3.219 \(\int (-4+4 x+x^2) (5-12 x+6 x^2+x^3) \, dx\)

Optimal. Leaf size=19 \[ \frac {1}{6} \left (x^3+6 x^2-12 x+5\right )^2 \]

[Out]

1/6*(x^3+6*x^2-12*x+5)^2

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Rubi [A]  time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {1588} \[ \frac {1}{6} \left (x^3+6 x^2-12 x+5\right )^2 \]

Antiderivative was successfully verified.

[In]

Int[(-4 + 4*x + x^2)*(5 - 12*x + 6*x^2 + x^3),x]

[Out]

(5 - 12*x + 6*x^2 + x^3)^2/6

Rule 1588

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[(Coeff[Pp, x, p]*x^(p - q
+ 1)*Qq^(m + 1))/((p + m*q + 1)*Coeff[Qq, x, q]), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int \left (-4+4 x+x^2\right ) \left (5-12 x+6 x^2+x^3\right ) \, dx &=\frac {1}{6} \left (5-12 x+6 x^2+x^3\right )^2\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 33, normalized size = 1.74 \[ \frac {x^6}{6}+2 x^5+2 x^4-\frac {67 x^3}{3}+34 x^2-20 x \]

Antiderivative was successfully verified.

[In]

Integrate[(-4 + 4*x + x^2)*(5 - 12*x + 6*x^2 + x^3),x]

[Out]

-20*x + 34*x^2 - (67*x^3)/3 + 2*x^4 + 2*x^5 + x^6/6

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fricas [A]  time = 0.77, size = 29, normalized size = 1.53 \[ \frac {1}{6} x^{6} + 2 x^{5} + 2 x^{4} - \frac {67}{3} x^{3} + 34 x^{2} - 20 x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+4*x-4)*(x^3+6*x^2-12*x+5),x, algorithm="fricas")

[Out]

1/6*x^6 + 2*x^5 + 2*x^4 - 67/3*x^3 + 34*x^2 - 20*x

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giac [A]  time = 0.35, size = 30, normalized size = 1.58 \[ \frac {5}{3} \, x^{3} + \frac {1}{6} \, {\left (x^{3} + 6 \, x^{2} - 12 \, x\right )}^{2} + 10 \, x^{2} - 20 \, x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+4*x-4)*(x^3+6*x^2-12*x+5),x, algorithm="giac")

[Out]

5/3*x^3 + 1/6*(x^3 + 6*x^2 - 12*x)^2 + 10*x^2 - 20*x

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maple [A]  time = 0.00, size = 30, normalized size = 1.58 \[ \frac {1}{6} x^{6}+2 x^{5}+2 x^{4}-\frac {67}{3} x^{3}+34 x^{2}-20 x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+4*x-4)*(x^3+6*x^2-12*x+5),x)

[Out]

1/6*x^6+2*x^5+2*x^4-67/3*x^3+34*x^2-20*x

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maxima [A]  time = 0.43, size = 17, normalized size = 0.89 \[ \frac {1}{6} \, {\left (x^{3} + 6 \, x^{2} - 12 \, x + 5\right )}^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+4*x-4)*(x^3+6*x^2-12*x+5),x, algorithm="maxima")

[Out]

1/6*(x^3 + 6*x^2 - 12*x + 5)^2

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mupad [B]  time = 0.03, size = 29, normalized size = 1.53 \[ \frac {x^6}{6}+2\,x^5+2\,x^4-\frac {67\,x^3}{3}+34\,x^2-20\,x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x + x^2 - 4)*(6*x^2 - 12*x + x^3 + 5),x)

[Out]

34*x^2 - 20*x - (67*x^3)/3 + 2*x^4 + 2*x^5 + x^6/6

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sympy [A]  time = 0.06, size = 29, normalized size = 1.53 \[ \frac {x^{6}}{6} + 2 x^{5} + 2 x^{4} - \frac {67 x^{3}}{3} + 34 x^{2} - 20 x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+4*x-4)*(x**3+6*x**2-12*x+5),x)

[Out]

x**6/6 + 2*x**5 + 2*x**4 - 67*x**3/3 + 34*x**2 - 20*x

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