Optimal. Leaf size=63 \[ \frac {1}{16} \tan ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x}}\right )-\frac {1}{16} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{x^4-x}}\right )+\frac {1}{24} \sqrt [4]{x^4-x} \left (4 x^5-x^2\right ) \]
________________________________________________________________________________________
Rubi [B] time = 0.12, antiderivative size = 127, normalized size of antiderivative = 2.02, number of steps used = 9, number of rules used = 9, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {2021, 2024, 2032, 329, 275, 331, 298, 203, 206} \begin {gather*} \frac {1}{6} \sqrt [4]{x^4-x} x^5-\frac {1}{24} \sqrt [4]{x^4-x} x^2+\frac {\left (x^3-1\right )^{3/4} x^{3/4} \tan ^{-1}\left (\frac {x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{16 \left (x^4-x\right )^{3/4}}-\frac {\left (x^3-1\right )^{3/4} x^{3/4} \tanh ^{-1}\left (\frac {x^{3/4}}{\sqrt [4]{x^3-1}}\right )}{16 \left (x^4-x\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 206
Rule 275
Rule 298
Rule 329
Rule 331
Rule 2021
Rule 2024
Rule 2032
Rubi steps
\begin {align*} \int x^4 \sqrt [4]{-x+x^4} \, dx &=\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {1}{8} \int \frac {x^5}{\left (-x+x^4\right )^{3/4}} \, dx\\ &=-\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {3}{32} \int \frac {x^2}{\left (-x+x^4\right )^{3/4}} \, dx\\ &=-\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {\left (3 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \int \frac {x^{5/4}}{\left (-1+x^3\right )^{3/4}} \, dx}{32 \left (-x+x^4\right )^{3/4}}\\ &=-\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {\left (3 x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^8}{\left (-1+x^{12}\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{8 \left (-x+x^4\right )^{3/4}}\\ &=-\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {\left (x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-1+x^4\right )^{3/4}} \, dx,x,x^{3/4}\right )}{8 \left (-x+x^4\right )^{3/4}}\\ &=-\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {\left (x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {x^2}{1-x^4} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{8 \left (-x+x^4\right )^{3/4}}\\ &=-\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}-\frac {\left (x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{16 \left (-x+x^4\right )^{3/4}}+\frac {\left (x^{3/4} \left (-1+x^3\right )^{3/4}\right ) \operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{16 \left (-x+x^4\right )^{3/4}}\\ &=-\frac {1}{24} x^2 \sqrt [4]{-x+x^4}+\frac {1}{6} x^5 \sqrt [4]{-x+x^4}+\frac {x^{3/4} \left (-1+x^3\right )^{3/4} \tan ^{-1}\left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{16 \left (-x+x^4\right )^{3/4}}-\frac {x^{3/4} \left (-1+x^3\right )^{3/4} \tanh ^{-1}\left (\frac {x^{3/4}}{\sqrt [4]{-1+x^3}}\right )}{16 \left (-x+x^4\right )^{3/4}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 56, normalized size = 0.89 \begin {gather*} \frac {x^2 \sqrt [4]{x \left (x^3-1\right )} \left (\, _2F_1\left (-\frac {1}{4},\frac {3}{4};\frac {7}{4};x^3\right )-\left (1-x^3\right )^{5/4}\right )}{6 \sqrt [4]{1-x^3}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.26, size = 63, normalized size = 1.00 \begin {gather*} \frac {1}{24} \sqrt [4]{-x+x^4} \left (-x^2+4 x^5\right )+\frac {1}{16} \tan ^{-1}\left (\frac {x}{\sqrt [4]{-x+x^4}}\right )-\frac {1}{16} \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 2.07, size = 99, normalized size = 1.57 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.24, size = 68, normalized size = 1.08 \begin {gather*} -\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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 5.26, size = 33, normalized size = 0.52
method | result | size |
meijerg | \(\frac {4 \mathrm {signum}\left (x^{3}-1\right )^{\frac {1}{4}} x^{\frac {21}{4}} \hypergeom \left (\left [-\frac {1}{4}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], x^{3}\right )}{21 \left (-\mathrm {signum}\left (x^{3}-1\right )\right )^{\frac {1}{4}}}\) | \(33\) |
trager | \(\frac {x^{2} \left (4 x^{3}-1\right ) \left (x^{4}-x \right )^{\frac {1}{4}}}{24}+\frac {\ln \left (-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x \sqrt {x^{4}-x}-2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}+2 x^{3}-1\right )}{32}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-2 \sqrt {x^{4}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x +2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{4}-x \right )^{\frac {3}{4}}+2 x^{2} \left (x^{4}-x \right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right )\right )}{32}\) | \(142\) |
risch | \(\frac {x^{2} \left (4 x^{3}-1\right ) \left (x \left (x^{3}-1\right )\right )^{\frac {1}{4}}}{24}+\frac {\left (\frac {\ln \left (-\frac {-2 x^{9}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{6}+5 x^{6}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}-4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} x^{3}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}-4 x^{3}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}+2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}+1}{\left (-1+x \right )^{2} \left (x^{2}+x +1\right )^{2}}\right )}{32}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-2 x^{9}-2 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}+5 x^{6}+2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}\, x^{3}+4 \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {3}{4}}-4 x^{3}-2 \sqrt {x^{12}-3 x^{9}+3 x^{6}-x^{3}}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) \left (x^{12}-3 x^{9}+3 x^{6}-x^{3}\right )^{\frac {1}{4}}+1}{\left (-1+x \right )^{2} \left (x^{2}+x +1\right )^{2}}\right )}{32}\right ) \left (x \left (x^{3}-1\right )\right )^{\frac {1}{4}} \left (x^{3} \left (x^{3}-1\right )^{3}\right )^{\frac {1}{4}}}{x \left (x^{3}-1\right )}\) | \(450\) |
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\right )}^{\frac {1}{4}} x^{4}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int x^4\,{\left (x^4-x\right )}^{1/4} \,d x \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 x^{4} \sqrt [4]{x \left (x - 1\right ) \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________