Optimal. Leaf size=51 \[ -\frac{A \log \left (a+b x^2\right )}{2 a^2}+\frac{A \log (x)}{a^2}+\frac{A b-a B}{2 a b \left (a+b x^2\right )} \]
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Rubi [A] time = 0.113601, antiderivative size = 51, 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 \[ -\frac{A \log \left (a+b x^2\right )}{2 a^2}+\frac{A \log (x)}{a^2}+\frac{A b-a B}{2 a b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x*(a + b*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 16.503, size = 44, normalized size = 0.86 \[ \frac{A \log{\left (x^{2} \right )}}{2 a^{2}} - \frac{A \log{\left (a + b x^{2} \right )}}{2 a^{2}} + \frac{A b - B a}{2 a b \left (a + b x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.0500623, size = 46, normalized size = 0.9 \[ \frac{\frac{a (A b-a B)}{b \left (a+b x^2\right )}-A \log \left (a+b x^2\right )+2 A \log (x)}{2 a^2} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x*(a + b*x^2)^2),x]
[Out]
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Maple [A] time = 0.017, size = 53, normalized size = 1. \[{\frac{A\ln \left ( x \right ) }{{a}^{2}}}-{\frac{A\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2}}}+{\frac{A}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{B}{2\,b \left ( b{x}^{2}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x/(b*x^2+a)^2,x)
[Out]
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Maxima [A] time = 1.34773, size = 69, normalized size = 1.35 \[ -\frac{B a - A b}{2 \,{\left (a b^{2} x^{2} + a^{2} b\right )}} - \frac{A \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac{A \log \left (x^{2}\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^2*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.233235, size = 95, normalized size = 1.86 \[ -\frac{B a^{2} - A a b +{\left (A b^{2} x^{2} + A a b\right )} \log \left (b x^{2} + a\right ) - 2 \,{\left (A b^{2} x^{2} + A a b\right )} \log \left (x\right )}{2 \,{\left (a^{2} b^{2} x^{2} + a^{3} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^2*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.19838, size = 46, normalized size = 0.9 \[ \frac{A \log{\left (x \right )}}{a^{2}} - \frac{A \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}} - \frac{- A b + B a}{2 a^{2} b + 2 a b^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.22716, size = 85, normalized size = 1.67 \[ \frac{A{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} - \frac{A{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2}} + \frac{A b^{2} x^{2} - B a^{2} + 2 \, A a b}{2 \,{\left (b x^{2} + a\right )} a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((b*x^2 + a)^2*x),x, algorithm="giac")
[Out]