3.440 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{13}} \, dx\)

Optimal. Leaf size=40 \[ \frac{b \left (a+b x^2\right )^5}{60 a^2 x^{10}}-\frac{\left (a+b x^2\right )^5}{12 a x^{12}} \]

[Out]

-(a + b*x^2)^5/(12*a*x^12) + (b*(a + b*x^2)^5)/(60*a^2*x^10)

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Rubi [A]  time = 0.0724243, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ \frac{b \left (a+b x^2\right )^5}{60 a^2 x^{10}}-\frac{\left (a+b x^2\right )^5}{12 a x^{12}} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^13,x]

[Out]

-(a + b*x^2)^5/(12*a*x^12) + (b*(a + b*x^2)^5)/(60*a^2*x^10)

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Rubi in Sympy [A]  time = 21.0023, size = 54, normalized size = 1.35 \[ - \frac{a^{4}}{12 x^{12}} - \frac{2 a^{3} b}{5 x^{10}} - \frac{3 a^{2} b^{2}}{4 x^{8}} - \frac{2 a b^{3}}{3 x^{6}} - \frac{b^{4}}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)**2/x**13,x)

[Out]

-a**4/(12*x**12) - 2*a**3*b/(5*x**10) - 3*a**2*b**2/(4*x**8) - 2*a*b**3/(3*x**6)
- b**4/(4*x**4)

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Mathematica [A]  time = 0.00694587, size = 56, normalized size = 1.4 \[ -\frac{a^4}{12 x^{12}}-\frac{2 a^3 b}{5 x^{10}}-\frac{3 a^2 b^2}{4 x^8}-\frac{2 a b^3}{3 x^6}-\frac{b^4}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^13,x]

[Out]

-a^4/(12*x^12) - (2*a^3*b)/(5*x^10) - (3*a^2*b^2)/(4*x^8) - (2*a*b^3)/(3*x^6) -
b^4/(4*x^4)

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Maple [A]  time = 0.008, size = 47, normalized size = 1.2 \[ -{\frac{3\,{a}^{2}{b}^{2}}{4\,{x}^{8}}}-{\frac{{a}^{4}}{12\,{x}^{12}}}-{\frac{2\,a{b}^{3}}{3\,{x}^{6}}}-{\frac{2\,{a}^{3}b}{5\,{x}^{10}}}-{\frac{{b}^{4}}{4\,{x}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)^2/x^13,x)

[Out]

-3/4*a^2*b^2/x^8-1/12*a^4/x^12-2/3*a*b^3/x^6-2/5*a^3*b/x^10-1/4*b^4/x^4

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Maxima [A]  time = 0.702922, size = 65, normalized size = 1.62 \[ -\frac{15 \, b^{4} x^{8} + 40 \, a b^{3} x^{6} + 45 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x^{2} + 5 \, a^{4}}{60 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^2/x^13,x, algorithm="maxima")

[Out]

-1/60*(15*b^4*x^8 + 40*a*b^3*x^6 + 45*a^2*b^2*x^4 + 24*a^3*b*x^2 + 5*a^4)/x^12

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Fricas [A]  time = 0.254856, size = 65, normalized size = 1.62 \[ -\frac{15 \, b^{4} x^{8} + 40 \, a b^{3} x^{6} + 45 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x^{2} + 5 \, a^{4}}{60 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^2/x^13,x, algorithm="fricas")

[Out]

-1/60*(15*b^4*x^8 + 40*a*b^3*x^6 + 45*a^2*b^2*x^4 + 24*a^3*b*x^2 + 5*a^4)/x^12

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Sympy [A]  time = 1.98, size = 51, normalized size = 1.27 \[ - \frac{5 a^{4} + 24 a^{3} b x^{2} + 45 a^{2} b^{2} x^{4} + 40 a b^{3} x^{6} + 15 b^{4} x^{8}}{60 x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)**2/x**13,x)

[Out]

-(5*a**4 + 24*a**3*b*x**2 + 45*a**2*b**2*x**4 + 40*a*b**3*x**6 + 15*b**4*x**8)/(
60*x**12)

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GIAC/XCAS [A]  time = 0.269104, size = 65, normalized size = 1.62 \[ -\frac{15 \, b^{4} x^{8} + 40 \, a b^{3} x^{6} + 45 \, a^{2} b^{2} x^{4} + 24 \, a^{3} b x^{2} + 5 \, a^{4}}{60 \, x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)^2/x^13,x, algorithm="giac")

[Out]

-1/60*(15*b^4*x^8 + 40*a*b^3*x^6 + 45*a^2*b^2*x^4 + 24*a^3*b*x^2 + 5*a^4)/x^12