Optimal. Leaf size=196 \[ -\frac {x^3 \left (15-2 x^2\right )}{10 \sqrt {5+x^4}}-\frac {1}{5} x \sqrt {5+x^4}+\frac {9 x \sqrt {5+x^4}}{2 \left (\sqrt {5}+x^2\right )}-\frac {9 \sqrt [4]{5} \left (\sqrt {5}+x^2\right ) \sqrt {\frac {5+x^4}{\left (\sqrt {5}+x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{2 \sqrt {5+x^4}}+\frac {\left (2+9 \sqrt {5}\right ) \left (\sqrt {5}+x^2\right ) \sqrt {\frac {5+x^4}{\left (\sqrt {5}+x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{4 \sqrt [4]{5} \sqrt {5+x^4}} \]
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Rubi [A]
time = 0.06, antiderivative size = 196, 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 = {1290, 1294,
1212, 226, 1210} \begin {gather*} \frac {\left (2+9 \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 )}{4 \sqrt [4]{5} \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 \text {ArcTan}\left (\frac {x}{\sqrt [4]{5}}\right )|\frac {1}{2}\right )}{2 \sqrt {x^4+5}}-\frac {1}{5} \sqrt {x^4+5} x+\frac {9 \sqrt {x^4+5} x}{2 \left (x^2+\sqrt {5}\right )}-\frac {\left (15-2 x^2\right ) x^3}{10 \sqrt {x^4+5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 1210
Rule 1212
Rule 1290
Rule 1294
Rubi steps
\begin {align*} \int \frac {x^4 \left (2+3 x^2\right )}{\left (5+x^4\right )^{3/2}} \, dx &=-\frac {x^3 \left (15-2 x^2\right )}{10 \sqrt {5+x^4}}+\frac {1}{10} \int \frac {x^2 \left (45-6 x^2\right )}{\sqrt {5+x^4}} \, dx\\ &=-\frac {x^3 \left (15-2 x^2\right )}{10 \sqrt {5+x^4}}-\frac {1}{5} x \sqrt {5+x^4}-\frac {1}{30} \int \frac {-30-135 x^2}{\sqrt {5+x^4}} \, dx\\ &=-\frac {x^3 \left (15-2 x^2\right )}{10 \sqrt {5+x^4}}-\frac {1}{5} x \sqrt {5+x^4}-\frac {1}{2} \left (9 \sqrt {5}\right ) \int \frac {1-\frac {x^2}{\sqrt {5}}}{\sqrt {5+x^4}} \, dx-\frac {1}{2} \left (-2-9 \sqrt {5}\right ) \int \frac {1}{\sqrt {5+x^4}} \, dx\\ &=-\frac {x^3 \left (15-2 x^2\right )}{10 \sqrt {5+x^4}}-\frac {1}{5} x \sqrt {5+x^4}+\frac {9 x \sqrt {5+x^4}}{2 \left (\sqrt {5}+x^2\right )}-\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 )}{2 \sqrt {5+x^4}}+\frac {\left (2+9 \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 \sqrt [4]{5} \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 = 10.03, size = 70, normalized size = 0.36 \begin {gather*} \frac {x \left (-1+3 x^2\right )}{\sqrt {5+x^4}}+\frac {x \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {x^4}{5}\right )}{\sqrt {5}}-\frac {3 x^3 \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-\frac {x^4}{5}\right )}{\sqrt {5}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.14, size = 168, normalized size = 0.86
method | result | size |
meijerg | \(\frac {3 \sqrt {5}\, x^{7} \hypergeom \left (\left [\frac {3}{2}, \frac {7}{4}\right ], \left [\frac {11}{4}\right ], -\frac {x^{4}}{5}\right )}{175}+\frac {2 \sqrt {5}\, x^{5} \hypergeom \left (\left [\frac {5}{4}, \frac {3}{2}\right ], \left [\frac {9}{4}\right ], -\frac {x^{4}}{5}\right )}{125}\) | \(40\) |
risch | \(-\frac {x \left (3 x^{2}+2\right )}{2 \sqrt {x^{4}+5}}+\frac {9 i \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )-\EllipticE \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )\right )}{10 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}+\frac {\sqrt {5}\, \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )}{25 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}\) | \(163\) |
elliptic | \(-\frac {2 \left (\frac {3}{4} x^{3}+\frac {1}{2} x \right )}{\sqrt {x^{4}+5}}+\frac {9 i \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )-\EllipticE \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )\right )}{10 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}+\frac {\sqrt {5}\, \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )}{25 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}\) | \(164\) |
default | \(-\frac {3 x^{3}}{2 \sqrt {x^{4}+5}}+\frac {9 i \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \left (\EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )-\EllipticE \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )\right )}{10 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}-\frac {x}{\sqrt {x^{4}+5}}+\frac {\sqrt {5}\, \sqrt {25-5 i \sqrt {5}\, x^{2}}\, \sqrt {25+5 i \sqrt {5}\, x^{2}}\, \EllipticF \left (\frac {x \sqrt {5}\, \sqrt {i \sqrt {5}}}{5}, i\right )}{25 \sqrt {i \sqrt {5}}\, \sqrt {x^{4}+5}}\) | \(168\) |
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 = 2.42, size = 75, 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 {3}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {x^{4} e^{i \pi }}{5}} \right )}}{100 \Gamma \left (\frac {11}{4}\right )} + \frac {\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 )}}{50 \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 = -0.01 \begin {gather*} \int \frac {x^4\,\left (3\,x^2+2\right )}{{\left (x^4+5\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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