Optimal. Leaf size=63 \[ \frac {1}{8} \sqrt [4]{x^4-x^3} (4 x-1)+\frac {3}{16} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right )-\frac {3}{16} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x^3}}\right ) \]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 122, normalized size of antiderivative = 1.94, number of steps used = 8, number of rules used = 8, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.615, Rules used = {2004, 2024, 2032, 63, 240, 212, 206, 203} \begin {gather*} \frac {1}{2} \sqrt [4]{x^4-x^3} x-\frac {1}{8} \sqrt [4]{x^4-x^3}-\frac {3 (x-1)^{3/4} x^{9/4} \tan ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{16 \left (x^4-x^3\right )^{3/4}}-\frac {3 (x-1)^{3/4} x^{9/4} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{16 \left (x^4-x^3\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 203
Rule 206
Rule 212
Rule 240
Rule 2004
Rule 2024
Rule 2032
Rubi steps
\begin {align*} \int \sqrt [4]{-x^3+x^4} \, dx &=\frac {1}{2} x \sqrt [4]{-x^3+x^4}-\frac {1}{8} \int \frac {x^3}{\left (-x^3+x^4\right )^{3/4}} \, dx\\ &=-\frac {1}{8} \sqrt [4]{-x^3+x^4}+\frac {1}{2} x \sqrt [4]{-x^3+x^4}-\frac {3}{32} \int \frac {x^2}{\left (-x^3+x^4\right )^{3/4}} \, dx\\ &=-\frac {1}{8} \sqrt [4]{-x^3+x^4}+\frac {1}{2} x \sqrt [4]{-x^3+x^4}-\frac {\left (3 (-1+x)^{3/4} x^{9/4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{32 \left (-x^3+x^4\right )^{3/4}}\\ &=-\frac {1}{8} \sqrt [4]{-x^3+x^4}+\frac {1}{2} x \sqrt [4]{-x^3+x^4}-\frac {\left (3 (-1+x)^{3/4} x^{9/4}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{8 \left (-x^3+x^4\right )^{3/4}}\\ &=-\frac {1}{8} \sqrt [4]{-x^3+x^4}+\frac {1}{2} x \sqrt [4]{-x^3+x^4}-\frac {\left (3 (-1+x)^{3/4} x^{9/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{8 \left (-x^3+x^4\right )^{3/4}}\\ &=-\frac {1}{8} \sqrt [4]{-x^3+x^4}+\frac {1}{2} x \sqrt [4]{-x^3+x^4}-\frac {\left (3 (-1+x)^{3/4} x^{9/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{16 \left (-x^3+x^4\right )^{3/4}}-\frac {\left (3 (-1+x)^{3/4} x^{9/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{16 \left (-x^3+x^4\right )^{3/4}}\\ &=-\frac {1}{8} \sqrt [4]{-x^3+x^4}+\frac {1}{2} x \sqrt [4]{-x^3+x^4}-\frac {3 (-1+x)^{3/4} x^{9/4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{16 \left (-x^3+x^4\right )^{3/4}}-\frac {3 (-1+x)^{3/4} x^{9/4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{16 \left (-x^3+x^4\right )^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 35, normalized size = 0.56 \begin {gather*} \frac {4 \left ((x-1) x^3\right )^{5/4} \, _2F_1\left (-\frac {3}{4},\frac {5}{4};\frac {9}{4};1-x\right )}{5 x^{15/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.25, size = 63, normalized size = 1.00 \begin {gather*} \frac {1}{8} (-1+4 x) \sqrt [4]{-x^3+x^4}+\frac {3}{16} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )-\frac {3}{16} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.68, size = 80, normalized size = 1.27 \begin {gather*} \frac {1}{8} \, {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}} {\left (4 \, x - 1\right )} - \frac {3}{16} \, \arctan \left (\frac {{\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - \frac {3}{32} \, \log \left (\frac {x + {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + \frac {3}{32} \, \log \left (-\frac {x - {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 68, normalized size = 1.08 \begin {gather*} -\frac {1}{8} \, {\left ({\left (-\frac {1}{x} + 1\right )}^{\frac {5}{4}} + 3 \, {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right )} x^{2} + \frac {3}{16} \, \arctan \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}}\right ) + \frac {3}{32} \, \log \left ({\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} + 1\right ) - \frac {3}{32} \, \log \left ({\left | {\left (-\frac {1}{x} + 1\right )}^{\frac {1}{4}} - 1 \right |}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.67, size = 27, normalized size = 0.43
method | result | size |
meijerg | \(\frac {4 \mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} x^{\frac {7}{4}} \hypergeom \left (\left [-\frac {1}{4}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], x\right )}{7 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}}}\) | \(27\) |
trager | \(\left (-\frac {1}{8}+\frac {x}{2}\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-\frac {3 \ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+2 x^{3}-x^{2}}{x^{2}}\right )}{32}+\frac {3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-2 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x +2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}}{x^{2}}\right )}{32}\) | \(165\) |
risch | \(\frac {\left (-1+4 x \right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{8}+\frac {\left (\frac {3 \ln \left (\frac {2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, x +2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}-2 x^{3}+2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x +5 x^{2}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}-4 x +1}{\left (-1+x \right )^{2}}\right )}{32}+\frac {3 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right )+5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}-4 \RootOf \left (\textit {\_Z}^{2}+1\right ) x +4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x +\RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}}{\left (-1+x \right )^{2}}\right )}{32}\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x \left (-1+x \right )^{3}\right )^{\frac {1}{4}}}{x \left (-1+x \right )}\) | \(397\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left (x^{4} - x^{3}\right )}^{\frac {1}{4}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.84, size = 27, normalized size = 0.43 \begin {gather*} \frac {4\,x\,{\left (x^4-x^3\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {7}{4};\ \frac {11}{4};\ x\right )}{7\,{\left (1-x\right )}^{1/4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt [4]{x^{4} - x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________