Optimal. Leaf size=50 \[ \frac{(b c-a d) \log \left (a+b x^2\right )}{2 a^2}-\frac{\log (x) (b c-a d)}{a^2}-\frac{c}{2 a x^2} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.120203, 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 \[ \frac{(b c-a d) \log \left (a+b x^2\right )}{2 a^2}-\frac{\log (x) (b c-a d)}{a^2}-\frac{c}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)/(x^3*(a + b*x^2)),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 16.0644, size = 44, normalized size = 0.88 \[ - \frac{c}{2 a x^{2}} + \frac{\left (a d - b c\right ) \log{\left (x^{2} \right )}}{2 a^{2}} - \frac{\left (a d - b c\right ) \log{\left (a + b x^{2} \right )}}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)/x**3/(b*x**2+a),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0423776, size = 49, normalized size = 0.98 \[ \frac{(b c-a d) \log \left (a+b x^2\right )}{2 a^2}+\frac{\log (x) (a d-b c)}{a^2}-\frac{c}{2 a x^2} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)/(x^3*(a + b*x^2)),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.01, size = 56, normalized size = 1.1 \[ -{\frac{c}{2\,a{x}^{2}}}+{\frac{\ln \left ( x \right ) d}{a}}-{\frac{bc\ln \left ( x \right ) }{{a}^{2}}}-{\frac{\ln \left ( b{x}^{2}+a \right ) d}{2\,a}}+{\frac{bc\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)/x^3/(b*x^2+a),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.35141, size = 65, normalized size = 1.3 \[ \frac{{\left (b c - a d\right )} \log \left (b x^{2} + a\right )}{2 \, a^{2}} - \frac{{\left (b c - a d\right )} \log \left (x^{2}\right )}{2 \, a^{2}} - \frac{c}{2 \, a x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)*x^3),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.242194, size = 65, normalized size = 1.3 \[ \frac{{\left (b c - a d\right )} x^{2} \log \left (b x^{2} + a\right ) - 2 \,{\left (b c - a d\right )} x^{2} \log \left (x\right ) - a c}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)*x^3),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 3.01857, size = 41, normalized size = 0.82 \[ - \frac{c}{2 a x^{2}} + \frac{\left (a d - b c\right ) \log{\left (x \right )}}{a^{2}} - \frac{\left (a d - b c\right ) \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)/x**3/(b*x**2+a),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.224113, size = 97, normalized size = 1.94 \[ -\frac{{\left (b c - a d\right )}{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} + \frac{{\left (b^{2} c - a b d\right )}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2} b} + \frac{b c x^{2} - a d x^{2} - a c}{2 \, a^{2} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)*x^3),x, algorithm="giac")
[Out]