Optimal. Leaf size=224 \[ \frac {4 \sqrt [4]{2} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) \operatorname {EllipticF}\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right ),\frac {1}{2}\right )}{15 \sqrt {3} x}-\frac {2}{15} \left (-3 x^2-2\right )^{3/4} x-\frac {8 \sqrt [4]{-3 x^2-2} x}{15 \left (\sqrt {-3 x^2-2}+\sqrt {2}\right )}-\frac {8 \sqrt [4]{2} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{15 \sqrt {3} x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {321, 230, 305, 220, 1196} \[ -\frac {2}{15} \left (-3 x^2-2\right )^{3/4} x-\frac {8 \sqrt [4]{-3 x^2-2} x}{15 \left (\sqrt {-3 x^2-2}+\sqrt {2}\right )}+\frac {4 \sqrt [4]{2} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{15 \sqrt {3} x}-\frac {8 \sqrt [4]{2} \sqrt {-\frac {x^2}{\left (\sqrt {-3 x^2-2}+\sqrt {2}\right )^2}} \left (\sqrt {-3 x^2-2}+\sqrt {2}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-3 x^2-2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{15 \sqrt {3} x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 220
Rule 230
Rule 305
Rule 321
Rule 1196
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt [4]{-2-3 x^2}} \, dx &=-\frac {2}{15} x \left (-2-3 x^2\right )^{3/4}-\frac {4}{15} \int \frac {1}{\sqrt [4]{-2-3 x^2}} \, dx\\ &=-\frac {2}{15} x \left (-2-3 x^2\right )^{3/4}+\frac {\left (4 \sqrt {\frac {2}{3}} \sqrt {-x^2}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2-3 x^2}\right )}{15 x}\\ &=-\frac {2}{15} x \left (-2-3 x^2\right )^{3/4}+\frac {\left (8 \sqrt {-x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2-3 x^2}\right )}{15 \sqrt {3} x}-\frac {\left (8 \sqrt {-x^2}\right ) \operatorname {Subst}\left (\int \frac {1-\frac {x^2}{\sqrt {2}}}{\sqrt {1+\frac {x^4}{2}}} \, dx,x,\sqrt [4]{-2-3 x^2}\right )}{15 \sqrt {3} x}\\ &=-\frac {2}{15} x \left (-2-3 x^2\right )^{3/4}-\frac {8 x \sqrt [4]{-2-3 x^2}}{15 \left (\sqrt {2}+\sqrt {-2-3 x^2}\right )}-\frac {8 \sqrt [4]{2} \sqrt {-\frac {x^2}{\left (\sqrt {2}+\sqrt {-2-3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2-3 x^2}\right ) E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2-3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{15 \sqrt {3} x}+\frac {4 \sqrt [4]{2} \sqrt {-\frac {x^2}{\left (\sqrt {2}+\sqrt {-2-3 x^2}\right )^2}} \left (\sqrt {2}+\sqrt {-2-3 x^2}\right ) F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{-2-3 x^2}}{\sqrt [4]{2}}\right )|\frac {1}{2}\right )}{15 \sqrt {3} x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 58, normalized size = 0.26 \[ \frac {2 x \left (-2^{3/4} \sqrt [4]{3 x^2+2} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {3}{2};-\frac {3 x^2}{2}\right )+3 x^2+2\right )}{15 \sqrt [4]{-3 x^2-2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ \frac {45 \, x {\rm integral}\left (\frac {16 \, {\left (-3 \, x^{2} - 2\right )}^{\frac {3}{4}}}{45 \, {\left (3 \, x^{4} + 2 \, x^{2}\right )}}, x\right ) - 2 \, {\left (3 \, x^{2} - 4\right )} {\left (-3 \, x^{2} - 2\right )}^{\frac {3}{4}}}{45 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (-3 \, x^{2} - 2\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.27, size = 41, normalized size = 0.18 \[ \frac {2 \left (-1\right )^{\frac {3}{4}} 2^{\frac {3}{4}} x \hypergeom \left (\left [\frac {1}{4}, \frac {1}{2}\right ], \left [\frac {3}{2}\right ], -\frac {3 x^{2}}{2}\right )}{15}+\frac {2 \left (3 x^{2}+2\right ) x}{15 \left (-3 x^{2}-2\right )^{\frac {1}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{2}}{{\left (-3 \, x^{2} - 2\right )}^{\frac {1}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^2}{{\left (-3\,x^2-2\right )}^{1/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 0.72, size = 34, normalized size = 0.15 \[ \frac {2^{\frac {3}{4}} x^{3} e^{- \frac {i \pi }{4}} {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{i \pi }}{2}} \right )}}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________