Optimal. Leaf size=394 \[ -\frac {x^{10}}{3 \sqrt {x^6+2}}+\frac {10}{21} \sqrt {x^6+2} x^4-\frac {80 \sqrt {x^6+2}}{21 \left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )}-\frac {80\ 2^{2/3} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )|-7-4 \sqrt {3}\right )}{21 \sqrt [4]{3} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}+\frac {40 \sqrt [6]{2} \sqrt {2-\sqrt {3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} E\left (\sin ^{-1}\left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )|-7-4 \sqrt {3}\right )}{7\ 3^{3/4} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}} \]
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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, 288, 321, 303, 218, 1877} \[ -\frac {x^{10}}{3 \sqrt {x^6+2}}+\frac {10}{21} \sqrt {x^6+2} x^4-\frac {80 \sqrt {x^6+2}}{21 \left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )}-\frac {80\ 2^{2/3} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} F\left (\sin ^{-1}\left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )|-7-4 \sqrt {3}\right )}{21 \sqrt [4]{3} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}}+\frac {40 \sqrt [6]{2} \sqrt {2-\sqrt {3}} \left (x^2+\sqrt [3]{2}\right ) \sqrt {\frac {x^4-\sqrt [3]{2} x^2+2^{2/3}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} E\left (\sin ^{-1}\left (\frac {x^2+\sqrt [3]{2} \left (1-\sqrt {3}\right )}{x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )}\right )|-7-4 \sqrt {3}\right )}{7\ 3^{3/4} \sqrt {\frac {x^2+\sqrt [3]{2}}{\left (x^2+\sqrt [3]{2} \left (1+\sqrt {3}\right )\right )^2}} \sqrt {x^6+2}} \]
Antiderivative was successfully verified.
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Rule 218
Rule 275
Rule 288
Rule 303
Rule 321
Rule 1877
Rubi steps
\begin {align*} \int \frac {x^{15}}{\left (2+x^6\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x^7}{\left (2+x^3\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {x^{10}}{3 \sqrt {2+x^6}}+\frac {5}{3} \operatorname {Subst}\left (\int \frac {x^4}{\sqrt {2+x^3}} \, dx,x,x^2\right )\\ &=-\frac {x^{10}}{3 \sqrt {2+x^6}}+\frac {10}{21} x^4 \sqrt {2+x^6}-\frac {40}{21} \operatorname {Subst}\left (\int \frac {x}{\sqrt {2+x^3}} \, dx,x,x^2\right )\\ &=-\frac {x^{10}}{3 \sqrt {2+x^6}}+\frac {10}{21} x^4 \sqrt {2+x^6}-\frac {40}{21} \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 (40\ 2^{5/6}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {2+x^3}} \, dx,x,x^2\right )}{21 \sqrt {2+\sqrt {3}}}\\ &=-\frac {x^{10}}{3 \sqrt {2+x^6}}+\frac {10}{21} x^4 \sqrt {2+x^6}-\frac {80 \sqrt {2+x^6}}{21 \left (\sqrt [3]{2} \left (1+\sqrt {3}\right )+x^2\right )}+\frac {40 \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 )}{7\ 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 {80\ 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 )}{21 \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 = 0.01, size = 54, normalized size = 0.14 \[ \frac {x^4 \left (10 \sqrt {2} \sqrt {x^6+2} \, _2F_1\left (\frac {2}{3},\frac {3}{2};\frac {5}{3};-\frac {x^6}{2}\right )+x^6-20\right )}{7 \sqrt {x^6+2}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.90, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{6} + 2} x^{15}}{x^{12} + 4 \, x^{6} + 4}, x\right ) \]
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 \[ \int \frac {x^{15}}{{\left (x^{6} + 2\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ -\frac {10 \sqrt {2}\, x^{4} \hypergeom \left (\left [\frac {1}{2}, \frac {2}{3}\right ], \left [\frac {5}{3}\right ], -\frac {x^{6}}{2}\right )}{21}+\frac {\left (3 x^{6}+20\right ) x^{4}}{21 \sqrt {x^{6}+2}} \]
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 \[ \int \frac {x^{15}}{{\left (x^{6} + 2\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {x^{15}}{{\left (x^6+2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.82, size = 36, normalized size = 0.09 \[ \frac {\sqrt {2} x^{16} \Gamma \left (\frac {8}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, \frac {8}{3} \\ \frac {11}{3} \end {matrix}\middle | {\frac {x^{6} e^{i \pi }}{2}} \right )}}{24 \Gamma \left (\frac {11}{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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