3.21.61 \(\int (1+x) (a+b x) (1-x+x^2) \, dx\)

Optimal. Leaf size=28 \[ \frac {a x^4}{4}+a x+\frac {b x^5}{5}+\frac {b x^2}{2} \]

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Rubi [A]  time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {771} \begin {gather*} \frac {a x^4}{4}+a x+\frac {b x^5}{5}+\frac {b x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)*(a + b*x)*(1 - x + x^2),x]

[Out]

a*x + (b*x^2)/2 + (a*x^4)/4 + (b*x^5)/5

Rule 771

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> In
t[ExpandIntegrand[(d + e*x)^m*(f + g*x)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && N
eQ[b^2 - 4*a*c, 0] && IntegerQ[p] && (GtQ[p, 0] || (EqQ[a, 0] && IntegerQ[m]))

Rubi steps

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

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Mathematica [A]  time = 0.00, size = 28, normalized size = 1.00 \begin {gather*} \frac {a x^4}{4}+a x+\frac {b x^5}{5}+\frac {b x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*x + (b*x^2)/2 + (a*x^4)/4 + (b*x^5)/5

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (1+x) (a+b x) \left (1-x+x^2\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.34, size = 22, normalized size = 0.79 \begin {gather*} \frac {1}{5} x^{5} b + \frac {1}{4} x^{4} a + \frac {1}{2} x^{2} b + x a \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*x^5*b + 1/4*x^4*a + 1/2*x^2*b + x*a

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giac [A]  time = 0.17, size = 22, normalized size = 0.79 \begin {gather*} \frac {1}{5} \, b x^{5} + \frac {1}{4} \, a x^{4} + \frac {1}{2} \, b x^{2} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*x^5 + 1/4*a*x^4 + 1/2*b*x^2 + a*x

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maple [A]  time = 0.04, size = 23, normalized size = 0.82 \begin {gather*} \frac {1}{5} b \,x^{5}+\frac {1}{4} a \,x^{4}+\frac {1}{2} b \,x^{2}+a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)*(b*x+a)*(x^2-x+1),x)

[Out]

a*x+1/2*b*x^2+1/4*a*x^4+1/5*b*x^5

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maxima [A]  time = 0.52, size = 22, normalized size = 0.79 \begin {gather*} \frac {1}{5} \, b x^{5} + \frac {1}{4} \, a x^{4} + \frac {1}{2} \, b x^{2} + a x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*b*x^5 + 1/4*a*x^4 + 1/2*b*x^2 + a*x

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mupad [B]  time = 0.04, size = 22, normalized size = 0.79 \begin {gather*} \frac {b\,x^5}{5}+\frac {a\,x^4}{4}+\frac {b\,x^2}{2}+a\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)*(a + b*x)*(x^2 - x + 1),x)

[Out]

a*x + (a*x^4)/4 + (b*x^2)/2 + (b*x^5)/5

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sympy [A]  time = 0.06, size = 22, normalized size = 0.79 \begin {gather*} \frac {a x^{4}}{4} + a x + \frac {b x^{5}}{5} + \frac {b x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(b*x+a)*(x**2-x+1),x)

[Out]

a*x**4/4 + a*x + b*x**5/5 + b*x**2/2

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