Optimal. Leaf size=66 \[ \frac{2 b^2 c x^3}{3 \left (b \sqrt{\frac{a^2}{b^2}+c x^2}+a\right )^{3/2}}+\frac{2 a x}{\sqrt{b \sqrt{\frac{a^2}{b^2}+c x^2}+a}} \]
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Rubi [A] time = 0.0329764, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2129} \[ \frac{2 b^2 c x^3}{3 \left (b \sqrt{\frac{a^2}{b^2}+c x^2}+a\right )^{3/2}}+\frac{2 a x}{\sqrt{b \sqrt{\frac{a^2}{b^2}+c x^2}+a}} \]
Antiderivative was successfully verified.
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Rule 2129
Rubi steps
\begin{align*} \int \sqrt{a+b \sqrt{\frac{a^2}{b^2}+c x^2}} \, dx &=\frac{2 b^2 c x^3}{3 \left (a+b \sqrt{\frac{a^2}{b^2}+c x^2}\right )^{3/2}}+\frac{2 a x}{\sqrt{a+b \sqrt{\frac{a^2}{b^2}+c x^2}}}\\ \end{align*}
Mathematica [A] time = 0.217532, size = 55, normalized size = 0.83 \[ \frac{2 b x \sqrt{\frac{a^2}{b^2}+c x^2}+4 a x}{3 \sqrt{b \sqrt{\frac{a^2}{b^2}+c x^2}+a}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.016, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a+b\sqrt{{\frac{{a}^{2}}{{b}^{2}}}+c{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sqrt{c x^{2} + \frac{a^{2}}{b^{2}}} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82889, size = 144, normalized size = 2.18 \begin{align*} \frac{2 \,{\left (b^{2} c x^{2} + a b \sqrt{\frac{b^{2} c x^{2} + a^{2}}{b^{2}}} - a^{2}\right )} \sqrt{b \sqrt{\frac{b^{2} c x^{2} + a^{2}}{b^{2}}} + a}}{3 \, b^{2} 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{\frac{a^{2}}{b^{2}} + c x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\sqrt{c x^{2} + \frac{a^{2}}{b^{2}}} b + a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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