Optimal. Leaf size=53 \[ \frac {1}{21} \left (x^6+2\right )^{7/2}-\frac {2}{5} \left (x^6+2\right )^{5/2}+\frac {4}{3} \left (x^6+2\right )^{3/2}-\frac {8 \sqrt {x^6+2}}{3} \]
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Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {1}{21} \left (x^6+2\right )^{7/2}-\frac {2}{5} \left (x^6+2\right )^{5/2}+\frac {4}{3} \left (x^6+2\right )^{3/2}-\frac {8 \sqrt {x^6+2}}{3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^{23}}{\sqrt {2+x^6}} \, dx &=\frac {1}{6} \operatorname {Subst}\left (\int \frac {x^3}{\sqrt {2+x}} \, dx,x,x^6\right )\\ &=\frac {1}{6} \operatorname {Subst}\left (\int \left (-\frac {8}{\sqrt {2+x}}+12 \sqrt {2+x}-6 (2+x)^{3/2}+(2+x)^{5/2}\right ) \, dx,x,x^6\right )\\ &=-\frac {8}{3} \sqrt {2+x^6}+\frac {4}{3} \left (2+x^6\right )^{3/2}-\frac {2}{5} \left (2+x^6\right )^{5/2}+\frac {1}{21} \left (2+x^6\right )^{7/2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 30, normalized size = 0.57 \[ \frac {1}{105} \sqrt {x^6+2} \left (5 x^{18}-12 x^{12}+32 x^6-128\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 26, normalized size = 0.49 \[ \frac {1}{105} \, {\left (5 \, x^{18} - 12 \, x^{12} + 32 \, x^{6} - 128\right )} \sqrt {x^{6} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 37, normalized size = 0.70 \[ \frac {1}{21} \, {\left (x^{6} + 2\right )}^{\frac {7}{2}} - \frac {2}{5} \, {\left (x^{6} + 2\right )}^{\frac {5}{2}} + \frac {4}{3} \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} - \frac {8}{3} \, \sqrt {x^{6} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 27, normalized size = 0.51 \[ \frac {\sqrt {x^{6}+2}\, \left (5 x^{18}-12 x^{12}+32 x^{6}-128\right )}{105} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.01, size = 37, normalized size = 0.70 \[ \frac {1}{21} \, {\left (x^{6} + 2\right )}^{\frac {7}{2}} - \frac {2}{5} \, {\left (x^{6} + 2\right )}^{\frac {5}{2}} + \frac {4}{3} \, {\left (x^{6} + 2\right )}^{\frac {3}{2}} - \frac {8}{3} \, \sqrt {x^{6} + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 25, normalized size = 0.47 \[ \sqrt {x^6+2}\,\left (\frac {x^{18}}{21}-\frac {4\,x^{12}}{35}+\frac {32\,x^6}{105}-\frac {128}{105}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.03, size = 54, normalized size = 1.02 \[ \frac {x^{18} \sqrt {x^{6} + 2}}{21} - \frac {4 x^{12} \sqrt {x^{6} + 2}}{35} + \frac {32 x^{6} \sqrt {x^{6} + 2}}{105} - \frac {128 \sqrt {x^{6} + 2}}{105} \]
Verification of antiderivative is not currently implemented for this CAS.
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