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

Optimal. Leaf size=4 \[ c \log (x) \]

[Out]

c*Log[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.00530692, antiderivative size = 4, 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 \[ c \log (x) \]

Antiderivative was successfully verified.

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

[Out]

c*Log[x]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 3.59288, size = 3, normalized size = 0.75 \[ c \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*log(x)

_______________________________________________________________________________________

Mathematica [A]  time = 0.000593888, size = 4, normalized size = 1. \[ c \log (x) \]

Antiderivative was successfully verified.

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

[Out]

c*Log[x]

_______________________________________________________________________________________

Maple [A]  time = 0.002, size = 5, normalized size = 1.3 \[ c\ln \left ( x \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*ln(x)

_______________________________________________________________________________________

Maxima [A]  time = 1.3403, size = 9, normalized size = 2.25 \[ \frac{1}{2} \, c \log \left (x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*c*log(x^2)

_______________________________________________________________________________________

Fricas [A]  time = 0.220173, size = 5, normalized size = 1.25 \[ c \log \left (x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*log(x)

_______________________________________________________________________________________

Sympy [A]  time = 0.115203, size = 3, normalized size = 0.75 \[ c \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*log(x)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.22222, size = 7, normalized size = 1.75 \[ c{\rm ln}\left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

c*ln(abs(x))