3.201 \(\int \frac{x}{\left (b x^2+c x^4\right )^2} \, dx\)

Optimal. Leaf size=49 \[ \frac{c \log \left (b+c x^2\right )}{b^3}-\frac{2 c \log (x)}{b^3}-\frac{c}{2 b^2 \left (b+c x^2\right )}-\frac{1}{2 b^2 x^2} \]

[Out]

-1/(2*b^2*x^2) - c/(2*b^2*(b + c*x^2)) - (2*c*Log[x])/b^3 + (c*Log[b + c*x^2])/b
^3

_______________________________________________________________________________________

Rubi [A]  time = 0.0922975, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{c \log \left (b+c x^2\right )}{b^3}-\frac{2 c \log (x)}{b^3}-\frac{c}{2 b^2 \left (b+c x^2\right )}-\frac{1}{2 b^2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(b*x^2 + c*x^4)^2,x]

[Out]

-1/(2*b^2*x^2) - c/(2*b^2*(b + c*x^2)) - (2*c*Log[x])/b^3 + (c*Log[b + c*x^2])/b
^3

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 14.7485, size = 46, normalized size = 0.94 \[ - \frac{c}{2 b^{2} \left (b + c x^{2}\right )} - \frac{1}{2 b^{2} x^{2}} - \frac{c \log{\left (x^{2} \right )}}{b^{3}} + \frac{c \log{\left (b + c x^{2} \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x/(c*x**4+b*x**2)**2,x)

[Out]

-c/(2*b**2*(b + c*x**2)) - 1/(2*b**2*x**2) - c*log(x**2)/b**3 + c*log(b + c*x**2
)/b**3

_______________________________________________________________________________________

Mathematica [A]  time = 0.061829, size = 41, normalized size = 0.84 \[ -\frac{b \left (\frac{c}{b+c x^2}+\frac{1}{x^2}\right )-2 c \log \left (b+c x^2\right )+4 c \log (x)}{2 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x/(b*x^2 + c*x^4)^2,x]

[Out]

-(b*(x^(-2) + c/(b + c*x^2)) + 4*c*Log[x] - 2*c*Log[b + c*x^2])/(2*b^3)

_______________________________________________________________________________________

Maple [A]  time = 0.019, size = 46, normalized size = 0.9 \[ -{\frac{1}{2\,{b}^{2}{x}^{2}}}-{\frac{c}{2\,{b}^{2} \left ( c{x}^{2}+b \right ) }}-2\,{\frac{c\ln \left ( x \right ) }{{b}^{3}}}+{\frac{c\ln \left ( c{x}^{2}+b \right ) }{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(c*x^4+b*x^2)^2,x)

[Out]

-1/2/b^2/x^2-1/2*c/b^2/(c*x^2+b)-2*c*ln(x)/b^3+c*ln(c*x^2+b)/b^3

_______________________________________________________________________________________

Maxima [A]  time = 0.702168, size = 70, normalized size = 1.43 \[ -\frac{2 \, c x^{2} + b}{2 \,{\left (b^{2} c x^{4} + b^{3} x^{2}\right )}} + \frac{c \log \left (c x^{2} + b\right )}{b^{3}} - \frac{c \log \left (x^{2}\right )}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(c*x^4 + b*x^2)^2,x, algorithm="maxima")

[Out]

-1/2*(2*c*x^2 + b)/(b^2*c*x^4 + b^3*x^2) + c*log(c*x^2 + b)/b^3 - c*log(x^2)/b^3

_______________________________________________________________________________________

Fricas [A]  time = 0.25666, size = 99, normalized size = 2.02 \[ -\frac{2 \, b c x^{2} + b^{2} - 2 \,{\left (c^{2} x^{4} + b c x^{2}\right )} \log \left (c x^{2} + b\right ) + 4 \,{\left (c^{2} x^{4} + b c x^{2}\right )} \log \left (x\right )}{2 \,{\left (b^{3} c x^{4} + b^{4} x^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(c*x^4 + b*x^2)^2,x, algorithm="fricas")

[Out]

-1/2*(2*b*c*x^2 + b^2 - 2*(c^2*x^4 + b*c*x^2)*log(c*x^2 + b) + 4*(c^2*x^4 + b*c*
x^2)*log(x))/(b^3*c*x^4 + b^4*x^2)

_______________________________________________________________________________________

Sympy [A]  time = 2.03763, size = 49, normalized size = 1. \[ - \frac{b + 2 c x^{2}}{2 b^{3} x^{2} + 2 b^{2} c x^{4}} - \frac{2 c \log{\left (x \right )}}{b^{3}} + \frac{c \log{\left (\frac{b}{c} + x^{2} \right )}}{b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(c*x**4+b*x**2)**2,x)

[Out]

-(b + 2*c*x**2)/(2*b**3*x**2 + 2*b**2*c*x**4) - 2*c*log(x)/b**3 + c*log(b/c + x*
*2)/b**3

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.271084, size = 68, normalized size = 1.39 \[ \frac{c{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{b^{3}} - \frac{2 \, c{\rm ln}\left ({\left | x \right |}\right )}{b^{3}} - \frac{2 \, c x^{2} + b}{2 \,{\left (c x^{4} + b x^{2}\right )} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(c*x^4 + b*x^2)^2,x, algorithm="giac")

[Out]

c*ln(abs(c*x^2 + b))/b^3 - 2*c*ln(abs(x))/b^3 - 1/2*(2*c*x^2 + b)/((c*x^4 + b*x^
2)*b^2)