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

Optimal. Leaf size=25 \[ \frac{a x^3}{3}+\frac{b x^5}{5}+\frac{c x^7}{7} \]

[Out]

(a*x^3)/3 + (b*x^5)/5 + (c*x^7)/7

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Rubi [A]  time = 0.0070952, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {14} \[ \frac{a x^3}{3}+\frac{b x^5}{5}+\frac{c x^7}{7} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^2 + c*x^4),x]

[Out]

(a*x^3)/3 + (b*x^5)/5 + (c*x^7)/7

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int x^2 \left (a+b x^2+c x^4\right ) \, dx &=\int \left (a x^2+b x^4+c x^6\right ) \, dx\\ &=\frac{a x^3}{3}+\frac{b x^5}{5}+\frac{c x^7}{7}\\ \end{align*}

Mathematica [A]  time = 0.0022796, size = 25, normalized size = 1. \[ \frac{a x^3}{3}+\frac{b x^5}{5}+\frac{c x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^2 + c*x^4),x]

[Out]

(a*x^3)/3 + (b*x^5)/5 + (c*x^7)/7

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Maple [A]  time = 0.039, size = 20, normalized size = 0.8 \begin{align*}{\frac{a{x}^{3}}{3}}+{\frac{b{x}^{5}}{5}}+{\frac{c{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a*x^3+1/5*b*x^5+1/7*c*x^7

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Maxima [A]  time = 0.975144, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{7} \, c x^{7} + \frac{1}{5} \, b x^{5} + \frac{1}{3} \, a x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c*x^7 + 1/5*b*x^5 + 1/3*a*x^3

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Fricas [A]  time = 1.29524, size = 47, normalized size = 1.88 \begin{align*} \frac{1}{7} x^{7} c + \frac{1}{5} x^{5} b + \frac{1}{3} x^{3} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*c + 1/5*x^5*b + 1/3*x^3*a

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Sympy [A]  time = 0.062342, size = 19, normalized size = 0.76 \begin{align*} \frac{a x^{3}}{3} + \frac{b x^{5}}{5} + \frac{c x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**3/3 + b*x**5/5 + c*x**7/7

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Giac [A]  time = 1.12102, size = 26, normalized size = 1.04 \begin{align*} \frac{1}{7} \, c x^{7} + \frac{1}{5} \, b x^{5} + \frac{1}{3} \, a x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*c*x^7 + 1/5*b*x^5 + 1/3*a*x^3