3.3 \(\int (a x^2+b x^3+c x^4) \, dx\)

Optimal. Leaf size=25 \[ \frac{a x^3}{3}+\frac{b x^4}{4}+\frac{c x^5}{5} \]

[Out]

(a*x^3)/3 + (b*x^4)/4 + (c*x^5)/5

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Rubi [A]  time = 0.00384, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \frac{a x^3}{3}+\frac{b x^4}{4}+\frac{c x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[a*x^2 + b*x^3 + c*x^4,x]

[Out]

(a*x^3)/3 + (b*x^4)/4 + (c*x^5)/5

Rubi steps

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

Mathematica [A]  time = 0.0000454, size = 25, normalized size = 1. \[ \frac{a x^3}{3}+\frac{b x^4}{4}+\frac{c x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[a*x^2 + b*x^3 + c*x^4,x]

[Out]

(a*x^3)/3 + (b*x^4)/4 + (c*x^5)/5

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Maple [A]  time = 0., size = 20, normalized size = 0.8 \begin{align*}{\frac{a{x}^{3}}{3}}+{\frac{b{x}^{4}}{4}}+{\frac{c{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(c*x^4+b*x^3+a*x^2,x)

[Out]

1/3*a*x^3+1/4*b*x^4+1/5*c*x^5

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Maxima [A]  time = 1.0763, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{5} \, c x^{5} + \frac{1}{4} \, b x^{4} + \frac{1}{3} \, a x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*x^5 + 1/4*b*x^4 + 1/3*a*x^3

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Fricas [A]  time = 1.36483, size = 47, normalized size = 1.88 \begin{align*} \frac{1}{5} x^{5} c + \frac{1}{4} x^{4} b + \frac{1}{3} x^{3} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*c + 1/4*x^4*b + 1/3*x^3*a

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Sympy [A]  time = 0.057533, size = 19, normalized size = 0.76 \begin{align*} \frac{a x^{3}}{3} + \frac{b x^{4}}{4} + \frac{c x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(c*x**4+b*x**3+a*x**2,x)

[Out]

a*x**3/3 + b*x**4/4 + c*x**5/5

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Giac [A]  time = 1.08135, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{5} \, c x^{5} + \frac{1}{4} \, b x^{4} + \frac{1}{3} \, a x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*x^5 + 1/4*b*x^4 + 1/3*a*x^3