Optimal. Leaf size=97 \[ \frac{c (2 b B-3 A c) \log \left (b+c x^2\right )}{2 b^4}-\frac{c \log (x) (2 b B-3 A c)}{b^4}-\frac{c (b B-A c)}{2 b^3 \left (b+c x^2\right )}-\frac{b B-2 A c}{2 b^3 x^2}-\frac{A}{4 b^2 x^4} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.234514, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{c (2 b B-3 A c) \log \left (b+c x^2\right )}{2 b^4}-\frac{c \log (x) (2 b B-3 A c)}{b^4}-\frac{c (b B-A c)}{2 b^3 \left (b+c x^2\right )}-\frac{b B-2 A c}{2 b^3 x^2}-\frac{A}{4 b^2 x^4} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x*(b*x^2 + c*x^4)^2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 26.9305, size = 94, normalized size = 0.97 \[ - \frac{A}{4 b^{2} x^{4}} + \frac{c \left (A c - B b\right )}{2 b^{3} \left (b + c x^{2}\right )} + \frac{2 A c - B b}{2 b^{3} x^{2}} + \frac{c \left (3 A c - 2 B b\right ) \log{\left (x^{2} \right )}}{2 b^{4}} - \frac{c \left (3 A c - 2 B b\right ) \log{\left (b + c x^{2} \right )}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x/(c*x**4+b*x**2)**2,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.184245, size = 85, normalized size = 0.88 \[ -\frac{\frac{A b^2}{x^4}+\frac{2 b c (b B-A c)}{b+c x^2}+\frac{2 b (b B-2 A c)}{x^2}+2 c (3 A c-2 b B) \log \left (b+c x^2\right )-4 c \log (x) (3 A c-2 b B)}{4 b^4} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x*(b*x^2 + c*x^4)^2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.022, size = 114, normalized size = 1.2 \[ -{\frac{A}{4\,{b}^{2}{x}^{4}}}+{\frac{Ac}{{b}^{3}{x}^{2}}}-{\frac{B}{2\,{b}^{2}{x}^{2}}}+3\,{\frac{A\ln \left ( x \right ){c}^{2}}{{b}^{4}}}-2\,{\frac{Bc\ln \left ( x \right ) }{{b}^{3}}}+{\frac{A{c}^{2}}{2\,{b}^{3} \left ( c{x}^{2}+b \right ) }}-{\frac{Bc}{2\,{b}^{2} \left ( c{x}^{2}+b \right ) }}-{\frac{3\,{c}^{2}\ln \left ( c{x}^{2}+b \right ) A}{2\,{b}^{4}}}+{\frac{c\ln \left ( c{x}^{2}+b \right ) B}{{b}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x/(c*x^4+b*x^2)^2,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.37669, size = 143, normalized size = 1.47 \[ -\frac{2 \,{\left (2 \, B b c - 3 \, A c^{2}\right )} x^{4} + A b^{2} +{\left (2 \, B b^{2} - 3 \, A b c\right )} x^{2}}{4 \,{\left (b^{3} c x^{6} + b^{4} x^{4}\right )}} + \frac{{\left (2 \, B b c - 3 \, A c^{2}\right )} \log \left (c x^{2} + b\right )}{2 \, b^{4}} - \frac{{\left (2 \, B b c - 3 \, A c^{2}\right )} \log \left (x^{2}\right )}{2 \, b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((c*x^4 + b*x^2)^2*x),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.21206, size = 208, normalized size = 2.14 \[ -\frac{2 \,{\left (2 \, B b^{2} c - 3 \, A b c^{2}\right )} x^{4} + A b^{3} +{\left (2 \, B b^{3} - 3 \, A b^{2} c\right )} x^{2} - 2 \,{\left ({\left (2 \, B b c^{2} - 3 \, A c^{3}\right )} x^{6} +{\left (2 \, B b^{2} c - 3 \, A b c^{2}\right )} x^{4}\right )} \log \left (c x^{2} + b\right ) + 4 \,{\left ({\left (2 \, B b c^{2} - 3 \, A c^{3}\right )} x^{6} +{\left (2 \, B b^{2} c - 3 \, A b c^{2}\right )} x^{4}\right )} \log \left (x\right )}{4 \,{\left (b^{4} c x^{6} + b^{5} x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((c*x^4 + b*x^2)^2*x),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [A] time = 2.41974, size = 100, normalized size = 1.03 \[ - \frac{A b^{2} + x^{4} \left (- 6 A c^{2} + 4 B b c\right ) + x^{2} \left (- 3 A b c + 2 B b^{2}\right )}{4 b^{4} x^{4} + 4 b^{3} c x^{6}} - \frac{c \left (- 3 A c + 2 B b\right ) \log{\left (x \right )}}{b^{4}} + \frac{c \left (- 3 A c + 2 B b\right ) \log{\left (\frac{b}{c} + x^{2} \right )}}{2 b^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)/x/(c*x**4+b*x**2)**2,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.212624, size = 203, normalized size = 2.09 \[ -\frac{{\left (2 \, B b c - 3 \, A c^{2}\right )}{\rm ln}\left (x^{2}\right )}{2 \, b^{4}} + \frac{{\left (2 \, B b c^{2} - 3 \, A c^{3}\right )}{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, b^{4} c} - \frac{2 \, B b c^{2} x^{2} - 3 \, A c^{3} x^{2} + 3 \, B b^{2} c - 4 \, A b c^{2}}{2 \,{\left (c x^{2} + b\right )} b^{4}} + \frac{6 \, B b c x^{4} - 9 \, A c^{2} x^{4} - 2 \, B b^{2} x^{2} + 4 \, A b c x^{2} - A b^{2}}{4 \, b^{4} x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/((c*x^4 + b*x^2)^2*x),x, algorithm="giac")
[Out]