Optimal. Leaf size=18 \[ \frac {4 \left (x^5+x^3\right )^{3/4}}{3 x^3} \]
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Rubi [A] time = 0.06, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {2036} \begin {gather*} \frac {4 \left (x^5+x^3\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2036
Rubi steps
\begin {align*} \int \frac {-1+x^2}{x \sqrt [4]{x^3+x^5}} \, dx &=\frac {4 \left (x^3+x^5\right )^{3/4}}{3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 20, normalized size = 1.11 \begin {gather*} \frac {4 \left (x^2+1\right )}{3 \sqrt [4]{x^5+x^3}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 18, normalized size = 1.00 \begin {gather*} \frac {4 \left (x^3+x^5\right )^{3/4}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 14, normalized size = 0.78 \begin {gather*} \frac {4 \, {\left (x^{5} + x^{3}\right )}^{\frac {3}{4}}}{3 \, x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 1}{{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 15, normalized size = 0.83
method | result | size |
trager | \(\frac {4 \left (x^{5}+x^{3}\right )^{\frac {3}{4}}}{3 x^{3}}\) | \(15\) |
gosper | \(\frac {\frac {4 x^{2}}{3}+\frac {4}{3}}{\left (x^{5}+x^{3}\right )^{\frac {1}{4}}}\) | \(17\) |
risch | \(\frac {\frac {4 x^{2}}{3}+\frac {4}{3}}{\left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}\) | \(19\) |
meijerg | \(\frac {4 \hypergeom \left (\left [-\frac {3}{8}, \frac {1}{4}\right ], \left [\frac {5}{8}\right ], -x^{2}\right )}{3 x^{\frac {3}{4}}}+\frac {4 \hypergeom \left (\left [\frac {1}{4}, \frac {5}{8}\right ], \left [\frac {13}{8}\right ], -x^{2}\right ) x^{\frac {5}{4}}}{5}\) | \(34\) |
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^{2} - 1}{{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} x}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 14, normalized size = 0.78 \begin {gather*} \frac {4\,{\left (x^5+x^3\right )}^{3/4}}{3\,x^3} \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 )}{x \sqrt [4]{x^{3} \left (x^{2} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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