Optimal. Leaf size=76 \[ \frac{(2 b c-a d) \log \left (a+b x^2\right )}{2 a^3}-\frac{\log (x) (2 b c-a d)}{a^3}-\frac{b c-a d}{2 a^2 \left (a+b x^2\right )}-\frac{c}{2 a^2 x^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.177952, antiderivative size = 76, 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 \[ \frac{(2 b c-a d) \log \left (a+b x^2\right )}{2 a^3}-\frac{\log (x) (2 b c-a d)}{a^3}-\frac{b c-a d}{2 a^2 \left (a+b x^2\right )}-\frac{c}{2 a^2 x^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)/(x^3*(a + b*x^2)^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 22.2363, size = 68, normalized size = 0.89 \[ - \frac{c}{2 a^{2} x^{2}} + \frac{a d - b c}{2 a^{2} \left (a + b x^{2}\right )} + \frac{\left (a d - 2 b c\right ) \log{\left (x^{2} \right )}}{2 a^{3}} - \frac{\left (a d - 2 b c\right ) \log{\left (a + b x^{2} \right )}}{2 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)/x**3/(b*x**2+a)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0833559, size = 64, normalized size = 0.84 \[ \frac{\frac{a (a d-b c)}{a+b x^2}+(2 b c-a d) \log \left (a+b x^2\right )+2 \log (x) (a d-2 b c)-\frac{a c}{x^2}}{2 a^3} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)/(x^3*(a + b*x^2)^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.02, size = 86, normalized size = 1.1 \[ -{\frac{c}{2\,{a}^{2}{x}^{2}}}+{\frac{\ln \left ( x \right ) d}{{a}^{2}}}-2\,{\frac{bc\ln \left ( x \right ) }{{a}^{3}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) d}{2\,{a}^{2}}}+{\frac{bc\ln \left ( b{x}^{2}+a \right ) }{{a}^{3}}}+{\frac{d}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{bc}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)/x^3/(b*x^2+a)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.34356, size = 105, normalized size = 1.38 \[ -\frac{{\left (2 \, b c - a d\right )} x^{2} + a c}{2 \,{\left (a^{2} b x^{4} + a^{3} x^{2}\right )}} + \frac{{\left (2 \, b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, a^{3}} - \frac{{\left (2 \, b c - a d\right )} \log \left (x^{2}\right )}{2 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^2*x^3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.231165, size = 165, normalized size = 2.17 \[ -\frac{a^{2} c +{\left (2 \, a b c - a^{2} d\right )} x^{2} -{\left ({\left (2 \, b^{2} c - a b d\right )} x^{4} +{\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \log \left (b x^{2} + a\right ) + 2 \,{\left ({\left (2 \, b^{2} c - a b d\right )} x^{4} +{\left (2 \, a b c - a^{2} d\right )} x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} b x^{4} + a^{4} x^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^2*x^3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 3.86831, size = 70, normalized size = 0.92 \[ \frac{- a c + x^{2} \left (a d - 2 b c\right )}{2 a^{3} x^{2} + 2 a^{2} b x^{4}} + \frac{\left (a d - 2 b c\right ) \log{\left (x \right )}}{a^{3}} - \frac{\left (a d - 2 b c\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)/x**3/(b*x**2+a)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.23943, size = 113, normalized size = 1.49 \[ -\frac{{\left (2 \, b c - a d\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{3}} - \frac{2 \, b c x^{2} - a d x^{2} + a c}{2 \,{\left (b x^{4} + a x^{2}\right )} a^{2}} + \frac{{\left (2 \, b^{2} c - a b d\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^2*x^3),x, algorithm="giac")
[Out]