### 3.2111 $$\int (a+b x+c x^2) \, dx$$

Optimal. Leaf size=20 $a x+\frac{b x^2}{2}+\frac{c x^3}{3}$

[Out]

a*x + (b*x^2)/2 + (c*x^3)/3

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Rubi [A]  time = 0.0039018, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 10, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0., Rules used = {} $a x+\frac{b x^2}{2}+\frac{c x^3}{3}$

Antiderivative was successfully veriﬁed.

[In]

Int[a + b*x + c*x^2,x]

[Out]

a*x + (b*x^2)/2 + (c*x^3)/3

Rubi steps

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

Mathematica [A]  time = 0.0000391, size = 20, normalized size = 1. $a x+\frac{b x^2}{2}+\frac{c x^3}{3}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[a + b*x + c*x^2,x]

[Out]

a*x + (b*x^2)/2 + (c*x^3)/3

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Maple [A]  time = 0.039, size = 17, normalized size = 0.9 \begin{align*} ax+{\frac{b{x}^{2}}{2}}+{\frac{c{x}^{3}}{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(c*x^2+b*x+a,x)

[Out]

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

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Maxima [A]  time = 1.01306, size = 22, normalized size = 1.1 \begin{align*} \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(c*x^2+b*x+a,x, algorithm="maxima")

[Out]

1/3*c*x^3 + 1/2*b*x^2 + a*x

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Fricas [A]  time = 1.71683, size = 39, normalized size = 1.95 \begin{align*} \frac{1}{3} x^{3} c + \frac{1}{2} x^{2} b + x a \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(c*x^2+b*x+a,x, algorithm="fricas")

[Out]

1/3*x^3*c + 1/2*x^2*b + x*a

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Sympy [A]  time = 0.054928, size = 15, normalized size = 0.75 \begin{align*} a x + \frac{b x^{2}}{2} + \frac{c x^{3}}{3} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(c*x**2+b*x+a,x)

[Out]

a*x + b*x**2/2 + c*x**3/3

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Giac [A]  time = 1.13015, size = 22, normalized size = 1.1 \begin{align*} \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(c*x^2+b*x+a,x, algorithm="giac")

[Out]

1/3*c*x^3 + 1/2*b*x^2 + a*x