3.2100 \(\int \left (a+b x+c x^2\right ) \, dx\)

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 = 0.0130694, 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. \[ a x+\frac{b x^2}{2}+\frac{c x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[a + b*x + c*x^2,x]

[Out]

a*x + (b*x^2)/2 + (c*x^3)/3

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(c*x**2+b*x+a,x)

[Out]

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

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Mathematica [A]  time = 0.0000812757, 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 = 0.001, size = 17, normalized size = 0.9 \[ ax+{\frac{b{x}^{2}}{2}}+{\frac{c{x}^{3}}{3}} \]

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 = 0.817472, size = 22, normalized size = 1.1 \[ \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x \]

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 = 0.180951, size = 1, normalized size = 0.05 \[ \frac{1}{3} x^{3} c + \frac{1}{2} x^{2} b + x a \]

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 = 0.06902, size = 15, normalized size = 0.75 \[ a x + \frac{b x^{2}}{2} + \frac{c x^{3}}{3} \]

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/XCAS [A]  time = 0.201113, size = 22, normalized size = 1.1 \[ \frac{1}{3} \, c x^{3} + \frac{1}{2} \, b x^{2} + a x \]

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