Optimal. Leaf size=40 \[ -\frac{\left (25-x^2\right )^{3/2}}{3 x^3}+\frac{\sqrt{25-x^2}}{x}+\sin ^{-1}\left (\frac{x}{5}\right ) \]
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Rubi [A] time = 0.0085835, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {277, 216} \[ -\frac{\left (25-x^2\right )^{3/2}}{3 x^3}+\frac{\sqrt{25-x^2}}{x}+\sin ^{-1}\left (\frac{x}{5}\right ) \]
Antiderivative was successfully verified.
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Rule 277
Rule 216
Rubi steps
\begin{align*} \int \frac{\left (25-x^2\right )^{3/2}}{x^4} \, dx &=-\frac{\left (25-x^2\right )^{3/2}}{3 x^3}-\int \frac{\sqrt{25-x^2}}{x^2} \, dx\\ &=\frac{\sqrt{25-x^2}}{x}-\frac{\left (25-x^2\right )^{3/2}}{3 x^3}+\int \frac{1}{\sqrt{25-x^2}} \, dx\\ &=\frac{\sqrt{25-x^2}}{x}-\frac{\left (25-x^2\right )^{3/2}}{3 x^3}+\sin ^{-1}\left (\frac{x}{5}\right )\\ \end{align*}
Mathematica [C] time = 0.0031677, size = 24, normalized size = 0.6 \[ -\frac{125 \, _2F_1\left (-\frac{3}{2},-\frac{3}{2};-\frac{1}{2};\frac{x^2}{25}\right )}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 58, normalized size = 1.5 \begin{align*} -{\frac{1}{75\,{x}^{3}} \left ( -{x}^{2}+25 \right ) ^{{\frac{5}{2}}}}+{\frac{2}{1875\,x} \left ( -{x}^{2}+25 \right ) ^{{\frac{5}{2}}}}+{\frac{2\,x}{1875} \left ( -{x}^{2}+25 \right ) ^{{\frac{3}{2}}}}+{\frac{x}{25}\sqrt{-{x}^{2}+25}}+\arcsin \left ({\frac{x}{5}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42977, size = 61, normalized size = 1.52 \begin{align*} \frac{1}{25} \, \sqrt{-x^{2} + 25} x + \frac{2 \,{\left (-x^{2} + 25\right )}^{\frac{3}{2}}}{75 \, x} - \frac{{\left (-x^{2} + 25\right )}^{\frac{5}{2}}}{75 \, x^{3}} + \arcsin \left (\frac{1}{5} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.01717, size = 109, normalized size = 2.72 \begin{align*} -\frac{6 \, x^{3} \arctan \left (\frac{\sqrt{-x^{2} + 25} - 5}{x}\right ) -{\left (4 \, x^{2} - 25\right )} \sqrt{-x^{2} + 25}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.56946, size = 32, normalized size = 0.8 \begin{align*} \operatorname{asin}{\left (\frac{x}{5} \right )} + \frac{4 \sqrt{25 - x^{2}}}{3 x} - \frac{25 \sqrt{25 - x^{2}}}{3 x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.0877, size = 104, normalized size = 2.6 \begin{align*} -\frac{x^{3}{\left (\frac{15 \,{\left (\sqrt{-x^{2} + 25} - 5\right )}^{2}}{x^{2}} - 1\right )}}{24 \,{\left (\sqrt{-x^{2} + 25} - 5\right )}^{3}} + \frac{5 \,{\left (\sqrt{-x^{2} + 25} - 5\right )}}{8 \, x} - \frac{{\left (\sqrt{-x^{2} + 25} - 5\right )}^{3}}{24 \, x^{3}} + \arcsin \left (\frac{1}{5} \, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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