3.49 \(\int \frac{\left (a+b x^2\right )^3}{x^8} \, dx\)

Optimal. Leaf size=39 \[ -\frac{a^3}{7 x^7}-\frac{3 a^2 b}{5 x^5}-\frac{a b^2}{x^3}-\frac{b^3}{x} \]

[Out]

-a^3/(7*x^7) - (3*a^2*b)/(5*x^5) - (a*b^2)/x^3 - b^3/x

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Rubi [A]  time = 0.0418237, antiderivative size = 39, 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 \[ -\frac{a^3}{7 x^7}-\frac{3 a^2 b}{5 x^5}-\frac{a b^2}{x^3}-\frac{b^3}{x} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^3/x^8,x]

[Out]

-a^3/(7*x^7) - (3*a^2*b)/(5*x^5) - (a*b^2)/x^3 - b^3/x

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Rubi in Sympy [A]  time = 7.57277, size = 34, normalized size = 0.87 \[ - \frac{a^{3}}{7 x^{7}} - \frac{3 a^{2} b}{5 x^{5}} - \frac{a b^{2}}{x^{3}} - \frac{b^{3}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**3/x**8,x)

[Out]

-a**3/(7*x**7) - 3*a**2*b/(5*x**5) - a*b**2/x**3 - b**3/x

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Mathematica [A]  time = 0.00694587, size = 39, normalized size = 1. \[ -\frac{a^3}{7 x^7}-\frac{3 a^2 b}{5 x^5}-\frac{a b^2}{x^3}-\frac{b^3}{x} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^3/x^8,x]

[Out]

-a^3/(7*x^7) - (3*a^2*b)/(5*x^5) - (a*b^2)/x^3 - b^3/x

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Maple [A]  time = 0.007, size = 36, normalized size = 0.9 \[ -{\frac{{a}^{3}}{7\,{x}^{7}}}-{\frac{3\,{a}^{2}b}{5\,{x}^{5}}}-{\frac{a{b}^{2}}{{x}^{3}}}-{\frac{{b}^{3}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3/x^8,x)

[Out]

-1/7*a^3/x^7-3/5*a^2*b/x^5-a*b^2/x^3-b^3/x

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Maxima [A]  time = 1.34556, size = 50, normalized size = 1.28 \[ -\frac{35 \, b^{3} x^{6} + 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} + 5 \, a^{3}}{35 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^8,x, algorithm="maxima")

[Out]

-1/35*(35*b^3*x^6 + 35*a*b^2*x^4 + 21*a^2*b*x^2 + 5*a^3)/x^7

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Fricas [A]  time = 0.220698, size = 50, normalized size = 1.28 \[ -\frac{35 \, b^{3} x^{6} + 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} + 5 \, a^{3}}{35 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^8,x, algorithm="fricas")

[Out]

-1/35*(35*b^3*x^6 + 35*a*b^2*x^4 + 21*a^2*b*x^2 + 5*a^3)/x^7

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Sympy [A]  time = 1.54212, size = 39, normalized size = 1. \[ - \frac{5 a^{3} + 21 a^{2} b x^{2} + 35 a b^{2} x^{4} + 35 b^{3} x^{6}}{35 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**3/x**8,x)

[Out]

-(5*a**3 + 21*a**2*b*x**2 + 35*a*b**2*x**4 + 35*b**3*x**6)/(35*x**7)

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GIAC/XCAS [A]  time = 0.206679, size = 50, normalized size = 1.28 \[ -\frac{35 \, b^{3} x^{6} + 35 \, a b^{2} x^{4} + 21 \, a^{2} b x^{2} + 5 \, a^{3}}{35 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^8,x, algorithm="giac")

[Out]

-1/35*(35*b^3*x^6 + 35*a*b^2*x^4 + 21*a^2*b*x^2 + 5*a^3)/x^7