3.2163 \(\int x^4 (3-4 x+x^2)^2 \, dx\)

Optimal. Leaf size=32 \[ \frac{x^9}{9}-x^8+\frac{22 x^7}{7}-4 x^6+\frac{9 x^5}{5} \]

[Out]

(9*x^5)/5 - 4*x^6 + (22*x^7)/7 - x^8 + x^9/9

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Rubi [A]  time = 0.0137426, 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} \[ \frac{x^9}{9}-x^8+\frac{22 x^7}{7}-4 x^6+\frac{9 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(3 - 4*x + x^2)^2,x]

[Out]

(9*x^5)/5 - 4*x^6 + (22*x^7)/7 - x^8 + x^9/9

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

Mathematica [A]  time = 0.0009931, size = 32, normalized size = 1. \[ \frac{x^9}{9}-x^8+\frac{22 x^7}{7}-4 x^6+\frac{9 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(3 - 4*x + x^2)^2,x]

[Out]

(9*x^5)/5 - 4*x^6 + (22*x^7)/7 - x^8 + x^9/9

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Maple [A]  time = 0.038, size = 27, normalized size = 0.8 \begin{align*}{\frac{9\,{x}^{5}}{5}}-4\,{x}^{6}+{\frac{22\,{x}^{7}}{7}}-{x}^{8}+{\frac{{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/5*x^5-4*x^6+22/7*x^7-x^8+1/9*x^9

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Maxima [A]  time = 0.971315, size = 35, normalized size = 1.09 \begin{align*} \frac{1}{9} \, x^{9} - x^{8} + \frac{22}{7} \, x^{7} - 4 \, x^{6} + \frac{9}{5} \, x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9 - x^8 + 22/7*x^7 - 4*x^6 + 9/5*x^5

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Fricas [A]  time = 1.4604, size = 59, normalized size = 1.84 \begin{align*} \frac{1}{9} x^{9} - x^{8} + \frac{22}{7} x^{7} - 4 x^{6} + \frac{9}{5} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9 - x^8 + 22/7*x^7 - 4*x^6 + 9/5*x^5

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Sympy [A]  time = 0.062411, size = 26, normalized size = 0.81 \begin{align*} \frac{x^{9}}{9} - x^{8} + \frac{22 x^{7}}{7} - 4 x^{6} + \frac{9 x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**9/9 - x**8 + 22*x**7/7 - 4*x**6 + 9*x**5/5

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Giac [A]  time = 1.10492, size = 35, normalized size = 1.09 \begin{align*} \frac{1}{9} \, x^{9} - x^{8} + \frac{22}{7} \, x^{7} - 4 \, x^{6} + \frac{9}{5} \, x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*x^9 - x^8 + 22/7*x^7 - 4*x^6 + 9/5*x^5