3.420 \(\int \frac{a^2+2 a b x^2+b^2 x^4}{x^8} \, dx\)

Optimal. Leaf size=30 \[ -\frac{a^2}{7 x^7}-\frac{2 a b}{5 x^5}-\frac{b^2}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0243965, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045 \[ -\frac{a^2}{7 x^7}-\frac{2 a b}{5 x^5}-\frac{b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^8,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.6315, size = 27, normalized size = 0.9 \[ - \frac{a^{2}}{7 x^{7}} - \frac{2 a b}{5 x^{5}} - \frac{b^{2}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b**2*x**4+2*a*b*x**2+a**2)/x**8,x)

[Out]

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

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Mathematica [A]  time = 0.00187926, size = 30, normalized size = 1. \[ -\frac{a^2}{7 x^7}-\frac{2 a b}{5 x^5}-\frac{b^2}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a^2 + 2*a*b*x^2 + b^2*x^4)/x^8,x]

[Out]

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

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Maple [A]  time = 0.007, size = 25, normalized size = 0.8 \[ -{\frac{{a}^{2}}{7\,{x}^{7}}}-{\frac{2\,ab}{5\,{x}^{5}}}-{\frac{{b}^{2}}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^4+2*a*b*x^2+a^2)/x^8,x)

[Out]

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

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Maxima [A]  time = 0.690185, size = 35, normalized size = 1.17 \[ -\frac{35 \, b^{2} x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)/x^8,x, algorithm="maxima")

[Out]

-1/105*(35*b^2*x^4 + 42*a*b*x^2 + 15*a^2)/x^7

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Fricas [A]  time = 0.258226, size = 35, normalized size = 1.17 \[ -\frac{35 \, b^{2} x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)/x^8,x, algorithm="fricas")

[Out]

-1/105*(35*b^2*x^4 + 42*a*b*x^2 + 15*a^2)/x^7

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Sympy [A]  time = 1.29084, size = 27, normalized size = 0.9 \[ - \frac{15 a^{2} + 42 a b x^{2} + 35 b^{2} x^{4}}{105 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b**2*x**4+2*a*b*x**2+a**2)/x**8,x)

[Out]

-(15*a**2 + 42*a*b*x**2 + 35*b**2*x**4)/(105*x**7)

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GIAC/XCAS [A]  time = 0.267973, size = 35, normalized size = 1.17 \[ -\frac{35 \, b^{2} x^{4} + 42 \, a b x^{2} + 15 \, a^{2}}{105 \, x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b^2*x^4 + 2*a*b*x^2 + a^2)/x^8,x, algorithm="giac")

[Out]

-1/105*(35*b^2*x^4 + 42*a*b*x^2 + 15*a^2)/x^7