Optimal. Leaf size=192 \[ \frac {10}{7} \sqrt {x^4+5} x+\frac {4 \sqrt {x^4+5} x}{x^2+\sqrt {5}}+\frac {\sqrt [4]{5} \left (14-5 \sqrt {5}\right ) \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{7 \sqrt {x^4+5}}-\frac {4 \sqrt [4]{5} \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{\sqrt {x^4+5}}+\frac {1}{35} \left (15 x^2+14\right ) \sqrt {x^4+5} x^3 \]
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Rubi [A] time = 0.10, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1274, 1280, 1198, 220, 1196} \[ \frac {1}{35} \left (15 x^2+14\right ) \sqrt {x^4+5} x^3+\frac {4 \sqrt {x^4+5} x}{x^2+\sqrt {5}}+\frac {10}{7} \sqrt {x^4+5} x+\frac {\sqrt [4]{5} \left (14-5 \sqrt {5}\right ) \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{7 \sqrt {x^4+5}}-\frac {4 \sqrt [4]{5} \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{\sqrt {x^4+5}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 1196
Rule 1198
Rule 1274
Rule 1280
Rubi steps
\begin {align*} \int x^2 \left (2+3 x^2\right ) \sqrt {5+x^4} \, dx &=\frac {1}{35} x^3 \left (14+15 x^2\right ) \sqrt {5+x^4}+\frac {2}{7} \int \frac {x^2 \left (14+15 x^2\right )}{\sqrt {5+x^4}} \, dx\\ &=\frac {10}{7} x \sqrt {5+x^4}+\frac {1}{35} x^3 \left (14+15 x^2\right ) \sqrt {5+x^4}-\frac {2}{21} \int \frac {75-42 x^2}{\sqrt {5+x^4}} \, dx\\ &=\frac {10}{7} x \sqrt {5+x^4}+\frac {1}{35} x^3 \left (14+15 x^2\right ) \sqrt {5+x^4}-\left (4 \sqrt {5}\right ) \int \frac {1-\frac {x^2}{\sqrt {5}}}{\sqrt {5+x^4}} \, dx-\frac {1}{7} \left (2 \left (25-14 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt {5+x^4}} \, dx\\ &=\frac {10}{7} x \sqrt {5+x^4}+\frac {4 x \sqrt {5+x^4}}{\sqrt {5}+x^2}+\frac {1}{35} x^3 \left (14+15 x^2\right ) \sqrt {5+x^4}-\frac {4 \sqrt [4]{5} \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 )}{\sqrt {5+x^4}}+\frac {\sqrt [4]{5} \left (14-5 \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 )}{7 \sqrt {5+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.02, size = 68, normalized size = 0.35 \[ \frac {1}{21} x \left (-45 \sqrt {5} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {x^4}{5}\right )+14 \sqrt {5} x^2 \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {x^4}{5}\right )+9 \left (x^4+5\right )^{3/2}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.66, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (3 \, x^{4} + 2 \, x^{2}\right )} \sqrt {x^{4} + 5}, x\right ) \]
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 \[ \int \sqrt {x^{4} + 5} {\left (3 \, x^{2} + 2\right )} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 180, normalized size = 0.94 \[ \frac {3 \sqrt {x^{4}+5}\, x^{5}}{7}+\frac {2 \sqrt {x^{4}+5}\, x^{3}}{5}+\frac {10 \sqrt {x^{4}+5}\, x}{7}-\frac {2 \sqrt {5}\, \sqrt {-5 i \sqrt {5}\, x^{2}+25}\, \sqrt {5 i \sqrt {5}\, x^{2}+25}\, \EllipticF \left (\frac {\sqrt {5}\, \sqrt {i \sqrt {5}}\, x}{5}, i\right )}{7 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}+\frac {4 i \sqrt {-5 i \sqrt {5}\, x^{2}+25}\, \sqrt {5 i \sqrt {5}\, x^{2}+25}\, \left (-\EllipticE \left (\frac {\sqrt {5}\, \sqrt {i \sqrt {5}}\, x}{5}, i\right )+\EllipticF \left (\frac {\sqrt {5}\, \sqrt {i \sqrt {5}}\, x}{5}, i\right )\right )}{5 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}} \]
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 \[ \int \sqrt {x^{4} + 5} {\left (3 \, x^{2} + 2\right )} x^{2}\,{d x} \]
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 \[ \int x^2\,\sqrt {x^4+5}\,\left (3\,x^2+2\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.14, size = 78, normalized size = 0.41 \[ \frac {3 \sqrt {5} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{5}} \right )}}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {\sqrt {5} x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{5}} \right )}}{2 \Gamma \left (\frac {7}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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