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

Optimal. Leaf size=17 \[ -\frac{(a+b x)^2}{2 a x^2} \]

[Out]

-(a + b*x)^2/(2*a*x^2)

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Rubi [A]  time = 0.0100279, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{(a+b x)^2}{2 a x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)/x^3,x]

[Out]

-(a + b*x)^2/(2*a*x^2)

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Rubi in Sympy [A]  time = 2.55554, size = 10, normalized size = 0.59 \[ - \frac{a}{2 x^{2}} - \frac{b}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)/x**3,x)

[Out]

-a/(2*x**2) - b/x

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Mathematica [A]  time = 0.00198773, size = 15, normalized size = 0.88 \[ -\frac{a}{2 x^2}-\frac{b}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)/x^3,x]

[Out]

-a/(2*x^2) - b/x

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Maple [A]  time = 0.008, size = 14, normalized size = 0.8 \[ -{\frac{b}{x}}-{\frac{a}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/x^3,x)

[Out]

-b/x-1/2*a/x^2

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Maxima [A]  time = 1.33047, size = 15, normalized size = 0.88 \[ -\frac{2 \, b x + a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*b*x + a)/x^2

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Fricas [A]  time = 0.191097, size = 15, normalized size = 0.88 \[ -\frac{2 \, b x + a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*b*x + a)/x^2

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Sympy [A]  time = 0.994592, size = 12, normalized size = 0.71 \[ - \frac{a + 2 b x}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)/x**3,x)

[Out]

-(a + 2*b*x)/(2*x**2)

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GIAC/XCAS [A]  time = 0.210818, size = 15, normalized size = 0.88 \[ -\frac{2 \, b x + a}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2*(2*b*x + a)/x^2