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

Optimal. Leaf size=6 \[ -\frac{c}{x} \]

[Out]

-(c/x)

_______________________________________________________________________________________

Rubi [A]  time = 0.00592865, antiderivative size = 6, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ -\frac{c}{x} \]

Antiderivative was successfully verified.

[In]  Int[(a*c + b*c*x^2)/(x^2*(a + b*x^2)),x]

[Out]

-(c/x)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.59483, size = 3, normalized size = 0.5 \[ - \frac{c}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c/x

_______________________________________________________________________________________

Mathematica [A]  time = 0.000589409, size = 6, normalized size = 1. \[ -\frac{c}{x} \]

Antiderivative was successfully verified.

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

[Out]

-(c/x)

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 7, normalized size = 1.2 \[ -{\frac{c}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*c*x^2+a*c)/x^2/(b*x^2+a),x)

[Out]

-c/x

_______________________________________________________________________________________

Maxima [A]  time = 1.34327, size = 8, normalized size = 1.33 \[ -\frac{c}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c/x

_______________________________________________________________________________________

Fricas [A]  time = 0.216027, size = 8, normalized size = 1.33 \[ -\frac{c}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c/x

_______________________________________________________________________________________

Sympy [A]  time = 0.122779, size = 3, normalized size = 0.5 \[ - \frac{c}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c/x

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.222543, size = 8, normalized size = 1.33 \[ -\frac{c}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c/x