### 3.1455 $$\int (a^2+2 a b x+b^2 x^2) \, dx$$

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 = 0.004962, 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., Rules used = {} $a^2 x+a b x^2+\frac{b^2 x^3}{3}$

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

Rubi steps

\begin{align*} \int \left (a^2+2 a b x+b^2 x^2\right ) \, dx &=a^2 x+a b x^2+\frac{b^2 x^3}{3}\\ \end{align*}

Mathematica [A]  time = 0.0000511, 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 = 0.038, size = 21, normalized size = 1. \begin{align*}{a}^{2}x+ab{x}^{2}+{\frac{{b}^{2}{x}^{3}}{3}} \end{align*}

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 = 1.17979, size = 27, normalized size = 1.23 \begin{align*} \frac{1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \end{align*}

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 = 1.64184, size = 42, normalized size = 1.91 \begin{align*} \frac{1}{3} x^{3} b^{2} + x^{2} b a + x a^{2} \end{align*}

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 = 0.064795, size = 19, normalized size = 0.86 \begin{align*} a^{2} x + a b x^{2} + \frac{b^{2} x^{3}}{3} \end{align*}

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 [A]  time = 1.11468, size = 27, normalized size = 1.23 \begin{align*} \frac{1}{3} \, b^{2} x^{3} + a b x^{2} + a^{2} x \end{align*}

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