Optimal. Leaf size=34 \[ \frac{2 x \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b \sqrt{c x^2}} \]
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Rubi [A] time = 0.0105151, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {254, 32} \[ \frac{2 x \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Rule 254
Rule 32
Rubi steps
\begin{align*} \int \sqrt{a+b \sqrt{c x^2}} \, dx &=\frac{x \operatorname{Subst}\left (\int \sqrt{a+b x} \, dx,x,\sqrt{c x^2}\right )}{\sqrt{c x^2}}\\ &=\frac{2 x \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b \sqrt{c x^2}}\\ \end{align*}
Mathematica [A] time = 0.007202, size = 34, normalized size = 1. \[ \frac{2 x \left (a+b \sqrt{c x^2}\right )^{3/2}}{3 b \sqrt{c x^2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.001, size = 27, normalized size = 0.8 \begin{align*}{\frac{2\,x}{3\,b} \left ( a+b\sqrt{c{x}^{2}} \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{c{x}^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.994868, size = 57, normalized size = 1.68 \begin{align*} \frac{{\left ({\left (c^{\frac{3}{2}} + c\right )} b x + a{\left (c + \sqrt{c}\right )}\right )} \sqrt{b \sqrt{c} x + a}}{{\left (c^{2} + 2 \, c^{\frac{3}{2}} + c\right )} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55586, size = 85, normalized size = 2.5 \begin{align*} \frac{2 \,{\left (b c x^{2} + \sqrt{c x^{2}} a\right )} \sqrt{\sqrt{c x^{2}} b + a}}{3 \, b c x} \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 \sqrt{a + b \sqrt{c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1316, size = 24, normalized size = 0.71 \begin{align*} \frac{2 \,{\left (b \sqrt{c} x + a\right )}^{\frac{3}{2}}}{3 \, b \sqrt{c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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