Optimal. Leaf size=23 \[ \frac {2 x \sqrt {2 x^3+x}}{2 x^2+1} \]
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Rubi [A] time = 0.08, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2056, 449} \begin {gather*} \frac {2 x \sqrt {2 x^3+x}}{2 x^2+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 449
Rule 2056
Rubi steps
\begin {align*} \int \frac {\left (3+2 x^2\right ) \sqrt {x+2 x^3}}{\left (1+2 x^2\right )^2} \, dx &=\frac {\sqrt {x+2 x^3} \int \frac {\sqrt {x} \left (3+2 x^2\right )}{\left (1+2 x^2\right )^{3/2}} \, dx}{\sqrt {x} \sqrt {1+2 x^2}}\\ &=\frac {2 x \sqrt {x+2 x^3}}{1+2 x^2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 x \sqrt {2 x^3+x}}{2 x^2+1} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 23, normalized size = 1.00 \begin {gather*} \frac {2 x \sqrt {x+2 x^3}}{1+2 x^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 21, normalized size = 0.91 \begin {gather*} \frac {2 \, \sqrt {2 \, x^{3} + x} x}{2 \, x^{2} + 1} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.31, size = 13, normalized size = 0.57 \begin {gather*} \frac {2}{\sqrt {\frac {2}{x} + \frac {1}{x^{3}}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.12, size = 17, normalized size = 0.74
method | result | size |
default | \(\frac {x^{2} \sqrt {2}}{\sqrt {\left (x^{2}+\frac {1}{2}\right ) x}}\) | \(17\) |
elliptic | \(\frac {x^{2} \sqrt {2}}{\sqrt {\left (x^{2}+\frac {1}{2}\right ) x}}\) | \(17\) |
gosper | \(\frac {2 x \sqrt {2 x^{3}+x}}{2 x^{2}+1}\) | \(22\) |
trager | \(\frac {2 x \sqrt {2 x^{3}+x}}{2 x^{2}+1}\) | \(22\) |
meijerg | \(2 \hypergeom \left (\left [\frac {3}{4}, \frac {3}{2}\right ], \left [\frac {7}{4}\right ], -2 x^{2}\right ) x^{\frac {3}{2}}+\frac {4 \hypergeom \left (\left [\frac {3}{2}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], -2 x^{2}\right ) x^{\frac {7}{2}}}{7}\) | \(34\) |
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 {\sqrt {2 \, x^{3} + x} {\left (2 \, x^{2} + 3\right )}}{{\left (2 \, x^{2} + 1\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 14, normalized size = 0.61 \begin {gather*} \frac {2\,x^2}{\sqrt {2\,x^3+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 {x \left (2 x^{2} + 1\right )} \left (2 x^{2} + 3\right )}{\left (2 x^{2} + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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