Optimal. Leaf size=38 \[ \frac{2 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{2} x\right ),-1\right )}{\sqrt{3}}+\sqrt{\frac{2}{3}} \sqrt{1-4 x^4} x \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.009185, antiderivative size = 38, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {248, 195, 221} \[ \sqrt{\frac{2}{3}} \sqrt{1-4 x^4} x+\frac{2 F\left (\left .\sin ^{-1}\left (\sqrt{2} x\right )\right |-1\right )}{\sqrt{3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 248
Rule 195
Rule 221
Rubi steps
\begin{align*} \int \sqrt{3-6 x^2} \sqrt{2+4 x^2} \, dx &=\int \sqrt{6-24 x^4} \, dx\\ &=\sqrt{\frac{2}{3}} x \sqrt{1-4 x^4}+4 \int \frac{1}{\sqrt{6-24 x^4}} \, dx\\ &=\sqrt{\frac{2}{3}} x \sqrt{1-4 x^4}+\frac{2 F\left (\left .\sin ^{-1}\left (\sqrt{2} x\right )\right |-1\right )}{\sqrt{3}}\\ \end{align*}
Mathematica [C] time = 0.0116507, size = 22, normalized size = 0.58 \[ \sqrt{6} x \, _2F_1\left (-\frac{1}{2},\frac{1}{4};\frac{5}{4};4 x^4\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.026, size = 75, normalized size = 2. \begin{align*} -{\frac{\sqrt{2}}{36\,{x}^{4}-9}\sqrt{-6\,{x}^{2}+3}\sqrt{2\,{x}^{2}+1} \left ( \sqrt{2}\sqrt{3}\sqrt{-6\,{x}^{2}+3}\sqrt{2\,{x}^{2}+1}{\it EllipticF} \left ( x\sqrt{2},i \right ) -12\,{x}^{5}+3\,x \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{4 \, x^{2} + 2} \sqrt{-6 \, x^{2} + 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{4 \, x^{2} + 2} \sqrt{-6 \, x^{2} + 3}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \sqrt{6} \int \sqrt{1 - 2 x^{2}} \sqrt{2 x^{2} + 1}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{4 \, x^{2} + 2} \sqrt{-6 \, x^{2} + 3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]