3.2165 \(\int x^2 (3-4 x+x^2)^2 \, dx\)

Optimal. Leaf size=32 \[ \frac{x^7}{7}-\frac{4 x^6}{3}+\frac{22 x^5}{5}-6 x^4+3 x^3 \]

[Out]

3*x^3 - 6*x^4 + (22*x^5)/5 - (4*x^6)/3 + x^7/7

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Rubi [A]  time = 0.0128501, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {698} \[ \frac{x^7}{7}-\frac{4 x^6}{3}+\frac{22 x^5}{5}-6 x^4+3 x^3 \]

Antiderivative was successfully verified.

[In]

Int[x^2*(3 - 4*x + x^2)^2,x]

[Out]

3*x^3 - 6*x^4 + (22*x^5)/5 - (4*x^6)/3 + x^7/7

Rule 698

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d +
e*x)^m*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*
e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

\begin{align*} \int x^2 \left (3-4 x+x^2\right )^2 \, dx &=\int \left (9 x^2-24 x^3+22 x^4-8 x^5+x^6\right ) \, dx\\ &=3 x^3-6 x^4+\frac{22 x^5}{5}-\frac{4 x^6}{3}+\frac{x^7}{7}\\ \end{align*}

Mathematica [A]  time = 0.000748, size = 32, normalized size = 1. \[ \frac{x^7}{7}-\frac{4 x^6}{3}+\frac{22 x^5}{5}-6 x^4+3 x^3 \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*(3 - 4*x + x^2)^2,x]

[Out]

3*x^3 - 6*x^4 + (22*x^5)/5 - (4*x^6)/3 + x^7/7

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Maple [A]  time = 0.04, size = 27, normalized size = 0.8 \begin{align*} 3\,{x}^{3}-6\,{x}^{4}+{\frac{22\,{x}^{5}}{5}}-{\frac{4\,{x}^{6}}{3}}+{\frac{{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x^2-4*x+3)^2,x)

[Out]

3*x^3-6*x^4+22/5*x^5-4/3*x^6+1/7*x^7

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Maxima [A]  time = 1.02264, size = 35, normalized size = 1.09 \begin{align*} \frac{1}{7} \, x^{7} - \frac{4}{3} \, x^{6} + \frac{22}{5} \, x^{5} - 6 \, x^{4} + 3 \, x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^2-4*x+3)^2,x, algorithm="maxima")

[Out]

1/7*x^7 - 4/3*x^6 + 22/5*x^5 - 6*x^4 + 3*x^3

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Fricas [A]  time = 1.44467, size = 62, normalized size = 1.94 \begin{align*} \frac{1}{7} x^{7} - \frac{4}{3} x^{6} + \frac{22}{5} x^{5} - 6 x^{4} + 3 x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^2-4*x+3)^2,x, algorithm="fricas")

[Out]

1/7*x^7 - 4/3*x^6 + 22/5*x^5 - 6*x^4 + 3*x^3

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Sympy [A]  time = 0.060042, size = 27, normalized size = 0.84 \begin{align*} \frac{x^{7}}{7} - \frac{4 x^{6}}{3} + \frac{22 x^{5}}{5} - 6 x^{4} + 3 x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*(x**2-4*x+3)**2,x)

[Out]

x**7/7 - 4*x**6/3 + 22*x**5/5 - 6*x**4 + 3*x**3

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Giac [A]  time = 1.10361, size = 35, normalized size = 1.09 \begin{align*} \frac{1}{7} \, x^{7} - \frac{4}{3} \, x^{6} + \frac{22}{5} \, x^{5} - 6 \, x^{4} + 3 \, x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^2-4*x+3)^2,x, algorithm="giac")

[Out]

1/7*x^7 - 4/3*x^6 + 22/5*x^5 - 6*x^4 + 3*x^3