Optimal. Leaf size=214 \[ -\frac {9 \sqrt {x^4+5}}{50 x}-\frac {\sqrt {x^4+5}}{15 x^3}+\frac {9 \sqrt {x^4+5} x}{50 \left (x^2+\sqrt {5}\right )}+\frac {\left (27-2 \sqrt {5}\right ) \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{60\ 5^{3/4} \sqrt {x^4+5}}-\frac {9 \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{10\ 5^{3/4} \sqrt {x^4+5}}+\frac {3 x^2+2}{10 \sqrt {x^4+5} x^3} \]
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Rubi [A] time = 0.11, antiderivative size = 214, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1278, 1282, 1198, 220, 1196} \[ \frac {9 \sqrt {x^4+5} x}{50 \left (x^2+\sqrt {5}\right )}-\frac {9 \sqrt {x^4+5}}{50 x}-\frac {\sqrt {x^4+5}}{15 x^3}+\frac {3 x^2+2}{10 \sqrt {x^4+5} x^3}+\frac {\left (27-2 \sqrt {5}\right ) \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{60\ 5^{3/4} \sqrt {x^4+5}}-\frac {9 \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{10\ 5^{3/4} \sqrt {x^4+5}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 1196
Rule 1198
Rule 1278
Rule 1282
Rubi steps
\begin {align*} \int \frac {2+3 x^2}{x^4 \left (5+x^4\right )^{3/2}} \, dx &=\frac {2+3 x^2}{10 x^3 \sqrt {5+x^4}}-\frac {1}{10} \int \frac {-10-9 x^2}{x^4 \sqrt {5+x^4}} \, dx\\ &=\frac {2+3 x^2}{10 x^3 \sqrt {5+x^4}}-\frac {\sqrt {5+x^4}}{15 x^3}+\frac {1}{150} \int \frac {135-10 x^2}{x^2 \sqrt {5+x^4}} \, dx\\ &=\frac {2+3 x^2}{10 x^3 \sqrt {5+x^4}}-\frac {\sqrt {5+x^4}}{15 x^3}-\frac {9 \sqrt {5+x^4}}{50 x}-\frac {1}{750} \int \frac {50-135 x^2}{\sqrt {5+x^4}} \, dx\\ &=\frac {2+3 x^2}{10 x^3 \sqrt {5+x^4}}-\frac {\sqrt {5+x^4}}{15 x^3}-\frac {9 \sqrt {5+x^4}}{50 x}-\frac {9 \int \frac {1-\frac {x^2}{\sqrt {5}}}{\sqrt {5+x^4}} \, dx}{10 \sqrt {5}}-\frac {1}{150} \left (10-27 \sqrt {5}\right ) \int \frac {1}{\sqrt {5+x^4}} \, dx\\ &=\frac {2+3 x^2}{10 x^3 \sqrt {5+x^4}}-\frac {\sqrt {5+x^4}}{15 x^3}-\frac {9 \sqrt {5+x^4}}{50 x}+\frac {9 x \sqrt {5+x^4}}{50 \left (\sqrt {5}+x^2\right )}-\frac {9 \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 )}{10\ 5^{3/4} \sqrt {5+x^4}}+\frac {\left (27-2 \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 )}{60\ 5^{3/4} \sqrt {5+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 54, normalized size = 0.25 \[ -\frac {2 \, _2F_1\left (-\frac {3}{4},\frac {3}{2};\frac {1}{4};-\frac {x^4}{5}\right )+9 x^2 \, _2F_1\left (-\frac {1}{4},\frac {3}{2};\frac {3}{4};-\frac {x^4}{5}\right )}{15 \sqrt {5} x^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 5} {\left (3 \, x^{2} + 2\right )}}{x^{12} + 10 \, x^{8} + 25 \, x^{4}}, x\right ) \]
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 \[ \int \frac {3 \, x^{2} + 2}{{\left (x^{4} + 5\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 192, normalized size = 0.90 \[ -\frac {3 x^{3}}{50 \sqrt {x^{4}+5}}-\frac {x}{25 \sqrt {x^{4}+5}}-\frac {\sqrt {5}\, \sqrt {-5 i \sqrt {5}\, x^{2}+25}\, \sqrt {5 i \sqrt {5}\, x^{2}+25}\, \EllipticF \left (\frac {\sqrt {5}\, \sqrt {i \sqrt {5}}\, x}{5}, i\right )}{375 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}-\frac {3 \sqrt {x^{4}+5}}{25 x}-\frac {2 \sqrt {x^{4}+5}}{75 x^{3}}+\frac {9 i \sqrt {-5 i \sqrt {5}\, x^{2}+25}\, \sqrt {5 i \sqrt {5}\, x^{2}+25}\, \left (-\EllipticE \left (\frac {\sqrt {5}\, \sqrt {i \sqrt {5}}\, x}{5}, i\right )+\EllipticF \left (\frac {\sqrt {5}\, \sqrt {i \sqrt {5}}\, x}{5}, i\right )\right )}{250 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}} \]
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 \[ \int \frac {3 \, x^{2} + 2}{{\left (x^{4} + 5\right )}^{\frac {3}{2}} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {3\,x^2+2}{x^4\,{\left (x^4+5\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 8.15, size = 80, normalized size = 0.37 \[ \frac {3 \sqrt {5} \Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{5}} \right )}}{100 x \Gamma \left (\frac {3}{4}\right )} + \frac {\sqrt {5} \Gamma \left (- \frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {3}{4}, \frac {3}{2} \\ \frac {1}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{5}} \right )}}{50 x^{3} \Gamma \left (\frac {1}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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