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

Optimal. Leaf size=17 \[ -\frac{(a+b x)^3}{3 a x^3} \]

[Out]

-(a + b*x)^3/(3*a*x^3)

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Rubi [A]  time = 0.0108487, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{(a+b x)^3}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^2/x^4,x]

[Out]

-(a + b*x)^3/(3*a*x^3)

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Rubi in Sympy [A]  time = 2.26892, size = 14, normalized size = 0.82 \[ - \frac{\left (a + b x\right )^{3}}{3 a x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x)**3/(3*a*x**3)

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Mathematica [A]  time = 0.011258, size = 26, normalized size = 1.53 \[ -\frac{a^2}{3 x^3}-\frac{a b}{x^2}-\frac{b^2}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^2/x^4,x]

[Out]

-a^2/(3*x^3) - (a*b)/x^2 - b^2/x

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Maple [A]  time = 0.007, size = 25, normalized size = 1.5 \[ -{\frac{{a}^{2}}{3\,{x}^{3}}}-{\frac{{b}^{2}}{x}}-{\frac{ab}{{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^2/x^4,x)

[Out]

-1/3*a^2/x^3-b^2/x-a*b/x^2

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Maxima [A]  time = 1.34092, size = 30, normalized size = 1.76 \[ -\frac{3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(3*b^2*x^2 + 3*a*b*x + a^2)/x^3

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Fricas [A]  time = 0.190737, size = 30, normalized size = 1.76 \[ -\frac{3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(3*b^2*x^2 + 3*a*b*x + a^2)/x^3

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Sympy [A]  time = 1.17441, size = 24, normalized size = 1.41 \[ - \frac{a^{2} + 3 a b x + 3 b^{2} x^{2}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**2/x**4,x)

[Out]

-(a**2 + 3*a*b*x + 3*b**2*x**2)/(3*x**3)

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GIAC/XCAS [A]  time = 0.215298, size = 30, normalized size = 1.76 \[ -\frac{3 \, b^{2} x^{2} + 3 \, a b x + a^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*(3*b^2*x^2 + 3*a*b*x + a^2)/x^3