Optimal. Leaf size=66 \[ \frac{4 c \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
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Rubi [A] time = 0.068717, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222 \[ \frac{4 c \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac{b+2 c x}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[1/((c + a/x^2 + b/x)^2*x^4),x]
[Out]
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Rubi in Sympy [A] time = 11.3361, size = 60, normalized size = 0.91 \[ \frac{4 c \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\left (- 4 a c + b^{2}\right )^{\frac{3}{2}}} - \frac{b + 2 c x}{\left (- 4 a c + b^{2}\right ) \left (a + b x + c x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c+a/x**2+b/x)**2/x**4,x)
[Out]
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Mathematica [A] time = 0.122845, size = 70, normalized size = 1.06 \[ -\frac{\frac{4 c \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\sqrt{4 a c-b^2}}+\frac{b+2 c x}{a+x (b+c x)}}{b^2-4 a c} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c + a/x^2 + b/x)^2*x^4),x]
[Out]
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Maple [A] time = 0.003, size = 68, normalized size = 1. \[{\frac{2\,cx+b}{ \left ( 4\,ac-{b}^{2} \right ) \left ( c{x}^{2}+bx+a \right ) }}+4\,{\frac{c}{ \left ( 4\,ac-{b}^{2} \right ) ^{3/2}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c+a/x^2+b/x)^2/x^4,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(1/((c + b/x + a/x^2)^2*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.277398, size = 1, normalized size = 0.02 \[ \left [-\frac{2 \,{\left (c^{2} x^{2} + b c x + a c\right )} \log \left (-\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x -{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) + \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{{\left (a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{4 \,{\left (c^{2} x^{2} + b c x + a c\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + \sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{{\left (a b^{2} - 4 \, a^{2} c +{\left (b^{2} c - 4 \, a c^{2}\right )} x^{2} +{\left (b^{3} - 4 \, a b c\right )} x\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c + b/x + a/x^2)^2*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.71092, size = 265, normalized size = 4.02 \[ - 2 c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{- 32 a^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 16 a b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 2 b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b c}{4 c^{2}} \right )} + 2 c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} \log{\left (x + \frac{32 a^{2} c^{3} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} - 16 a b^{2} c^{2} \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b^{4} c \sqrt{- \frac{1}{\left (4 a c - b^{2}\right )^{3}}} + 2 b c}{4 c^{2}} \right )} + \frac{b + 2 c x}{4 a^{2} c - a b^{2} + x^{2} \left (4 a c^{2} - b^{2} c\right ) + x \left (4 a b c - b^{3}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c+a/x**2+b/x)**2/x**4,x)
[Out]
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GIAC/XCAS [A] time = 0.277385, size = 103, normalized size = 1.56 \[ -\frac{4 \, c \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} - 4 \, a c\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, c x + b}{{\left (c x^{2} + b x + a\right )}{\left (b^{2} - 4 \, a c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c + b/x + a/x^2)^2*x^4),x, algorithm="giac")
[Out]