Optimal. Leaf size=394 \[ \frac {5 \sqrt {x^6+2}}{24 \left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )}-\frac {5 \sqrt {x^6+2}}{24 x^2}+\frac {1}{6 x^2 \sqrt {x^6+2}}+\frac {5 \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )|-7-4 \sqrt {3}\right )}{12 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}-\frac {5 \sqrt {2-\sqrt {3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} E\left (\sin ^{-1}\left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )|-7-4 \sqrt {3}\right )}{8\ 2^{5/6} 3^{3/4} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}} \]
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Rubi [A] time = 0.22, antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {275, 290, 325, 303, 218, 1877} \[ \frac {5 \sqrt {x^6+2}}{24 \left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )}-\frac {5 \sqrt {x^6+2}}{24 x^2}+\frac {1}{6 x^2 \sqrt {x^6+2}}+\frac {5 \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )|-7-4 \sqrt {3}\right )}{12 \sqrt [3]{2} \sqrt [4]{3} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}-\frac {5 \sqrt {2-\sqrt {3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} E\left (\sin ^{-1}\left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )|-7-4 \sqrt {3}\right )}{8\ 2^{5/6} 3^{3/4} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}} \]
Antiderivative was successfully verified.
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Rule 218
Rule 275
Rule 290
Rule 303
Rule 325
Rule 1877
Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^2 \left (2+x^3\right )^{3/2}} \, dx,x,x^2\right )\\ &=\frac {1}{6 x^2 \sqrt {2+x^6}}+\frac {5}{12} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {2+x^3}} \, dx,x,x^2\right )\\ &=\frac {1}{6 x^2 \sqrt {2+x^6}}-\frac {5 \sqrt {2+x^6}}{24 x^2}+\frac {5}{48} \operatorname {Subst}\left (\int \frac {x}{\sqrt {2+x^3}} \, dx,x,x^2\right )\\ &=\frac {1}{6 x^2 \sqrt {2+x^6}}-\frac {5 \sqrt {2+x^6}}{24 x^2}+\frac {5}{48} \operatorname {Subst}\left (\int \frac {\sqrt [3]{2} \left (1-\sqrt {3}\right )+x}{\sqrt {2+x^3}} \, dx,x,x^2\right )+\frac {5 \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+x^3}} \, dx,x,x^2\right )}{24 \sqrt [6]{2} \sqrt {2+\sqrt {3}}}\\ &=\frac {1}{6 x^2 \sqrt {2+x^6}}-\frac {5 \sqrt {2+x^6}}{24 x^2}+\frac {5 \sqrt {2+x^6}}{24 \left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )}-\frac {5 \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 )}{8\ 2^{5/6} 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 {5 \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 )}{12 \sqrt [3]{2} \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*}
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Mathematica [C] time = 0.03, size = 29, normalized size = 0.07 \[ -\frac {\, _2F_1\left (-\frac {1}{3},\frac {3}{2};\frac {2}{3};-\frac {x^6}{2}\right )}{4 \sqrt {2} x^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{6} + 2}}{x^{15} + 4 \, x^{9} + 4 \, x^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{6} + 2\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 40, normalized size = 0.10 \[ \frac {5 \sqrt {2}\, x^{4} \hypergeom \left (\left [\frac {1}{2}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -\frac {x^{6}}{2}\right )}{192}-\frac {5 x^{6}+6}{24 \sqrt {x^{6}+2}\, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x^{6} + 2\right )}^{\frac {3}{2}} x^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^3\,{\left (x^6+2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.68, size = 39, normalized size = 0.10 \[ \frac {\sqrt {2} \Gamma \left (- \frac {1}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{3}, \frac {3}{2} \\ \frac {2}{3} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{24 x^{2} \Gamma \left (\frac {2}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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