Optimal. Leaf size=65 \[ \frac{1}{6} \sqrt{5-x^2} x^5-\frac{5}{24} \sqrt{5-x^2} x^3-\frac{25}{16} \sqrt{5-x^2} x+\frac{125}{16} \sin ^{-1}\left (\frac{x}{\sqrt{5}}\right ) \]
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Rubi [A] time = 0.0173211, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {279, 321, 216} \[ \frac{1}{6} \sqrt{5-x^2} x^5-\frac{5}{24} \sqrt{5-x^2} x^3-\frac{25}{16} \sqrt{5-x^2} x+\frac{125}{16} \sin ^{-1}\left (\frac{x}{\sqrt{5}}\right ) \]
Antiderivative was successfully verified.
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Rule 279
Rule 321
Rule 216
Rubi steps
\begin{align*} \int x^4 \sqrt{5-x^2} \, dx &=\frac{1}{6} x^5 \sqrt{5-x^2}+\frac{5}{6} \int \frac{x^4}{\sqrt{5-x^2}} \, dx\\ &=-\frac{5}{24} x^3 \sqrt{5-x^2}+\frac{1}{6} x^5 \sqrt{5-x^2}+\frac{25}{8} \int \frac{x^2}{\sqrt{5-x^2}} \, dx\\ &=-\frac{25}{16} x \sqrt{5-x^2}-\frac{5}{24} x^3 \sqrt{5-x^2}+\frac{1}{6} x^5 \sqrt{5-x^2}+\frac{125}{16} \int \frac{1}{\sqrt{5-x^2}} \, dx\\ &=-\frac{25}{16} x \sqrt{5-x^2}-\frac{5}{24} x^3 \sqrt{5-x^2}+\frac{1}{6} x^5 \sqrt{5-x^2}+\frac{125}{16} \sin ^{-1}\left (\frac{x}{\sqrt{5}}\right )\\ \end{align*}
Mathematica [A] time = 0.0218373, size = 40, normalized size = 0.62 \[ \frac{1}{48} \left (x \sqrt{5-x^2} \left (8 x^4-10 x^2-75\right )+375 \sin ^{-1}\left (\frac{x}{\sqrt{5}}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 49, normalized size = 0.8 \begin{align*} -{\frac{{x}^{3}}{6} \left ( -{x}^{2}+5 \right ) ^{{\frac{3}{2}}}}-{\frac{5\,x}{8} \left ( -{x}^{2}+5 \right ) ^{{\frac{3}{2}}}}+{\frac{25\,x}{16}\sqrt{-{x}^{2}+5}}+{\frac{125}{16}\arcsin \left ({\frac{x\sqrt{5}}{5}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54181, size = 65, normalized size = 1. \begin{align*} -\frac{1}{6} \,{\left (-x^{2} + 5\right )}^{\frac{3}{2}} x^{3} - \frac{5}{8} \,{\left (-x^{2} + 5\right )}^{\frac{3}{2}} x + \frac{25}{16} \, \sqrt{-x^{2} + 5} x + \frac{125}{16} \, \arcsin \left (\frac{1}{5} \, \sqrt{5} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.7991, size = 107, normalized size = 1.65 \begin{align*} \frac{1}{48} \,{\left (8 \, x^{5} - 10 \, x^{3} - 75 \, x\right )} \sqrt{-x^{2} + 5} - \frac{125}{16} \, \arctan \left (\frac{\sqrt{-x^{2} + 5}}{x}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.53878, size = 155, normalized size = 2.38 \begin{align*} \begin{cases} \frac{i x^{7}}{6 \sqrt{x^{2} - 5}} - \frac{25 i x^{5}}{24 \sqrt{x^{2} - 5}} - \frac{25 i x^{3}}{48 \sqrt{x^{2} - 5}} + \frac{125 i x}{16 \sqrt{x^{2} - 5}} - \frac{125 i \operatorname{acosh}{\left (\frac{\sqrt{5} x}{5} \right )}}{16} & \text{for}\: \frac{\left |{x^{2}}\right |}{5} > 1 \\- \frac{x^{7}}{6 \sqrt{5 - x^{2}}} + \frac{25 x^{5}}{24 \sqrt{5 - x^{2}}} + \frac{25 x^{3}}{48 \sqrt{5 - x^{2}}} - \frac{125 x}{16 \sqrt{5 - x^{2}}} + \frac{125 \operatorname{asin}{\left (\frac{\sqrt{5} x}{5} \right )}}{16} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.06507, size = 49, normalized size = 0.75 \begin{align*} \frac{1}{48} \,{\left (2 \,{\left (4 \, x^{2} - 5\right )} x^{2} - 75\right )} \sqrt{-x^{2} + 5} x + \frac{125}{16} \, \arcsin \left (\frac{1}{5} \, \sqrt{5} x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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