Integrand size = 13, antiderivative size = 55 \[ \int x \sqrt [4]{-x+x^4} \, dx=\frac {1}{3} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} \arctan \left (\frac {x}{\sqrt [4]{-x+x^4}}\right )-\frac {1}{6} \text {arctanh}\left (\frac {x}{\sqrt [4]{-x+x^4}}\right ) \] Output:
1/3*x^2*(x^4-x)^(1/4)+1/6*arctan(x/(x^4-x)^(1/4))-1/6*arctanh(x/(x^4-x)^(1 /4))
Time = 6.89 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.45 \[ \int x \sqrt [4]{-x+x^4} \, dx=\frac {x^{3/4} \left (-1+x^3\right )^{3/4} \left (2 x^{9/4} \sqrt [4]{-1+x^3}+\arctan \left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )-\text {arctanh}\left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )\right )}{6 \left (x \left (-1+x^3\right )\right )^{3/4}} \] Input:
Integrate[x*(-x + x^4)^(1/4),x]
Output:
(x^(3/4)*(-1 + x^3)^(3/4)*(2*x^(9/4)*(-1 + x^3)^(1/4) + ArcTan[x^(3/4)/(-1 + x^3)^(1/4)] - ArcTanh[x^(3/4)/(-1 + x^3)^(1/4)]))/(6*(x*(-1 + x^3))^(3/ 4))
Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.55, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {1927, 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.
\(\displaystyle \int x \sqrt [4]{x^4-x} \, dx\) |
\(\Big \downarrow \) 1927 |
\(\displaystyle \frac {1}{3} x^2 \sqrt [4]{x^4-x}-\frac {1}{4} \int \frac {x^2}{\left (x^4-x\right )^{3/4}}dx\) |
\(\Big \downarrow \) 1938 |
\(\displaystyle \frac {1}{3} x^2 \sqrt [4]{x^4-x}-\frac {x^{3/4} \left (x^3-1\right )^{3/4} \int \frac {x^{5/4}}{\left (x^3-1\right )^{3/4}}dx}{4 \left (x^4-x\right )^{3/4}}\) |
\(\Big \downarrow \) 851 |
\(\displaystyle \frac {1}{3} x^2 \sqrt [4]{x^4-x}-\frac {x^{3/4} \left (x^3-1\right )^{3/4} \int \frac {x^2}{\left (x^3-1\right )^{3/4}}d\sqrt [4]{x}}{\left (x^4-x\right )^{3/4}}\) |
\(\Big \downarrow \) 807 |
\(\displaystyle \frac {1}{3} x^2 \sqrt [4]{x^4-x}-\frac {x^{3/4} \left (x^3-1\right )^{3/4} \int \frac {\sqrt {x}}{(x-1)^{3/4}}dx^{3/4}}{3 \left (x^4-x\right )^{3/4}}\) |
\(\Big \downarrow \) 854 |
\(\displaystyle \frac {1}{3} x^2 \sqrt [4]{x^4-x}-\frac {x^{3/4} \left (x^3-1\right )^{3/4} \int \frac {\sqrt {x}}{1-x}d\frac {x^{3/4}}{\sqrt [4]{x-1}}}{3 \left (x^4-x\right )^{3/4}}\) |
\(\Big \downarrow \) 827 |
\(\displaystyle \frac {1}{3} x^2 \sqrt [4]{x^4-x}-\frac {x^{3/4} \left (x^3-1\right )^{3/4} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x}}d\frac {x^{3/4}}{\sqrt [4]{x-1}}-\frac {1}{2} \int \frac {1}{\sqrt {x}+1}d\frac {x^{3/4}}{\sqrt [4]{x-1}}\right )}{3 \left (x^4-x\right )^{3/4}}\) |
\(\Big \downarrow \) 216 |
\(\displaystyle \frac {1}{3} x^2 \sqrt [4]{x^4-x}-\frac {x^{3/4} \left (x^3-1\right )^{3/4} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x}}d\frac {x^{3/4}}{\sqrt [4]{x-1}}-\frac {1}{2} \arctan \left (\frac {x^{3/4}}{\sqrt [4]{x-1}}\right )\right )}{3 \left (x^4-x\right )^{3/4}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle \frac {1}{3} x^2 \sqrt [4]{x^4-x}-\frac {x^{3/4} \left (x^3-1\right )^{3/4} \left (\frac {1}{2} \text {arctanh}\left (\frac {x^{3/4}}{\sqrt [4]{x-1}}\right )-\frac {1}{2} \arctan \left (\frac {x^{3/4}}{\sqrt [4]{x-1}}\right )\right )}{3 \left (x^4-x\right )^{3/4}}\) |
Input:
Int[x*(-x + x^4)^(1/4),x]
Output:
(x^2*(-x + x^4)^(1/4))/3 - (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))/(3*(-x + x^4)^(3/4))
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^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 = 3.88 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.60
method | result | size |
meijerg | \(\frac {4 \operatorname {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {9}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {3}{4}\right ], \left [\frac {7}{4}\right ], x^{3}\right )}{9 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {1}{4}}}\) | \(33\) |
pseudoelliptic | \(\frac {x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}}{3}-\frac {\ln \left (\frac {x +\left (x^{4}-x \right )^{\frac {1}{4}}}{x}\right )}{12}-\frac {\arctan \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}}{x}\right )}{6}+\frac {\ln \left (\frac {\left (x^{4}-x \right )^{\frac {1}{4}}-x}{x}\right )}{12}\) | \(70\) |
