\left (x^{4}-x \right )^{\frac {1}{4}}+x \right )}^{2}}\) | \(132\) |
trager | \(\frac {x^{2} \left (4 x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {1}{4}}}{24}+\frac {\ln \left (-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x \sqrt {x^{4}-x}-2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+2 x^{3}-1\right )}{32}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \sqrt {x^{4}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{32}\) | \(140\) |
risch | \(\frac {x^{2} \left (4 x^{3}-1\right ) {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}}}{24}+\frac {\left (-\frac {\ln \left (\frac {2 x^{9}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{6}-5 x^{6}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}-4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{3}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}-1}{\left (x^{2}+x +1\right )^{2} \left (x -1\right )^{2}}\right )}{32}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 x^{9}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{6}-5 x^{6}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{3}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}-1}{\left (x^{2}+x +1\right )^{2} \left (x -1\right )^{2}}\right )}{32}\right ) {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (x^{3} \left (x^{3}-1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{3}-1\right )}\) | \(450\) |
Time = 1.18 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.57 \[ \int x^4 \sqrt [4]{-x+x^4} \, dx=\frac {1}{24} \, {\left (4 \, x^{5} - x^{2}\right )} {\left (x^{4} - x\right )}^{\frac {1}{4}} - \frac {1}{32} \, \arctan \left (2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}}\right ) + \frac {1}{32} \, \log \left (2 \, x^{3} - 2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{4} - x} x - 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}} - 1\right ) \]
1/24*(4*x^5 - x^2)*(x^4 - x)^(1/4) - 1/32*arctan(2*(x^4 - x)^(1/4)*x^2 + 2 *(x^4 - x)^(3/4)) + 1/32*log(2*x^3 - 2*(x^4 - x)^(1/4)*x^2 + 2*sqrt(x^4 - x)*x - 2*(x^4 - x)^(3/4) - 1)
\[ \int x^4 \sqrt [4]{-x+x^4} \, dx=\int x^{4} \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \]
\[ \int x^4 \sqrt [4]{-x+x^4} \, dx=\int { {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{4} \,d x } \]
Time = 0.26 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.08 \[ \int x^4 \sqrt [4]{-x+x^4} \, dx=\frac {1}{24} \, {\left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {5}{4}} + 3 \, {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right )} x^{6} - \frac {1}{16} \, \arctan \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{32} \, \log \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{32} \, \log \left ({\left | {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
1/24*((-1/x^3 + 1)^(5/4) + 3*(-1/x^3 + 1)^(1/4))*x^6 - 1/16*arctan((-1/x^3 + 1)^(1/4)) - 1/32*log((-1/x^3 + 1)^(1/4) + 1) + 1/32*log(abs((-1/x^3 + 1 )^(1/4) - 1))
Timed out. \[ \int x^4 \sqrt [4]{-x+x^4} \, dx=\int x^4\,{\left (x^4-x\right )}^{1/4} \,d x \]