Optimal. Leaf size=75 \[ \frac{2 a+b x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
[Out]
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Rubi [A] time = 0.119919, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 a+b x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[x^5/(a*x + b*x^3 + c*x^5)^2,x]
[Out]
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Rubi in Sympy [A] time = 18.3567, size = 65, normalized size = 0.87 \[ - \frac{b \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} + \frac{2 a + b x^{2}}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**5/(c*x**5+b*x**3+a*x)**2,x)
[Out]
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Mathematica [A] time = 0.122188, size = 79, normalized size = 1.05 \[ \frac{2 a+b x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac{b \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[x^5/(a*x + b*x^3 + c*x^5)^2,x]
[Out]
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Maple [A] time = 0.005, size = 77, normalized size = 1. \[{\frac{-b{x}^{2}-2\,a}{ \left ( 8\,ac-2\,{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) }}-{b\arctan \left ({(2\,c{x}^{2}+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ) \left ( 4\,ac-{b}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^5/(c*x^5+b*x^3+a*x)^2,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^5 + b*x^3 + a*x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.307436, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (b c x^{4} + b^{2} x^{2} + a b\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) -{\left (b x^{2} + 2 \, a\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, \frac{2 \,{\left (b c x^{4} + b^{2} x^{2} + a b\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left (b x^{2} + 2 \, a\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(c*x^5 + b*x^3 + a*x)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.89395, size = 267, normalized size = 3.56 \[ \frac{b \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x^{2} + \frac{- 16 a^{2} b c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 8 a b^{3} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - b^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b^{2}}{2 b c} \right )}}{2} - \frac{b \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x^{2} + \frac{16 a^{2} b c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 8 a b^{3} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b^{5} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b^{2}}{2 b c} \right )}}{2} - \frac{2 a + b x^{2}}{8 a^{2} c - 2 a b^{2} + x^{4} \left (8 a c^{2} - 2 b^{2} c\right ) + x^{2} \left (8 a b c - 2 b^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**5/(c*x**5+b*x**3+a*x)**2,x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^5/(c*x^5 + b*x^3 + a*x)^2,x, algorithm="giac")
[Out]