Optimal. Leaf size=376 \[ \frac{4\ 2^{2/3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right ),-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}}-\frac{x^4}{3 \sqrt{x^6+2}}+\frac{4 \sqrt{x^6+2}}{3 \left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )}-\frac{2 \sqrt [6]{2} \sqrt{2-\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} E\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.191485, antiderivative size = 376, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {275, 288, 303, 218, 1877} \[ -\frac{x^4}{3 \sqrt{x^6+2}}+\frac{4 \sqrt{x^6+2}}{3 \left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )}+\frac{4\ 2^{2/3} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}}-\frac{2 \sqrt [6]{2} \sqrt{2-\sqrt{3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt{\frac{x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} E\left (\sin ^{-1}\left (\frac{x^2+\sqrt [3]{2} \left (1-\sqrt{3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt{\frac{x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt{3}\right )\right )^2}} \sqrt{x^6+2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 275
Rule 288
Rule 303
Rule 218
Rule 1877
Rubi steps
\begin{align*} \int \frac{x^9}{\left (2+x^6\right )^{3/2}} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{x^4}{\left (2+x^3\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac{x^4}{3 \sqrt{2+x^6}}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{x}{\sqrt{2+x^3}} \, dx,x,x^2\right )\\ &=-\frac{x^4}{3 \sqrt{2+x^6}}+\frac{2}{3} \operatorname{Subst}\left (\int \frac{\sqrt [3]{2} \left (1-\sqrt{3}\right )+x}{\sqrt{2+x^3}} \, dx,x,x^2\right )+\frac{\left (2\ 2^{5/6}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2+x^3}} \, dx,x,x^2\right )}{3 \sqrt{2+\sqrt{3}}}\\ &=-\frac{x^4}{3 \sqrt{2+x^6}}+\frac{4 \sqrt{2+x^6}}{3 \left (\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2\right )}-\frac{2 \sqrt [6]{2} \sqrt{2-\sqrt{3}} \left (\sqrt [3]{2}+x^2\right ) \sqrt{\frac{2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2\right )^2}} E\left (\sin ^{-1}\left (\frac{\sqrt [3]{2} \left (1-\sqrt{3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2}\right )|-7-4 \sqrt{3}\right )}{3^{3/4} \sqrt{\frac{\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2\right )^2}} \sqrt{2+x^6}}+\frac{4\ 2^{2/3} \left (\sqrt [3]{2}+x^2\right ) \sqrt{\frac{2^{2/3}-\sqrt [3]{2} x^2+x^4}{\left (\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2\right )^2}} F\left (\sin ^{-1}\left (\frac{\sqrt [3]{2} \left (1-\sqrt{3}\right )+x^2}{\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2}\right )|-7-4 \sqrt{3}\right )}{3 \sqrt [4]{3} \sqrt{\frac{\sqrt [3]{2}+x^2}{\left (\sqrt [3]{2} \left (1+\sqrt{3}\right )+x^2\right )^2}} \sqrt{2+x^6}}\\ \end{align*}
Mathematica [C] time = 0.0114482, size = 41, normalized size = 0.11 \[ \frac{x^4}{\sqrt{x^6+2}}-\frac{x^4 \, _2F_1\left (\frac{2}{3},\frac{3}{2};\frac{5}{3};-\frac{x^6}{2}\right )}{\sqrt{2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.026, size = 33, normalized size = 0.1 \begin{align*} -{\frac{{x}^{4}}{3}{\frac{1}{\sqrt{{x}^{6}+2}}}}+{\frac{{x}^{4}\sqrt{2}}{6}{\mbox{$_2$F$_1$}({\frac{1}{2}},{\frac{2}{3}};\,{\frac{5}{3}};\,-{\frac{{x}^{6}}{2}})}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{9}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x^{6} + 2} x^{9}}{x^{12} + 4 \, x^{6} + 4}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.946132, size = 36, normalized size = 0.1 \begin{align*} \frac{\sqrt{2} x^{10} \Gamma \left (\frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{5}{3} \\ \frac{8}{3} \end{matrix}\middle |{\frac{x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac{8}{3}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{9}}{{\left (x^{6} + 2\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]