Optimal. Leaf size=65 \[ \frac {16\ 2^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right ),2\right )}{63 \sqrt {3}}-\frac {8}{63} \sqrt [4]{2-3 x^2} x-\frac {2}{21} \sqrt [4]{2-3 x^2} x^3 \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 65, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {321, 232} \[ -\frac {2}{21} \sqrt [4]{2-3 x^2} x^3-\frac {8}{63} \sqrt [4]{2-3 x^2} x+\frac {16\ 2^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{63 \sqrt {3}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 232
Rule 321
Rubi steps
\begin {align*} \int \frac {x^4}{\left (2-3 x^2\right )^{3/4}} \, dx &=-\frac {2}{21} x^3 \sqrt [4]{2-3 x^2}+\frac {4}{7} \int \frac {x^2}{\left (2-3 x^2\right )^{3/4}} \, dx\\ &=-\frac {8}{63} x \sqrt [4]{2-3 x^2}-\frac {2}{21} x^3 \sqrt [4]{2-3 x^2}+\frac {16}{63} \int \frac {1}{\left (2-3 x^2\right )^{3/4}} \, dx\\ &=-\frac {8}{63} x \sqrt [4]{2-3 x^2}-\frac {2}{21} x^3 \sqrt [4]{2-3 x^2}+\frac {16\ 2^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right )\right |2\right )}{63 \sqrt {3}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 54, normalized size = 0.83 \[ -\frac {2}{189} \left (3 x \sqrt [4]{2-3 x^2} \left (3 x^2+4\right )-8\ 2^{3/4} \sqrt {3} \operatorname {EllipticF}\left (\frac {1}{2} \sin ^{-1}\left (\sqrt {\frac {3}{2}} x\right ),2\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (-3 \, x^{2} + 2\right )}^{\frac {1}{4}} x^{4}}{3 \, x^{2} - 2}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{{\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.03, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{\left (-3 x^{2}+2\right )^{\frac {3}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4}}{{\left (-3 \, x^{2} + 2\right )}^{\frac {3}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^4}{{\left (2-3\,x^2\right )}^{3/4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [C] time = 0.77, size = 29, normalized size = 0.45 \[ \frac {\sqrt [4]{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {3 x^{2} e^{2 i \pi }}{2}} \right )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________