Optimal. Leaf size=49 \[ \frac{b \log \left (a+b x^2\right )}{a^3}-\frac{2 b \log (x)}{a^3}-\frac{b}{2 a^2 \left (a+b x^2\right )}-\frac{1}{2 a^2 x^2} \]
[Out]
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Rubi [A] time = 0.087268, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{b \log \left (a+b x^2\right )}{a^3}-\frac{2 b \log (x)}{a^3}-\frac{b}{2 a^2 \left (a+b x^2\right )}-\frac{1}{2 a^2 x^2} \]
Antiderivative was successfully verified.
[In] Int[1/(x^3*(a + b*x^2)^2),x]
[Out]
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Rubi in Sympy [A] time = 12.2934, size = 46, normalized size = 0.94 \[ - \frac{b}{2 a^{2} \left (a + b x^{2}\right )} - \frac{1}{2 a^{2} x^{2}} - \frac{b \log{\left (x^{2} \right )}}{a^{3}} + \frac{b \log{\left (a + b x^{2} \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/x**3/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.0723158, size = 41, normalized size = 0.84 \[ -\frac{a \left (\frac{b}{a+b x^2}+\frac{1}{x^2}\right )-2 b \log \left (a+b x^2\right )+4 b \log (x)}{2 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[1/(x^3*(a + b*x^2)^2),x]
[Out]
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Maple [A] time = 0.018, size = 46, normalized size = 0.9 \[ -{\frac{1}{2\,{a}^{2}{x}^{2}}}-{\frac{b}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}-2\,{\frac{b\ln \left ( x \right ) }{{a}^{3}}}+{\frac{b\ln \left ( b{x}^{2}+a \right ) }{{a}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/x^3/(b*x^2+a)^2,x)
[Out]
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Maxima [A] time = 1.33551, size = 70, normalized size = 1.43 \[ -\frac{2 \, b x^{2} + a}{2 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} + \frac{b \log \left (b x^{2} + a\right )}{a^{3}} - \frac{b \log \left (x^{2}\right )}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*x^3),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213269, size = 99, normalized size = 2.02 \[ -\frac{2 \, a b x^{2} + a^{2} - 2 \,{\left (b^{2} x^{4} + a b x^{2}\right )} \log \left (b x^{2} + a\right ) + 4 \,{\left (b^{2} x^{4} + a b x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} b x^{4} + a^{4} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*x^3),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.04719, size = 49, normalized size = 1. \[ - \frac{a + 2 b x^{2}}{2 a^{3} x^{2} + 2 a^{2} b x^{4}} - \frac{2 b \log{\left (x \right )}}{a^{3}} + \frac{b \log{\left (\frac{a}{b} + x^{2} \right )}}{a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/x**3/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.221261, size = 69, normalized size = 1.41 \[ -\frac{b{\rm ln}\left (x^{2}\right )}{a^{3}} + \frac{b{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{a^{3}} - \frac{2 \, b x^{2} + a}{2 \,{\left (b x^{4} + a x^{2}\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((b*x^2 + a)^2*x^3),x, algorithm="giac")
[Out]