Optimal. Leaf size=41 \[ \frac {2 x^3}{3 \left (5+\sqrt {25+x^2}\right )^{3/2}}+\frac {10 x}{\sqrt {5+\sqrt {25+x^2}}} \]
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Rubi [A]
time = 0.00, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2154}
\begin {gather*} \frac {10 x}{\sqrt {\sqrt {x^2+25}+5}}+\frac {2 x^3}{3 \left (\sqrt {x^2+25}+5\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2154
Rubi steps
\begin {align*} \int \sqrt {5+\sqrt {25+x^2}} \, dx &=\frac {2 x^3}{3 \left (5+\sqrt {25+x^2}\right )^{3/2}}+\frac {10 x}{\sqrt {5+\sqrt {25+x^2}}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 31, normalized size = 0.76 \begin {gather*} \frac {2 x \left (10+\sqrt {25+x^2}\right )}{3 \sqrt {5+\sqrt {25+x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 3 vs. order
2.
time = 0.03, size = 64, normalized size = 1.56
method | result | size |
meijerg | \(-\frac {5 \sqrt {5}\, \left (-\frac {32 \sqrt {\pi }\, \sqrt {2}\, x^{3} \cosh \left (\frac {3 \arcsinh \left (\frac {x}{5}\right )}{2}\right )}{375}-\frac {8 \sqrt {\pi }\, \sqrt {2}\, \left (-\frac {4}{1875} x^{4}-\frac {2}{75} x^{2}+\frac {2}{3}\right ) \sinh \left (\frac {3 \arcsinh \left (\frac {x}{5}\right )}{2}\right )}{\sqrt {\frac {x^{2}}{25}+1}}\right )}{8 \sqrt {\pi }}\) | \(64\) |
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 [A]
time = 0.36, size = 30, normalized size = 0.73 \begin {gather*} \frac {2 \, {\left (x^{2} + 5 \, \sqrt {x^{2} + 25} - 25\right )} \sqrt {\sqrt {x^{2} + 25} + 5}}{3 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 197 vs.
\(2 (36) = 72\).
time = 0.63, size = 197, normalized size = 4.80 \begin {gather*} - \frac {\sqrt {2} x^{3} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {x^{2} + 25} \sqrt {\sqrt {x^{2} + 25} + 5} + 60 \pi \sqrt {\sqrt {x^{2} + 25} + 5}} - \frac {15 \sqrt {2} x \sqrt {x^{2} + 25} \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {x^{2} + 25} \sqrt {\sqrt {x^{2} + 25} + 5} + 60 \pi \sqrt {\sqrt {x^{2} + 25} + 5}} - \frac {75 \sqrt {2} x \Gamma \left (- \frac {1}{4}\right ) \Gamma \left (\frac {1}{4}\right )}{12 \pi \sqrt {x^{2} + 25} \sqrt {\sqrt {x^{2} + 25} + 5} + 60 \pi \sqrt {\sqrt {x^{2} + 25} + 5}} \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 = -0.02 \begin {gather*} \int \sqrt {\sqrt {x^2+25}+5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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