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 = 0.0164506, antiderivative size = 25, normalized size of antiderivative = 1., 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 6720
Rule 191
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*}
Mathematica [A] time = 0.0193384, size = 25, normalized size = 1. \[ \frac{x \left (a^2+x^2\right )}{a^2 \sqrt{\left (a^2+x^2\right )^3}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 24, normalized size = 1. \begin{align*}{\frac{x \left ({a}^{2}+{x}^{2} \right ) }{{a}^{2}}{\frac{1}{\sqrt{ \left ({a}^{2}+{x}^{2} \right ) ^{3}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07124, size = 19, normalized size = 0.76 \begin{align*} \frac{x}{\sqrt{a^{2} + x^{2}} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.65692, size = 131, normalized size = 5.24 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{\left (a^{2} + x^{2}\right )^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11909, size = 19, normalized size = 0.76 \begin{align*} \frac{x}{\sqrt{a^{2} + x^{2}} a^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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