Optimal. Leaf size=242 \[ \frac{5 x^3}{3}-\frac{21}{512} \sqrt{22721 \sqrt{3}-34271} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{21}{512} \sqrt{22721 \sqrt{3}-34271} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{\left (835 x^2+1468\right ) x}{64 \left (x^4+2 x^2+3\right )}+\frac{25 \left (5 x^2+3\right ) x}{16 \left (x^4+2 x^2+3\right )^2}-27 x-\frac{21}{256} \sqrt{34271+22721 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{21}{256} \sqrt{34271+22721 \sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
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Rubi [A] time = 0.757398, antiderivative size = 242, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 8, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.258 \[ \frac{5 x^3}{3}-\frac{21}{512} \sqrt{22721 \sqrt{3}-34271} \log \left (x^2-\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )+\frac{21}{512} \sqrt{22721 \sqrt{3}-34271} \log \left (x^2+\sqrt{2 \left (\sqrt{3}-1\right )} x+\sqrt{3}\right )-\frac{\left (835 x^2+1468\right ) x}{64 \left (x^4+2 x^2+3\right )}+\frac{25 \left (5 x^2+3\right ) x}{16 \left (x^4+2 x^2+3\right )^2}-27 x-\frac{21}{256} \sqrt{34271+22721 \sqrt{3}} \tan ^{-1}\left (\frac{\sqrt{2 \left (\sqrt{3}-1\right )}-2 x}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right )+\frac{21}{256} \sqrt{34271+22721 \sqrt{3}} \tan ^{-1}\left (\frac{2 x+\sqrt{2 \left (\sqrt{3}-1\right )}}{\sqrt{2 \left (1+\sqrt{3}\right )}}\right ) \]
Antiderivative was successfully verified.
[In] Int[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^3,x]
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Rubi in Sympy [A] time = 53.1182, size = 360, normalized size = 1.49 \[ \frac{5 x^{3}}{3} + \frac{x \left (96000 x^{2} + 57600\right )}{12288 \left (x^{4} + 2 x^{2} + 3\right )^{2}} - \frac{x \left (246251520 x^{2} + 432930816\right )}{18874368 \left (x^{4} + 2 x^{2} + 3\right )} - 27 x - \frac{\sqrt{6} \left (- 424230912 \sqrt{3} + 966131712\right ) \log{\left (x^{2} - \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{226492416 \sqrt{-1 + \sqrt{3}}} + \frac{\sqrt{6} \left (- 424230912 \sqrt{3} + 966131712\right ) \log{\left (x^{2} + \sqrt{2} x \sqrt{-1 + \sqrt{3}} + \sqrt{3} \right )}}{226492416 \sqrt{-1 + \sqrt{3}}} + \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 848461824 \sqrt{3} + 1932263424\right )}{2} + 1932263424 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x - \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{113246208 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} + \frac{\sqrt{3} \left (- \frac{\sqrt{2} \sqrt{-1 + \sqrt{3}} \left (- 848461824 \sqrt{3} + 1932263424\right )}{2} + 1932263424 \sqrt{2} \sqrt{-1 + \sqrt{3}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \left (x + \frac{\sqrt{-2 + 2 \sqrt{3}}}{2}\right )}{\sqrt{1 + \sqrt{3}}} \right )}}{113246208 \sqrt{-1 + \sqrt{3}} \sqrt{1 + \sqrt{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**8*(5*x**6+3*x**4+x**2+4)/(x**4+2*x**2+3)**3,x)
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Mathematica [C] time = 0.391872, size = 155, normalized size = 0.64 \[ \frac{5 x^3}{3}-\frac{\left (835 x^2+1468\right ) x}{64 \left (x^4+2 x^2+3\right )}+\frac{25 \left (5 x^2+3\right ) x}{16 \left (x^4+2 x^2+3\right )^2}-27 x+\frac{21 \left (137 \sqrt{2}-175 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1-i \sqrt{2}}}\right )}{128 \sqrt{2-2 i \sqrt{2}}}+\frac{21 \left (137 \sqrt{2}+175 i\right ) \tan ^{-1}\left (\frac{x}{\sqrt{1+i \sqrt{2}}}\right )}{128 \sqrt{2+2 i \sqrt{2}}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^8*(4 + x^2 + 3*x^4 + 5*x^6))/(3 + 2*x^2 + x^4)^3,x]
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Maple [B] time = 0.033, size = 426, normalized size = 1.8 \[{\frac{5\,{x}^{3}}{3}}-27\,x+{\frac{1}{ \left ({x}^{4}+2\,{x}^{2}+3 \right ) ^{2}} \left ( -{\frac{835\,{x}^{7}}{64}}-{\frac{1569\,{x}^{5}}{32}}-{\frac{4941\,{x}^{3}}{64}}-{\frac{513\,x}{8}} \right ) }-{\frac{693\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}+{\frac{3675\,\ln \left ({x}^{2}+\sqrt{3}+x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -1386+1386\,\sqrt{3} \right ) \sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-7350+7350\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{273\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x+\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{693\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}\sqrt{3}}{1024}}-{\frac{3675\,\ln \left ({x}^{2}+\sqrt{3}-x\sqrt{-2+2\,\sqrt{3}} \right ) \sqrt{-2+2\,\sqrt{3}}}{1024}}+{\frac{ \left ( -1386+1386\,\sqrt{3} \right ) \sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }-{\frac{-7350+7350\,\sqrt{3}}{512\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) }+{\frac{273\,\sqrt{3}}{8\,\sqrt{2+2\,\sqrt{3}}}\arctan \left ({\frac{2\,x-\sqrt{-2+2\,\sqrt{3}}}{\sqrt{2+2\,\sqrt{3}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^8*(5*x^6+3*x^4+x^2+4)/(x^4+2*x^2+3)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \frac{5}{3} \, x^{3} - 27 \, x - \frac{835 \, x^{7} + 3138 \, x^{5} + 4941 \, x^{3} + 4104 \, x}{64 \,{\left (x^{8} + 4 \, x^{6} + 10 \, x^{4} + 12 \, x^{2} + 9\right )}} + \frac{21}{64} \, \int \frac{137 \, x^{2} + 312}{x^{4} + 2 \, x^{2} + 3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^8/(x^4 + 2*x^2 + 3)^3,x, algorithm="maxima")
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Fricas [A] time = 0.316244, size = 1188, normalized size = 4.91 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^8/(x^4 + 2*x^2 + 3)^3,x, algorithm="fricas")
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Sympy [A] time = 2.19193, size = 80, normalized size = 0.33 \[ \frac{5 x^{3}}{3} - 27 x - \frac{835 x^{7} + 3138 x^{5} + 4941 x^{3} + 4104 x}{64 x^{8} + 256 x^{6} + 640 x^{4} + 768 x^{2} + 576} + 21 \operatorname{RootSum}{\left (17179869184 t^{4} + 8983937024 t^{2} + 1548731523, \left ( t \mapsto t \log{\left (- \frac{1107296256 t^{3}}{310800559} + \frac{438857984 t}{310800559} + x \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**8*(5*x**6+3*x**4+x**2+4)/(x**4+2*x**2+3)**3,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (5 \, x^{6} + 3 \, x^{4} + x^{2} + 4\right )} x^{8}}{{\left (x^{4} + 2 \, x^{2} + 3\right )}^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((5*x^6 + 3*x^4 + x^2 + 4)*x^8/(x^4 + 2*x^2 + 3)^3,x, algorithm="giac")
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