3.228 \(\int \frac{\left (a+b x^3\right )^2}{x^6} \, dx\)

Optimal. Leaf size=23 \[ -\frac{a^2}{5 x^5}-\frac{a b}{x^2}+b^2 x \]

[Out]

-a^2/(5*x^5) - (a*b)/x^2 + b^2*x

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Rubi [A]  time = 0.0280289, 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 \[ -\frac{a^2}{5 x^5}-\frac{a b}{x^2}+b^2 x \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^3)^2/x^6,x]

[Out]

-a^2/(5*x^5) - (a*b)/x^2 + b^2*x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{2}}{5 x^{5}} - \frac{a b}{x^{2}} + \int b^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**2/x**6,x)

[Out]

-a**2/(5*x**5) - a*b/x**2 + Integral(b**2, x)

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Mathematica [A]  time = 0.00125369, size = 23, normalized size = 1. \[ -\frac{a^2}{5 x^5}-\frac{a b}{x^2}+b^2 x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^3)^2/x^6,x]

[Out]

-a^2/(5*x^5) - (a*b)/x^2 + b^2*x

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Maple [A]  time = 0.007, size = 22, normalized size = 1. \[ -{\frac{{a}^{2}}{5\,{x}^{5}}}-{\frac{ab}{{x}^{2}}}+{b}^{2}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^2/x^6,x)

[Out]

-1/5*a^2/x^5-a*b/x^2+b^2*x

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Maxima [A]  time = 1.44031, size = 30, normalized size = 1.3 \[ b^{2} x - \frac{5 \, a b x^{3} + a^{2}}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2/x^6,x, algorithm="maxima")

[Out]

b^2*x - 1/5*(5*a*b*x^3 + a^2)/x^5

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Fricas [A]  time = 0.256298, size = 35, normalized size = 1.52 \[ \frac{5 \, b^{2} x^{6} - 5 \, a b x^{3} - a^{2}}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2/x^6,x, algorithm="fricas")

[Out]

1/5*(5*b^2*x^6 - 5*a*b*x^3 - a^2)/x^5

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Sympy [A]  time = 1.30997, size = 20, normalized size = 0.87 \[ b^{2} x - \frac{a^{2} + 5 a b x^{3}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**2/x**6,x)

[Out]

b**2*x - (a**2 + 5*a*b*x**3)/(5*x**5)

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GIAC/XCAS [A]  time = 0.243837, size = 30, normalized size = 1.3 \[ b^{2} x - \frac{5 \, a b x^{3} + a^{2}}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^3 + a)^2/x^6,x, algorithm="giac")

[Out]

b^2*x - 1/5*(5*a*b*x^3 + a^2)/x^5