3.181 \(\int \frac{x^3}{b x^2+c x^4} \, dx\)

Optimal. Leaf size=15 \[ \frac{\log \left (b+c x^2\right )}{2 c} \]

[Out]

Log[b + c*x^2]/(2*c)

_______________________________________________________________________________________

Rubi [A]  time = 0.0197065, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\log \left (b+c x^2\right )}{2 c} \]

Antiderivative was successfully verified.

[In]  Int[x^3/(b*x^2 + c*x^4),x]

[Out]

Log[b + c*x^2]/(2*c)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.18605, size = 10, normalized size = 0.67 \[ \frac{\log{\left (b + c x^{2} \right )}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3/(c*x**4+b*x**2),x)

[Out]

log(b + c*x**2)/(2*c)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0034123, size = 15, normalized size = 1. \[ \frac{\log \left (b+c x^2\right )}{2 c} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3/(b*x^2 + c*x^4),x]

[Out]

Log[b + c*x^2]/(2*c)

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 14, normalized size = 0.9 \[{\frac{\ln \left ( c{x}^{2}+b \right ) }{2\,c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(c*x^4+b*x^2),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.688025, size = 18, normalized size = 1.2 \[ \frac{\log \left (c x^{2} + b\right )}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(c*x^4 + b*x^2),x, algorithm="maxima")

[Out]

1/2*log(c*x^2 + b)/c

_______________________________________________________________________________________

Fricas [A]  time = 0.250013, size = 18, normalized size = 1.2 \[ \frac{\log \left (c x^{2} + b\right )}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(c*x^4 + b*x^2),x, algorithm="fricas")

[Out]

1/2*log(c*x^2 + b)/c

_______________________________________________________________________________________

Sympy [A]  time = 0.237187, size = 10, normalized size = 0.67 \[ \frac{\log{\left (b + c x^{2} \right )}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3/(c*x**4+b*x**2),x)

[Out]

log(b + c*x**2)/(2*c)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.270813, size = 19, normalized size = 1.27 \[ \frac{{\rm ln}\left ({\left | c x^{2} + b \right |}\right )}{2 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^3/(c*x^4 + b*x^2),x, algorithm="giac")

[Out]

1/2*ln(abs(c*x^2 + b))/c