Optimal. Leaf size=40 \[ \frac {\sqrt {25-x^2}}{x}-\frac {\left (25-x^2\right )^{3/2}}{3 x^3}+\sin ^{-1}\left (\frac {x}{5}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.00, 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 = {283, 222}
\begin {gather*} \text {ArcSin}\left (\frac {x}{5}\right )+\frac {\sqrt {25-x^2}}{x}-\frac {\left (25-x^2\right )^{3/2}}{3 x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 222
Rule 283
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*}
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Mathematica [A]
time = 0.06, size = 46, normalized size = 1.15 \begin {gather*} \frac {\sqrt {25-x^2} \left (-25+4 x^2\right )}{3 x^3}-2 \tan ^{-1}\left (\frac {\sqrt {25-x^2}}{5+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 58, normalized size = 1.45
method | result | size |
risch | \(-\frac {4 x^{4}-125 x^{2}+625}{3 x^{3} \sqrt {-x^{2}+25}}+\arcsin \left (\frac {x}{5}\right )\) | \(32\) |
meijerg | \(-\frac {3 i \left (-\frac {1000 i \sqrt {\pi }\, \left (1-\frac {4 x^{2}}{25}\right ) \sqrt {-\frac {x^{2}}{25}+1}}{9 x^{3}}+\frac {8 i \sqrt {\pi }\, \arcsin \left (\frac {x}{5}\right )}{3}\right )}{8 \sqrt {\pi }}\) | \(43\) |
trager | \(\frac {\left (4 x^{2}-25\right ) \sqrt {-x^{2}+25}}{3 x^{3}}+\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\RootOf \left (\textit {\_Z}^{2}+1\right ) x +\sqrt {-x^{2}+25}\right )\) | \(50\) |
default | \(-\frac {\left (-x^{2}+25\right )^{\frac {5}{2}}}{75 x^{3}}+\frac {2 \left (-x^{2}+25\right )^{\frac {5}{2}}}{1875 x}+\frac {2 x \left (-x^{2}+25\right )^{\frac {3}{2}}}{1875}+\frac {\sqrt {-x^{2}+25}\, x}{25}+\arcsin \left (\frac {x}{5}\right )\) | \(58\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 1.43, size = 45, normalized size = 1.12 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.76, size = 45, normalized size = 1.12 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.44, size = 32, normalized size = 0.80 \begin {gather*} \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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 77 vs.
\(2 (32) = 64\).
time = 0.67, size = 77, normalized size = 1.92 \begin {gather*} -\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 {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.04, size = 33, normalized size = 0.82 \begin {gather*} \mathrm {asin}\left (\frac {x}{5}\right )+\frac {4\,\sqrt {25-x^2}}{3\,x}-\frac {25\,\sqrt {25-x^2}}{3\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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