Optimal. Leaf size=18 \[ \frac {2 \left (x^3-x\right )^{3/2}}{3 x^3} \]
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Rubi [C] time = 0.10, antiderivative size = 96, normalized size of antiderivative = 5.33, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2048, 2025, 2011, 329, 222} \begin {gather*} \frac {2 \sqrt {x^3-x}}{3}-\frac {\sqrt {2} \sqrt {x-1} \sqrt {x} \sqrt {x+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x-1}}\right )|\frac {1}{2}\right )}{3 \sqrt {x^3-x}}-\frac {2 \sqrt {x^3-x}}{3 x^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 222
Rule 329
Rule 2011
Rule 2025
Rule 2048
Rubi steps
\begin {align*} \int \frac {-1+x^4}{x^2 \sqrt {-x+x^3}} \, dx &=\frac {2}{3} \sqrt {-x+x^3}-\int \frac {1}{x^2 \sqrt {-x+x^3}} \, dx\\ &=\frac {2}{3} \sqrt {-x+x^3}-\frac {2 \sqrt {-x+x^3}}{3 x^2}-\frac {1}{3} \int \frac {1}{\sqrt {-x+x^3}} \, dx\\ &=\frac {2}{3} \sqrt {-x+x^3}-\frac {2 \sqrt {-x+x^3}}{3 x^2}-\frac {\left (\sqrt {x} \sqrt {-1+x^2}\right ) \int \frac {1}{\sqrt {x} \sqrt {-1+x^2}} \, dx}{3 \sqrt {-x+x^3}}\\ &=\frac {2}{3} \sqrt {-x+x^3}-\frac {2 \sqrt {-x+x^3}}{3 x^2}-\frac {\left (2 \sqrt {x} \sqrt {-1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {-1+x^4}} \, dx,x,\sqrt {x}\right )}{3 \sqrt {-x+x^3}}\\ &=\frac {2}{3} \sqrt {-x+x^3}-\frac {2 \sqrt {-x+x^3}}{3 x^2}-\frac {\sqrt {2} \sqrt {-1+x} \sqrt {x} \sqrt {1+x} F\left (\sin ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-1+x}}\right )|\frac {1}{2}\right )}{3 \sqrt {-x+x^3}}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 \left (x \left (x^2-1\right )\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 18, normalized size = 1.00 \begin {gather*} \frac {2 \left (-x+x^3\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 19, normalized size = 1.06 \begin {gather*} \frac {2 \, \sqrt {x^{3} - x} {\left (x^{2} - 1\right )}}{3 \, x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.54, size = 25, normalized size = 1.39 \begin {gather*} \frac {2}{3} \, \sqrt {x^{3} - x} - \frac {2}{3} \, \sqrt {\frac {1}{x} - \frac {1}{x^{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 20, normalized size = 1.11
method | result | size |
trager | \(\frac {2 \left (x^{2}-1\right ) \sqrt {x^{3}-x}}{3 x^{2}}\) | \(20\) |
risch | \(\frac {\frac {2}{3} x^{4}-\frac {4}{3} x^{2}+\frac {2}{3}}{x \sqrt {x \left (x^{2}-1\right )}}\) | \(25\) |
gosper | \(\frac {2 \left (x^{2}-1\right ) \left (-1+x \right ) \left (1+x \right )}{3 \sqrt {x^{3}-x}\, x}\) | \(26\) |
default | \(\frac {2 \sqrt {x^{3}-x}}{3}-\frac {2 \sqrt {x^{3}-x}}{3 x^{2}}\) | \(27\) |
elliptic | \(\frac {2 \sqrt {x^{3}-x}}{3}-\frac {2 \sqrt {x^{3}-x}}{3 x^{2}}\) | \(27\) |
meijerg | \(\frac {2 \sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \hypergeom \left (\left [\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {9}{4}\right ], x^{2}\right ) x^{\frac {5}{2}}}{5 \sqrt {\mathrm {signum}\left (x^{2}-1\right )}}+\frac {2 \sqrt {-\mathrm {signum}\left (x^{2}-1\right )}\, \hypergeom \left (\left [-\frac {3}{4}, \frac {1}{2}\right ], \left [\frac {1}{4}\right ], x^{2}\right )}{3 \sqrt {\mathrm {signum}\left (x^{2}-1\right )}\, x^{\frac {3}{2}}}\) | \(66\) |
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*} \int \frac {x^{4} - 1}{\sqrt {x^{3} - x} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.01, size = 1, normalized size = 0.06 \begin {gather*} 0 \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 \frac {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}{x^{2} \sqrt {x \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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