3.1891 \(\int (a+b x+c x^2+d x^3) \, dx\)

Optimal. Leaf size=28 \[ a x+\frac{b x^2}{2}+\frac{c x^3}{3}+\frac{d x^4}{4} \]

[Out]

a*x + (b*x^2)/2 + (c*x^3)/3 + (d*x^4)/4

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Rubi [A]  time = 0.0046075, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ a x+\frac{b x^2}{2}+\frac{c x^3}{3}+\frac{d x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[a + b*x + c*x^2 + d*x^3,x]

[Out]

a*x + (b*x^2)/2 + (c*x^3)/3 + (d*x^4)/4

Rubi steps

\begin{align*} \int \left (a+b x+c x^2+d x^3\right ) \, dx &=a x+\frac{b x^2}{2}+\frac{c x^3}{3}+\frac{d x^4}{4}\\ \end{align*}

Mathematica [A]  time = 0.0000379, size = 28, normalized size = 1. \[ a x+\frac{b x^2}{2}+\frac{c x^3}{3}+\frac{d x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*x + c*x^2 + d*x^3,x]

[Out]

a*x + (b*x^2)/2 + (c*x^3)/3 + (d*x^4)/4

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Maple [A]  time = 0., size = 23, normalized size = 0.8 \begin{align*} ax+{\frac{b{x}^{2}}{2}}+{\frac{c{x}^{3}}{3}}+{\frac{d{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(d*x^3+c*x^2+b*x+a,x)

[Out]

a*x+1/2*b*x^2+1/3*c*x^3+1/4*d*x^4

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Maxima [A]  time = 0.990462, size = 30, normalized size = 1.07 \begin{align*} \frac{1}{4} \, d x^{4} + \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(d*x^3+c*x^2+b*x+a,x, algorithm="maxima")

[Out]

1/4*d*x^4 + 1/3*c*x^3 + 1/2*b*x^2 + a*x

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Fricas [A]  time = 1.66902, size = 55, normalized size = 1.96 \begin{align*} \frac{1}{4} x^{4} d + \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(d*x^3+c*x^2+b*x+a,x, algorithm="fricas")

[Out]

1/4*x^4*d + 1/3*x^3*c + 1/2*x^2*b + x*a

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Sympy [A]  time = 0.05424, size = 22, normalized size = 0.79 \begin{align*} a x + \frac{b x^{2}}{2} + \frac{c x^{3}}{3} + \frac{d x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(d*x**3+c*x**2+b*x+a,x)

[Out]

a*x + b*x**2/2 + c*x**3/3 + d*x**4/4

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Giac [A]  time = 1.05218, size = 30, normalized size = 1.07 \begin{align*} \frac{1}{4} \, d x^{4} + \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(d*x^3+c*x^2+b*x+a,x, algorithm="giac")

[Out]

1/4*d*x^4 + 1/3*c*x^3 + 1/2*b*x^2 + a*x