3.18.80 \(\int (a+b x+c x^2) \, dx\)

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

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Rubi [A]  time = 0.00, antiderivative size = 20, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 0, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} a x+\frac {b x^2}{2}+\frac {c x^3}{3} \end {gather*}

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*}

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Mathematica [A]  time = 0.00, size = 20, normalized size = 1.00 \begin {gather*} a x+\frac {b x^2}{2}+\frac {c x^3}{3} \end {gather*}

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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a+b x+c x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[a + b*x + c*x^2,x]

[Out]

IntegrateAlgebraic[a + b*x + c*x^2, x]

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fricas [A]  time = 0.33, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{3} x^{3} c + \frac {1}{2} x^{2} b + x a \end {gather*}

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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giac [A]  time = 0.15, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x \end {gather*}

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

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maple [A]  time = 0.04, size = 17, normalized size = 0.85 \begin {gather*} \frac {1}{3} c \,x^{3}+\frac {1}{2} b \,x^{2}+a x \end {gather*}

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 = 1.08, size = 16, normalized size = 0.80 \begin {gather*} \frac {1}{3} \, c x^{3} + \frac {1}{2} \, b x^{2} + a x \end {gather*}

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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mupad [B]  time = 0.03, size = 16, normalized size = 0.80 \begin {gather*} \frac {c\,x^3}{3}+\frac {b\,x^2}{2}+a\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(a + b*x + c*x^2,x)

[Out]

a*x + (b*x^2)/2 + (c*x^3)/3

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sympy [A]  time = 0.06, size = 15, normalized size = 0.75 \begin {gather*} a x + \frac {b x^{2}}{2} + \frac {c x^{3}}{3} \end {gather*}

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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