3.1458 \(\int \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx\)

Optimal. Leaf size=14 \[ \frac{(a+b x)^5}{5 b} \]

[Out]

(a + b*x)^5/(5*b)

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Rubi [A]  time = 0.0100311, antiderivative size = 14, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{(a+b x)^5}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x)^5/(5*b)

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Rubi in Sympy [A]  time = 1.89059, size = 29, normalized size = 2.07 \[ \frac{\left (2 a + 2 b x\right ) \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{2}}{10 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(2*a + 2*b*x)*(a**2 + 2*a*b*x + b**2*x**2)**2/(10*b)

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Mathematica [A]  time = 0.00269074, size = 14, normalized size = 1. \[ \frac{(a+b x)^5}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x)^5/(5*b)

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Maple [B]  time = 0.002, size = 43, normalized size = 3.1 \[{\frac{{x}^{5}{b}^{4}}{5}}+{x}^{4}a{b}^{3}+2\,{x}^{3}{a}^{2}{b}^{2}+2\,{a}^{3}b{x}^{2}+{a}^{4}x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b^2*x^2+2*a*b*x+a^2)^2,x)

[Out]

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

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Maxima [A]  time = 0.678413, size = 72, normalized size = 5.14 \[ \frac{1}{5} \, b^{4} x^{5} + a b^{3} x^{4} + \frac{4}{3} \, a^{2} b^{2} x^{3} + a^{4} x + \frac{2}{3} \,{\left (b^{2} x^{3} + 3 \, a b x^{2}\right )} a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.178448, size = 1, normalized size = 0.07 \[ \frac{1}{5} x^{5} b^{4} + x^{4} b^{3} a + 2 x^{3} b^{2} a^{2} + 2 x^{2} b a^{3} + x a^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 0.099241, size = 42, normalized size = 3. \[ a^{4} x + 2 a^{3} b x^{2} + 2 a^{2} b^{2} x^{3} + a b^{3} x^{4} + \frac{b^{4} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.210097, size = 57, normalized size = 4.07 \[ \frac{1}{5} \, b^{4} x^{5} + a b^{3} x^{4} + 2 \, a^{2} b^{2} x^{3} + 2 \, a^{3} b x^{2} + a^{4} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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