3.119 \(\int (a+8 x-8 x^2+4 x^3-x^4) \, dx\)

Optimal. Leaf size=26 \[ a x-\frac{x^5}{5}+x^4-\frac{8 x^3}{3}+4 x^2 \]

[Out]

a*x + 4*x^2 - (8*x^3)/3 + x^4 - x^5/5

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Rubi [A]  time = 0.00396, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ a x-\frac{x^5}{5}+x^4-\frac{8 x^3}{3}+4 x^2 \]

Antiderivative was successfully verified.

[In]

Int[a + 8*x - 8*x^2 + 4*x^3 - x^4,x]

[Out]

a*x + 4*x^2 - (8*x^3)/3 + x^4 - x^5/5

Rubi steps

\begin{align*} \int \left (a+8 x-8 x^2+4 x^3-x^4\right ) \, dx &=a x+4 x^2-\frac{8 x^3}{3}+x^4-\frac{x^5}{5}\\ \end{align*}

Mathematica [A]  time = 0.0000663, size = 26, normalized size = 1. \[ a x-\frac{x^5}{5}+x^4-\frac{8 x^3}{3}+4 x^2 \]

Antiderivative was successfully verified.

[In]

Integrate[a + 8*x - 8*x^2 + 4*x^3 - x^4,x]

[Out]

a*x + 4*x^2 - (8*x^3)/3 + x^4 - x^5/5

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Maple [A]  time = 0., size = 23, normalized size = 0.9 \begin{align*} ax+4\,{x}^{2}-{\frac{8\,{x}^{3}}{3}}+{x}^{4}-{\frac{{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-x^4+4*x^3-8*x^2+a+8*x,x)

[Out]

a*x+4*x^2-8/3*x^3+x^4-1/5*x^5

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Maxima [A]  time = 1.06695, size = 30, normalized size = 1.15 \begin{align*} -\frac{1}{5} \, x^{5} + x^{4} - \frac{8}{3} \, x^{3} + a x + 4 \, x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-x^4+4*x^3-8*x^2+a+8*x,x, algorithm="maxima")

[Out]

-1/5*x^5 + x^4 - 8/3*x^3 + a*x + 4*x^2

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Fricas [A]  time = 1.27597, size = 54, normalized size = 2.08 \begin{align*} -\frac{1}{5} x^{5} + x^{4} - \frac{8}{3} x^{3} + 4 x^{2} + x a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-x^4+4*x^3-8*x^2+a+8*x,x, algorithm="fricas")

[Out]

-1/5*x^5 + x^4 - 8/3*x^3 + 4*x^2 + x*a

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Sympy [A]  time = 0.056983, size = 22, normalized size = 0.85 \begin{align*} a x - \frac{x^{5}}{5} + x^{4} - \frac{8 x^{3}}{3} + 4 x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-x**4+4*x**3-8*x**2+a+8*x,x)

[Out]

a*x - x**5/5 + x**4 - 8*x**3/3 + 4*x**2

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Giac [A]  time = 1.10444, size = 30, normalized size = 1.15 \begin{align*} -\frac{1}{5} \, x^{5} + x^{4} - \frac{8}{3} \, x^{3} + a x + 4 \, x^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-x^4+4*x^3-8*x^2+a+8*x,x, algorithm="giac")

[Out]

-1/5*x^5 + x^4 - 8/3*x^3 + a*x + 4*x^2