Optimal. Leaf size=16 \[ \frac {2 \left (x^3+1\right )^{5/2}}{5 x^5} \]
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Rubi [A] time = 0.01, 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 = {449} \begin {gather*} \frac {2 \left (x^3+1\right )^{5/2}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rubi steps
\begin {align*} \int \frac {\left (-2+x^3\right ) \left (1+x^3\right )^{3/2}}{x^6} \, dx &=\frac {2 \left (1+x^3\right )^{5/2}}{5 x^5}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 \left (x^3+1\right )^{5/2}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.06, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 \left (1+x^3\right )^{5/2}}{5 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 22, normalized size = 1.38 \begin {gather*} \frac {2 \, {\left (x^{6} + 2 \, x^{3} + 1\right )} \sqrt {x^{3} + 1}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.50, size = 27, normalized size = 1.69 \begin {gather*} \frac {2}{5} \, \sqrt {x^{3} + 1} x + \frac {2}{5} \, \sqrt {\frac {1}{x} + \frac {1}{x^{4}}} {\left (\frac {1}{x^{3}} + 2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 23, normalized size = 1.44
method | result | size |
trager | \(\frac {2 \left (x^{6}+2 x^{3}+1\right ) \sqrt {x^{3}+1}}{5 x^{5}}\) | \(23\) |
gosper | \(\frac {2 \left (1+x \right ) \left (x^{2}-x +1\right ) \left (x^{3}+1\right )^{\frac {3}{2}}}{5 x^{5}}\) | \(24\) |
risch | \(\frac {\frac {2}{5} x^{9}+\frac {6}{5} x^{6}+\frac {6}{5} x^{3}+\frac {2}{5}}{x^{5} \sqrt {x^{3}+1}}\) | \(28\) |
meijerg | \(\frac {2 \hypergeom \left (\left [-\frac {5}{3}, -\frac {3}{2}\right ], \left [-\frac {2}{3}\right ], -x^{3}\right )}{5 x^{5}}-\frac {\hypergeom \left (\left [-\frac {3}{2}, -\frac {2}{3}\right ], \left [\frac {1}{3}\right ], -x^{3}\right )}{2 x^{2}}\) | \(34\) |
default | \(\frac {2 \sqrt {x^{3}+1}}{5 x^{5}}+\frac {4 \sqrt {x^{3}+1}}{5 x^{2}}+\frac {2 x \sqrt {x^{3}+1}}{5}\) | \(36\) |
elliptic | \(\frac {2 \sqrt {x^{3}+1}}{5 x^{5}}+\frac {4 \sqrt {x^{3}+1}}{5 x^{2}}+\frac {2 x \sqrt {x^{3}+1}}{5}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.51, size = 30, normalized size = 1.88 \begin {gather*} \frac {2 \, {\left (x^{6} + 2 \, x^{3} + 1\right )} \sqrt {x^{2} - x + 1} \sqrt {x + 1}}{5 \, x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 12, normalized size = 0.75 \begin {gather*} \frac {2\,{\left (x^3+1\right )}^{5/2}}{5\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.89, size = 105, normalized size = 6.56 \begin {gather*} \frac {x \Gamma \left (\frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{3} \\ \frac {4}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 \Gamma \left (\frac {4}{3}\right )} - \frac {\Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {1}{2} \\ \frac {1}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x^{2} \Gamma \left (\frac {1}{3}\right )} - \frac {2 \Gamma \left (- \frac {5}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {5}{3}, - \frac {1}{2} \\ - \frac {2}{3} \end {matrix}\middle | {x^{3} e^{i \pi }} \right )}}{3 x^{5} \Gamma \left (- \frac {2}{3}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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