3.88 \(\int (-1+x) (-1+2 x+3 x^2)^2 \, dx\)

Optimal. Leaf size=39 \[ \frac{3 x^6}{2}+\frac{3 x^5}{5}-\frac{7 x^4}{2}-\frac{2 x^3}{3}+\frac{5 x^2}{2}-x \]

[Out]

-x + (5*x^2)/2 - (2*x^3)/3 - (7*x^4)/2 + (3*x^5)/5 + (3*x^6)/2

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Rubi [A]  time = 0.0157301, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {631} \[ \frac{3 x^6}{2}+\frac{3 x^5}{5}-\frac{7 x^4}{2}-\frac{2 x^3}{3}+\frac{5 x^2}{2}-x \]

Antiderivative was successfully verified.

[In]

Int[(-1 + x)*(-1 + 2*x + 3*x^2)^2,x]

[Out]

-x + (5*x^2)/2 - (2*x^3)/3 - (7*x^4)/2 + (3*x^5)/5 + (3*x^6)/2

Rule 631

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

Rubi steps

\begin{align*} \int (-1+x) \left (-1+2 x+3 x^2\right )^2 \, dx &=\int \left (-1+5 x-2 x^2-14 x^3+3 x^4+9 x^5\right ) \, dx\\ &=-x+\frac{5 x^2}{2}-\frac{2 x^3}{3}-\frac{7 x^4}{2}+\frac{3 x^5}{5}+\frac{3 x^6}{2}\\ \end{align*}

Mathematica [A]  time = 0.0014211, size = 39, normalized size = 1. \[ \frac{3 x^6}{2}+\frac{3 x^5}{5}-\frac{7 x^4}{2}-\frac{2 x^3}{3}+\frac{5 x^2}{2}-x \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + x)*(-1 + 2*x + 3*x^2)^2,x]

[Out]

-x + (5*x^2)/2 - (2*x^3)/3 - (7*x^4)/2 + (3*x^5)/5 + (3*x^6)/2

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Maple [A]  time = 0., size = 30, normalized size = 0.8 \begin{align*} -x+{\frac{5\,{x}^{2}}{2}}-{\frac{2\,{x}^{3}}{3}}-{\frac{7\,{x}^{4}}{2}}+{\frac{3\,{x}^{5}}{5}}+{\frac{3\,{x}^{6}}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+x)*(3*x^2+2*x-1)^2,x)

[Out]

-x+5/2*x^2-2/3*x^3-7/2*x^4+3/5*x^5+3/2*x^6

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Maxima [A]  time = 0.931304, size = 39, normalized size = 1. \begin{align*} \frac{3}{2} \, x^{6} + \frac{3}{5} \, x^{5} - \frac{7}{2} \, x^{4} - \frac{2}{3} \, x^{3} + \frac{5}{2} \, x^{2} - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)*(3*x^2+2*x-1)^2,x, algorithm="maxima")

[Out]

3/2*x^6 + 3/5*x^5 - 7/2*x^4 - 2/3*x^3 + 5/2*x^2 - x

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Fricas [A]  time = 1.59602, size = 72, normalized size = 1.85 \begin{align*} \frac{3}{2} x^{6} + \frac{3}{5} x^{5} - \frac{7}{2} x^{4} - \frac{2}{3} x^{3} + \frac{5}{2} x^{2} - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)*(3*x^2+2*x-1)^2,x, algorithm="fricas")

[Out]

3/2*x^6 + 3/5*x^5 - 7/2*x^4 - 2/3*x^3 + 5/2*x^2 - x

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Sympy [A]  time = 0.058886, size = 34, normalized size = 0.87 \begin{align*} \frac{3 x^{6}}{2} + \frac{3 x^{5}}{5} - \frac{7 x^{4}}{2} - \frac{2 x^{3}}{3} + \frac{5 x^{2}}{2} - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)*(3*x**2+2*x-1)**2,x)

[Out]

3*x**6/2 + 3*x**5/5 - 7*x**4/2 - 2*x**3/3 + 5*x**2/2 - x

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Giac [A]  time = 1.06343, size = 39, normalized size = 1. \begin{align*} \frac{3}{2} \, x^{6} + \frac{3}{5} \, x^{5} - \frac{7}{2} \, x^{4} - \frac{2}{3} \, x^{3} + \frac{5}{2} \, x^{2} - x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+x)*(3*x^2+2*x-1)^2,x, algorithm="giac")

[Out]

3/2*x^6 + 3/5*x^5 - 7/2*x^4 - 2/3*x^3 + 5/2*x^2 - x