Optimal. Leaf size=71 \[ -\frac{(3 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{x (A b-a B)}{2 a^2 \left (a+b x^2\right )}-\frac{A}{a^2 x} \]
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Rubi [A] time = 0.148648, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{(3 A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}-\frac{x (A b-a B)}{2 a^2 \left (a+b x^2\right )}-\frac{A}{a^2 x} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^2*(a + b*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 19.3615, size = 61, normalized size = 0.86 \[ - \frac{A}{a^{2} x} - \frac{x \left (A b - B a\right )}{2 a^{2} \left (a + b x^{2}\right )} - \frac{\left (3 A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{5}{2}} \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**2/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.0556774, size = 70, normalized size = 0.99 \[ \frac{(a B-3 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{5/2} \sqrt{b}}+\frac{x (a B-A b)}{2 a^2 \left (a+b x^2\right )}-\frac{A}{a^2 x} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^2*(a + b*x^2)^2),x]
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Maple [A] time = 0.014, size = 85, normalized size = 1.2 \[ -{\frac{A}{{a}^{2}x}}-{\frac{Axb}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{xB}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{3\,Ab}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{B}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/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((B*x^2 + A)/((b*x^2 + a)^2*x^2),x, algorithm="maxima")
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Fricas [A] time = 0.243319, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{3} +{\left (B a^{2} - 3 \, A a b\right )} x\right )} \log \left (-\frac{2 \, a b x -{\left (b x^{2} - a\right )} \sqrt{-a b}}{b x^{2} + a}\right ) - 2 \,{\left ({\left (B a - 3 \, A b\right )} x^{2} - 2 \, A a\right )} \sqrt{-a b}}{4 \,{\left (a^{2} b x^{3} + a^{3} x\right )} \sqrt{-a b}}, \frac{{\left ({\left (B a b - 3 \, A b^{2}\right )} x^{3} +{\left (B a^{2} - 3 \, A a b\right )} x\right )} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) +{\left ({\left (B a - 3 \, A b\right )} x^{2} - 2 \, A a\right )} \sqrt{a b}}{2 \,{\left (a^{2} b x^{3} + a^{3} x\right )} \sqrt{a b}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^2*x^2),x, algorithm="fricas")
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Sympy [A] time = 2.50535, size = 114, normalized size = 1.61 \[ - \frac{\sqrt{- \frac{1}{a^{5} b}} \left (- 3 A b + B a\right ) \log{\left (- a^{3} \sqrt{- \frac{1}{a^{5} b}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{5} b}} \left (- 3 A b + B a\right ) \log{\left (a^{3} \sqrt{- \frac{1}{a^{5} b}} + x \right )}}{4} + \frac{- 2 A a + x^{2} \left (- 3 A b + B a\right )}{2 a^{3} x + 2 a^{2} b x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x**2/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.222502, size = 84, normalized size = 1.18 \[ \frac{{\left (B a - 3 \, A b\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{2}} + \frac{B a x^{2} - 3 \, A b x^{2} - 2 \, A a}{2 \,{\left (b x^{3} + a x\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^2*x^2),x, algorithm="giac")
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