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

Optimal. Leaf size=48 \[ -\frac{a^4}{x}+4 a^3 b x+2 a^2 b^2 x^3+\frac{4}{5} a b^3 x^5+\frac{b^4 x^7}{7} \]

[Out]

-(a^4/x) + 4*a^3*b*x + 2*a^2*b^2*x^3 + (4*a*b^3*x^5)/5 + (b^4*x^7)/7

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Rubi [A]  time = 0.0639931, antiderivative size = 48, 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 \[ -\frac{a^4}{x}+4 a^3 b x+2 a^2 b^2 x^3+\frac{4}{5} a b^3 x^5+\frac{b^4 x^7}{7} \]

Antiderivative was successfully verified.

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

[Out]

-(a^4/x) + 4*a^3*b*x + 2*a^2*b^2*x^3 + (4*a*b^3*x^5)/5 + (b^4*x^7)/7

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Rubi in Sympy [A]  time = 17.4721, size = 44, normalized size = 0.92 \[ - \frac{a^{4}}{x} + 4 a^{3} b x + 2 a^{2} b^{2} x^{3} + \frac{4 a b^{3} x^{5}}{5} + \frac{b^{4} x^{7}}{7} \]

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**2,x)

[Out]

-a**4/x + 4*a**3*b*x + 2*a**2*b**2*x**3 + 4*a*b**3*x**5/5 + b**4*x**7/7

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Mathematica [A]  time = 0.0130249, size = 48, normalized size = 1. \[ -\frac{a^4}{x}+4 a^3 b x+2 a^2 b^2 x^3+\frac{4}{5} a b^3 x^5+\frac{b^4 x^7}{7} \]

Antiderivative was successfully verified.

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

[Out]

-(a^4/x) + 4*a^3*b*x + 2*a^2*b^2*x^3 + (4*a*b^3*x^5)/5 + (b^4*x^7)/7

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Maple [A]  time = 0.005, size = 45, normalized size = 0.9 \[ -{\frac{{a}^{4}}{x}}+4\,{a}^{3}bx+2\,{a}^{2}{b}^{2}{x}^{3}+{\frac{4\,a{b}^{3}{x}^{5}}{5}}+{\frac{{b}^{4}{x}^{7}}{7}} \]

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^2,x)

[Out]

-a^4/x+4*a^3*b*x+2*a^2*b^2*x^3+4/5*a*b^3*x^5+1/7*b^4*x^7

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Maxima [A]  time = 0.694275, size = 59, normalized size = 1.23 \[ \frac{1}{7} \, b^{4} x^{7} + \frac{4}{5} \, a b^{3} x^{5} + 2 \, a^{2} b^{2} x^{3} + 4 \, a^{3} b x - \frac{a^{4}}{x} \]

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^2,x, algorithm="maxima")

[Out]

1/7*b^4*x^7 + 4/5*a*b^3*x^5 + 2*a^2*b^2*x^3 + 4*a^3*b*x - a^4/x

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Fricas [A]  time = 0.258635, size = 65, normalized size = 1.35 \[ \frac{5 \, b^{4} x^{8} + 28 \, a b^{3} x^{6} + 70 \, a^{2} b^{2} x^{4} + 140 \, a^{3} b x^{2} - 35 \, a^{4}}{35 \, x} \]

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^2,x, algorithm="fricas")

[Out]

1/35*(5*b^4*x^8 + 28*a*b^3*x^6 + 70*a^2*b^2*x^4 + 140*a^3*b*x^2 - 35*a^4)/x

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Sympy [A]  time = 1.05458, size = 44, normalized size = 0.92 \[ - \frac{a^{4}}{x} + 4 a^{3} b x + 2 a^{2} b^{2} x^{3} + \frac{4 a b^{3} x^{5}}{5} + \frac{b^{4} x^{7}}{7} \]

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**2,x)

[Out]

-a**4/x + 4*a**3*b*x + 2*a**2*b**2*x**3 + 4*a*b**3*x**5/5 + b**4*x**7/7

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GIAC/XCAS [A]  time = 0.267815, size = 59, normalized size = 1.23 \[ \frac{1}{7} \, b^{4} x^{7} + \frac{4}{5} \, a b^{3} x^{5} + 2 \, a^{2} b^{2} x^{3} + 4 \, a^{3} b x - \frac{a^{4}}{x} \]

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^2,x, algorithm="giac")

[Out]

1/7*b^4*x^7 + 4/5*a*b^3*x^5 + 2*a^2*b^2*x^3 + 4*a^3*b*x - a^4/x