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

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Rubi [A]  time = 0.01, antiderivative size = 32, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 32, normalized size = 1.00 \[ \frac {x^7}{7}-\frac {4 x^6}{3}+\frac {22 x^5}{5}-6 x^4+3 x^3 \]

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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fricas [A]  time = 0.73, size = 26, normalized size = 0.81 \[ \frac {1}{7} x^{7} - \frac {4}{3} x^{6} + \frac {22}{5} x^{5} - 6 x^{4} + 3 x^{3} \]

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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giac [A]  time = 0.18, size = 26, normalized size = 0.81 \[ \frac {1}{7} \, x^{7} - \frac {4}{3} \, x^{6} + \frac {22}{5} \, x^{5} - 6 \, x^{4} + 3 \, x^{3} \]

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

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maple [A]  time = 0.04, size = 27, normalized size = 0.84 \[ \frac {1}{7} x^{7}-\frac {4}{3} x^{6}+\frac {22}{5} x^{5}-6 x^{4}+3 x^{3} \]

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 = 1.03, size = 26, normalized size = 0.81 \[ \frac {1}{7} \, x^{7} - \frac {4}{3} \, x^{6} + \frac {22}{5} \, x^{5} - 6 \, x^{4} + 3 \, x^{3} \]

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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mupad [B]  time = 0.02, size = 26, normalized size = 0.81 \[ \frac {x^7}{7}-\frac {4\,x^6}{3}+\frac {22\,x^5}{5}-6\,x^4+3\,x^3 \]

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*x^5)/5 - (4*x^6)/3 + x^7/7

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sympy [A]  time = 0.07, size = 27, normalized size = 0.84 \[ \frac {x^{7}}{7} - \frac {4 x^{6}}{3} + \frac {22 x^{5}}{5} - 6 x^{4} + 3 x^{3} \]

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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