3.443 \(\int \frac{\left (a^2+2 a b x^2+b^2 x^4\right )^2}{x^{16}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{a^4}{15 x^{15}}-\frac{4 a^3 b}{13 x^{13}}-\frac{6 a^2 b^2}{11 x^{11}}-\frac{4 a b^3}{9 x^9}-\frac{b^4}{7 x^7} \]

[Out]

-a^4/(15*x^15) - (4*a^3*b)/(13*x^13) - (6*a^2*b^2)/(11*x^11) - (4*a*b^3)/(9*x^9)
- b^4/(7*x^7)

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Rubi [A]  time = 0.0673062, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ -\frac{a^4}{15 x^{15}}-\frac{4 a^3 b}{13 x^{13}}-\frac{6 a^2 b^2}{11 x^{11}}-\frac{4 a b^3}{9 x^9}-\frac{b^4}{7 x^7} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^16,x]

[Out]

-a^4/(15*x^15) - (4*a^3*b)/(13*x^13) - (6*a^2*b^2)/(11*x^11) - (4*a*b^3)/(9*x^9)
- b^4/(7*x^7)

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Rubi in Sympy [A]  time = 17.6708, size = 54, normalized size = 0.96 \[ - \frac{a^{4}}{15 x^{15}} - \frac{4 a^{3} b}{13 x^{13}} - \frac{6 a^{2} b^{2}}{11 x^{11}} - \frac{4 a b^{3}}{9 x^{9}} - \frac{b^{4}}{7 x^{7}} \]

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**16,x)

[Out]

-a**4/(15*x**15) - 4*a**3*b/(13*x**13) - 6*a**2*b**2/(11*x**11) - 4*a*b**3/(9*x*
*9) - b**4/(7*x**7)

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Mathematica [A]  time = 0.0117933, size = 56, normalized size = 1. \[ -\frac{a^4}{15 x^{15}}-\frac{4 a^3 b}{13 x^{13}}-\frac{6 a^2 b^2}{11 x^{11}}-\frac{4 a b^3}{9 x^9}-\frac{b^4}{7 x^7} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)^2/x^16,x]

[Out]

-a^4/(15*x^15) - (4*a^3*b)/(13*x^13) - (6*a^2*b^2)/(11*x^11) - (4*a*b^3)/(9*x^9)
- b^4/(7*x^7)

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Maple [A]  time = 0.008, size = 47, normalized size = 0.8 \[ -{\frac{{a}^{4}}{15\,{x}^{15}}}-{\frac{4\,{a}^{3}b}{13\,{x}^{13}}}-{\frac{6\,{a}^{2}{b}^{2}}{11\,{x}^{11}}}-{\frac{4\,a{b}^{3}}{9\,{x}^{9}}}-{\frac{{b}^{4}}{7\,{x}^{7}}} \]

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^16,x)

[Out]

-1/15*a^4/x^15-4/13*a^3*b/x^13-6/11*a^2*b^2/x^11-4/9*a*b^3/x^9-1/7*b^4/x^7

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Maxima [A]  time = 0.695141, size = 65, normalized size = 1.16 \[ -\frac{6435 \, b^{4} x^{8} + 20020 \, a b^{3} x^{6} + 24570 \, a^{2} b^{2} x^{4} + 13860 \, a^{3} b x^{2} + 3003 \, a^{4}}{45045 \, x^{15}} \]

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^16,x, algorithm="maxima")

[Out]

-1/45045*(6435*b^4*x^8 + 20020*a*b^3*x^6 + 24570*a^2*b^2*x^4 + 13860*a^3*b*x^2 +
3003*a^4)/x^15

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Fricas [A]  time = 0.251345, size = 65, normalized size = 1.16 \[ -\frac{6435 \, b^{4} x^{8} + 20020 \, a b^{3} x^{6} + 24570 \, a^{2} b^{2} x^{4} + 13860 \, a^{3} b x^{2} + 3003 \, a^{4}}{45045 \, x^{15}} \]

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^16,x, algorithm="fricas")

[Out]

-1/45045*(6435*b^4*x^8 + 20020*a*b^3*x^6 + 24570*a^2*b^2*x^4 + 13860*a^3*b*x^2 +
3003*a^4)/x^15

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Sympy [A]  time = 2.09507, size = 51, normalized size = 0.91 \[ - \frac{3003 a^{4} + 13860 a^{3} b x^{2} + 24570 a^{2} b^{2} x^{4} + 20020 a b^{3} x^{6} + 6435 b^{4} x^{8}}{45045 x^{15}} \]

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**16,x)

[Out]

-(3003*a**4 + 13860*a**3*b*x**2 + 24570*a**2*b**2*x**4 + 20020*a*b**3*x**6 + 643
5*b**4*x**8)/(45045*x**15)

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GIAC/XCAS [A]  time = 0.268397, size = 65, normalized size = 1.16 \[ -\frac{6435 \, b^{4} x^{8} + 20020 \, a b^{3} x^{6} + 24570 \, a^{2} b^{2} x^{4} + 13860 \, a^{3} b x^{2} + 3003 \, a^{4}}{45045 \, x^{15}} \]

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^16,x, algorithm="giac")

[Out]

-1/45045*(6435*b^4*x^8 + 20020*a*b^3*x^6 + 24570*a^2*b^2*x^4 + 13860*a^3*b*x^2 +
3003*a^4)/x^15