Integrand size = 17, antiderivative size = 73 \[ \int x^2 \sqrt [4]{-x^3+x^4} \, dx=\frac {\left (-77-44 x-32 x^2+384 x^3\right ) \sqrt [4]{-x^3+x^4}}{1536}+\frac {77 \arctan \left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{1024}-\frac {77 \text {arctanh}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{1024} \]
1/1536*(384*x^3-32*x^2-44*x-77)*(x^4-x^3)^(1/4)+77/1024*arctan(x/(x^4-x^3) ^(1/4))-77/1024*arctanh(x/(x^4-x^3)^(1/4))
Time = 0.28 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.16 \[ \int x^2 \sqrt [4]{-x^3+x^4} \, dx=\frac {(-1+x)^{3/4} x^{9/4} \left (2 \sqrt [4]{-1+x} x^{3/4} \left (-77-44 x-32 x^2+384 x^3\right )+231 \arctan \left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-231 \text {arctanh}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )}{3072 \left ((-1+x) x^3\right )^{3/4}} \]
((-1 + x)^(3/4)*x^(9/4)*(2*(-1 + x)^(1/4)*x^(3/4)*(-77 - 44*x - 32*x^2 + 3 84*x^3) + 231*ArcTan[((-1 + x)/x)^(-1/4)] - 231*ArcTanh[((-1 + x)/x)^(-1/4 )]))/(3072*((-1 + x)*x^3)^(3/4))
Leaf count is larger than twice the leaf count of optimal. \(151\) vs. \(2(73)=146\).
Time = 0.37 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.588, Rules used = {1927, 1930, 1930, 1930, 1938, 73, 770, 756, 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.
| \(\displaystyle \int x^2 \sqrt [4]{x^4-x^3} \, dx\) |
| \(\Big \downarrow \) 1927 |
| \(\displaystyle \frac {1}{4} x^3 \sqrt [4]{x^4-x^3}-\frac {1}{16} \int \frac {x^5}{\left (x^4-x^3\right )^{3/4}}dx\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle \frac {1}{16} \left (-\frac {11}{12} \int \frac {x^4}{\left (x^4-x^3\right )^{3/4}}dx-\frac {1}{3} \sqrt [4]{x^4-x^3} x^2\right )+\frac {1}{4} \sqrt [4]{x^4-x^3} x^3\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle \frac {1}{16} \left (-\frac {11}{12} \left (\frac {7}{8} \int \frac {x^3}{\left (x^4-x^3\right )^{3/4}}dx+\frac {1}{2} \sqrt [4]{x^4-x^3} x\right )-\frac {1}{3} \sqrt [4]{x^4-x^3} x^2\right )+\frac {1}{4} \sqrt [4]{x^4-x^3} x^3\) |
| \(\Big \downarrow \) 1930 |
| \(\displaystyle \frac {1}{16} \left (-\frac {11}{12} \left (\frac {7}{8} \left (\frac {3}{4} \int \frac {x^2}{\left (x^4-x^3\right )^{3/4}}dx+\sqrt [4]{x^4-x^3}\right )+\frac {1}{2} \sqrt [4]{x^4-x^3} x\right )-\frac {1}{3} \sqrt [4]{x^4-x^3} x^2\right )+\frac {1}{4} \sqrt [4]{x^4-x^3} x^3\) |
| \(\Big \downarrow \) 1938 |
| \(\displaystyle \frac {1}{16} \left (-\frac {11}{12} \left (\frac {7}{8} \left (\frac {3 (x-1)^{3/4} x^{9/4} \int \frac {1}{(x-1)^{3/4} \sqrt [4]{x}}dx}{4 \left (x^4-x^3\right )^{3/4}}+\sqrt [4]{x^4-x^3}\right )+\frac {1}{2} \sqrt [4]{x^4-x^3} x\right )-\frac {1}{3} \sqrt [4]{x^4-x^3} x^2\right )+\frac {1}{4} \sqrt [4]{x^4-x^3} x^3\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {1}{16} \left (-\frac {11}{12} \left (\frac {7}{8} \left (\frac {3 (x-1)^{3/4} x^{9/4} \int \frac {1}{\sqrt [4]{x}}d\sqrt [4]{x-1}}{\left (x^4-x^3\right )^{3/4}}+\sqrt [4]{x^4-x^3}\right )+\frac {1}{2} \sqrt [4]{x^4-x^3} x\right )-\frac {1}{3} \sqrt [4]{x^4-x^3} x^2\right )+\frac {1}{4} \sqrt [4]{x^4-x^3} x^3\) |
| \(\Big \downarrow \) 770 |
