3.39 \(\int \frac{\left (a+b x^2\right )^3}{x^{11}} \, dx\)

Optimal. Leaf size=40 \[ \frac{b \left (a+b x^2\right )^4}{40 a^2 x^8}-\frac{\left (a+b x^2\right )^4}{10 a x^{10}} \]

[Out]

-(a + b*x^2)^4/(10*a*x^10) + (b*(a + b*x^2)^4)/(40*a^2*x^8)

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Rubi [A]  time = 0.0563301, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{b \left (a+b x^2\right )^4}{40 a^2 x^8}-\frac{\left (a+b x^2\right )^4}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^3/x^11,x]

[Out]

-(a + b*x^2)^4/(10*a*x^10) + (b*(a + b*x^2)^4)/(40*a^2*x^8)

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Rubi in Sympy [A]  time = 9.3241, size = 39, normalized size = 0.98 \[ - \frac{a^{3}}{10 x^{10}} - \frac{3 a^{2} b}{8 x^{8}} - \frac{a b^{2}}{2 x^{6}} - \frac{b^{3}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**3/x**11,x)

[Out]

-a**3/(10*x**10) - 3*a**2*b/(8*x**8) - a*b**2/(2*x**6) - b**3/(4*x**4)

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Mathematica [A]  time = 0.00714586, size = 43, normalized size = 1.08 \[ -\frac{a^3}{10 x^{10}}-\frac{3 a^2 b}{8 x^8}-\frac{a b^2}{2 x^6}-\frac{b^3}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^3/x^11,x]

[Out]

-a^3/(10*x^10) - (3*a^2*b)/(8*x^8) - (a*b^2)/(2*x^6) - b^3/(4*x^4)

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Maple [A]  time = 0.009, size = 36, normalized size = 0.9 \[ -{\frac{{a}^{3}}{10\,{x}^{10}}}-{\frac{a{b}^{2}}{2\,{x}^{6}}}-{\frac{{b}^{3}}{4\,{x}^{4}}}-{\frac{3\,{a}^{2}b}{8\,{x}^{8}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^3/x^11,x)

[Out]

-1/10*a^3/x^10-1/2*a*b^2/x^6-1/4*b^3/x^4-3/8*a^2*b/x^8

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Maxima [A]  time = 1.34767, size = 50, normalized size = 1.25 \[ -\frac{10 \, b^{3} x^{6} + 20 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} + 4 \, a^{3}}{40 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^11,x, algorithm="maxima")

[Out]

-1/40*(10*b^3*x^6 + 20*a*b^2*x^4 + 15*a^2*b*x^2 + 4*a^3)/x^10

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Fricas [A]  time = 0.193905, size = 50, normalized size = 1.25 \[ -\frac{10 \, b^{3} x^{6} + 20 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} + 4 \, a^{3}}{40 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^11,x, algorithm="fricas")

[Out]

-1/40*(10*b^3*x^6 + 20*a*b^2*x^4 + 15*a^2*b*x^2 + 4*a^3)/x^10

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Sympy [A]  time = 1.70705, size = 39, normalized size = 0.98 \[ - \frac{4 a^{3} + 15 a^{2} b x^{2} + 20 a b^{2} x^{4} + 10 b^{3} x^{6}}{40 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**3/x**11,x)

[Out]

-(4*a**3 + 15*a**2*b*x**2 + 20*a*b**2*x**4 + 10*b**3*x**6)/(40*x**10)

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GIAC/XCAS [A]  time = 0.20664, size = 50, normalized size = 1.25 \[ -\frac{10 \, b^{3} x^{6} + 20 \, a b^{2} x^{4} + 15 \, a^{2} b x^{2} + 4 \, a^{3}}{40 \, x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^3/x^11,x, algorithm="giac")

[Out]

-1/40*(10*b^3*x^6 + 20*a*b^2*x^4 + 15*a^2*b*x^2 + 4*a^3)/x^10