3.480 \(\int (-1+5 x) (-1-x+x^2)^2 \, dx\)

Optimal. Leaf size=39 \[ \frac {5 x^6}{6}-\frac {11 x^5}{5}-\frac {3 x^4}{4}+\frac {11 x^3}{3}+\frac {3 x^2}{2}-x \]

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Rubi [A]  time = 0.01, antiderivative size = 39, normalized size of antiderivative = 1.00, 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} \[ \frac {5 x^6}{6}-\frac {11 x^5}{5}-\frac {3 x^4}{4}+\frac {11 x^3}{3}+\frac {3 x^2}{2}-x \]

Antiderivative was successfully verified.

[In]

Int[(-1 + 5*x)*(-1 - x + x^2)^2,x]

[Out]

-x + (3*x^2)/2 + (11*x^3)/3 - (3*x^4)/4 - (11*x^5)/5 + (5*x^6)/6

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+5 x) \left (-1-x+x^2\right )^2 \, dx &=\int \left (-1+3 x+11 x^2-3 x^3-11 x^4+5 x^5\right ) \, dx\\ &=-x+\frac {3 x^2}{2}+\frac {11 x^3}{3}-\frac {3 x^4}{4}-\frac {11 x^5}{5}+\frac {5 x^6}{6}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 39, normalized size = 1.00 \[ \frac {5 x^6}{6}-\frac {11 x^5}{5}-\frac {3 x^4}{4}+\frac {11 x^3}{3}+\frac {3 x^2}{2}-x \]

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 5*x)*(-1 - x + x^2)^2,x]

[Out]

-x + (3*x^2)/2 + (11*x^3)/3 - (3*x^4)/4 - (11*x^5)/5 + (5*x^6)/6

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IntegrateAlgebraic [A]  time = 0.01, size = 30, normalized size = 0.77 \[ \frac {1}{60} x \left (50 x^5-132 x^4-45 x^3+220 x^2+90 x-60\right ) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-1 + 5*x)*(-1 - x + x^2)^2,x]

[Out]

(x*(-60 + 90*x + 220*x^2 - 45*x^3 - 132*x^4 + 50*x^5))/60

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fricas [A]  time = 0.99, size = 29, normalized size = 0.74 \[ \frac {5}{6} x^{6} - \frac {11}{5} x^{5} - \frac {3}{4} x^{4} + \frac {11}{3} x^{3} + \frac {3}{2} x^{2} - x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+5*x)*(x^2-x-1)^2,x, algorithm="fricas")

[Out]

5/6*x^6 - 11/5*x^5 - 3/4*x^4 + 11/3*x^3 + 3/2*x^2 - x

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giac [A]  time = 0.58, size = 29, normalized size = 0.74 \[ \frac {5}{6} \, x^{6} - \frac {11}{5} \, x^{5} - \frac {3}{4} \, x^{4} + \frac {11}{3} \, x^{3} + \frac {3}{2} \, x^{2} - x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+5*x)*(x^2-x-1)^2,x, algorithm="giac")

[Out]

5/6*x^6 - 11/5*x^5 - 3/4*x^4 + 11/3*x^3 + 3/2*x^2 - x

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maple [A]  time = 0.36, size = 30, normalized size = 0.77

method	result	size
gosper	\(-x +\frac {3}{2} x^{2}+\frac {11}{3} x^{3}-\frac {3}{4} x^{4}-\frac {11}{5} x^{5}+\frac {5}{6} x^{6}\)	\(30\)
default	\(-x +\frac {3}{2} x^{2}+\frac {11}{3} x^{3}-\frac {3}{4} x^{4}-\frac {11}{5} x^{5}+\frac {5}{6} x^{6}\)	\(30\)
norman	\(-x +\frac {3}{2} x^{2}+\frac {11}{3} x^{3}-\frac {3}{4} x^{4}-\frac {11}{5} x^{5}+\frac {5}{6} x^{6}\)	\(30\)
risch	\(-x +\frac {3}{2} x^{2}+\frac {11}{3} x^{3}-\frac {3}{4} x^{4}-\frac {11}{5} x^{5}+\frac {5}{6} x^{6}\)	\(30\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-1+5*x)*(x^2-x-1)^2,x,method=_RETURNVERBOSE)

[Out]

-x+3/2*x^2+11/3*x^3-3/4*x^4-11/5*x^5+5/6*x^6

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maxima [A]  time = 0.46, size = 29, normalized size = 0.74 \[ \frac {5}{6} \, x^{6} - \frac {11}{5} \, x^{5} - \frac {3}{4} \, x^{4} + \frac {11}{3} \, x^{3} + \frac {3}{2} \, x^{2} - x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+5*x)*(x^2-x-1)^2,x, algorithm="maxima")

[Out]

5/6*x^6 - 11/5*x^5 - 3/4*x^4 + 11/3*x^3 + 3/2*x^2 - x

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mupad [B]  time = 0.03, size = 29, normalized size = 0.74 \[ \frac {5\,x^6}{6}-\frac {11\,x^5}{5}-\frac {3\,x^4}{4}+\frac {11\,x^3}{3}+\frac {3\,x^2}{2}-x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x - 1)*(x - x^2 + 1)^2,x)

[Out]

(3*x^2)/2 - x + (11*x^3)/3 - (3*x^4)/4 - (11*x^5)/5 + (5*x^6)/6

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sympy [A]  time = 0.06, size = 34, normalized size = 0.87 \[ \frac {5 x^{6}}{6} - \frac {11 x^{5}}{5} - \frac {3 x^{4}}{4} + \frac {11 x^{3}}{3} + \frac {3 x^{2}}{2} - x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-1+5*x)*(x**2-x-1)**2,x)

[Out]

5*x**6/6 - 11*x**5/5 - 3*x**4/4 + 11*x**3/3 + 3*x**2/2 - x

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