3.47 \(\int \frac{\left (a+b x^2\right )^3}{x^4} \, dx\)

Optimal. Leaf size=37 \[ -\frac{a^3}{3 x^3}-\frac{3 a^2 b}{x}+3 a b^2 x+\frac{b^3 x^3}{3} \]

[Out]

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

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Rubi [A]  time = 0.0404206, antiderivative size = 37, 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 \[ -\frac{a^3}{3 x^3}-\frac{3 a^2 b}{x}+3 a b^2 x+\frac{b^3 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^3/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 7.12991, size = 32, normalized size = 0.86 \[ - \frac{a^{3}}{3 x^{3}} - \frac{3 a^{2} b}{x} + 3 a b^{2} x + \frac{b^{3} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**3/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.00699355, size = 37, normalized size = 1. \[ -\frac{a^3}{3 x^3}-\frac{3 a^2 b}{x}+3 a b^2 x+\frac{b^3 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^3/x^4,x]

[Out]

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

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Maple [A]  time = 0.009, size = 34, normalized size = 0.9 \[ -{\frac{{a}^{3}}{3\,{x}^{3}}}-3\,{\frac{{a}^{2}b}{x}}+3\,a{b}^{2}x+{\frac{{b}^{3}{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3/x^4,x)

[Out]

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

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Maxima [A]  time = 1.34259, size = 46, normalized size = 1.24 \[ \frac{1}{3} \, b^{3} x^{3} + 3 \, a b^{2} x - \frac{9 \, a^{2} b x^{2} + a^{3}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^4,x, algorithm="maxima")

[Out]

1/3*b^3*x^3 + 3*a*b^2*x - 1/3*(9*a^2*b*x^2 + a^3)/x^3

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Fricas [A]  time = 0.205746, size = 49, normalized size = 1.32 \[ \frac{b^{3} x^{6} + 9 \, a b^{2} x^{4} - 9 \, a^{2} b x^{2} - a^{3}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^4,x, algorithm="fricas")

[Out]

1/3*(b^3*x^6 + 9*a*b^2*x^4 - 9*a^2*b*x^2 - a^3)/x^3

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Sympy [A]  time = 1.20199, size = 34, normalized size = 0.92 \[ 3 a b^{2} x + \frac{b^{3} x^{3}}{3} - \frac{a^{3} + 9 a^{2} b x^{2}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**3/x**4,x)

[Out]

3*a*b**2*x + b**3*x**3/3 - (a**3 + 9*a**2*b*x**2)/(3*x**3)

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GIAC/XCAS [A]  time = 0.20782, size = 46, normalized size = 1.24 \[ \frac{1}{3} \, b^{3} x^{3} + 3 \, a b^{2} x - \frac{9 \, a^{2} b x^{2} + a^{3}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^4,x, algorithm="giac")

[Out]

1/3*b^3*x^3 + 3*a*b^2*x - 1/3*(9*a^2*b*x^2 + a^3)/x^3