Optimal. Leaf size=140 \[ -\frac{1}{4} \sqrt{-x^2-4 x-3} x+\frac{5}{2} \sqrt{-x^2-4 x-3}+\frac{\tan ^{-1}\left (\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right )}{2 \sqrt{2}}-\frac{5}{4} \tanh ^{-1}\left (\frac{x}{\sqrt{-x^2-4 x-3}}\right )+\frac{11}{2} \sin ^{-1}(x+2) \]
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Rubi [A] time = 1.03351, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 24, number of rules used = 14, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ -\frac{1}{4} \sqrt{-x^2-4 x-3} x+\frac{5}{2} \sqrt{-x^2-4 x-3}+\frac{\tan ^{-1}\left (\frac{1-\frac{x+3}{\sqrt{-x^2-4 x-3}}}{\sqrt{2}}\right )}{2 \sqrt{2}}-\frac{\tan ^{-1}\left (\frac{\frac{x+3}{\sqrt{-x^2-4 x-3}}+1}{\sqrt{2}}\right )}{2 \sqrt{2}}-\frac{5}{4} \tanh ^{-1}\left (\frac{x}{\sqrt{-x^2-4 x-3}}\right )+\frac{11}{2} \sin ^{-1}(x+2) \]
Antiderivative was successfully verified.
[In] Int[x^4/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 121.949, size = 153, normalized size = 1.09 \[ - \frac{x \sqrt{- x^{2} - 4 x - 3}}{4} + \frac{5 \sqrt{- x^{2} - 4 x - 3}}{2} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{3 \left (\frac{x}{3} + 1\right )}{2 \sqrt{- x^{2} - 4 x - 3}} - \frac{1}{2}\right ) \right )}}{4} - \frac{\sqrt{2} \operatorname{atan}{\left (\sqrt{2} \left (\frac{3 \left (\frac{x}{3} + 1\right )}{2 \sqrt{- x^{2} - 4 x - 3}} + \frac{1}{2}\right ) \right )}}{4} + \frac{11 \operatorname{atan}{\left (- \frac{- 2 x - 4}{2 \sqrt{- x^{2} - 4 x - 3}} \right )}}{2} - \frac{5 \operatorname{atanh}{\left (\frac{x}{\sqrt{- x^{2} - 4 x - 3}} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4/(2*x**2+4*x+3)/(-x**2-4*x-3)**(1/2),x)
[Out]
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Mathematica [C] time = 6.29382, size = 1101, normalized size = 7.86 \[ \sqrt{-x^2-4 x-3} \left (\frac{5}{2}-\frac{x}{4}\right )+\frac{11}{2} \sin ^{-1}(x+2)+\frac{i \left (-7 i+4 \sqrt{2}\right ) \tan ^{-1}\left (\frac{78 i \sqrt{2} x^4+224 x^4+162 i \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x^3+20 i \sqrt{2} x^3+1276 x^3+648 i \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x^2-727 i \sqrt{2} x^2+2236 x^2+891 i \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x-1168 i \sqrt{2} x+1316 x+486 i \sqrt{1-2 i \sqrt{2}} \sqrt{-x^2-4 x-3}-471 i \sqrt{2}+132}{128 \sqrt{2} x^4+66 i x^4+544 \sqrt{2} x^3+208 i x^3+514 \sqrt{2} x^2+685 i x^2+104 \sqrt{2} x+1396 i x+6 \sqrt{2}+885 i}\right )}{8 \sqrt{1-2 i \sqrt{2}}}-\frac{\left (7 i+4 \sqrt{2}\right ) \tanh ^{-1}\left (\frac{78 \sqrt{2} x^4+224 i x^4+162 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x^3+20 \sqrt{2} x^3+1276 i x^3+648 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x^2-727 \sqrt{2} x^2+2236 i x^2+891 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3} x-1168 \sqrt{2} x+1316 i x+486 \sqrt{1+2 i \sqrt{2}} \sqrt{-x^2-4 x-3}-471 \sqrt{2}+132 i}{128 \sqrt{2} x^4-66 i x^4+544 \sqrt{2} x^3-208 i x^3+514 \sqrt{2} x^2-685 i x^2+104 \sqrt{2} x-1396 i x+6 \sqrt{2}-885 i}\right )}{8 \sqrt{1+2 i \sqrt{2}}}-\frac{\left (7 i+4 \sqrt{2}\right ) \log \left (\left (-2 i x+\sqrt{2}-2 i\right )^2 \left (2 i x+\sqrt{2}+2 i\right )^2\right )}{16 \sqrt{1+2 i \sqrt{2}}}-\frac{\left (-7 i+4 \sqrt{2}\right ) \log \left (\left (-2 i x+\sqrt{2}-2 i\right )^2 \left (2 i x+\sqrt{2}+2 i\right )^2\right )}{16 \sqrt{1-2 i \sqrt{2}}}+\frac{\left (-7 i+4 \sqrt{2}\right ) \log \left (\left (2 x^2+4 x+3\right ) \left (2 i \sqrt{2} x^2+2 x^2-2 \sqrt{2 \left (1-2 i \sqrt{2}\right )} \sqrt{-x^2-4 x-3} x+8 i \sqrt{2} x+4 x-2 \sqrt{2 \left (1-2 i \sqrt{2}\right )} \sqrt{-x^2-4 x-3}+6 i \sqrt{2}+3\right )\right )}{16 \sqrt{1-2 i \sqrt{2}}}+\frac{\left (7 i+4 \sqrt{2}\right ) \log \left (\left (2 x^2+4 x+3\right ) \left (-2 i \sqrt{2} x^2+2 x^2-2 \sqrt{2 \left (1+2 i \sqrt{2}\right )} \sqrt{-x^2-4 x-3} x-8 i \sqrt{2} x+4 x-2 \sqrt{2 \left (1+2 i \sqrt{2}\right )} \sqrt{-x^2-4 x-3}-6 i \sqrt{2}+3\right )\right )}{16 \sqrt{1+2 i \sqrt{2}}} \]
Antiderivative was successfully verified.
