3.6.92 \(\int x^7 (1+x) (1+2 x+x^2)^5 \, dx\) [592]

Optimal. Leaf size=73 \[ -\frac {1}{12} (1+x)^{12}+\frac {7}{13} (1+x)^{13}-\frac {3}{2} (1+x)^{14}+\frac {7}{3} (1+x)^{15}-\frac {35}{16} (1+x)^{16}+\frac {21}{17} (1+x)^{17}-\frac {7}{18} (1+x)^{18}+\frac {1}{19} (1+x)^{19} \]

[Out]

-1/12*(1+x)^12+7/13*(1+x)^13-3/2*(1+x)^14+7/3*(1+x)^15-35/16*(1+x)^16+21/17*(1+x)^17-7/18*(1+x)^18+1/19*(1+x)^
19

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Rubi [A]
time = 0.02, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {27, 45} \begin {gather*} \frac {1}{19} (x+1)^{19}-\frac {7}{18} (x+1)^{18}+\frac {21}{17} (x+1)^{17}-\frac {35}{16} (x+1)^{16}+\frac {7}{3} (x+1)^{15}-\frac {3}{2} (x+1)^{14}+\frac {7}{13} (x+1)^{13}-\frac {1}{12} (x+1)^{12} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

-1/12*(1 + x)^12 + (7*(1 + x)^13)/13 - (3*(1 + x)^14)/2 + (7*(1 + x)^15)/3 - (35*(1 + x)^16)/16 + (21*(1 + x)^
17)/17 - (7*(1 + x)^18)/18 + (1 + x)^19/19

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 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin {align*} \int x^7 (1+x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^7 (1+x)^{11} \, dx\\ &=\int \left (-(1+x)^{11}+7 (1+x)^{12}-21 (1+x)^{13}+35 (1+x)^{14}-35 (1+x)^{15}+21 (1+x)^{16}-7 (1+x)^{17}+(1+x)^{18}\right ) \, dx\\ &=-\frac {1}{12} (1+x)^{12}+\frac {7}{13} (1+x)^{13}-\frac {3}{2} (1+x)^{14}+\frac {7}{3} (1+x)^{15}-\frac {35}{16} (1+x)^{16}+\frac {21}{17} (1+x)^{17}-\frac {7}{18} (1+x)^{18}+\frac {1}{19} (1+x)^{19}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 79, normalized size = 1.08 \begin {gather*} \frac {x^8}{8}+\frac {11 x^9}{9}+\frac {11 x^{10}}{2}+15 x^{11}+\frac {55 x^{12}}{2}+\frac {462 x^{13}}{13}+33 x^{14}+22 x^{15}+\frac {165 x^{16}}{16}+\frac {55 x^{17}}{17}+\frac {11 x^{18}}{18}+\frac {x^{19}}{19} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7*(1 + x)*(1 + 2*x + x^2)^5,x]

[Out]

x^8/8 + (11*x^9)/9 + (11*x^10)/2 + 15*x^11 + (55*x^12)/2 + (462*x^13)/13 + 33*x^14 + 22*x^15 + (165*x^16)/16 +
(55*x^17)/17 + (11*x^18)/18 + x^19/19

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Maple [A]
time = 0.68, size = 62, normalized size = 0.85

