Optimal. Leaf size=74 \[ \frac{b x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{a x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]
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Rubi [A] time = 0.0136244, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {1088} \[ \frac{b x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}+\frac{a x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2} \]
Antiderivative was successfully verified.
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Rule 1088
Rubi steps
\begin{align*} \int \sqrt{a^2+2 a b x^2+b^2 x^4} \, dx &=\frac{\sqrt{a^2+2 a b x^2+b^2 x^4} \int \left (2 a b+2 b^2 x^2\right ) \, dx}{2 a b+2 b^2 x^2}\\ &=\frac{a x \sqrt{a^2+2 a b x^2+b^2 x^4}}{a+b x^2}+\frac{b x^3 \sqrt{a^2+2 a b x^2+b^2 x^4}}{3 \left (a+b x^2\right )}\\ \end{align*}
Mathematica [A] time = 0.0068423, size = 36, normalized size = 0.49 \[ \frac{\sqrt{\left (a+b x^2\right )^2} \left (3 a x+b x^3\right )}{3 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 33, normalized size = 0.5 \begin{align*}{\frac{x \left ( b{x}^{2}+3\,a \right ) }{3\,b{x}^{2}+3\,a}\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00162, size = 14, normalized size = 0.19 \begin{align*} \frac{1}{3} \, b x^{3} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.49377, size = 23, normalized size = 0.31 \begin{align*} \frac{1}{3} \, b x^{3} + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.088153, size = 8, normalized size = 0.11 \begin{align*} a x + \frac{b x^{3}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1003, size = 27, normalized size = 0.36 \begin{align*} \frac{1}{3} \,{\left (b x^{3} + 3 \, a x\right )} \mathrm{sgn}\left (b x^{2} + a\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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