[In] Integrate[x^4/(Sqrt[-3 - 4*x - x^2]*(3 + 4*x + 2*x^2)),x]
[Out]
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Maple [A] time = 0.024, size = 159, normalized size = 1.1 \[{\frac{11\,\arcsin \left ( 2+x \right ) }{2}}+{\frac{5}{2}\sqrt{-{x}^{2}-4\,x-3}}-{\frac{x}{4}\sqrt{-{x}^{2}-4\,x-3}}+{\frac{\sqrt{3}\sqrt{4}}{24}\sqrt{3\,{\frac{{x}^{2}}{ \left ( -3/2-x \right ) ^{2}}}-12} \left ( \sqrt{2}\arctan \left ({\frac{\sqrt{2}}{6}\sqrt{3\,{\frac{{x}^{2}}{ \left ( -3/2-x \right ) ^{2}}}-12}} \right ) +5\,{\it Artanh} \left ( 3\,{\frac{x}{-3/2-x}{\frac{1}{\sqrt{3\,{\frac{{x}^{2}}{ \left ( -3/2-x \right ) ^{2}}}-12}}}} \right ) \right ){\frac{1}{\sqrt{{1 \left ({{x}^{2} \left ( -{\frac{3}{2}}-x \right ) ^{-2}}-4 \right ) \left ( 1+{x \left ( -{\frac{3}{2}}-x \right ) ^{-1}} \right ) ^{-2}}}}} \left ( 1+{x \left ( -{\frac{3}{2}}-x \right ) ^{-1}} \right ) ^{-1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(2*x^2+4*x+3)/(-x^2-4*x-3)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (2 \, x^{2} + 4 \, x + 3\right )} \sqrt{-x^{2} - 4 \, x - 3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((2*x^2 + 4*x + 3)*sqrt(-x^2 - 4*x - 3)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.307625, size = 242, normalized size = 1.73 \[ -\frac{1}{32} \, \sqrt{2}{\left (4 \, \sqrt{2} \sqrt{-x^{2} - 4 \, x - 3}{\left (x - 10\right )} - 88 \, \sqrt{2} \arctan \left (\frac{x + 2}{\sqrt{-x^{2} - 4 \, x - 3}}\right ) - 5 \, \sqrt{2} \log \left (-\frac{2 \, \sqrt{-x^{2} - 4 \, x - 3} x + 4 \, x + 3}{x^{2}}\right ) + 5 \, \sqrt{2} \log \left (\frac{2 \, \sqrt{-x^{2} - 4 \, x - 3} x - 4 \, x - 3}{x^{2}}\right ) - 4 \, \arctan \left (\frac{\sqrt{2} x + 3 \, \sqrt{2} \sqrt{-x^{2} - 4 \, x - 3}}{2 \,{\left (2 \, x + 3\right )}}\right ) - 4 \, \arctan \left (-\frac{\sqrt{2} x - 3 \, \sqrt{2} \sqrt{-x^{2} - 4 \, x - 3}}{2 \,{\left (2 \, x + 3\right )}}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((2*x^2 + 4*x + 3)*sqrt(-x^2 - 4*x - 3)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{- \left (x + 1\right ) \left (x + 3\right )} \left (2 x^{2} + 4 x + 3\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(2*x**2+4*x+3)/(-x**2-4*x-3)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.290451, size = 254, normalized size = 1.81 \[ -\frac{1}{4} \, \sqrt{-x^{2} - 4 \, x - 3}{\left (x - 10\right )} + \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\frac{3 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + 1\right )}\right ) + \frac{1}{4} \, \sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\frac{\sqrt{-x^{2} - 4 \, x - 3} - 1}{x + 2} + 1\right )}\right ) + \frac{11}{2} \, \arcsin \left (x + 2\right ) - \frac{5}{8} \,{\rm ln}\left (\frac{2 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac{3 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 1\right ) + \frac{5}{8} \,{\rm ln}\left (\frac{2 \,{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}}{x + 2} + \frac{{\left (\sqrt{-x^{2} - 4 \, x - 3} - 1\right )}^{2}}{{\left (x + 2\right )}^{2}} + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((2*x^2 + 4*x + 3)*sqrt(-x^2 - 4*x - 3)),x, algorithm="giac")
[Out]