Optimal. Leaf size=73 \[ \frac {\left (-77-44 x-32 x^2+384 x^3\right ) \sqrt [4]{-x^3+x^4}}{1536}+\frac {77 \text {ArcTan}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{1024}-\frac {77 \tanh ^{-1}\left (\frac {x}{\sqrt [4]{-x^3+x^4}}\right )}{1024} \]
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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(162\) vs. \(2(73)=146\).
time = 0.12, antiderivative size = 162, normalized size of antiderivative = 2.22, number of steps
used = 10, number of rules used = 8, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.471, Rules used = {2046, 2049,
2057, 65, 246, 218, 212, 209} \begin {gather*} -\frac {77 (x-1)^{3/4} x^{9/4} \text {ArcTan}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{1024 \left (x^4-x^3\right )^{3/4}}+\frac {1}{4} \sqrt [4]{x^4-x^3} x^3-\frac {11}{384} \sqrt [4]{x^4-x^3} x-\frac {77 \sqrt [4]{x^4-x^3}}{1536}-\frac {77 (x-1)^{3/4} x^{9/4} \tanh ^{-1}\left (\frac {\sqrt [4]{x-1}}{\sqrt [4]{x}}\right )}{1024 \left (x^4-x^3\right )^{3/4}}-\frac {1}{48} \sqrt [4]{x^4-x^3} x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 209
Rule 212
Rule 218
Rule 246
Rule 2046
Rule 2049
Rule 2057
Rubi steps
\begin {align*} \int x^2 \sqrt [4]{-x^3+x^4} \, dx &=\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {1}{16} \int \frac {x^5}{\left (-x^3+x^4\right )^{3/4}} \, dx\\ &=-\frac {1}{48} x^2 \sqrt [4]{-x^3+x^4}+\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {11}{192} \int \frac {x^4}{\left (-x^3+x^4\right )^{3/4}} \, dx\\ &=-\frac {11}{384} x \sqrt [4]{-x^3+x^4}-\frac {1}{48} x^2 \sqrt [4]{-x^3+x^4}+\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {77 \int \frac {x^3}{\left (-x^3+x^4\right )^{3/4}} \, dx}{1536}\\ &=-\frac {77 \sqrt [4]{-x^3+x^4}}{1536}-\frac {11}{384} x \sqrt [4]{-x^3+x^4}-\frac {1}{48} x^2 \sqrt [4]{-x^3+x^4}+\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {77 \int \frac {x^2}{\left (-x^3+x^4\right )^{3/4}} \, dx}{2048}\\ &=-\frac {77 \sqrt [4]{-x^3+x^4}}{1536}-\frac {11}{384} x \sqrt [4]{-x^3+x^4}-\frac {1}{48} x^2 \sqrt [4]{-x^3+x^4}+\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {\left (77 (-1+x)^{3/4} x^{9/4}\right ) \int \frac {1}{(-1+x)^{3/4} \sqrt [4]{x}} \, dx}{2048 \left (-x^3+x^4\right )^{3/4}}\\ &=-\frac {77 \sqrt [4]{-x^3+x^4}}{1536}-\frac {11}{384} x \sqrt [4]{-x^3+x^4}-\frac {1}{48} x^2 \sqrt [4]{-x^3+x^4}+\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {\left (77 (-1+x)^{3/4} x^{9/4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt [4]{1+x^4}} \, dx,x,\sqrt [4]{-1+x}\right )}{512 \left (-x^3+x^4\right )^{3/4}}\\ &=-\frac {77 \sqrt [4]{-x^3+x^4}}{1536}-\frac {11}{384} x \sqrt [4]{-x^3+x^4}-\frac {1}{48} x^2 \sqrt [4]{-x^3+x^4}+\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {\left (77 (-1+x)^{3/4} x^{9/4}\right ) \text {Subst}\left (\int \frac {1}{1-x^4} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{512 \left (-x^3+x^4\right )^{3/4}}\\ &=-\frac {77 \sqrt [4]{-x^3+x^4}}{1536}-\frac {11}{384} x \sqrt [4]{-x^3+x^4}-\frac {1}{48} x^2 \sqrt [4]{-x^3+x^4}+\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {\left (77 (-1+x)^{3/4} x^{9/4}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{1024 \left (-x^3+x^4\right )^{3/4}}-\frac {\left (77 (-1+x)^{3/4} x^{9/4}\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{1024 \left (-x^3+x^4\right )^{3/4}}\\ &=-\frac {77 \sqrt [4]{-x^3+x^4}}{1536}-\frac {11}{384} x \sqrt [4]{-x^3+x^4}-\frac {1}{48} x^2 \sqrt [4]{-x^3+x^4}+\frac {1}{4} x^3 \sqrt [4]{-x^3+x^4}-\frac {77 (-1+x)^{3/4} x^{9/4} \tan ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{1024 \left (-x^3+x^4\right )^{3/4}}-\frac {77 (-1+x)^{3/4} x^{9/4} \tanh ^{-1}\left (\frac {\sqrt [4]{-1+x}}{\sqrt [4]{x}}\right )}{1024 \left (-x^3+x^4\right )^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 85, normalized size = 1.16 \begin {gather*} \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 \text {ArcTan}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )-231 \tanh ^{-1}\left (\frac {1}{\sqrt [4]{\frac {-1+x}{x}}}\right )\right )}{3072 \left ((-1+x) x^3\right )^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 1.42, size = 27, normalized size = 0.37
method | result | size |
meijerg | \(\frac {4 \mathrm {signum}\left (-1+x \right )^{\frac {1}{4}} x^{\frac {15}{4}} \hypergeom \left (\left [-\frac {1}{4}, \frac {15}{4}\right ], \left [\frac {19}{4}\right ], x\right )}{15 \left (-\mathrm {signum}\left (-1+x \right )\right )^{\frac {1}{4}}}\) | \(27\) |
trager | \(\left (\frac {1}{4} x^{3}-\frac {1}{48} x^{2}-\frac {11}{384} x -\frac {77}{1536}\right ) \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+\frac {77 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-2 \sqrt {x^{4}-x^{3}}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x +2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}-2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}}{x^{2}}\right )}{2048}-\frac {77 \ln \left (\frac {2 \left (x^{4}-x^{3}\right )^{\frac {3}{4}}+2 \sqrt {x^{4}-x^{3}}\, x +2 x^{2} \left (x^{4}-x^{3}\right )^{\frac {1}{4}}+2 x^{3}-x^{2}}{x^{2}}\right )}{2048}\) | \(175\) |
risch | \(\frac {\left (384 x^{3}-32 x^{2}-44 x -77\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}}}{1536}+\frac {\left (-\frac {77 \ln \left (\frac {2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}+2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, x +2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+2 x^{3}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}-4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x -5 x^{2}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}+4 x -1}{\left (-1+x \right )^{2}}\right )}{2048}+\frac {77 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right ) x -2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{3}+2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {3}{4}}-2 \sqrt {x^{4}-3 x^{3}+3 x^{2}-x}\, \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x^{2}+5 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}+4 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}} x -4 x \RootOf \left (\textit {\_Z}^{2}+1\right )-2 \left (x^{4}-3 x^{3}+3 x^{2}-x \right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{\left (-1+x \right )^{2}}\right )}{2048}\right ) \left (x^{3} \left (-1+x \right )\right )^{\frac {1}{4}} \left (x \left (-1+x \right )^{3}\right )^{\frac {1}{4}}}{x \left (-1+x \right )}\) | \(407\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 90, normalized size = 1.23 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{2} \sqrt [4]{x^{3} \left (x - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.43, size = 106, normalized size = 1.45 \begin {gather*} \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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^2\,{\left (x^4-x^3\right )}^{1/4} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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