Optimal. Leaf size=44 \[ -\frac {1}{15} \left (2+2 x^2+x^4\right )^{3/2}+\frac {1}{10} \left (1+x^2\right )^2 \left (2+2 x^2+x^4\right )^{3/2} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {1261, 706, 643}
\begin {gather*} \frac {1}{10} \left (x^2+1\right )^2 \left (x^4+2 x^2+2\right )^{3/2}-\frac {1}{15} \left (x^4+2 x^2+2\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 643
Rule 706
Rule 1261
Rubi steps
\begin {align*} \int x \left (1+x^2\right )^3 \sqrt {2+2 x^2+x^4} \, dx &=\frac {1}{2} \text {Subst}\left (\int (1+x)^3 \sqrt {2+2 x+x^2} \, dx,x,x^2\right )\\ &=\frac {1}{10} \left (1+x^2\right )^2 \left (2+2 x^2+x^4\right )^{3/2}-\frac {1}{5} \text {Subst}\left (\int (1+x) \sqrt {2+2 x+x^2} \, dx,x,x^2\right )\\ &=-\frac {1}{15} \left (2+2 x^2+x^4\right )^{3/2}+\frac {1}{10} \left (1+x^2\right )^2 \left (2+2 x^2+x^4\right )^{3/2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.12, size = 40, normalized size = 0.91 \begin {gather*} \frac {1}{30} \sqrt {2+2 x^2+x^4} \left (2+14 x^2+19 x^4+12 x^6+3 x^8\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.38, size = 50, normalized size = 1.14
method | result | size |
gosper | \(\frac {\left (x^{4}+2 x^{2}+2\right )^{\frac {3}{2}} \left (3 x^{4}+6 x^{2}+1\right )}{30}\) | \(27\) |
elliptic | \(\frac {\left (x^{4}+2 x^{2}+2\right )^{\frac {3}{2}} \left (3 x^{4}+6 x^{2}+1\right )}{30}\) | \(27\) |
trager | \(\left (\frac {1}{10} x^{8}+\frac {2}{5} x^{6}+\frac {19}{30} x^{4}+\frac {7}{15} x^{2}+\frac {1}{15}\right ) \sqrt {x^{4}+2 x^{2}+2}\) | \(36\) |
risch | \(\frac {\left (3 x^{8}+12 x^{6}+19 x^{4}+14 x^{2}+2\right ) \sqrt {x^{4}+2 x^{2}+2}}{30}\) | \(37\) |
default | \(\frac {x^{4} \left (x^{4}+2 x^{2}+2\right )^{\frac {3}{2}}}{10}+\frac {x^{2} \left (x^{4}+2 x^{2}+2\right )^{\frac {3}{2}}}{5}+\frac {\left (x^{4}+2 x^{2}+2\right )^{\frac {3}{2}}}{30}\) | \(50\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.51, size = 49, normalized size = 1.11 \begin {gather*} \frac {1}{10} \, {\left (x^{4} + 2 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{4} + \frac {1}{5} \, {\left (x^{4} + 2 \, x^{2} + 2\right )}^{\frac {3}{2}} x^{2} + \frac {1}{30} \, {\left (x^{4} + 2 \, x^{2} + 2\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 36, normalized size = 0.82 \begin {gather*} \frac {1}{30} \, {\left (3 \, x^{8} + 12 \, x^{6} + 19 \, x^{4} + 14 \, x^{2} + 2\right )} \sqrt {x^{4} + 2 \, x^{2} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 94 vs.
\(2 (36) = 72\).
time = 0.15, size = 94, normalized size = 2.14 \begin {gather*} \frac {x^{8} \sqrt {x^{4} + 2 x^{2} + 2}}{10} + \frac {2 x^{6} \sqrt {x^{4} + 2 x^{2} + 2}}{5} + \frac {19 x^{4} \sqrt {x^{4} + 2 x^{2} + 2}}{30} + \frac {7 x^{2} \sqrt {x^{4} + 2 x^{2} + 2}}{15} + \frac {\sqrt {x^{4} + 2 x^{2} + 2}}{15} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 2.04, size = 29, normalized size = 0.66 \begin {gather*} \frac {1}{10} \, {\left (x^{4} + 2 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {1}{6} \, {\left (x^{4} + 2 \, x^{2} + 2\right )}^{\frac {3}{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 0.09, size = 26, normalized size = 0.59 \begin {gather*} \frac {{\left (x^4+2\,x^2+2\right )}^{3/2}\,\left (3\,x^4+6\,x^2+1\right )}{30} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________