3.9 \(\int \frac{a+b x^2}{x^4} \, dx\)

Optimal. Leaf size=15 \[ -\frac{a}{3 x^3}-\frac{b}{x} \]

[Out]

-a/(3*x^3) - b/x

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Rubi [A]  time = 0.0152293, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a}{3 x^3}-\frac{b}{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)/x^4,x]

[Out]

-a/(3*x^3) - b/x

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Rubi in Sympy [A]  time = 3.05136, size = 10, normalized size = 0.67 \[ - \frac{a}{3 x^{3}} - \frac{b}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a/(3*x**3) - b/x

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Mathematica [A]  time = 0.0033547, size = 15, normalized size = 1. \[ -\frac{a}{3 x^3}-\frac{b}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)/x^4,x]

[Out]

-a/(3*x^3) - b/x

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Maple [A]  time = 0.006, size = 14, normalized size = 0.9 \[ -{\frac{a}{3\,{x}^{3}}}-{\frac{b}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)/x^4,x)

[Out]

-1/3*a/x^3-b/x

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Maxima [A]  time = 1.34456, size = 18, normalized size = 1.2 \[ -\frac{3 \, b x^{2} + a}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(3*b*x^2 + a)/x^3

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Fricas [A]  time = 0.217099, size = 18, normalized size = 1.2 \[ -\frac{3 \, b x^{2} + a}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(3*b*x^2 + a)/x^3

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Sympy [A]  time = 1.10062, size = 14, normalized size = 0.93 \[ - \frac{a + 3 b x^{2}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)/x**4,x)

[Out]

-(a + 3*b*x**2)/(3*x**3)

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GIAC/XCAS [A]  time = 0.211536, size = 18, normalized size = 1.2 \[ -\frac{3 \, b x^{2} + a}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(3*b*x^2 + a)/x^3