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

Optimal. Leaf size=50 \[ -\frac{a^4}{3 x^3}-\frac{4 a^3 b}{x}+6 a^2 b^2 x+\frac{4}{3} a b^3 x^3+\frac{b^4 x^5}{5} \]

[Out]

-a^4/(3*x^3) - (4*a^3*b)/x + 6*a^2*b^2*x + (4*a*b^3*x^3)/3 + (b^4*x^5)/5

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Rubi [A]  time = 0.0663837, antiderivative size = 50, 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 \[ -\frac{a^4}{3 x^3}-\frac{4 a^3 b}{x}+6 a^2 b^2 x+\frac{4}{3} a b^3 x^3+\frac{b^4 x^5}{5} \]

Antiderivative was successfully verified.

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

[Out]

-a^4/(3*x^3) - (4*a^3*b)/x + 6*a^2*b^2*x + (4*a*b^3*x^3)/3 + (b^4*x^5)/5

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Rubi in Sympy [A]  time = 17.3557, size = 46, normalized size = 0.92 \[ - \frac{a^{4}}{3 x^{3}} - \frac{4 a^{3} b}{x} + 6 a^{2} b^{2} x + \frac{4 a b^{3} x^{3}}{3} + \frac{b^{4} x^{5}}{5} \]

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**4,x)

[Out]

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

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Mathematica [A]  time = 0.0105393, size = 50, normalized size = 1. \[ -\frac{a^4}{3 x^3}-\frac{4 a^3 b}{x}+6 a^2 b^2 x+\frac{4}{3} a b^3 x^3+\frac{b^4 x^5}{5} \]

Antiderivative was successfully verified.

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

[Out]

-a^4/(3*x^3) - (4*a^3*b)/x + 6*a^2*b^2*x + (4*a*b^3*x^3)/3 + (b^4*x^5)/5

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Maple [A]  time = 0.008, size = 45, normalized size = 0.9 \[ -{\frac{{a}^{4}}{3\,{x}^{3}}}-4\,{\frac{{a}^{3}b}{x}}+6\,{a}^{2}{b}^{2}x+{\frac{4\,a{b}^{3}{x}^{3}}{3}}+{\frac{{b}^{4}{x}^{5}}{5}} \]

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^4,x)

[Out]

-1/3*a^4/x^3-4*a^3*b/x+6*a^2*b^2*x+4/3*a*b^3*x^3+1/5*b^4*x^5

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Maxima [A]  time = 0.703725, size = 61, normalized size = 1.22 \[ \frac{1}{5} \, b^{4} x^{5} + \frac{4}{3} \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x - \frac{12 \, a^{3} b x^{2} + a^{4}}{3 \, x^{3}} \]

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^4,x, algorithm="maxima")

[Out]

1/5*b^4*x^5 + 4/3*a*b^3*x^3 + 6*a^2*b^2*x - 1/3*(12*a^3*b*x^2 + a^4)/x^3

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Fricas [A]  time = 0.256249, size = 65, normalized size = 1.3 \[ \frac{3 \, b^{4} x^{8} + 20 \, a b^{3} x^{6} + 90 \, a^{2} b^{2} x^{4} - 60 \, a^{3} b x^{2} - 5 \, a^{4}}{15 \, x^{3}} \]

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^4,x, algorithm="fricas")

[Out]

1/15*(3*b^4*x^8 + 20*a*b^3*x^6 + 90*a^2*b^2*x^4 - 60*a^3*b*x^2 - 5*a^4)/x^3

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Sympy [A]  time = 1.17604, size = 48, normalized size = 0.96 \[ 6 a^{2} b^{2} x + \frac{4 a b^{3} x^{3}}{3} + \frac{b^{4} x^{5}}{5} - \frac{a^{4} + 12 a^{3} b x^{2}}{3 x^{3}} \]

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**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.268362, size = 61, normalized size = 1.22 \[ \frac{1}{5} \, b^{4} x^{5} + \frac{4}{3} \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x - \frac{12 \, a^{3} b x^{2} + a^{4}}{3 \, x^{3}} \]

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^4,x, algorithm="giac")

[Out]

1/5*b^4*x^5 + 4/3*a*b^3*x^3 + 6*a^2*b^2*x - 1/3*(12*a^3*b*x^2 + a^4)/x^3