Optimal. Leaf size=173 \[ -\frac {2 \sqrt {5+x^4}}{5 x}+\frac {2 x \sqrt {5+x^4}}{5 \left (\sqrt {5}+x^2\right )}-\frac {2 \left (\sqrt {5}+x^2\right ) \sqrt {\frac {5+x^4}{\left (\sqrt {5}+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{5^{3/4} \sqrt {5+x^4}}+\frac {\left (2+3 \sqrt {5}\right ) \left (\sqrt {5}+x^2\right ) \sqrt {\frac {5+x^4}{\left (\sqrt {5}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{2\ 5^{3/4} \sqrt {5+x^4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1296, 1212,
226, 1210} \begin {gather*} \frac {\left (2+3 \sqrt {5}\right ) \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} F\left (2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{2\ 5^{3/4} \sqrt {x^4+5}}-\frac {2 \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} E\left (2 \text {ArcTan}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{5^{3/4} \sqrt {x^4+5}}-\frac {2 \sqrt {x^4+5}}{5 x}+\frac {2 \sqrt {x^4+5} x}{5 \left (x^2+\sqrt {5}\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 1210
Rule 1212
Rule 1296
Rubi steps
\begin {align*} \int \frac {2+3 x^2}{x^2 \sqrt {5+x^4}} \, dx &=-\frac {2 \sqrt {5+x^4}}{5 x}-\frac {1}{5} \int \frac {-15-2 x^2}{\sqrt {5+x^4}} \, dx\\ &=-\frac {2 \sqrt {5+x^4}}{5 x}-\frac {2 \int \frac {1-\frac {x^2}{\sqrt {5}}}{\sqrt {5+x^4}} \, dx}{\sqrt {5}}+\frac {1}{5} \left (15+2 \sqrt {5}\right ) \int \frac {1}{\sqrt {5+x^4}} \, dx\\ &=-\frac {2 \sqrt {5+x^4}}{5 x}+\frac {2 x \sqrt {5+x^4}}{5 \left (\sqrt {5}+x^2\right )}-\frac {2 \left (\sqrt {5}+x^2\right ) \sqrt {\frac {5+x^4}{\left (\sqrt {5}+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{5^{3/4} \sqrt {5+x^4}}+\frac {\left (2+3 \sqrt {5}\right ) \left (\sqrt {5}+x^2\right ) \sqrt {\frac {5+x^4}{\left (\sqrt {5}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{2\ 5^{3/4} \sqrt {5+x^4}}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 10.12, size = 81, normalized size = 0.47 \begin {gather*} \frac {1}{5} \left (-\frac {2 \sqrt {5+x^4}}{x}-2 (-1)^{3/4} \sqrt [4]{5} E\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {1}{5}} x\right )\right |-1\right )-\sqrt [4]{-5} \left (-2 i+3 \sqrt {5}\right ) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-\frac {1}{5}} x\right )\right |-1\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.14, size = 158, normalized size = 0.91
method | result | size |
meijerg | \(-\frac {2 \sqrt {5}\, \hypergeom \left (\left [-\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{4}\right ], -\frac {x^{4}}{5}\right )}{5 x}+\frac {3 \sqrt {5}\, x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {5}{4}\right ], -\frac {x^{4}}{5}\right )}{5}\) | \(38\) |
default | \(\frac {3 \sqrt {5}\, \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )}{25 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}-\frac {2 \sqrt {x^{4}+5}}{5 x}+\frac {2 i \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )-\EllipticE \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )\right )}{25 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}\) | \(158\) |
risch | \(\frac {3 \sqrt {5}\, \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )}{25 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}-\frac {2 \sqrt {x^{4}+5}}{5 x}+\frac {2 i \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )-\EllipticE \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )\right )}{25 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}\) | \(158\) |
elliptic | \(\frac {3 \sqrt {5}\, \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )}{25 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}-\frac {2 \sqrt {x^{4}+5}}{5 x}+\frac {2 i \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )-\EllipticE \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )\right )}{25 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}\) | \(158\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = 0.85, size = 75, normalized size = 0.43 \begin {gather*} \frac {3 \sqrt {5} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{5}} \right )}}{20 \Gamma \left (\frac {5}{4}\right )} + \frac {\sqrt {5} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{5}} \right )}}{10 x \Gamma \left (\frac {3}{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 = 0.50, size = 48, normalized size = 0.28 \begin {gather*} \frac {3\,\sqrt {5}\,x\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ -\frac {x^4}{5}\right )}{5}-\frac {2\,\sqrt {\frac {5}{x^4}+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {3}{4};\ \frac {7}{4};\ -\frac {5}{x^4}\right )}{3\,x\,\sqrt {x^4+5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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