Optimal. Leaf size=43 \[ -\frac {1}{5} \sqrt {16-x^4} x^3-\frac {96}{5} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right )+\frac {96}{5} E\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {321, 307, 221, 1181, 21, 424} \[ -\frac {1}{5} \sqrt {16-x^4} x^3-\frac {96}{5} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right )+\frac {96}{5} E\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 21
Rule 221
Rule 307
Rule 321
Rule 424
Rule 1181
Rubi steps
\begin {align*} \int \frac {x^6}{\sqrt {16-x^4}} \, dx &=-\frac {1}{5} x^3 \sqrt {16-x^4}+\frac {48}{5} \int \frac {x^2}{\sqrt {16-x^4}} \, dx\\ &=-\frac {1}{5} x^3 \sqrt {16-x^4}-\frac {192}{5} \int \frac {1}{\sqrt {16-x^4}} \, dx+\frac {192}{5} \int \frac {1+\frac {x^2}{4}}{\sqrt {16-x^4}} \, dx\\ &=-\frac {1}{5} x^3 \sqrt {16-x^4}-\frac {96}{5} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right )+\frac {192}{5} \int \frac {1+\frac {x^2}{4}}{\sqrt {4-x^2} \sqrt {4+x^2}} \, dx\\ &=-\frac {1}{5} x^3 \sqrt {16-x^4}-\frac {96}{5} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right )+\frac {48}{5} \int \frac {\sqrt {4+x^2}}{\sqrt {4-x^2}} \, dx\\ &=-\frac {1}{5} x^3 \sqrt {16-x^4}+\frac {96}{5} E\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right )-\frac {96}{5} F\left (\left .\sin ^{-1}\left (\frac {x}{2}\right )\right |-1\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.01, size = 38, normalized size = 0.88 \[ -\frac {1}{5} x^3 \left (\sqrt {16-x^4}-4 \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};\frac {x^4}{16}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-x^{4} + 16} x^{6}}{x^{4} - 16}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{\sqrt {-x^{4} + 16}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 58, normalized size = 1.35 \[ -\frac {\sqrt {-x^{4}+16}\, x^{3}}{5}-\frac {96 \sqrt {-x^{2}+4}\, \sqrt {x^{2}+4}\, \left (-\EllipticE \left (\frac {x}{2}, i\right )+\EllipticF \left (\frac {x}{2}, i\right )\right )}{5 \sqrt {-x^{4}+16}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{6}}{\sqrt {-x^{4} + 16}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x^6}{\sqrt {16-x^4}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.99, size = 32, normalized size = 0.74 \[ \frac {x^{7} \Gamma \left (\frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {7}{4} \\ \frac {11}{4} \end {matrix}\middle | {\frac {x^{4} e^{2 i \pi }}{16}} \right )}}{16 \Gamma \left (\frac {11}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________