Optimal. Leaf size=74 \[ \frac{2 c \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
[Out]
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Rubi [A] time = 0.119195, antiderivative size = 74, 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 \[ \frac{2 c \tanh ^{-1}\left (\frac{b+2 c x^2}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{b+2 c x^2}{2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]
Antiderivative was successfully verified.
[In] Int[x^3/(a*x + b*x^3 + c*x^5)^2,x]
[Out]
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Rubi in Sympy [A] time = 14.8741, size = 66, normalized size = 0.89 \[ \frac{2 c \operatorname{atanh}{\left (\frac{b + 2 c x^{2}}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{b + 2 c x^{2}}{2 \left (- 4 a c + b^{2}\right ) \left (a + b x^{2} + c x^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3/(c*x**5+b*x**3+a*x)**2,x)
[Out]
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Mathematica [A] time = 0.144012, size = 79, normalized size = 1.07 \[ -\frac{\frac{4 c \tan ^{-1}\left (\frac{b+2 c x^2}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{b+2 c x^2}{a+b x^2+c x^4}}{2 \left (b^2-4 a c\right )} \]
Antiderivative was successfully verified.
[In] Integrate[x^3/(a*x + b*x^3 + c*x^5)^2,x]
[Out]
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Maple [A] time = 0.005, size = 75, normalized size = 1. \[{\frac{2\,c{x}^{2}+b}{ \left ( 8\,ac-2\,{b}^{2} \right ) \left ( c{x}^{4}+b{x}^{2}+a \right ) }}+2\,{\frac{c}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,c{x}^{2}+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3/(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^3/(c*x^5 + b*x^3 + a*x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.284356, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \,{\left (c^{2} x^{4} + b c x^{2} + a c\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} -{\left (2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) +{\left (2 \, c x^{2} + b\right )} \sqrt{b^{2} - 4 \, a c}}{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{4 \,{\left (c^{2} x^{4} + b c x^{2} + a c\right )} \arctan \left (-\frac{{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) +{\left (2 \, c x^{2} + b\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \,{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} x^{4} + a b^{2} - 4 \, a^{2} c +{\left (b^{3} - 4 \, a b c\right )} x^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^3/(c*x^5 + b*x^3 + a*x)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.74371, size = 267, normalized size = 3.61 \[ - c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x^{2} + \frac{- 16 a^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 8 a b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b c}{2 c^{2}} \right )} + c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x^{2} + \frac{16 a^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 8 a b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + b c}{2 c^{2}} \right )} + \frac{b + 2 c x^{2}}{8 a^{2} c - 2 a b^{2} + x^{4} \left (8 a c^{2} - 2 b^{2} c\right ) + x^{2} \left (8 a b c - 2 b^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3/(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^3/(c*x^5 + b*x^3 + a*x)^2,x, algorithm="giac")
[Out]