Optimal. Leaf size=16 \[ \frac {4 \sqrt [4]{x^5+x^3}}{x} \]
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Rubi [A] time = 0.14, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2056, 449} \begin {gather*} \frac {4 \sqrt [4]{x^5+x^3}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rule 2056
Rubi steps
\begin {align*} \int \frac {\left (-1+x^2\right ) \sqrt [4]{x^3+x^5}}{x^2 \left (1+x^2\right )} \, dx &=\frac {\sqrt [4]{x^3+x^5} \int \frac {-1+x^2}{x^{5/4} \left (1+x^2\right )^{3/4}} \, dx}{x^{3/4} \sqrt [4]{1+x^2}}\\ &=\frac {4 \sqrt [4]{x^3+x^5}}{x}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{x^5+x^3}}{x} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.08, size = 16, normalized size = 1.00 \begin {gather*} \frac {4 \sqrt [4]{x^3+x^5}}{x} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 14, normalized size = 0.88 \begin {gather*} \frac {4 \, {\left (x^{5} + x^{3}\right )}^{\frac {1}{4}}}{x} \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 {{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 1\right )} x^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 15, normalized size = 0.94
method | result | size |
gosper | \(\frac {4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{x}\) | \(15\) |
trager | \(\frac {4 \left (x^{5}+x^{3}\right )^{\frac {1}{4}}}{x}\) | \(15\) |
risch | \(\frac {4 \left (x^{3} \left (x^{2}+1\right )\right )^{\frac {1}{4}}}{x}\) | \(17\) |
meijerg | \(\frac {4 \hypergeom \left (\left [-\frac {1}{8}, \frac {3}{4}\right ], \left [\frac {7}{8}\right ], -x^{2}\right )}{x^{\frac {1}{4}}}+\frac {4 \hypergeom \left (\left [\frac {3}{4}, \frac {7}{8}\right ], \left [\frac {15}{8}\right ], -x^{2}\right ) x^{\frac {7}{4}}}{7}\) | \(34\) |
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 {{\left (x^{5} + x^{3}\right )}^{\frac {1}{4}} {\left (x^{2} - 1\right )}}{{\left (x^{2} + 1\right )} x^{2}}\,{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 = 0.88 \begin {gather*} \frac {4\,{\left (x^5+x^3\right )}^{1/4}}{x} \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 {\sqrt [4]{x^{3} \left (x^{2} + 1\right )} \left (x - 1\right ) \left (x + 1\right )}{x^{2} \left (x^{2} + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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