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

Optimal. Leaf size=20 \[ a x+\frac{b x^3}{3}+\frac{c x^5}{5} \]

[Out]

a*x + (b*x^3)/3 + (c*x^5)/5

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Rubi [A]  time = 0.0034928, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 0, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ a x+\frac{b x^3}{3}+\frac{c x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[a + b*x^2 + c*x^4,x]

[Out]

a*x + (b*x^3)/3 + (c*x^5)/5

Rubi steps

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

Mathematica [A]  time = 0.0000322, size = 20, normalized size = 1. \[ a x+\frac{b x^3}{3}+\frac{c x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[a + b*x^2 + c*x^4,x]

[Out]

a*x + (b*x^3)/3 + (c*x^5)/5

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Maple [A]  time = 0.04, size = 17, normalized size = 0.9 \begin{align*} ax+{\frac{b{x}^{3}}{3}}+{\frac{c{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.970358, size = 22, normalized size = 1.1 \begin{align*} \frac{1}{5} \, c x^{5} + \frac{1}{3} \, b x^{3} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*x^5 + 1/3*b*x^3 + a*x

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Fricas [A]  time = 1.24892, size = 39, normalized size = 1.95 \begin{align*} \frac{1}{5} x^{5} c + \frac{1}{3} x^{3} b + x a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*c + 1/3*x^3*b + x*a

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Sympy [A]  time = 0.058773, size = 15, normalized size = 0.75 \begin{align*} a x + \frac{b x^{3}}{3} + \frac{c x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + b*x**3/3 + c*x**5/5

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Giac [A]  time = 1.20711, size = 22, normalized size = 1.1 \begin{align*} \frac{1}{5} \, c x^{5} + \frac{1}{3} \, b x^{3} + a x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*c*x^5 + 1/3*b*x^3 + a*x