Optimal. Leaf size=53 \[ \frac{1}{21} \left (x^6+2\right )^{7/2}-\frac{2}{5} \left (x^6+2\right )^{5/2}+\frac{4}{3} \left (x^6+2\right )^{3/2}-\frac{8 \sqrt{x^6+2}}{3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0212099, antiderivative size = 53, normalized size of antiderivative = 1., 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.
[In]
[Out]
Rule 266
Rule 43
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*}
Mathematica [A] time = 0.0087013, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.005, size = 27, normalized size = 0.5 \begin{align*}{\frac{5\,{x}^{18}-12\,{x}^{12}+32\,{x}^{6}-128}{105}\sqrt{{x}^{6}+2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.02291, size = 50, normalized size = 0.94 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.41826, size = 74, normalized size = 1.4 \begin{align*} \frac{1}{105} \,{\left (5 \, x^{18} - 12 \, x^{12} + 32 \, x^{6} - 128\right )} \sqrt{x^{6} + 2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 12.0197, size = 54, normalized size = 1.02 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18287, size = 50, normalized size = 0.94 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]