Optimal. Leaf size=112 \[ -\frac{x \left (-\frac{a^2 f}{b^2}+\frac{b c}{a}+\frac{a e}{b}-d\right )}{2 a \left (a+b x^2\right )}-\frac{c}{a^2 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^3 f-a^2 b e-a b^2 d+3 b^3 c\right )}{2 a^{5/2} b^{5/2}}+\frac{f x}{b^2} \]
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Rubi [A] time = 0.322123, antiderivative size = 112, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{x \left (-\frac{a^2 f}{b^2}+\frac{b c}{a}+\frac{a e}{b}-d\right )}{2 a \left (a+b x^2\right )}-\frac{c}{a^2 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^3 f-a^2 b e-a b^2 d+3 b^3 c\right )}{2 a^{5/2} b^{5/2}}+\frac{f x}{b^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^2*(a + b*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 140.885, size = 122, normalized size = 1.09 \[ \frac{f x}{b^{2}} - \frac{x \left (\frac{a^{3} f}{x^{2}} - \frac{a^{2} b e}{x^{2}} + \frac{a b^{2} d}{x^{2}} - \frac{b^{3} c}{x^{2}}\right )}{2 a b^{3} \left (a + b x^{2}\right )} - \frac{a^{2} f - a b e + b^{2} d}{a b^{3} x} - \frac{\left (3 a^{2} f - 2 a b e + b^{2} d\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{3}{2}} b^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**6+e*x**4+d*x**2+c)/x**2/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.114736, size = 115, normalized size = 1.03 \[ -\frac{c}{a^2 x}+\frac{x \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{2 a^2 b^2 \left (a+b x^2\right )}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (3 a^3 f-a^2 b e-a b^2 d+3 b^3 c\right )}{2 a^{5/2} b^{5/2}}+\frac{f x}{b^2} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^2*(a + b*x^2)^2),x]
[Out]
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Maple [A] time = 0.017, size = 165, normalized size = 1.5 \[{\frac{fx}{{b}^{2}}}-{\frac{c}{x{a}^{2}}}+{\frac{axf}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{ex}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{dx}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{bxc}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,af}{2\,{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{e}{2\,b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{d}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,bc}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^6+e*x^4+d*x^2+c)/x^2/(b*x^2+a)^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((f*x^6 + e*x^4 + d*x^2 + c)/((b*x^2 + a)^2*x^2),x, algorithm="maxima")
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Fricas [A] time = 0.234765, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (3 \, b^{4} c - a b^{3} d - a^{2} b^{2} e + 3 \, a^{3} b f\right )} x^{3} +{\left (3 \, a b^{3} c - a^{2} b^{2} d - a^{3} b e + 3 \, a^{4} f\right )} x\right )} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) + 2 \,{\left (2 \, a^{2} b f x^{4} - 2 \, a b^{2} c -{\left (3 \, b^{3} c - a b^{2} d + a^{2} b e - 3 \, a^{3} f\right )} x^{2}\right )} \sqrt{-a b}}{4 \,{\left (a^{2} b^{3} x^{3} + a^{3} b^{2} x\right )} \sqrt{-a b}}, -\frac{{\left ({\left (3 \, b^{4} c - a b^{3} d - a^{2} b^{2} e + 3 \, a^{3} b f\right )} x^{3} +{\left (3 \, a b^{3} c - a^{2} b^{2} d - a^{3} b e + 3 \, a^{4} f\right )} x\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) -{\left (2 \, a^{2} b f x^{4} - 2 \, a b^{2} c -{\left (3 \, b^{3} c - a b^{2} d + a^{2} b e - 3 \, a^{3} f\right )} x^{2}\right )} \sqrt{a b}}{2 \,{\left (a^{2} b^{3} x^{3} + a^{3} b^{2} x\right )} \sqrt{a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^6 + e*x^4 + d*x^2 + c)/((b*x^2 + a)^2*x^2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 9.2556, size = 197, normalized size = 1.76 \[ \frac{\sqrt{- \frac{1}{a^{5} b^{5}}} \left (3 a^{3} f - a^{2} b e - a b^{2} d + 3 b^{3} c\right ) \log{\left (- a^{3} b^{2} \sqrt{- \frac{1}{a^{5} b^{5}}} + x \right )}}{4} - \frac{\sqrt{- \frac{1}{a^{5} b^{5}}} \left (3 a^{3} f - a^{2} b e - a b^{2} d + 3 b^{3} c\right ) \log{\left (a^{3} b^{2} \sqrt{- \frac{1}{a^{5} b^{5}}} + x \right )}}{4} + \frac{- 2 a b^{2} c + x^{2} \left (a^{3} f - a^{2} b e + a b^{2} d - 3 b^{3} c\right )}{2 a^{3} b^{2} x + 2 a^{2} b^{3} x^{3}} + \frac{f x}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x**6+e*x**4+d*x**2+c)/x**2/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.216176, size = 165, normalized size = 1.47 \[ \frac{f x}{b^{2}} - \frac{{\left (3 \, b^{3} c - a b^{2} d + 3 \, a^{3} f - a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2} b^{2}} - \frac{3 \, b^{3} c x^{2} - a b^{2} d x^{2} - a^{3} f x^{2} + a^{2} b x^{2} e + 2 \, a b^{2} c}{2 \,{\left (b x^{3} + a x\right )} a^{2} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x^6 + e*x^4 + d*x^2 + c)/((b*x^2 + a)^2*x^2),x, algorithm="giac")
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