Optimal. Leaf size=29 \[ \frac{1}{6} \left (16-x^4\right )^{3/2}-8 \sqrt{16-x^4} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0136172, antiderivative size = 29, normalized size of antiderivative = 1., 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 = {266, 43} \[ \frac{1}{6} \left (16-x^4\right )^{3/2}-8 \sqrt{16-x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{x^7}{\sqrt{16-x^4}} \, dx &=\frac{1}{4} \operatorname{Subst}\left (\int \frac{x}{\sqrt{16-x}} \, dx,x,x^4\right )\\ &=\frac{1}{4} \operatorname{Subst}\left (\int \left (\frac{16}{\sqrt{16-x}}-\sqrt{16-x}\right ) \, dx,x,x^4\right )\\ &=-8 \sqrt{16-x^4}+\frac{1}{6} \left (16-x^4\right )^{3/2}\\ \end{align*}
Mathematica [A] time = 0.0068138, size = 20, normalized size = 0.69 \[ -\frac{1}{6} \sqrt{16-x^4} \left (x^4+32\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.006, size = 28, normalized size = 1. \begin{align*}{\frac{ \left ( -2+x \right ) \left ( 2+x \right ) \left ({x}^{2}+4 \right ) \left ({x}^{4}+32 \right ) }{6}{\frac{1}{\sqrt{-{x}^{4}+16}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02771, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{6} \,{\left (-x^{4} + 16\right )}^{\frac{3}{2}} - 8 \, \sqrt{-x^{4} + 16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.44237, size = 45, normalized size = 1.55 \begin{align*} -\frac{1}{6} \,{\left (x^{4} + 32\right )} \sqrt{-x^{4} + 16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.665099, size = 26, normalized size = 0.9 \begin{align*} - \frac{x^{4} \sqrt{16 - x^{4}}}{6} - \frac{16 \sqrt{16 - x^{4}}}{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18428, size = 31, normalized size = 1.07 \begin{align*} \frac{1}{6} \,{\left (-x^{4} + 16\right )}^{\frac{3}{2}} - 8 \, \sqrt{-x^{4} + 16} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]