3.227 \(\int \frac{\left (a+b x^3\right )^2}{x^5} \, dx\)

Optimal. Leaf size=28 \[ -\frac{a^2}{4 x^4}-\frac{2 a b}{x}+\frac{b^2 x^2}{2} \]

[Out]

-a^2/(4*x^4) - (2*a*b)/x + (b^2*x^2)/2

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Rubi [A]  time = 0.0290321, antiderivative size = 28, 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 \[ -\frac{a^2}{4 x^4}-\frac{2 a b}{x}+\frac{b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^2/x^5,x]

[Out]

-a^2/(4*x^4) - (2*a*b)/x + (b^2*x^2)/2

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{4 x^{4}} - \frac{2 a b}{x} + b^{2} \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**2/x**5,x)

[Out]

-a**2/(4*x**4) - 2*a*b/x + b**2*Integral(x, x)

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Mathematica [A]  time = 0.00172151, size = 28, normalized size = 1. \[ -\frac{a^2}{4 x^4}-\frac{2 a b}{x}+\frac{b^2 x^2}{2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^2/x^5,x]

[Out]

-a^2/(4*x^4) - (2*a*b)/x + (b^2*x^2)/2

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Maple [A]  time = 0.009, size = 25, normalized size = 0.9 \[ -{\frac{{a}^{2}}{4\,{x}^{4}}}-2\,{\frac{ab}{x}}+{\frac{{b}^{2}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^2/x^5,x)

[Out]

-1/4*a^2/x^4-2*a*b/x+1/2*b^2*x^2

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Maxima [A]  time = 1.43037, size = 34, normalized size = 1.21 \[ \frac{1}{2} \, b^{2} x^{2} - \frac{8 \, a b x^{3} + a^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2/x^5,x, algorithm="maxima")

[Out]

1/2*b^2*x^2 - 1/4*(8*a*b*x^3 + a^2)/x^4

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Fricas [A]  time = 0.268495, size = 35, normalized size = 1.25 \[ \frac{2 \, b^{2} x^{6} - 8 \, a b x^{3} - a^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2/x^5,x, algorithm="fricas")

[Out]

1/4*(2*b^2*x^6 - 8*a*b*x^3 - a^2)/x^4

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Sympy [A]  time = 1.2407, size = 24, normalized size = 0.86 \[ \frac{b^{2} x^{2}}{2} - \frac{a^{2} + 8 a b x^{3}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**2/x**5,x)

[Out]

b**2*x**2/2 - (a**2 + 8*a*b*x**3)/(4*x**4)

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GIAC/XCAS [A]  time = 0.229314, size = 34, normalized size = 1.21 \[ \frac{1}{2} \, b^{2} x^{2} - \frac{8 \, a b x^{3} + a^{2}}{4 \, x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2/x^5,x, algorithm="giac")

[Out]

1/2*b^2*x^2 - 1/4*(8*a*b*x^3 + a^2)/x^4