| \(\displaystyle \frac {1}{16} \left (-\frac {11}{12} \left (\frac {7}{8} \left (\frac {3 (x-1)^{3/4} x^{9/4} \int \frac {1}{2-x}d\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}}{\left (x^4-x^3\right )^{3/4}}+\sqrt [4]{x^4-x^3}\right )+\frac {1}{2} \sqrt [4]{x^4-x^3} x\right )-\frac {1}{3} \sqrt [4]{x^4-x^3} x^2\right )+\frac {1}{4} \sqrt [4]{x^4-x^3} x^3\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle \frac {1}{16} \left (-\frac {11}{12} \left (\frac {7}{8} \left (\frac {3 (x-1)^{3/4} x^{9/4} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x-1}}d\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}+\frac {1}{2} \int \frac {1}{\sqrt {x-1}+1}d\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{\left (x^4-x^3\right )^{3/4}}+\sqrt [4]{x^4-x^3}\right )+\frac {1}{2} \sqrt [4]{x^4-x^3} x\right )-\frac {1}{3} \sqrt [4]{x^4-x^3} x^2\right )+\frac {1}{4} \sqrt [4]{x^4-x^3} x^3\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {1}{16} \left (-\frac {11}{12} \left (\frac {7}{8} \left (\frac {3 (x-1)^{3/4} x^{9/4} \left (\frac {1}{2} \int \frac {1}{1-\sqrt {x-1}}d\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}+\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )\right )}{\left (x^4-x^3\right )^{3/4}}+\sqrt [4]{x^4-x^3}\right )+\frac {1}{2} \sqrt [4]{x^4-x^3} x\right )-\frac {1}{3} \sqrt [4]{x^4-x^3} x^2\right )+\frac {1}{4} \sqrt [4]{x^4-x^3} x^3\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{16} \left (-\frac {11}{12} \left (\frac {7}{8} \left (\frac {3 (x-1)^{3/4} x^{9/4} \left (\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )+\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )\right )}{\left (x^4-x^3\right )^{3/4}}+\sqrt [4]{x^4-x^3}\right )+\frac {1}{2} \sqrt [4]{x^4-x^3} x\right )-\frac {1}{3} \sqrt [4]{x^4-x^3} x^2\right )+\frac {1}{4} \sqrt [4]{x^4-x^3} x^3\) |
(x^3*(-x^3 + x^4)^(1/4))/4 + (-1/3*(x^2*(-x^3 + x^4)^(1/4)) - (11*((x*(-x^ 3 + x^4)^(1/4))/2 + (7*((-x^3 + x^4)^(1/4) + (3*(-1 + x)^(3/4)*x^(9/4)*(Ar cTan[(-1 + x)^(1/4)/x^(1/4)]/2 + ArcTanh[(-1 + x)^(1/4)/x^(1/4)]/2))/(-x^3 + x^4)^(3/4)))/8))/12)/16
3.10.59.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2 ]], s = Denominator[Rt[-a/b, 2]]}, Simp[r/(2*a) Int[1/(r - s*x^2), x], x] + Simp[r/(2*a) Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ[a /b, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + 1/n) Subst[In t[1/(1 - b*x^n)^(p + 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)] && IntegerQ[p + 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 = 1.43 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.37
| method | result | size |
| meijerg | \(\frac {4 \operatorname {signum}\left (x -1\right )^{\frac {1}{4}} x^{\frac {15}{4}} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, \frac {15}{4}\right ], \left [\frac {19}{4}\right ], x\right )}{15 \left (-\operatorname {signum}\left (x -1\right )\right )^{\frac {1}{4}}}\) | \(27\) |
| pseudoelliptic | \(\frac {x^{12} \left (1536 \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}} x^{3}-128 x^{2} \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}-176 x \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}+231 \ln \left (\frac {\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}-x}{x}\right )-462 \arctan \left (\frac {\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}{x}\right )-231 \ln \left (\frac {\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}+x}{x}\right )-308 \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}\right )}{6144 {\left (\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}-x \right )}^{4} \left (x^{2}+\sqrt {x^{3} \left (x -1\right )}\right )^{4} {\left (\left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}+x \right )}^{4}}\) | \(155\) |
