3.1444 \(\int \left (a^2+2 a b x+b^2 x^2\right ) \, dx\)

Optimal. Leaf size=22 \[ a^2 x+a b x^2+\frac{b^2 x^3}{3} \]

[Out]

a^2*x + a*b*x^2 + (b^2*x^3)/3

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Rubi [A]  time = 0.0169338, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0. \[ a^2 x+a b x^2+\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[a^2 + 2*a*b*x + b^2*x^2,x]

[Out]

a^2*x + a*b*x^2 + (b^2*x^3)/3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 a b \int x\, dx + \frac{b^{2} x^{3}}{3} + \int a^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a*b*Integral(x, x) + b**2*x**3/3 + Integral(a**2, x)

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Mathematica [A]  time = 0.0000867154, size = 22, normalized size = 1. \[ a^2 x+a b x^2+\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

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

[Out]

a^2*x + a*b*x^2 + (b^2*x^3)/3

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Maple [A]  time = 0.001, size = 21, normalized size = 1. \[{a}^{2}x+ab{x}^{2}+{\frac{{b}^{2}{x}^{3}}{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(b^2*x^2+2*a*b*x+a^2,x)

[Out]

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

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Maxima [A]  time = 0.682515, size = 27, normalized size = 1.23 \[ \frac{1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*b^2*x^3 + a*b*x^2 + a^2*x

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Fricas [A]  time = 0.177512, size = 1, normalized size = 0.05 \[ \frac{1}{3} x^{3} b^{2} + x^{2} b a + x a^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*x^3*b^2 + x^2*b*a + x*a^2

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Sympy [A]  time = 0.076543, size = 19, normalized size = 0.86 \[ a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x + a*b*x**2 + b**2*x**3/3

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GIAC/XCAS [A]  time = 0.209654, size = 27, normalized size = 1.23 \[ \frac{1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*b^2*x^3 + a*b*x^2 + a^2*x