Optimal. Leaf size=16 \[ \frac {3 \left (1+x^4\right )^{5/3}}{5 x^5} \]
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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {460}
\begin {gather*} \frac {3 \left (x^4+1\right )^{5/3}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 460
Rubi steps
\begin {align*} \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^6} \, dx &=\frac {3 \left (1+x^4\right )^{5/3}}{5 x^5}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 16, normalized size = 1.00 \begin {gather*} \frac {3 \left (1+x^4\right )^{5/3}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.27, size = 13, normalized size = 0.81
method | result | size |
gosper | \(\frac {3 \left (x^{4}+1\right )^{\frac {5}{3}}}{5 x^{5}}\) | \(13\) |
trager | \(\frac {3 \left (x^{4}+1\right )^{\frac {5}{3}}}{5 x^{5}}\) | \(13\) |
risch | \(\frac {\frac {3}{5} x^{8}+\frac {6}{5} x^{4}+\frac {3}{5}}{x^{5} \left (x^{4}+1\right )^{\frac {1}{3}}}\) | \(23\) |
meijerg | \(\frac {3 \hypergeom \left (\left [-\frac {5}{4}, -\frac {2}{3}\right ], \left [-\frac {1}{4}\right ], -x^{4}\right )}{5 x^{5}}-\frac {\hypergeom \left (\left [-\frac {2}{3}, -\frac {1}{4}\right ], \left [\frac {3}{4}\right ], -x^{4}\right )}{x}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 12, normalized size = 0.75 \begin {gather*} \frac {3 \, {\left (x^{4} + 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.33, size = 12, normalized size = 0.75 \begin {gather*} \frac {3 \, {\left (x^{4} + 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 1.46, size = 73, normalized size = 4.56 \begin {gather*} \frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{4} \\ \frac {3}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} - \frac {3 \Gamma \left (- \frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{4}, - \frac {2}{3} \\ - \frac {1}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x^{5} \Gamma \left (- \frac {1}{4}\right )} \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*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.19, size = 27, normalized size = 1.69 \begin {gather*} \frac {3\,{\left (x^4+1\right )}^{2/3}+3\,x^4\,{\left (x^4+1\right )}^{2/3}}{5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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