Optimal. Leaf size=59 \[ \frac{\sqrt{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{A b-a B}{a^2 x}-\frac{A}{3 a x^3} \]
[Out]
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Rubi [A] time = 0.104534, antiderivative size = 59, 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 \[ \frac{\sqrt{b} (A b-a B) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{A b-a B}{a^2 x}-\frac{A}{3 a x^3} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^4*(a + b*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 14.6906, size = 49, normalized size = 0.83 \[ - \frac{A}{3 a x^{3}} + \frac{A b - B a}{a^{2} x} + \frac{\sqrt{b} \left (A b - B a\right ) \operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{a^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**4/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.0912105, size = 60, normalized size = 1.02 \[ -\frac{\sqrt{b} (a B-A b) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{A b-a B}{a^2 x}-\frac{A}{3 a x^3} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^4*(a + b*x^2)),x]
[Out]
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Maple [A] time = 0.009, size = 72, normalized size = 1.2 \[ -{\frac{A}{3\,a{x}^{3}}}+{\frac{Ab}{{a}^{2}x}}-{\frac{B}{ax}}+{\frac{{b}^{2}A}{{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{Bb}{a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^4/(b*x^2+a),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)*x^4),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.23766, size = 1, normalized size = 0.02 \[ \left [-\frac{3 \,{\left (B a - A b\right )} x^{3} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right ) + 6 \,{\left (B a - A b\right )} x^{2} + 2 \, A a}{6 \, a^{2} x^{3}}, -\frac{3 \,{\left (B a - A b\right )} x^{3} \sqrt{\frac{b}{a}} \arctan \left (\frac{b x}{a \sqrt{\frac{b}{a}}}\right ) + 3 \,{\left (B a - A b\right )} x^{2} + A a}{3 \, a^{2} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)*x^4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.26094, size = 129, normalized size = 2.19 \[ \frac{\sqrt{- \frac{b}{a^{5}}} \left (- A b + B a\right ) \log{\left (- \frac{a^{3} \sqrt{- \frac{b}{a^{5}}} \left (- A b + B a\right )}{- A b^{2} + B a b} + x \right )}}{2} - \frac{\sqrt{- \frac{b}{a^{5}}} \left (- A b + B a\right ) \log{\left (\frac{a^{3} \sqrt{- \frac{b}{a^{5}}} \left (- A b + B a\right )}{- A b^{2} + B a b} + x \right )}}{2} - \frac{A a + x^{2} \left (- 3 A b + 3 B a\right )}{3 a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x**4/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.225285, size = 77, normalized size = 1.31 \[ -\frac{{\left (B a b - A b^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} - \frac{3 \, B a x^{2} - 3 \, A b x^{2} + A a}{3 \, a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)*x^4),x, algorithm="giac")
[Out]