### 3.2164 $$\int x^3 (3-4 x+x^2)^2 \, dx$$

Optimal. Leaf size=36 $\frac{x^8}{8}-\frac{8 x^7}{7}+\frac{11 x^6}{3}-\frac{24 x^5}{5}+\frac{9 x^4}{4}$

[Out]

(9*x^4)/4 - (24*x^5)/5 + (11*x^6)/3 - (8*x^7)/7 + x^8/8

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Rubi [A]  time = 0.0125975, antiderivative size = 36, 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} $\frac{x^8}{8}-\frac{8 x^7}{7}+\frac{11 x^6}{3}-\frac{24 x^5}{5}+\frac{9 x^4}{4}$

Antiderivative was successfully veriﬁed.

[In]

Int[x^3*(3 - 4*x + x^2)^2,x]

[Out]

(9*x^4)/4 - (24*x^5)/5 + (11*x^6)/3 - (8*x^7)/7 + x^8/8

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^3 \left (3-4 x+x^2\right )^2 \, dx &=\int \left (9 x^3-24 x^4+22 x^5-8 x^6+x^7\right ) \, dx\\ &=\frac{9 x^4}{4}-\frac{24 x^5}{5}+\frac{11 x^6}{3}-\frac{8 x^7}{7}+\frac{x^8}{8}\\ \end{align*}

Mathematica [A]  time = 0.0006064, size = 36, normalized size = 1. $\frac{x^8}{8}-\frac{8 x^7}{7}+\frac{11 x^6}{3}-\frac{24 x^5}{5}+\frac{9 x^4}{4}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[x^3*(3 - 4*x + x^2)^2,x]

[Out]

(9*x^4)/4 - (24*x^5)/5 + (11*x^6)/3 - (8*x^7)/7 + x^8/8

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Maple [A]  time = 0.039, size = 27, normalized size = 0.8 \begin{align*}{\frac{9\,{x}^{4}}{4}}-{\frac{24\,{x}^{5}}{5}}+{\frac{11\,{x}^{6}}{3}}-{\frac{8\,{x}^{7}}{7}}+{\frac{{x}^{8}}{8}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

9/4*x^4-24/5*x^5+11/3*x^6-8/7*x^7+1/8*x^8

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Maxima [A]  time = 0.99327, size = 35, normalized size = 0.97 \begin{align*} \frac{1}{8} \, x^{8} - \frac{8}{7} \, x^{7} + \frac{11}{3} \, x^{6} - \frac{24}{5} \, x^{5} + \frac{9}{4} \, x^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8 - 8/7*x^7 + 11/3*x^6 - 24/5*x^5 + 9/4*x^4

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Fricas [A]  time = 1.48796, size = 69, normalized size = 1.92 \begin{align*} \frac{1}{8} x^{8} - \frac{8}{7} x^{7} + \frac{11}{3} x^{6} - \frac{24}{5} x^{5} + \frac{9}{4} x^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8 - 8/7*x^7 + 11/3*x^6 - 24/5*x^5 + 9/4*x^4

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Sympy [A]  time = 0.059099, size = 31, normalized size = 0.86 \begin{align*} \frac{x^{8}}{8} - \frac{8 x^{7}}{7} + \frac{11 x^{6}}{3} - \frac{24 x^{5}}{5} + \frac{9 x^{4}}{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**8/8 - 8*x**7/7 + 11*x**6/3 - 24*x**5/5 + 9*x**4/4

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Giac [A]  time = 1.12255, size = 35, normalized size = 0.97 \begin{align*} \frac{1}{8} \, x^{8} - \frac{8}{7} \, x^{7} + \frac{11}{3} \, x^{6} - \frac{24}{5} \, x^{5} + \frac{9}{4} \, x^{4} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*x^8 - 8/7*x^7 + 11/3*x^6 - 24/5*x^5 + 9/4*x^4