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

Optimal. Leaf size=23 \[ -\frac{a^2}{3 x^3}-\frac{2 a b}{x}+b^2 x \]

[Out]

-a^2/(3*x^3) - (2*a*b)/x + b^2*x

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Rubi [A]  time = 0.0304979, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ -\frac{a^2}{3 x^3}-\frac{2 a b}{x}+b^2 x \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^2/x^4,x]

[Out]

-a^2/(3*x^3) - (2*a*b)/x + b^2*x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{3 x^{3}} - \frac{2 a b}{x} + \int b^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00149176, size = 23, normalized size = 1. \[ -\frac{a^2}{3 x^3}-\frac{2 a b}{x}+b^2 x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^2/x^4,x]

[Out]

-a^2/(3*x^3) - (2*a*b)/x + b^2*x

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Maple [A]  time = 0.007, size = 22, normalized size = 1. \[ -{\frac{{a}^{2}}{3\,{x}^{3}}}-2\,{\frac{ab}{x}}+{b}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^2/x^4,x)

[Out]

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

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Maxima [A]  time = 1.33237, size = 30, normalized size = 1.3 \[ b^{2} x - \frac{6 \, a b x^{2} + a^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^2*x - 1/3*(6*a*b*x^2 + a^2)/x^3

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Fricas [A]  time = 0.193146, size = 35, normalized size = 1.52 \[ \frac{3 \, b^{2} x^{4} - 6 \, a b x^{2} - a^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(3*b^2*x^4 - 6*a*b*x^2 - a^2)/x^3

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Sympy [A]  time = 1.13156, size = 20, normalized size = 0.87 \[ b^{2} x - \frac{a^{2} + 6 a b x^{2}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**2/x**4,x)

[Out]

b**2*x - (a**2 + 6*a*b*x**2)/(3*x**3)

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GIAC/XCAS [A]  time = 0.207674, size = 30, normalized size = 1.3 \[ b^{2} x - \frac{6 \, a b x^{2} + a^{2}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^2*x - 1/3*(6*a*b*x^2 + a^2)/x^3