Optimal. Leaf size=208 \[ \frac {20}{21} x \sqrt {5+x^4}+\frac {2}{3} x^3 \sqrt {5+x^4}-\frac {10 x \sqrt {5+x^4}}{\sqrt {5}+x^2}+\frac {1}{21} x^5 \left (6+7 x^2\right ) \sqrt {5+x^4}+\frac {10 \sqrt [4]{5} \left (\sqrt {5}+x^2\right ) \sqrt {\frac {5+x^4}{\left (\sqrt {5}+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{\sqrt {5+x^4}}-\frac {5 \sqrt [4]{5} \left (21+2 \sqrt {5}\right ) \left (\sqrt {5}+x^2\right ) \sqrt {\frac {5+x^4}{\left (\sqrt {5}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{21 \sqrt {5+x^4}} \]
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Rubi [A]
time = 0.08, antiderivative size = 208, 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 = {1288, 1294,
1212, 226, 1210} \begin {gather*} -\frac {5 \sqrt [4]{5} \left (21+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 \text {ArcTan}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{21 \sqrt {x^4+5}}+\frac {10 \sqrt [4]{5} \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 )}{\sqrt {x^4+5}}+\frac {20}{21} \sqrt {x^4+5} x+\frac {2}{3} \sqrt {x^4+5} x^3-\frac {10 \sqrt {x^4+5} x}{x^2+\sqrt {5}}+\frac {1}{21} \left (7 x^2+6\right ) \sqrt {x^4+5} x^5 \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 1210
Rule 1212
Rule 1288
Rule 1294
Rubi steps
\begin {align*} \int x^4 \left (2+3 x^2\right ) \sqrt {5+x^4} \, dx &=\frac {1}{21} x^5 \left (6+7 x^2\right ) \sqrt {5+x^4}+\frac {10}{63} \int \frac {x^4 \left (18+21 x^2\right )}{\sqrt {5+x^4}} \, dx\\ &=\frac {2}{3} x^3 \sqrt {5+x^4}+\frac {1}{21} x^5 \left (6+7 x^2\right ) \sqrt {5+x^4}-\frac {2}{63} \int \frac {x^2 \left (315-90 x^2\right )}{\sqrt {5+x^4}} \, dx\\ &=\frac {20}{21} x \sqrt {5+x^4}+\frac {2}{3} x^3 \sqrt {5+x^4}+\frac {1}{21} x^5 \left (6+7 x^2\right ) \sqrt {5+x^4}+\frac {2}{189} \int \frac {-450-945 x^2}{\sqrt {5+x^4}} \, dx\\ &=\frac {20}{21} x \sqrt {5+x^4}+\frac {2}{3} x^3 \sqrt {5+x^4}+\frac {1}{21} x^5 \left (6+7 x^2\right ) \sqrt {5+x^4}+\left (10 \sqrt {5}\right ) \int \frac {1-\frac {x^2}{\sqrt {5}}}{\sqrt {5+x^4}} \, dx-\frac {1}{21} \left (10 \left (10+21 \sqrt {5}\right )\right ) \int \frac {1}{\sqrt {5+x^4}} \, dx\\ &=\frac {20}{21} x \sqrt {5+x^4}+\frac {2}{3} x^3 \sqrt {5+x^4}-\frac {10 x \sqrt {5+x^4}}{\sqrt {5}+x^2}+\frac {1}{21} x^5 \left (6+7 x^2\right ) \sqrt {5+x^4}+\frac {10 \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 {5 \sqrt [4]{5} \left (21+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 )}{21 \sqrt {5+x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 4.50, size = 82, normalized size = 0.39 \begin {gather*} \frac {1}{21} x \left (6 \left (5+x^4\right )^{3/2}+7 x^2 \left (5+x^4\right )^{3/2}-30 \sqrt {5} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {5}{4};-\frac {x^4}{5}\right )-35 \sqrt {5} x^2 \, _2F_1\left (-\frac {1}{2},\frac {3}{4};\frac {7}{4};-\frac {x^4}{5}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.16, size = 192, normalized size = 0.92
method | result | size |
meijerg | \(\frac {3 \sqrt {5}\, x^{7} \hypergeom \left (\left [-\frac {1}{2}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], -\frac {x^{4}}{5}\right )}{7}+\frac {2 \sqrt {5}\, x^{5} \hypergeom \left (\left [-\frac {1}{2}, \frac {5}{4}\right ], \left [\frac {9}{4}\right ], -\frac {x^{4}}{5}\right )}{5}\) | \(40\) |
risch | \(\frac {x \left (7 x^{6}+6 x^{4}+14 x^{2}+20\right ) \sqrt {x^{4}+5}}{21}-\frac {2 i \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )-\EllipticE \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )\right )}{\sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}-\frac {4 \sqrt {5}\, \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )}{21 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}\) | \(173\) |
default | \(\frac {x^{7} \sqrt {x^{4}+5}}{3}+\frac {2 x^{3} \sqrt {x^{4}+5}}{3}-\frac {2 i \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )-\EllipticE \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )\right )}{\sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}+\frac {2 x^{5} \sqrt {x^{4}+5}}{7}+\frac {20 x \sqrt {x^{4}+5}}{21}-\frac {4 \sqrt {5}\, \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )}{21 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}\) | \(192\) |
elliptic | \(\frac {x^{7} \sqrt {x^{4}+5}}{3}+\frac {2 x^{3} \sqrt {x^{4}+5}}{3}-\frac {2 i \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )-\EllipticE \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )\right )}{\sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}+\frac {2 x^{5} \sqrt {x^{4}+5}}{7}+\frac {20 x \sqrt {x^{4}+5}}{21}-\frac {4 \sqrt {5}\, \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )}{21 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}\) | \(192\) |
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 = 1.11, size = 78, normalized size = 0.38 \begin {gather*} \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 )}}{4 \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 )}}{2 \Gamma \left (\frac {9}{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 [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^4\,\sqrt {x^4+5}\,\left (3\,x^2+2\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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