| trager | \(\left (\frac {1}{4} x^{3}-\frac {1}{48} x^{2}-\frac {11}{384} x -\frac {77}{1536}\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+\frac {77 \ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-2 x^{3}+x^{2}}{x^{2}}\right )}{2048}+\frac {77 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \sqrt {x^{4}-x^{3}}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}}{x^{2}}\right )}{2048}\) | \(173\) |
| risch | \(\frac {\left (384 x^{3}-32 x^{2}-44 x -77\right ) \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}}}{1536}+\frac {\left (-\frac {77 \ln \left (\frac {2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}+2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, x +2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+2 x^{3}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x -5 x^{2}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}+4 x -1}{\left (x -1\right )^{2}}\right )}{2048}+\frac {77 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x -4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (x -1\right )^{2}}\right )}{2048}\right ) \left (x^{3} \left (x -1\right )\right )^{\frac {1}{4}} \left (x \left (x -1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x -1\right )}\) | \(407\) |
Time = 0.26 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.23 \[ \int x^2 \sqrt [4]{-x^3+x^4} \, dx=\frac {1}{1536} \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (384 \, x^{3} - 32 \, x^{2} - 44 \, x - 77\right )} - \frac {77}{1024} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {77}{2048} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {77}{2048} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \]
1/1536*(x^4 - x^3)^(1/4)*(384*x^3 - 32*x^2 - 44*x - 77) - 77/1024*arctan(( x^4 - x^3)^(1/4)/x) - 77/2048*log((x + (x^4 - x^3)^(1/4))/x) + 77/2048*log (-(x - (x^4 - x^3)^(1/4))/x)
\[ \int x^2 \sqrt [4]{-x^3+x^4} \, dx=\int x^{2} \sqrt [4]{x^{3} \left (x - 1\right )}\, dx \]
\[ \int x^2 \sqrt [4]{-x^3+x^4} \, dx=\int { {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} x^{2} \,d x } \]
Time = 0.27 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.45 \[ \int x^2 \sqrt [4]{-x^3+x^4} \, dx=-\frac {1}{1536} \, {\left (77 \, {\left (\frac {1}{x} - 1\right )}^{3} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 275 \, {\left (\frac {1}{x} - 1\right )}^{2} {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 351 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} - 231 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{4} - \frac {77}{1024} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) - \frac {77}{2048} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) + \frac {77}{2048} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \]
-1/1536*(77*(1/x - 1)^3*(-1/x + 1)^(1/4) + 275*(1/x - 1)^2*(-1/x + 1)^(1/4 ) - 351*(-1/x + 1)^(5/4) - 231*(-1/x + 1)^(1/4))*x^4 - 77/1024*arctan((-1/ x + 1)^(1/4)) - 77/2048*log((-1/x + 1)^(1/4) + 1) + 77/2048*log(abs((-1/x + 1)^(1/4) - 1))
Timed out. \[ \int x^2 \sqrt [4]{-x^3+x^4} \, dx=\int x^2\,{\left (x^4-x^3\right )}^{1/4} \,d x \]