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

Optimal. Leaf size=34 \[ -\frac{a^3}{5 x^5}-\frac{a^2 b}{x^3}-\frac{3 a b^2}{x}+b^3 x \]

[Out]

-a^3/(5*x^5) - (a^2*b)/x^3 - (3*a*b^2)/x + b^3*x

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Rubi [A]  time = 0.0400347, antiderivative size = 34, 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 \[ -\frac{a^3}{5 x^5}-\frac{a^2 b}{x^3}-\frac{3 a b^2}{x}+b^3 x \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^3/x^6,x]

[Out]

-a^3/(5*x^5) - (a^2*b)/x^3 - (3*a*b^2)/x + b^3*x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{3}}{5 x^{5}} - \frac{a^{2} b}{x^{3}} - \frac{3 a b^{2}}{x} + \int b^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**3/(5*x**5) - a**2*b/x**3 - 3*a*b**2/x + Integral(b**3, x)

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Mathematica [A]  time = 0.00920143, size = 34, normalized size = 1. \[ -\frac{a^3}{5 x^5}-\frac{a^2 b}{x^3}-\frac{3 a b^2}{x}+b^3 x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^3/x^6,x]

[Out]

-a^3/(5*x^5) - (a^2*b)/x^3 - (3*a*b^2)/x + b^3*x

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Maple [A]  time = 0.008, size = 33, normalized size = 1. \[ -{\frac{{a}^{3}}{5\,{x}^{5}}}-{\frac{{a}^{2}b}{{x}^{3}}}-3\,{\frac{a{b}^{2}}{x}}+{b}^{3}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3/x^6,x)

[Out]

-1/5*a^3/x^5-a^2*b/x^3-3*a*b^2/x+b^3*x

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Maxima [A]  time = 1.35443, size = 45, normalized size = 1.32 \[ b^{3} x - \frac{15 \, a b^{2} x^{4} + 5 \, a^{2} b x^{2} + a^{3}}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^3*x - 1/5*(15*a*b^2*x^4 + 5*a^2*b*x^2 + a^3)/x^5

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Fricas [A]  time = 0.209635, size = 50, normalized size = 1.47 \[ \frac{5 \, b^{3} x^{6} - 15 \, a b^{2} x^{4} - 5 \, a^{2} b x^{2} - a^{3}}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*(5*b^3*x^6 - 15*a*b^2*x^4 - 5*a^2*b*x^2 - a^3)/x^5

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Sympy [A]  time = 1.40911, size = 32, normalized size = 0.94 \[ b^{3} x - \frac{a^{3} + 5 a^{2} b x^{2} + 15 a b^{2} x^{4}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**3/x**6,x)

[Out]

b**3*x - (a**3 + 5*a**2*b*x**2 + 15*a*b**2*x**4)/(5*x**5)

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GIAC/XCAS [A]  time = 0.208819, size = 45, normalized size = 1.32 \[ b^{3} x - \frac{15 \, a b^{2} x^{4} + 5 \, a^{2} b x^{2} + a^{3}}{5 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b^3*x - 1/5*(15*a*b^2*x^4 + 5*a^2*b*x^2 + a^3)/x^5