3.14 \(\int \frac{x^2}{a x+b x^3} \, dx\)

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

[Out]

Log[a + b*x^2]/(2*b)

_______________________________________________________________________________________

Rubi [A]  time = 0.0152597, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{\log \left (a+b x^2\right )}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a*x + b*x^3),x]

[Out]

Log[a + b*x^2]/(2*b)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.3114, size = 10, normalized size = 0.67 \[ \frac{\log{\left (a + b x^{2} \right )}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a + b*x**2)/(2*b)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0029752, size = 15, normalized size = 1. \[ \frac{\log \left (a+b x^2\right )}{2 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a*x + b*x^3),x]

[Out]

Log[a + b*x^2]/(2*b)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^3+a*x),x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.40561, size = 18, normalized size = 1.2 \[ \frac{\log \left (b x^{2} + a\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(b*x^2 + a)/b

_______________________________________________________________________________________

Fricas [A]  time = 0.200772, size = 18, normalized size = 1.2 \[ \frac{\log \left (b x^{2} + a\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(b*x^2 + a)/b

_______________________________________________________________________________________

Sympy [A]  time = 0.244588, size = 10, normalized size = 0.67 \[ \frac{\log{\left (a + b x^{2} \right )}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x**3+a*x),x)

[Out]

log(a + b*x**2)/(2*b)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.215477, size = 19, normalized size = 1.27 \[ \frac{{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*ln(abs(b*x^2 + a))/b