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

Optimal. Leaf size=44 \[ \frac {a x^7}{7}+\frac {a x^4}{2}+a x+\frac {b x^8}{8}+\frac {2 b x^5}{5}+\frac {b x^2}{2} \]

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Rubi [A]  time = 0.03, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {771} \begin {gather*} \frac {a x^7}{7}+\frac {a x^4}{2}+a x+\frac {b x^8}{8}+\frac {2 b x^5}{5}+\frac {b x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*x + (b*x^2)/2 + (a*x^4)/2 + (2*b*x^5)/5 + (a*x^7)/7 + (b*x^8)/8

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)^2 (a+b x) \left (1-x+x^2\right )^2 \, dx &=\int \left (a+b x+2 a x^3+2 b x^4+a x^6+b x^7\right ) \, dx\\ &=a x+\frac {b x^2}{2}+\frac {a x^4}{2}+\frac {2 b x^5}{5}+\frac {a x^7}{7}+\frac {b x^8}{8}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 44, normalized size = 1.00 \begin {gather*} \frac {a x^7}{7}+\frac {a x^4}{2}+a x+\frac {b x^8}{8}+\frac {2 b x^5}{5}+\frac {b x^2}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

a*x + (b*x^2)/2 + (a*x^4)/2 + (2*b*x^5)/5 + (a*x^7)/7 + (b*x^8)/8

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.35, size = 34, normalized size = 0.77 \begin {gather*} \frac {1}{8} x^{8} b + \frac {1}{7} x^{7} a + \frac {2}{5} x^{5} b + \frac {1}{2} 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)^2*(b*x+a)*(x^2-x+1)^2,x, algorithm="fricas")

[Out]

1/8*x^8*b + 1/7*x^7*a + 2/5*x^5*b + 1/2*x^4*a + 1/2*x^2*b + x*a

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giac [A]  time = 0.17, size = 34, normalized size = 0.77 \begin {gather*} \frac {1}{8} \, b x^{8} + \frac {1}{7} \, a x^{7} + \frac {2}{5} \, b x^{5} + \frac {1}{2} \, 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)^2*(b*x+a)*(x^2-x+1)^2,x, algorithm="giac")

[Out]

1/8*b*x^8 + 1/7*a*x^7 + 2/5*b*x^5 + 1/2*a*x^4 + 1/2*b*x^2 + a*x

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maple [A]  time = 0.04, size = 35, normalized size = 0.80 \begin {gather*} \frac {1}{8} b \,x^{8}+\frac {1}{7} a \,x^{7}+\frac {2}{5} b \,x^{5}+\frac {1}{2} 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)^2*(b*x+a)*(x^2-x+1)^2,x)

[Out]

a*x+1/2*b*x^2+1/2*a*x^4+2/5*b*x^5+1/7*a*x^7+1/8*b*x^8

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maxima [A]  time = 0.55, size = 34, normalized size = 0.77 \begin {gather*} \frac {1}{8} \, b x^{8} + \frac {1}{7} \, a x^{7} + \frac {2}{5} \, b x^{5} + \frac {1}{2} \, 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)^2*(b*x+a)*(x^2-x+1)^2,x, algorithm="maxima")

[Out]

1/8*b*x^8 + 1/7*a*x^7 + 2/5*b*x^5 + 1/2*a*x^4 + 1/2*b*x^2 + a*x

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mupad [B]  time = 0.02, size = 34, normalized size = 0.77 \begin {gather*} \frac {b\,x^8}{8}+\frac {a\,x^7}{7}+\frac {2\,b\,x^5}{5}+\frac {a\,x^4}{2}+\frac {b\,x^2}{2}+a\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x + (a*x^4)/2 + (a*x^7)/7 + (b*x^2)/2 + (2*b*x^5)/5 + (b*x^8)/8

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sympy [A]  time = 0.07, size = 37, normalized size = 0.84 \begin {gather*} \frac {a x^{7}}{7} + \frac {a x^{4}}{2} + a x + \frac {b x^{8}}{8} + \frac {2 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)**2*(b*x+a)*(x**2-x+1)**2,x)

[Out]

a*x**7/7 + a*x**4/2 + a*x + b*x**8/8 + 2*b*x**5/5 + b*x**2/2

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