trager | \(\frac {x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}}{3}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (2 \sqrt {x^{4}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )\right )}{12}-\frac {\ln \left (-2 \left (x^{4}-x \right )^{\frac {3}{4}}-2 x \sqrt {x^{4}-x}-2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}-2 x^{3}+1\right )}{12}\) | \(133\) |
risch | \(\frac {x^{2} {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}}}{3}+\frac {\left (-\frac {\ln \left (\frac {2 x^{9}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{6}-5 x^{6}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}-4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{3}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}-1}{\left (-1+x \right )^{2} \left (x^{2}+x +1\right )^{2}}\right )}{12}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 x^{9}-2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{6}-5 x^{6}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}+4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}+4 x^{3}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}-1}{\left (-1+x \right )^{2} \left (x^{2}+x +1\right )^{2}}\right )}{12}\right ) {\left (x \left (x^{3}-1\right )\right )}^{\frac {1}{4}} \left (x^{3} \left (x^{3}-1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{3}-1\right )}\) | \(443\) |
Input:
int(x*(x^4-x)^(1/4),x,method=_RETURNVERBOSE)
Output:
4/9*signum(x^3-1)^(1/4)/(-signum(x^3-1))^(1/4)*x^(9/4)*hypergeom([-1/4,3/4 ],[7/4],x^3)
Leaf count of result is larger than twice the leaf count of optimal. 91 vs. \(2 (43) = 86\).
Time = 0.58 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.65 \[ \int x \sqrt [4]{-x+x^4} \, dx=\frac {1}{3} \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} - \frac {1}{12} \, \arctan \left (2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}}\right ) + \frac {1}{12} \, \log \left (2 \, x^{3} - 2 \, {\left (x^{4} - x\right )}^{\frac {1}{4}} x^{2} + 2 \, \sqrt {x^{4} - x} x - 2 \, {\left (x^{4} - x\right )}^{\frac {3}{4}} - 1\right ) \] Input:
integrate(x*(x^4-x)^(1/4),x, algorithm="fricas")
Output:
1/3*(x^4 - x)^(1/4)*x^2 - 1/12*arctan(2*(x^4 - x)^(1/4)*x^2 + 2*(x^4 - x)^ (3/4)) + 1/12*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)
\[ \int x \sqrt [4]{-x+x^4} \, dx=\int x \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \] Input:
integrate(x*(x**4-x)**(1/4),x)
Output:
Integral(x*(x*(x - 1)*(x**2 + x + 1))**(1/4), x)
\[ \int x \sqrt [4]{-x+x^4} \, dx=\int { {\left (x^{4} - x\right )}^{\frac {1}{4}} x \,d x } \] Input:
integrate(x*(x^4-x)^(1/4),x, algorithm="maxima")
Output:
integrate((x^4 - x)^(1/4)*x, x)
Time = 0.33 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.02 \[ \int x \sqrt [4]{-x+x^4} \, dx=\frac {1}{3} \, x^{3} {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} - \frac {1}{6} \, \arctan \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}}\right ) - \frac {1}{12} \, \log \left ({\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {1}{12} \, \log \left ({\left | {\left (-\frac {1}{x^{3}} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \] Input:
integrate(x*(x^4-x)^(1/4),x, algorithm="giac")
Output:
1/3*x^3*(-1/x^3 + 1)^(1/4) - 1/6*arctan((-1/x^3 + 1)^(1/4)) - 1/12*log((-1 /x^3 + 1)^(1/4) + 1) + 1/12*log(abs((-1/x^3 + 1)^(1/4) - 1))
Timed out. \[ \int x \sqrt [4]{-x+x^4} \, dx=\int x\,{\left (x^4-x\right )}^{1/4} \,d x \] Input:
int(x*(x^4 - x)^(1/4),x)
Output:
int(x*(x^4 - x)^(1/4), x)
\[ \int x \sqrt [4]{-x+x^4} \, dx=\frac {x^{\frac {9}{4}} \left (x^{3}-1\right )^{\frac {1}{4}}}{3}-\frac {\left (\int \frac {x^{\frac {5}{4}}}{\left (x^{3}-1\right )^{\frac {3}{4}}}d x \right )}{4} \] Input:
int(x*(x^4-x)^(1/4),x)
Output:
(4*x**(1/4)*(x**3 - 1)**(1/4)*x**2 - 3*int((x**(1/4)*(x**3 - 1)**(1/4)*x)/ (x**3 - 1),x))/12