Optimal. Leaf size=185 \[ \frac {2}{3} \sqrt {x^4+5} x+\frac {3}{5} \sqrt {x^4+5} x^3-\frac {9 \sqrt {x^4+5} x}{x^2+\sqrt {5}}-\frac {\sqrt [4]{5} \left (27+2 \sqrt {5}\right ) \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{6 \sqrt {x^4+5}}+\frac {9 \sqrt [4]{5} \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{\sqrt {x^4+5}} \]
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Rubi [A] time = 0.09, antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1280, 1198, 220, 1196} \[ \frac {3}{5} \sqrt {x^4+5} x^3-\frac {9 \sqrt {x^4+5} x}{x^2+\sqrt {5}}+\frac {2}{3} \sqrt {x^4+5} x-\frac {\sqrt [4]{5} \left (27+2 \sqrt {5}\right ) \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{6 \sqrt {x^4+5}}+\frac {9 \sqrt [4]{5} \left (x^2+\sqrt {5}\right ) \sqrt {\frac {x^4+5}{\left (x^2+\sqrt {5}\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{\sqrt {x^4+5}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 1196
Rule 1198
Rule 1280
Rubi steps
\begin {align*} \int \frac {x^4 \left (2+3 x^2\right )}{\sqrt {5+x^4}} \, dx &=\frac {3}{5} x^3 \sqrt {5+x^4}-\frac {1}{5} \int \frac {x^2 \left (45-10 x^2\right )}{\sqrt {5+x^4}} \, dx\\ &=\frac {2}{3} x \sqrt {5+x^4}+\frac {3}{5} x^3 \sqrt {5+x^4}+\frac {1}{15} \int \frac {-50-135 x^2}{\sqrt {5+x^4}} \, dx\\ &=\frac {2}{3} x \sqrt {5+x^4}+\frac {3}{5} x^3 \sqrt {5+x^4}+\left (9 \sqrt {5}\right ) \int \frac {1-\frac {x^2}{\sqrt {5}}}{\sqrt {5+x^4}} \, dx-\frac {1}{3} \left (10+27 \sqrt {5}\right ) \int \frac {1}{\sqrt {5+x^4}} \, dx\\ &=\frac {2}{3} x \sqrt {5+x^4}+\frac {3}{5} x^3 \sqrt {5+x^4}-\frac {9 x \sqrt {5+x^4}}{\sqrt {5}+x^2}+\frac {9 \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 (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 )}{6 \sqrt {5+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.03, size = 74, normalized size = 0.40 \[ \frac {1}{15} x \left (-10 \sqrt {5} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {x^4}{5}\right )-9 \sqrt {5} x^2 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {x^4}{5}\right )+\left (9 x^2+10\right ) \sqrt {x^4+5}\right ) \]
Antiderivative was successfully verified.
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fricas [F] time = 0.68, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {3 \, x^{6} + 2 \, x^{4}}{\sqrt {x^{4} + 5}}, x\right ) \]
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 \[ \int \frac {{\left (3 \, x^{2} + 2\right )} x^{4}}{\sqrt {x^{4} + 5}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.02, size = 168, normalized size = 0.91 \[ \frac {3 \sqrt {x^{4}+5}\, x^{3}}{5}+\frac {2 \sqrt {x^{4}+5}\, x}{3}-\frac {2 \sqrt {5}\, \sqrt {-5 i \sqrt {5}\, x^{2}+25}\, \sqrt {5 i \sqrt {5}\, x^{2}+25}\, \EllipticF \left (\frac {\sqrt {5}\, \sqrt {i \sqrt {5}}\, x}{5}, i\right )}{15 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}-\frac {9 i \sqrt {-5 i \sqrt {5}\, x^{2}+25}\, \sqrt {5 i \sqrt {5}\, x^{2}+25}\, \left (-\EllipticE \left (\frac {\sqrt {5}\, \sqrt {i \sqrt {5}}\, x}{5}, i\right )+\EllipticF \left (\frac {\sqrt {5}\, \sqrt {i \sqrt {5}}\, x}{5}, i\right )\right )}{5 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}} \]
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 \[ \int \frac {{\left (3 \, x^{2} + 2\right )} x^{4}}{\sqrt {x^{4} + 5}}\,{d x} \]
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 \[ \int \frac {x^4\,\left (3\,x^2+2\right )}{\sqrt {x^4+5}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 2.55, size = 75, normalized size = 0.41 \[ \frac {3 \sqrt {5} x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{5}} \right )}}{20 \Gamma \left (\frac {11}{4}\right )} + \frac {\sqrt {5} x^{5} \Gamma \left (\frac {5}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {5}{4} \\ \frac {9}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{5}} \right )}}{10 \Gamma \left (\frac {9}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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