Optimal. Leaf size=79 \[ \frac {\sqrt {x^6-x^4+2} \left (2 x^{12}-14 x^{10}-3 x^8+8 x^6-28 x^4+8\right )}{6 x^6 \left (x^6+2\right )}-\frac {5}{2} \tan ^{-1}\left (\frac {x^2}{\sqrt {x^6-x^4+2}}\right ) \]
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Rubi [F] time = 4.39, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-4+x^6\right ) \left (2-x^4+x^6\right )^{5/2}}{x^7 \left (2+x^6\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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\begin {align*} \int \frac {\left (-4+x^6\right ) \left (2-x^4+x^6\right )^{5/2}}{x^7 \left (2+x^6\right )^2} \, dx &=\int \left (-\frac {\left (2-x^4+x^6\right )^{5/2}}{x^7}+\frac {5 \left (2-x^4+x^6\right )^{5/2}}{4 x}-\frac {3 x^5 \left (2-x^4+x^6\right )^{5/2}}{2 \left (2+x^6\right )^2}-\frac {5 x^5 \left (2-x^4+x^6\right )^{5/2}}{4 \left (2+x^6\right )}\right ) \, dx\\ &=\frac {5}{4} \int \frac {\left (2-x^4+x^6\right )^{5/2}}{x} \, dx-\frac {5}{4} \int \frac {x^5 \left (2-x^4+x^6\right )^{5/2}}{2+x^6} \, dx-\frac {3}{2} \int \frac {x^5 \left (2-x^4+x^6\right )^{5/2}}{\left (2+x^6\right )^2} \, dx-\int \frac {\left (2-x^4+x^6\right )^{5/2}}{x^7} \, dx\\ &=\frac {5}{8} \operatorname {Subst}\left (\int \frac {\left (2-x^2+x^3\right )^{5/2}}{x} \, dx,x,x^2\right )-\frac {3}{4} \operatorname {Subst}\left (\int \frac {x^2 \left (2-x^2+x^3\right )^{5/2}}{\left (2+x^3\right )^2} \, dx,x,x^2\right )-\frac {5}{4} \int \left (\frac {x^2 \left (2-x^4+x^6\right )^{5/2}}{2 \left (-i \sqrt {2}+x^3\right )}+\frac {x^2 \left (2-x^4+x^6\right )^{5/2}}{2 \left (i \sqrt {2}+x^3\right )}\right ) \, dx-\int \frac {\left (2-x^4+x^6\right )^{5/2}}{x^7} \, dx\\ &=-\left (\frac {5}{8} \int \frac {x^2 \left (2-x^4+x^6\right )^{5/2}}{-i \sqrt {2}+x^3} \, dx\right )-\frac {5}{8} \int \frac {x^2 \left (2-x^4+x^6\right )^{5/2}}{i \sqrt {2}+x^3} \, dx+\frac {5}{8} \operatorname {Subst}\left (\int \frac {\left (\frac {52}{27}-\frac {x}{3}+x^3\right )^{5/2}}{\frac {1}{3}+x} \, dx,x,\frac {1}{3} \left (-1+3 x^2\right )\right )-\frac {3}{4} \operatorname {Subst}\left (\int \frac {x^2 \left (2-x^2+x^3\right )^{5/2}}{\left (2+x^3\right )^2} \, dx,x,x^2\right )-\int \frac {\left (2-x^4+x^6\right )^{5/2}}{x^7} \, dx\\ &=-\left (\frac {5}{8} \int \left (-\frac {\left (2-x^4+x^6\right )^{5/2}}{3 \left (\sqrt [6]{-2}-x\right )}-\frac {\left (2-x^4+x^6\right )^{5/2}}{3 \left (-\sqrt [6]{-2} \sqrt [3]{-1}-x\right )}-\frac {\left (2-x^4+x^6\right )^{5/2}}{3 \left (\sqrt [6]{-2} (-1)^{2/3}-x\right )}\right ) \, dx\right )-\frac {5}{8} \int \left (\frac {\left (2-x^4+x^6\right )^{5/2}}{3 \left (\sqrt [6]{-2}+x\right )}+\frac {\left (2-x^4+x^6\right )^{5/2}}{3 \left (-\sqrt [6]{-2} \sqrt [3]{-1}+x\right )}+\frac {\left (2-x^4+x^6\right )^{5/2}}{3 \left (\sqrt [6]{-2} (-1)^{2/3}+x\right )}\right ) \, dx-\frac {3}{4} \operatorname {Subst}\left (\int \frac {x^2 \left (2-x^2+x^3\right )^{5/2}}{\left (2+x^3\right )^2} \, dx,x,x^2\right )+\frac {\left (1215 \left (2-x^4+x^6\right )^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{5/2} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{5/2}}{\frac {1}{3}+x} \, dx,x,\frac {1}{3} \left (-1+3 x^2\right )\right )}{8 \left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x^2\right )^{5/2} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) \left (-1+3 x^2\right )}{\sqrt [3]{26-15 \sqrt {3}}}+\left (-1+3 x^2\right )^2\right )^{5/2}}-\int \frac {\left (2-x^4+x^6\right )^{5/2}}{x^7} \, dx\\ &=\frac {5}{24} \int \frac {\left (2-x^4+x^6\right )^{5/2}}{\sqrt [6]{-2}-x} \, dx+\frac {5}{24} \int \frac {\left (2-x^4+x^6\right )^{5/2}}{-\sqrt [6]{-2} \sqrt [3]{-1}-x} \, dx+\frac {5}{24} \int \frac {\left (2-x^4+x^6\right )^{5/2}}{\sqrt [6]{-2} (-1)^{2/3}-x} \, dx-\frac {5}{24} \int \frac {\left (2-x^4+x^6\right )^{5/2}}{\sqrt [6]{-2}+x} \, dx-\frac {5}{24} \int \frac {\left (2-x^4+x^6\right )^{5/2}}{-\sqrt [6]{-2} \sqrt [3]{-1}+x} \, dx-\frac {5}{24} \int \frac {\left (2-x^4+x^6\right )^{5/2}}{\sqrt [6]{-2} (-1)^{2/3}+x} \, dx-\frac {3}{4} \operatorname {Subst}\left (\int \frac {x^2 \left (2-x^2+x^3\right )^{5/2}}{\left (2+x^3\right )^2} \, dx,x,x^2\right )+\frac {\left (1215 \left (2-x^4+x^6\right )^{5/2}\right ) \operatorname {Subst}\left (\int \frac {\left (\frac {1+\left (26-15 \sqrt {3}\right )^{2/3}}{3 \sqrt [3]{26-15 \sqrt {3}}}+x\right )^{5/2} \left (\frac {1}{9} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}\right )-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) x}{3 \sqrt [3]{26-15 \sqrt {3}}}+x^2\right )^{5/2}}{\frac {1}{3}+x} \, dx,x,\frac {1}{3} \left (-1+3 x^2\right )\right )}{8 \left (\frac {1}{3} \left (-1+\frac {1}{\sqrt [3]{26-15 \sqrt {3}}}+\sqrt [3]{26-15 \sqrt {3}}\right )+x^2\right )^{5/2} \left (-1+\frac {1}{\left (26-15 \sqrt {3}\right )^{2/3}}+\left (26-15 \sqrt {3}\right )^{2/3}-\frac {\left (1+\left (26-15 \sqrt {3}\right )^{2/3}\right ) \left (-1+3 x^2\right )}{\sqrt [3]{26-15 \sqrt {3}}}+\left (-1+3 x^2\right )^2\right )^{5/2}}-\int \frac {\left (2-x^4+x^6\right )^{5/2}}{x^7} \, dx\\ \end {align*}
