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

Optimal. Leaf size=30 \[ \frac{a^2 x^3}{3}+\frac{2}{5} a b x^5+\frac{b^2 x^7}{7} \]

[Out]

(a^2*x^3)/3 + (2*a*b*x^5)/5 + (b^2*x^7)/7

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Rubi [A]  time = 0.0090823, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {14} \[ \frac{a^2 x^3}{3}+\frac{2}{5} a b x^5+\frac{b^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Int[x^2*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

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

Mathematica [A]  time = 0.000928, size = 30, normalized size = 1. \[ \frac{a^2 x^3}{3}+\frac{2}{5} a b x^5+\frac{b^2 x^7}{7} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

(a^2*x^3)/3 + (2*a*b*x^5)/5 + (b^2*x^7)/7

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Maple [A]  time = 0.04, size = 25, normalized size = 0.8 \begin{align*}{\frac{{a}^{2}{x}^{3}}{3}}+{\frac{2\,ab{x}^{5}}{5}}+{\frac{{b}^{2}{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*a^2*x^3+2/5*a*b*x^5+1/7*b^2*x^7

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Maxima [A]  time = 0.985624, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{7} \, b^{2} x^{7} + \frac{2}{5} \, a b x^{5} + \frac{1}{3} \, a^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^2*x^7 + 2/5*a*b*x^5 + 1/3*a^2*x^3

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Fricas [A]  time = 1.2728, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{7} x^{7} b^{2} + \frac{2}{5} x^{5} b a + \frac{1}{3} x^{3} a^{2} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*b^2 + 2/5*x^5*b*a + 1/3*x^3*a^2

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Sympy [A]  time = 0.060454, size = 26, normalized size = 0.87 \begin{align*} \frac{a^{2} x^{3}}{3} + \frac{2 a b x^{5}}{5} + \frac{b^{2} x^{7}}{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**3/3 + 2*a*b*x**5/5 + b**2*x**7/7

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Giac [A]  time = 1.11288, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{7} \, b^{2} x^{7} + \frac{2}{5} \, a b x^{5} + \frac{1}{3} \, a^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*b^2*x^7 + 2/5*a*b*x^5 + 1/3*a^2*x^3