Optimal. Leaf size=21 \[ \frac {\left (x^8-2 x^4-1\right )^{3/2}}{6 x^6} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1590} \begin {gather*} \frac {\left (x^8-2 x^4-1\right )^{3/2}}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1590
Rubi steps
\begin {align*} \int \frac {\left (1+x^8\right ) \sqrt {-1-2 x^4+x^8}}{x^7} \, dx &=\frac {\left (-1-2 x^4+x^8\right )^{3/2}}{6 x^6}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 21, normalized size = 1.00 \begin {gather*} \frac {\left (x^8-2 x^4-1\right )^{3/2}}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [A] time = 0.62, size = 21, normalized size = 1.00 \begin {gather*} \frac {\left (-1-2 x^4+x^8\right )^{3/2}}{6 x^6} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 17, normalized size = 0.81 \begin {gather*} \frac {{\left (x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{2}}}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{8} - 2 \, x^{4} - 1} {\left (x^{8} + 1\right )}}{x^{7}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.09, size = 18, normalized size = 0.86
method | result | size |
gosper | \(\frac {\left (x^{8}-2 x^{4}-1\right )^{\frac {3}{2}}}{6 x^{6}}\) | \(18\) |
trager | \(\frac {\left (x^{8}-2 x^{4}-1\right )^{\frac {3}{2}}}{6 x^{6}}\) | \(18\) |
risch | \(\frac {x^{16}-4 x^{12}+2 x^{8}+4 x^{4}+1}{6 x^{6} \sqrt {x^{8}-2 x^{4}-1}}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.64, size = 17, normalized size = 0.81 \begin {gather*} \frac {{\left (x^{8} - 2 \, x^{4} - 1\right )}^{\frac {3}{2}}}{6 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.36, size = 17, normalized size = 0.81 \begin {gather*} \frac {{\left (x^8-2\,x^4-1\right )}^{3/2}}{6\,x^6} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x^{8} + 1\right ) \sqrt {x^{8} - 2 x^{4} - 1}}{x^{7}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________