3.424 \(\int x^3 \left (a^2+2 a b x^2+b^2 x^4\right )^2 \, dx\)

Optimal. Leaf size=34 \[ \frac{\left (a+b x^2\right )^6}{12 b^2}-\frac{a \left (a+b x^2\right )^5}{10 b^2} \]

[Out]

-(a*(a + b*x^2)^5)/(10*b^2) + (a + b*x^2)^6/(12*b^2)

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Rubi [A]  time = 0.1025, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\left (a+b x^2\right )^6}{12 b^2}-\frac{a \left (a+b x^2\right )^5}{10 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

-(a*(a + b*x^2)^5)/(10*b^2) + (a + b*x^2)^6/(12*b^2)

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Rubi in Sympy [A]  time = 16.5193, size = 27, normalized size = 0.79 \[ - \frac{a \left (a + b x^{2}\right )^{5}}{10 b^{2}} + \frac{\left (a + b x^{2}\right )^{6}}{12 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

-a*(a + b*x**2)**5/(10*b**2) + (a + b*x**2)**6/(12*b**2)

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Mathematica [A]  time = 0.00362381, size = 56, normalized size = 1.65 \[ \frac{a^4 x^4}{4}+\frac{2}{3} a^3 b x^6+\frac{3}{4} a^2 b^2 x^8+\frac{2}{5} a b^3 x^{10}+\frac{b^4 x^{12}}{12} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a^2 + 2*a*b*x^2 + b^2*x^4)^2,x]

[Out]

(a^4*x^4)/4 + (2*a^3*b*x^6)/3 + (3*a^2*b^2*x^8)/4 + (2*a*b^3*x^10)/5 + (b^4*x^12
)/12

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Maple [A]  time = 0.001, size = 47, normalized size = 1.4 \[{\frac{{b}^{4}{x}^{12}}{12}}+{\frac{2\,a{b}^{3}{x}^{10}}{5}}+{\frac{3\,{a}^{2}{b}^{2}{x}^{8}}{4}}+{\frac{2\,{a}^{3}b{x}^{6}}{3}}+{\frac{{a}^{4}{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b^2*x^4+2*a*b*x^2+a^2)^2,x)

[Out]

1/12*b^4*x^12+2/5*a*b^3*x^10+3/4*a^2*b^2*x^8+2/3*a^3*b*x^6+1/4*a^4*x^4

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Maxima [A]  time = 0.68829, size = 62, normalized size = 1.82 \[ \frac{1}{12} \, b^{4} x^{12} + \frac{2}{5} \, a b^{3} x^{10} + \frac{3}{4} \, a^{2} b^{2} x^{8} + \frac{2}{3} \, a^{3} b x^{6} + \frac{1}{4} \, a^{4} x^{4} \]

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^3,x, algorithm="maxima")

[Out]

1/12*b^4*x^12 + 2/5*a*b^3*x^10 + 3/4*a^2*b^2*x^8 + 2/3*a^3*b*x^6 + 1/4*a^4*x^4

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Fricas [A]  time = 0.242047, size = 1, normalized size = 0.03 \[ \frac{1}{12} x^{12} b^{4} + \frac{2}{5} x^{10} b^{3} a + \frac{3}{4} x^{8} b^{2} a^{2} + \frac{2}{3} x^{6} b a^{3} + \frac{1}{4} x^{4} a^{4} \]

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^3,x, algorithm="fricas")

[Out]

1/12*x^12*b^4 + 2/5*x^10*b^3*a + 3/4*x^8*b^2*a^2 + 2/3*x^6*b*a^3 + 1/4*x^4*a^4

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Sympy [A]  time = 0.10781, size = 53, normalized size = 1.56 \[ \frac{a^{4} x^{4}}{4} + \frac{2 a^{3} b x^{6}}{3} + \frac{3 a^{2} b^{2} x^{8}}{4} + \frac{2 a b^{3} x^{10}}{5} + \frac{b^{4} x^{12}}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b**2*x**4+2*a*b*x**2+a**2)**2,x)

[Out]

a**4*x**4/4 + 2*a**3*b*x**6/3 + 3*a**2*b**2*x**8/4 + 2*a*b**3*x**10/5 + b**4*x**
12/12

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GIAC/XCAS [A]  time = 0.267913, size = 62, normalized size = 1.82 \[ \frac{1}{12} \, b^{4} x^{12} + \frac{2}{5} \, a b^{3} x^{10} + \frac{3}{4} \, a^{2} b^{2} x^{8} + \frac{2}{3} \, a^{3} b x^{6} + \frac{1}{4} \, a^{4} x^{4} \]

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^3,x, algorithm="giac")

[Out]

1/12*b^4*x^12 + 2/5*a*b^3*x^10 + 3/4*a^2*b^2*x^8 + 2/3*a^3*b*x^6 + 1/4*a^4*x^4