Optimal. Leaf size=51 \[ -\frac{c \log \left (a+b x^2\right )}{2 a^2}+\frac{c \log (x)}{a^2}+\frac{b c-a d}{2 a b \left (a+b x^2\right )} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.113667, 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 \[ -\frac{c \log \left (a+b x^2\right )}{2 a^2}+\frac{c \log (x)}{a^2}+\frac{b c-a d}{2 a b \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
[In] Int[(c + d*x^2)/(x*(a + b*x^2)^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 16.6897, size = 44, normalized size = 0.86 \[ - \frac{a d - b c}{2 a b \left (a + b x^{2}\right )} + \frac{c \log{\left (x^{2} \right )}}{2 a^{2}} - \frac{c \log{\left (a + b x^{2} \right )}}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x**2+c)/x/(b*x**2+a)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0488924, size = 46, normalized size = 0.9 \[ \frac{\frac{a (b c-a d)}{b \left (a+b x^2\right )}-c \log \left (a+b x^2\right )+2 c \log (x)}{2 a^2} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*x^2)/(x*(a + b*x^2)^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.017, size = 53, normalized size = 1. \[{\frac{c\ln \left ( x \right ) }{{a}^{2}}}-{\frac{c\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{2}}}-{\frac{d}{2\,b \left ( b{x}^{2}+a \right ) }}+{\frac{c}{2\,a \left ( b{x}^{2}+a \right ) }} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x^2+c)/x/(b*x^2+a)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.34692, size = 69, normalized size = 1.35 \[ \frac{b c - a d}{2 \,{\left (a b^{2} x^{2} + a^{2} b\right )}} - \frac{c \log \left (b x^{2} + a\right )}{2 \, a^{2}} + \frac{c \log \left (x^{2}\right )}{2 \, a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^2*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.231858, size = 96, normalized size = 1.88 \[ \frac{a b c - a^{2} d -{\left (b^{2} c x^{2} + a b c\right )} \log \left (b x^{2} + a\right ) + 2 \,{\left (b^{2} c x^{2} + a b c\right )} \log \left (x\right )}{2 \,{\left (a^{2} b^{2} x^{2} + a^{3} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^2*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 2.2005, size = 46, normalized size = 0.9 \[ - \frac{a d - b c}{2 a^{2} b + 2 a b^{2} x^{2}} + \frac{c \log{\left (x \right )}}{a^{2}} - \frac{c \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x**2+c)/x/(b*x**2+a)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.251333, size = 85, normalized size = 1.67 \[ \frac{c{\rm ln}\left (x^{2}\right )}{2 \, a^{2}} - \frac{c{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{2}} + \frac{b^{2} c x^{2} + 2 \, a b c - a^{2} d}{2 \,{\left (b x^{2} + a\right )} a^{2} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)/((b*x^2 + a)^2*x),x, algorithm="giac")
[Out]