Optimal. Leaf size=14 \[ -\frac {4 \sqrt [4]{1+x^3}}{x} \]
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Rubi [A]
time = 0.00, antiderivative size = 14, 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 {4 \sqrt [4]{x^3+1}}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 460
Rubi steps
\begin {align*} \int \frac {4+x^3}{x^2 \left (1+x^3\right )^{3/4}} \, dx &=-\frac {4 \sqrt [4]{1+x^3}}{x}\\ \end {align*}
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Mathematica [A]
time = 1.76, size = 14, normalized size = 1.00 \begin {gather*} -\frac {4 \sqrt [4]{1+x^3}}{x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 13, normalized size = 0.93
method | result | size |
trager | \(-\frac {4 \left (x^{3}+1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
risch | \(-\frac {4 \left (x^{3}+1\right )^{\frac {1}{4}}}{x}\) | \(13\) |
gosper | \(-\frac {4 \left (1+x \right ) \left (x^{2}-x +1\right )}{x \left (x^{3}+1\right )^{\frac {3}{4}}}\) | \(24\) |
meijerg | \(\frac {x^{2} \hypergeom \left (\left [\frac {2}{3}, \frac {3}{4}\right ], \left [\frac {5}{3}\right ], -x^{3}\right )}{2}-\frac {4 \hypergeom \left (\left [-\frac {1}{3}, \frac {3}{4}\right ], \left [\frac {2}{3}\right ], -x^{3}\right )}{x}\) | \(34\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 20, normalized size = 1.43 \begin {gather*} -\frac {4 \, {\left (x^{2} - x + 1\right )}^{\frac {1}{4}} {\left (x + 1\right )}^{\frac {1}{4}}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 12, normalized size = 0.86 \begin {gather*} -\frac {4 \, {\left (x^{3} + 1\right )}^{\frac {1}{4}}}{x} \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 = 0.97, size = 63, normalized size = 4.50 \begin {gather*} \frac {x^{2} \Gamma \left (\frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {2}{3}, \frac {3}{4} \\ \frac {5}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {5}{3}\right )} + \frac {4 \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {3}{4} \\ \frac {2}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x \Gamma \left (\frac {2}{3}\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 = 0.07, size = 12, normalized size = 0.86 \begin {gather*} -\frac {4\,{\left (x^3+1\right )}^{1/4}}{x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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