3.426 \(\int x \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx\)

Optimal. Leaf size=16 \[ \frac{\left (a+b x^2\right )^5}{10 b} \]

[Out]

(a + b*x^2)^5/(10*b)

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Rubi [A]  time = 0.017018, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\left (a+b x^2\right )^5}{10 b} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x^2)^5/(10*b)

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Rubi in Sympy [A]  time = 6.66098, size = 10, normalized size = 0.62 \[ \frac{\left (a + b x^{2}\right )^{5}}{10 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x**2)**5/(10*b)

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Mathematica [A]  time = 0.0042337, size = 16, normalized size = 1. \[ \frac{\left (a+b x^2\right )^5}{10 b} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x^2)^5/(10*b)

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Maple [B]  time = 0.001, size = 45, normalized size = 2.8 \[{\frac{{b}^{4}{x}^{10}}{10}}+{\frac{a{b}^{3}{x}^{8}}{2}}+{a}^{2}{b}^{2}{x}^{6}+{a}^{3}b{x}^{4}+{\frac{{a}^{4}{x}^{2}}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*b^4*x^10+1/2*a*b^3*x^8+a^2*b^2*x^6+a^3*b*x^4+1/2*a^4*x^2

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Maxima [A]  time = 0.702543, size = 59, normalized size = 3.69 \[ \frac{1}{10} \, b^{4} x^{10} + \frac{1}{2} \, a b^{3} x^{8} + a^{2} b^{2} x^{6} + a^{3} b x^{4} + \frac{1}{2} \, a^{4} x^{2} \]

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,x, algorithm="maxima")

[Out]

1/10*b^4*x^10 + 1/2*a*b^3*x^8 + a^2*b^2*x^6 + a^3*b*x^4 + 1/2*a^4*x^2

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Fricas [A]  time = 0.241912, size = 1, normalized size = 0.06 \[ \frac{1}{10} x^{10} b^{4} + \frac{1}{2} x^{8} b^{3} a + x^{6} b^{2} a^{2} + x^{4} b a^{3} + \frac{1}{2} x^{2} a^{4} \]

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,x, algorithm="fricas")

[Out]

1/10*x^10*b^4 + 1/2*x^8*b^3*a + x^6*b^2*a^2 + x^4*b*a^3 + 1/2*x^2*a^4

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Sympy [A]  time = 0.108162, size = 44, normalized size = 2.75 \[ \frac{a^{4} x^{2}}{2} + a^{3} b x^{4} + a^{2} b^{2} x^{6} + \frac{a b^{3} x^{8}}{2} + \frac{b^{4} x^{10}}{10} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**4*x**2/2 + a**3*b*x**4 + a**2*b**2*x**6 + a*b**3*x**8/2 + b**4*x**10/10

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GIAC/XCAS [A]  time = 0.267687, size = 59, normalized size = 3.69 \[ \frac{1}{10} \, b^{4} x^{10} + \frac{1}{2} \, a b^{3} x^{8} + a^{2} b^{2} x^{6} + a^{3} b x^{4} + \frac{1}{2} \, a^{4} x^{2} \]

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,x, algorithm="giac")

[Out]

1/10*b^4*x^10 + 1/2*a*b^3*x^8 + a^2*b^2*x^6 + a^3*b*x^4 + 1/2*a^4*x^2