Optimal. Leaf size=82 \[ \frac{b c-a d}{a^2 x}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{5/2} b^{3/2}}-\frac{c}{3 a x^3}+\frac{f x}{b} \]
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Rubi [A] time = 0.174724, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{b c-a d}{a^2 x}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^3 (-f)+a^2 b e-a b^2 d+b^3 c\right )}{a^{5/2} b^{3/2}}-\frac{c}{3 a x^3}+\frac{f x}{b} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2 + e*x^4 + f*x^6)/(x^4*(a + b*x^2)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\int f\, dx}{b} - \frac{c}{3 a x^{3}} - \frac{a d - b c}{a^{2} x} - \frac{\left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{5}{2}} b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((f*x**6+e*x**4+d*x**2+c)/x**4/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.176651, size = 83, normalized size = 1.01 \[ \frac{b c-a d}{a^2 x}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ) \left (a^3 f-a^2 b e+a b^2 d-b^3 c\right )}{a^{5/2} b^{3/2}}-\frac{c}{3 a x^3}+\frac{f x}{b} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2 + e*x^4 + f*x^6)/(x^4*(a + b*x^2)),x]
[Out]
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Maple [A] time = 0.01, size = 115, normalized size = 1.4 \[{\frac{fx}{b}}-{\frac{c}{3\,a{x}^{3}}}-{\frac{d}{ax}}+{\frac{bc}{x{a}^{2}}}-{\frac{af}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{e\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{bd}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{{b}^{2}c}{{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((f*x^6+e*x^4+d*x^2+c)/x^4/(b*x^2+a),x)
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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)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233999, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left (3 \, a^{2} f x^{4} - a b c + 3 \,{\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt{-a b}}{6 \, \sqrt{-a b} a^{2} b x^{3}}, \frac{3 \,{\left (b^{3} c - a b^{2} d + a^{2} b e - a^{3} f\right )} x^{3} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left (3 \, a^{2} f x^{4} - a b c + 3 \,{\left (b^{2} c - a b d\right )} x^{2}\right )} \sqrt{a b}}{3 \, \sqrt{a b} a^{2} b x^{3}}\right ] \]
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)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 4.4074, size = 151, normalized size = 1.84 \[ \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (- a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (a^{3} f - a^{2} b e + a b^{2} d - b^{3} c\right ) \log{\left (a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} + x \right )}}{2} + \frac{f x}{b} - \frac{a c + x^{2} \left (3 a d - 3 b c\right )}{3 a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((f*x**6+e*x**4+d*x**2+c)/x**4/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.21612, size = 109, normalized size = 1.33 \[ \frac{f x}{b} + \frac{{\left (b^{3} c - a b^{2} d - a^{3} f + a^{2} b e\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2} b} + \frac{3 \, b c x^{2} - 3 \, a d x^{2} - a c}{3 \, a^{2} x^{3}} \]
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)*x^4),x, algorithm="giac")
[Out]