Optimal. Leaf size=25 \[ \frac {x \left (a^2+x^2\right )}{a^2 \sqrt {\left (a^2+x^2\right )^3}} \]
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Rubi [A] time = 0.02, antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {6720, 191} \[ \frac {x \left (a^2+x^2\right )}{a^2 \sqrt {\left (a^2+x^2\right )^3}} \]
Antiderivative was successfully verified.
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Rule 191
Rule 6720
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {\left (a^2+x^2\right )^3}} \, dx &=\frac {\left (a^2+x^2\right )^{3/2} \int \frac {1}{\left (a^2+x^2\right )^{3/2}} \, dx}{\sqrt {\left (a^2+x^2\right )^3}}\\ &=\frac {x \left (a^2+x^2\right )}{a^2 \sqrt {\left (a^2+x^2\right )^3}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 25, normalized size = 1.00 \[ \frac {x \left (a^2+x^2\right )}{a^2 \sqrt {\left (a^2+x^2\right )^3}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.73, size = 64, normalized size = 2.56 \[ \frac {a^{4} + 2 \, a^{2} x^{2} + x^{4} + \sqrt {a^{6} + 3 \, a^{4} x^{2} + 3 \, a^{2} x^{4} + x^{6}} x}{a^{6} + 2 \, a^{4} x^{2} + a^{2} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.42, size = 14, normalized size = 0.56 \[ \frac {x}{\sqrt {a^{2} + x^{2}} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 24, normalized size = 0.96 \[ \frac {\left (a^{2}+x^{2}\right ) x}{\sqrt {\left (a^{2}+x^{2}\right )^{3}}\, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.90, size = 14, normalized size = 0.56 \[ \frac {x}{\sqrt {a^{2} + x^{2}} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.51, size = 25, normalized size = 1.00 \[ \frac {x\,\sqrt {{\left (a^2+x^2\right )}^3}}{a^2\,{\left (a^2+x^2\right )}^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {\left (a^{2} + x^{2}\right )^{3}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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