Optimal. Leaf size=90 \[ \frac{\sqrt{b} (5 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2}}+\frac{b x (A b-a B)}{2 a^3 \left (a+b x^2\right )}+\frac{2 A b-a B}{a^3 x}-\frac{A}{3 a^2 x^3} \]
[Out]
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Rubi [A] time = 0.262274, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\sqrt{b} (5 A b-3 a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2}}+\frac{b x (A b-a B)}{2 a^3 \left (a+b x^2\right )}+\frac{2 A b-a B}{a^3 x}-\frac{A}{3 a^2 x^3} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^4*(a + b*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 40.404, size = 80, normalized size = 0.89 \[ - \frac{A}{3 a^{2} x^{3}} + \frac{b x \left (A b - B a\right )}{2 a^{3} \left (a + b x^{2}\right )} + \frac{2 A b - B a}{a^{3} x} + \frac{\sqrt{b} \left (5 A b - 3 B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2 a^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**4/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.132887, size = 90, normalized size = 1. \[ -\frac{\sqrt{b} (3 a B-5 A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2}}-\frac{b x (a B-A b)}{2 a^3 \left (a+b x^2\right )}+\frac{2 A b-a B}{a^3 x}-\frac{A}{3 a^2 x^3} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^4*(a + b*x^2)^2),x]
[Out]
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Maple [A] time = 0.017, size = 110, normalized size = 1.2 \[ -{\frac{A}{3\,{a}^{2}{x}^{3}}}+2\,{\frac{Ab}{{a}^{3}x}}-{\frac{B}{{a}^{2}x}}+{\frac{Ax{b}^{2}}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{bBx}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{5\,{b}^{2}A}{2\,{a}^{3}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{3\,Bb}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^4/(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^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222186, size = 1, normalized size = 0.01 \[ \left [-\frac{6 \,{\left (3 \, B a b - 5 \, A b^{2}\right )} x^{4} + 4 \, A a^{2} + 4 \,{\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2} + 3 \,{\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{5} +{\left (3 \, B a^{2} - 5 \, A a b\right )} x^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{12 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}, -\frac{3 \,{\left (3 \, B a b - 5 \, A b^{2}\right )} x^{4} + 2 \, A a^{2} + 2 \,{\left (3 \, B a^{2} - 5 \, A a b\right )} x^{2} + 3 \,{\left ({\left (3 \, B a b - 5 \, A b^{2}\right )} x^{5} +{\left (3 \, B a^{2} - 5 \, A a b\right )} x^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right )}{6 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^2*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.17059, size = 184, normalized size = 2.04 \[ \frac{\sqrt{- \frac{b}{a^{7}}} \left (- 5 A b + 3 B a\right ) \log{\left (- \frac{a^{4} \sqrt{- \frac{b}{a^{7}}} \left (- 5 A b + 3 B a\right )}{- 5 A b^{2} + 3 B a b} + x \right )}}{4} - \frac{\sqrt{- \frac{b}{a^{7}}} \left (- 5 A b + 3 B a\right ) \log{\left (\frac{a^{4} \sqrt{- \frac{b}{a^{7}}} \left (- 5 A b + 3 B a\right )}{- 5 A b^{2} + 3 B a b} + x \right )}}{4} - \frac{2 A a^{2} + x^{4} \left (- 15 A b^{2} + 9 B a b\right ) + x^{2} \left (- 10 A a b + 6 B a^{2}\right )}{6 a^{4} x^{3} + 6 a^{3} b x^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x**4/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.230397, size = 115, normalized size = 1.28 \[ -\frac{{\left (3 \, B a b - 5 \, A b^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{3}} - \frac{B a b x - A b^{2} x}{2 \,{\left (b x^{2} + a\right )} a^{3}} - \frac{3 \, B a x^{2} - 6 \, A b x^{2} + A a}{3 \, a^{3} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^2*x^4),x, algorithm="giac")
[Out]