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

Optimal. Leaf size=17 \[ \frac{a x^3}{3}+\frac{b x^5}{5} \]

[Out]

(a*x^3)/3 + (b*x^5)/5

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Rubi [A]  time = 0.0048836, antiderivative size = 17, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {14} \[ \frac{a x^3}{3}+\frac{b x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a + b*x^2),x]

[Out]

(a*x^3)/3 + (b*x^5)/5

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

Mathematica [A]  time = 0.000975, size = 17, normalized size = 1. \[ \frac{a x^3}{3}+\frac{b x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a + b*x^2),x]

[Out]

(a*x^3)/3 + (b*x^5)/5

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Maple [A]  time = 0.001, size = 14, normalized size = 0.8 \begin{align*}{\frac{a{x}^{3}}{3}}+{\frac{b{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.21825, size = 18, normalized size = 1.06 \begin{align*} \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*(b*x^2+a),x, algorithm="maxima")

[Out]

1/5*b*x^5 + 1/3*a*x^3

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Fricas [A]  time = 1.3667, size = 31, normalized size = 1.82 \begin{align*} \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*(b*x^2+a),x, algorithm="fricas")

[Out]

1/5*x^5*b + 1/3*x^3*a

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Sympy [A]  time = 0.055765, size = 12, normalized size = 0.71 \begin{align*} \frac{a x^{3}}{3} + \frac{b x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x**3/3 + b*x**5/5

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Giac [A]  time = 2.38683, size = 18, normalized size = 1.06 \begin{align*} \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*(b*x^2+a),x, algorithm="giac")

[Out]

1/5*b*x^5 + 1/3*a*x^3