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

Optimal. Leaf size=9 \[ \frac {1}{12} (x+1)^{12} \]

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Rubi [A]  time = 0.00, antiderivative size = 9, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {27, 32} \begin {gather*} \frac {1}{12} (x+1)^{12} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

(1 + x)^12/12

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {align*} \int (1+x) \left (1+2 x+x^2\right )^5 \, dx &=\int (1+x)^{11} \, dx\\ &=\frac {1}{12} (1+x)^{12}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 9, normalized size = 1.00 \begin {gather*} \frac {1}{12} (x+1)^{12} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(1 + x)^12/12

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int (1+x) \left (1+2 x+x^2\right )^5 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.36, size = 55, normalized size = 6.11 \begin {gather*} \frac {1}{12} x^{12} + x^{11} + \frac {11}{2} x^{10} + \frac {55}{3} x^{9} + \frac {165}{4} x^{8} + 66 x^{7} + 77 x^{6} + 66 x^{5} + \frac {165}{4} x^{4} + \frac {55}{3} x^{3} + \frac {11}{2} x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*x^12 + x^11 + 11/2*x^10 + 55/3*x^9 + 165/4*x^8 + 66*x^7 + 77*x^6 + 66*x^5 + 165/4*x^4 + 55/3*x^3 + 11/2*x
^2 + x

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giac [B]  time = 0.15, size = 62, normalized size = 6.89 \begin {gather*} \frac {1}{12} \, {\left (x^{2} + 2 \, x\right )}^{6} + \frac {1}{2} \, {\left (x^{2} + 2 \, x\right )}^{5} + \frac {5}{4} \, {\left (x^{2} + 2 \, x\right )}^{4} + \frac {5}{3} \, {\left (x^{2} + 2 \, x\right )}^{3} + \frac {5}{4} \, {\left (x^{2} + 2 \, x\right )}^{2} + \frac {1}{2} \, x^{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(x^2 + 2*x)^6 + 1/2*(x^2 + 2*x)^5 + 5/4*(x^2 + 2*x)^4 + 5/3*(x^2 + 2*x)^3 + 5/4*(x^2 + 2*x)^2 + 1/2*x^2 +
x

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maple [B]  time = 0.04, size = 56, normalized size = 6.22 \begin {gather*} \frac {1}{12} x^{12}+x^{11}+\frac {11}{2} x^{10}+\frac {55}{3} x^{9}+\frac {165}{4} x^{8}+66 x^{7}+77 x^{6}+66 x^{5}+\frac {165}{4} x^{4}+\frac {55}{3} x^{3}+\frac {11}{2} x^{2}+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x+1)*(x^2+2*x+1)^5,x)

[Out]

1/12*x^12+x^11+11/2*x^10+55/3*x^9+165/4*x^8+66*x^7+77*x^6+66*x^5+165/4*x^4+55/3*x^3+11/2*x^2+x

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maxima [A]  time = 0.67, size = 12, normalized size = 1.33 \begin {gather*} \frac {1}{12} \, {\left (x^{2} + 2 \, x + 1\right )}^{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(x^2 + 2*x + 1)^6

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mupad [B]  time = 0.06, size = 55, normalized size = 6.11 \begin {gather*} \frac {x^{12}}{12}+x^{11}+\frac {11\,x^{10}}{2}+\frac {55\,x^9}{3}+\frac {165\,x^8}{4}+66\,x^7+77\,x^6+66\,x^5+\frac {165\,x^4}{4}+\frac {55\,x^3}{3}+\frac {11\,x^2}{2}+x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x + 1)*(2*x + x^2 + 1)^5,x)

[Out]

x + (11*x^2)/2 + (55*x^3)/3 + (165*x^4)/4 + 66*x^5 + 77*x^6 + 66*x^7 + (165*x^8)/4 + (55*x^9)/3 + (11*x^10)/2
+ x^11 + x^12/12

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sympy [B]  time = 0.08, size = 65, normalized size = 7.22 \begin {gather*} \frac {x^{12}}{12} + x^{11} + \frac {11 x^{10}}{2} + \frac {55 x^{9}}{3} + \frac {165 x^{8}}{4} + 66 x^{7} + 77 x^{6} + 66 x^{5} + \frac {165 x^{4}}{4} + \frac {55 x^{3}}{3} + \frac {11 x^{2}}{2} + x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**12/12 + x**11 + 11*x**10/2 + 55*x**9/3 + 165*x**8/4 + 66*x**7 + 77*x**6 + 66*x**5 + 165*x**4/4 + 55*x**3/3
+ 11*x**2/2 + x

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