Optimal. Leaf size=177 \[ -\frac {\sqrt {x^4+5} x}{5 \left (x^2+\sqrt {5}\right )}-\frac {\left (15-2 x^2\right ) x}{10 \sqrt {x^4+5}}-\frac {\left (2-3 \sqrt {5}\right ) \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{4\ 5^{3/4} \sqrt {x^4+5}}+\frac {\left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{5^{3/4} \sqrt {x^4+5}} \]
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Rubi [A] time = 0.07, antiderivative size = 177, 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 = {1276, 1198, 220, 1196} \[ -\frac {\sqrt {x^4+5} x}{5 \left (x^2+\sqrt {5}\right )}-\frac {\left (15-2 x^2\right ) x}{10 \sqrt {x^4+5}}-\frac {\left (2-3 \sqrt {5}\right ) \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{4\ 5^{3/4} \sqrt {x^4+5}}+\frac {\left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{5^{3/4} \sqrt {x^4+5}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 1196
Rule 1198
Rule 1276
Rubi steps
\begin {align*} \int \frac {x^2 \left (2+3 x^2\right )}{\left (5+x^4\right )^{3/2}} \, dx &=-\frac {x \left (15-2 x^2\right )}{10 \sqrt {5+x^4}}+\frac {1}{10} \int \frac {15-2 x^2}{\sqrt {5+x^4}} \, dx\\ &=-\frac {x \left (15-2 x^2\right )}{10 \sqrt {5+x^4}}+\frac {\int \frac {1-\frac {x^2}{\sqrt {5}}}{\sqrt {5+x^4}} \, dx}{\sqrt {5}}+\frac {1}{10} \left (15-2 \sqrt {5}\right ) \int \frac {1}{\sqrt {5+x^4}} \, dx\\ &=-\frac {x \left (15-2 x^2\right )}{10 \sqrt {5+x^4}}-\frac {x \sqrt {5+x^4}}{5 \left (\sqrt {5}+x^2\right )}+\frac {\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 )}{4\ 5^{3/4} \sqrt {5+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 68, normalized size = 0.38 \[ \frac {1}{150} x \left (45 \sqrt {5} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {x^4}{5}\right )+4 \sqrt {5} x^2 \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\frac {x^4}{5}\right )-\frac {225}{\sqrt {x^4+5}}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (3 \, x^{4} + 2 \, x^{2}\right )} \sqrt {x^{4} + 5}}{x^{8} + 10 \, x^{4} + 25}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x^{2} + 2\right )} x^{2}}{{\left (x^{4} + 5\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 168, normalized size = 0.95 \[ \frac {x^{3}}{5 \sqrt {x^{4}+5}}-\frac {3 x}{2 \sqrt {x^{4}+5}}+\frac {3 \sqrt {5}\, \sqrt {-5 i \sqrt {5}\, x^{2}+25}\, \sqrt {5 i \sqrt {5}\, x^{2}+25}\, \EllipticF \left (\frac {\sqrt {5}\, \sqrt {i \sqrt {5}}\, x}{5}, i\right )}{50 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}-\frac {i \sqrt {-5 i \sqrt {5}\, x^{2}+25}\, \sqrt {5 i \sqrt {5}\, x^{2}+25}\, \left (-\EllipticE \left (\frac {\sqrt {5}\, \sqrt {i \sqrt {5}}\, x}{5}, i\right )+\EllipticF \left (\frac {\sqrt {5}\, \sqrt {i \sqrt {5}}\, x}{5}, i\right )\right )}{25 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}} \]
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 \[ \int \frac {{\left (3 \, x^{2} + 2\right )} x^{2}}{{\left (x^{4} + 5\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,\left (3\,x^2+2\right )}{{\left (x^4+5\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 5.09, size = 75, normalized size = 0.42 \[ \frac {3 \sqrt {5} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {5}{4}, \frac {3}{2} \\ \frac {9}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{5}} \right )}}{100 \Gamma \left (\frac {9}{4}\right )} + \frac {\sqrt {5} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{5}} \right )}}{50 \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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