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Mathematica [C] time = 1.70, size = 354, normalized size = 4.48 \begin {gather*} \frac {1}{6} \sqrt {x^6-x^4+2} \left (\frac {4}{x^6}-\frac {14}{x^2}-\frac {3 x^2}{x^6+2}+2\right )+\frac {\left (\frac {1}{2}+i\right ) \sqrt {i x^2+(1-i)} \sqrt {(-3-4 i) \left (x^4+i x^2-(1-i)\right )} \left (i F\left (\sin ^{-1}\left (\frac {\sqrt {(-2-i) \left (x^2-(1-i)\right )}}{\sqrt {5}}\right )|\frac {1}{2}+i\right )+\sqrt [3]{2} \left (\frac {3 \sqrt [6]{-1} \Pi \left (-\frac {2-i}{(-1+i)+\sqrt [3]{-2}};\sin ^{-1}\left (\frac {\sqrt {(-2-i) \left (x^2-(1-i)\right )}}{\sqrt {5}}\right )|\frac {1}{2}+i\right )}{\left (\sqrt [3]{-2}+(-1+i)\right ) \left (1+\sqrt [3]{-1}\right )^2}-\frac {i \Pi \left (\frac {2-i}{(1-i)+\sqrt [3]{2}};\sin ^{-1}\left (\frac {\sqrt {(-2-i) \left (x^2-(1-i)\right )}}{\sqrt {5}}\right )|\frac {1}{2}+i\right )}{\sqrt [3]{2}+(1-i)}-\frac {2 \sqrt [6]{-1} \Pi \left (\frac {2-i}{(1-i)+(-1)^{2/3} \sqrt [3]{2}};\sin ^{-1}\left (\frac {\sqrt {(-2-i) \left (x^2-(1-i)\right )}}{\sqrt {5}}\right )|\frac {1}{2}+i\right )}{(-2+2 i)+\sqrt [3]{2}-i \sqrt [3]{2} \sqrt {3}}\right )\right )}{\sqrt {2} \sqrt {x^6-x^4+2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 3.09, size = 79, normalized size = 1.00 \begin {gather*} \frac {\sqrt {x^6-x^4+2} \left (2 x^{12}-14 x^{10}-3 x^8+8 x^6-28 x^4+8\right )}{6 x^6 \left (x^6+2\right )}-\frac {5}{2} \tan ^{-1}\left (\frac {x^2}{\sqrt {x^6-x^4+2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 96, normalized size = 1.22 \begin {gather*} -\frac {15 \, {\left (x^{12} + 2 \, x^{6}\right )} \arctan \left (\frac {2 \, \sqrt {x^{6} - x^{4} + 2} x^{2}}{x^{6} - 2 \, x^{4} + 2}\right ) - 2 \, {\left (2 \, x^{12} - 14 \, x^{10} - 3 \, x^{8} + 8 \, x^{6} - 28 \, x^{4} + 8\right )} \sqrt {x^{6} - x^{4} + 2}}{12 \, {\left (x^{12} + 2 \, x^{6}\right )}} \end {gather*}
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 \begin {gather*} \int \frac {{\left (x^{6} - x^{4} + 2\right )}^{\frac {5}{2}} {\left (x^{6} - 4\right )}}{{\left (x^{6} + 2\right )}^{2} x^{7}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.51, size = 133, normalized size = 1.68 \begin {gather*} \frac {2 x^{18}-16 x^{16}+11 x^{14}+15 x^{12}-64 x^{10}+22 x^{8}+24 x^{6}-64 x^{4}+16}{6 x^{6} \sqrt {x^{6}-x^{4}+2}\, \left (x^{6}+2\right )}-\frac {5 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{6}-2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x^{4}+2 \sqrt {x^{6}-x^{4}+2}\, x^{2}+2 \RootOf \left (\textit {\_Z}^{2}+1\right )}{x^{6}+2}\right )}{4} \end {gather*}
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 \begin {gather*} \int \frac {{\left (x^{6} - x^{4} + 2\right )}^{\frac {5}{2}} {\left (x^{6} - 4\right )}}{{\left (x^{6} + 2\right )}^{2} x^{7}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\left (x^6-4\right )\,{\left (x^6-x^4+2\right )}^{5/2}}{x^7\,{\left (x^6+2\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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