3.126 \(\int \frac{a c+b c x^2}{x^3 \left (a+b x^2\right )} \, dx\)

Optimal. Leaf size=8 \[ -\frac{c}{2 x^2} \]

[Out]

-c/(2*x^2)

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Rubi [A]  time = 0.00625471, antiderivative size = 8, 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 \[ -\frac{c}{2 x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a*c + b*c*x^2)/(x^3*(a + b*x^2)),x]

[Out]

-c/(2*x^2)

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Rubi in Sympy [A]  time = 3.57448, size = 7, normalized size = 0.88 \[ - \frac{c}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c/(2*x**2)

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Mathematica [A]  time = 0.000651485, size = 8, normalized size = 1. \[ -\frac{c}{2 x^2} \]

Antiderivative was successfully verified.

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

[Out]

-c/(2*x^2)

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Maple [A]  time = 0., size = 7, normalized size = 0.9 \[ -{\frac{c}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*c*x^2+a*c)/x^3/(b*x^2+a),x)

[Out]

-1/2*c/x^2

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Maxima [A]  time = 1.3463, size = 8, normalized size = 1. \[ -\frac{c}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*c/x^2

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Fricas [A]  time = 0.217426, size = 8, normalized size = 1. \[ -\frac{c}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*c/x^2

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Sympy [A]  time = 0.115672, size = 7, normalized size = 0.88 \[ - \frac{c}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-c/(2*x**2)

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GIAC/XCAS [A]  time = 0.224218, size = 8, normalized size = 1. \[ -\frac{c}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*c/x^2