Optimal. Leaf size=130 \[ -\frac{\left (2 a c+b^2\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x^2 \left (2 a c+b^2\right )+3 a b}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )} \]
[Out]
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Rubi [A] time = 0.250301, antiderivative size = 130, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{\left (2 a c+b^2\right ) \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac{x^2 \left (2 a+b x^2\right )}{4 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac{x^2 \left (2 a c+b^2\right )+3 a b}{2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
[In] Int[x^5/(a + b*x^2 + c*x^4)^3,x]
[Out]
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Rubi in Sympy [A] time = 29.8488, size = 119, normalized size = 0.92 \[ \frac{x^{2} \left (2 a + b x^{2}\right )}{4 \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )^{2}} + \frac{6 a b + x^{2} \left (4 a c + 2 b^{2}\right )}{4 \left (- 4 a c + b^{2}\right )^{2} \left (a + b x^{2} + c x^{4}\right )} - \frac{\left (2 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(c*x**4+b*x**2+a)**3,x)
[Out]
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Mathematica [A] time = 0.259719, size = 145, normalized size = 1.12 \[ \frac{1}{4} \left (\frac{4 \left (2 a c+b^2\right ) \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{5/2}}+\frac{\left (2 a c+b^2\right ) \left (b+2 c x^2\right )}{c \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac{a \left (b-2 c x^2\right )+b^2 x^2}{c \left (4 a c-b^2\right ) \left (a+b x^2+c x^4\right )^2}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[x^5/(a + b*x^2 + c*x^4)^3,x]
[Out]
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Maple [B] time = 0.019, size = 270, normalized size = 2.1 \[{\frac{1}{2\, \left ( c{x}^{4}+b{x}^{2}+a \right ) ^{2}} \left ({\frac{c \left ( 2\,ac+{b}^{2} \right ){x}^{6}}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}}+{\frac{3\,b \left ( 2\,ac+{b}^{2} \right ){x}^{4}}{32\,{a}^{2}{c}^{2}-16\,a{b}^{2}c+2\,{b}^{4}}}-{\frac{a \left ( 2\,ac-5\,{b}^{2} \right ){x}^{2}}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}}+3\,{\frac{{a}^{2}b}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}} \right ) }+2\,{\frac{ac}{ \left ( 16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+{\frac{{b}^{2}}{16\,{a}^{2}{c}^{2}-8\,a{b}^{2}c+{b}^{4}}\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(c*x^4+b*x^2+a)^3,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(c*x^4 + b*x^2 + a)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.272466, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(c*x^4 + b*x^2 + a)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 35.7218, size = 580, normalized size = 4.46 \[ - \frac{\sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) \log{\left (x^{2} + \frac{- 64 a^{3} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) + 48 a^{2} b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) - 12 a b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) + 2 a b c + b^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) + b^{3}}{4 a c^{2} + 2 b^{2} c} \right )}}{2} + \frac{\sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) \log{\left (x^{2} + \frac{64 a^{3} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) - 48 a^{2} b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) + 12 a b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) + 2 a b c - b^{6} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{5}}} \left (2 a c + b^{2}\right ) + b^{3}}{4 a c^{2} + 2 b^{2} c} \right )}}{2} + \frac{6 a^{2} b + x^{6} \left (4 a c^{2} + 2 b^{2} c\right ) + x^{4} \left (6 a b c + 3 b^{3}\right ) + x^{2} \left (- 4 a^{2} c + 10 a b^{2}\right )}{64 a^{4} c^{2} - 32 a^{3} b^{2} c + 4 a^{2} b^{4} + x^{8} \left (64 a^{2} c^{4} - 32 a b^{2} c^{3} + 4 b^{4} c^{2}\right ) + x^{6} \left (128 a^{2} b c^{3} - 64 a b^{3} c^{2} + 8 b^{5} c\right ) + x^{4} \left (128 a^{3} c^{3} - 24 a b^{4} c + 4 b^{6}\right ) + x^{2} \left (128 a^{3} b c^{2} - 64 a^{2} b^{3} c + 8 a b^{5}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(c*x**4+b*x**2+a)**3,x)
[Out]
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GIAC/XCAS [A] time = 15.6234, size = 217, normalized size = 1.67 \[ \frac{{\left (b^{2} + 2 \, a c\right )} \arctan \left (\frac{2 \, c x^{2} + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{2 \, b^{2} c x^{6} + 4 \, a c^{2} x^{6} + 3 \, b^{3} x^{4} + 6 \, a b c x^{4} + 10 \, a b^{2} x^{2} - 4 \, a^{2} c x^{2} + 6 \, a^{2} b}{4 \,{\left (c x^{4} + b x^{2} + a\right )}^{2}{\left (b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(c*x^4 + b*x^2 + a)^3,x, algorithm="giac")
[Out]