Optimal. Leaf size=50 \[ \frac{(A b-a B) \log \left (a+b x^2\right )}{2 a^2}-\frac{\log (x) (A b-a B)}{a^2}-\frac{A}{2 a x^2} \]
[Out]
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Rubi [A] time = 0.120778, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{(A b-a B) \log \left (a+b x^2\right )}{2 a^2}-\frac{\log (x) (A b-a B)}{a^2}-\frac{A}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^3*(a + b*x^2)),x]
[Out]
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Rubi in Sympy [A] time = 16.0316, size = 44, normalized size = 0.88 \[ - \frac{A}{2 a x^{2}} - \frac{\left (A b - B a\right ) \log{\left (x^{2} \right )}}{2 a^{2}} + \frac{\left (A b - B a\right ) \log{\left (a + b x^{2} \right )}}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**3/(b*x**2+a),x)
[Out]
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Mathematica [A] time = 0.0343761, size = 49, normalized size = 0.98 \[ \frac{(A b-a B) \log \left (a+b x^2\right )}{2 a^2}+\frac{\log (x) (a B-A b)}{a^2}-\frac{A}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^3*(a + b*x^2)),x]
[Out]
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Maple [A] time = 0.01, size = 56, normalized size = 1.1 \[ -{\frac{A}{2\,a{x}^{2}}}-{\frac{A\ln \left ( x \right ) b}{{a}^{2}}}+{\frac{\ln \left ( x \right ) B}{a}}+{\frac{\ln \left ( b{x}^{2}+a \right ) Ab}{2\,{a}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) B}{2\,a}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^3/(b*x^2+a),x)
[Out]
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Maxima [A] time = 1.34712, size = 65, normalized size = 1.3 \[ -\frac{{\left (B a - A b\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac{{\left (B a - A b\right )} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac{A}{2 \, a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.227302, size = 63, normalized size = 1.26 \[ -\frac{{\left (B a - A b\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \,{\left (B a - A b\right )} x^{2} \log \left (x\right ) + A a}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 3.0413, size = 41, normalized size = 0.82 \[ - \frac{A}{2 a x^{2}} + \frac{\left (- A b + B a\right ) \log{\left (x \right )}}{a^{2}} - \frac{\left (- A b + B a\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x**3/(b*x**2+a),x)
[Out]
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GIAC/XCAS [A] time = 0.228604, size = 96, normalized size = 1.92 \[ \frac{{\left (B a - A b\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} - \frac{{\left (B a b - A b^{2}\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b} - \frac{B a x^{2} - A b x^{2} + A a}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)*x^3),x, algorithm="giac")
[Out]