method	result	size
gosper	\(\frac {x^{8} \left (31824 x^{11}+369512 x^{10}+1956240 x^{9}+6235515 x^{8}[phone] x^{7}[phone] x^{6}[phone] x^{5}[phone] x^{4}+9069840 x^{3}+3325608 x^{2}+739024 x +75582\right )}{604656}\)	\(61\)
default	\(\frac {1}{8} x^{8}+\frac {11}{9} x^{9}+\frac {11}{2} x^{10}+15 x^{11}+\frac {55}{2} x^{12}+\frac {462}{13} x^{13}+33 x^{14}+22 x^{15}+\frac {165}{16} x^{16}+\frac {55}{17} x^{17}+\frac {11}{18} x^{18}+\frac {1}{19} x^{19}\)	\(62\)
norman	\(\frac {1}{8} x^{8}+\frac {11}{9} x^{9}+\frac {11}{2} x^{10}+15 x^{11}+\frac {55}{2} x^{12}+\frac {462}{13} x^{13}+33 x^{14}+22 x^{15}+\frac {165}{16} x^{16}+\frac {55}{17} x^{17}+\frac {11}{18} x^{18}+\frac {1}{19} x^{19}\)	\(62\)
risch	\(\frac {1}{8} x^{8}+\frac {11}{9} x^{9}+\frac {11}{2} x^{10}+15 x^{11}+\frac {55}{2} x^{12}+\frac {462}{13} x^{13}+33 x^{14}+22 x^{15}+\frac {165}{16} x^{16}+\frac {55}{17} x^{17}+\frac {11}{18} x^{18}+\frac {1}{19} x^{19}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(x+1)*(x^2+2*x+1)^5,x,method=_RETURNVERBOSE)

[Out]

1/8*x^8+11/9*x^9+11/2*x^10+15*x^11+55/2*x^12+462/13*x^13+33*x^14+22*x^15+165/16*x^16+55/17*x^17+11/18*x^18+1/1
9*x^19

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Maxima [A]
time = 0.28, size = 61, normalized size = 0.84 \begin {gather*} \frac {1}{19} \, x^{19} + \frac {11}{18} \, x^{18} + \frac {55}{17} \, x^{17} + \frac {165}{16} \, x^{16} + 22 \, x^{15} + 33 \, x^{14} + \frac {462}{13} \, x^{13} + \frac {55}{2} \, x^{12} + 15 \, x^{11} + \frac {11}{2} \, x^{10} + \frac {11}{9} \, x^{9} + \frac {1}{8} \, x^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(1+x)*(x^2+2*x+1)^5,x, algorithm="maxima")

[Out]

1/19*x^19 + 11/18*x^18 + 55/17*x^17 + 165/16*x^16 + 22*x^15 + 33*x^14 + 462/13*x^13 + 55/2*x^12 + 15*x^11 + 11
/2*x^10 + 11/9*x^9 + 1/8*x^8

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Fricas [A]
time = 3.02, size = 61, normalized size = 0.84 \begin {gather*} \frac {1}{19} \, x^{19} + \frac {11}{18} \, x^{18} + \frac {55}{17} \, x^{17} + \frac {165}{16} \, x^{16} + 22 \, x^{15} + 33 \, x^{14} + \frac {462}{13} \, x^{13} + \frac {55}{2} \, x^{12} + 15 \, x^{11} + \frac {11}{2} \, x^{10} + \frac {11}{9} \, x^{9} + \frac {1}{8} \, x^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(1+x)*(x^2+2*x+1)^5,x, algorithm="fricas")

[Out]

1/19*x^19 + 11/18*x^18 + 55/17*x^17 + 165/16*x^16 + 22*x^15 + 33*x^14 + 462/13*x^13 + 55/2*x^12 + 15*x^11 + 11
/2*x^10 + 11/9*x^9 + 1/8*x^8

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Sympy [A]
time = 0.01, size = 71, normalized size = 0.97 \begin {gather*} \frac {x^{19}}{19} + \frac {11 x^{18}}{18} + \frac {55 x^{17}}{17} + \frac {165 x^{16}}{16} + 22 x^{15} + 33 x^{14} + \frac {462 x^{13}}{13} + \frac {55 x^{12}}{2} + 15 x^{11} + \frac {11 x^{10}}{2} + \frac {11 x^{9}}{9} + \frac {x^{8}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(1+x)*(x**2+2*x+1)**5,x)

[Out]

x**19/19 + 11*x**18/18 + 55*x**17/17 + 165*x**16/16 + 22*x**15 + 33*x**14 + 462*x**13/13 + 55*x**12/2 + 15*x**
11 + 11*x**10/2 + 11*x**9/9 + x**8/8

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Giac [A]
time = 1.06, size = 61, normalized size = 0.84 \begin {gather*} \frac {1}{19} \, x^{19} + \frac {11}{18} \, x^{18} + \frac {55}{17} \, x^{17} + \frac {165}{16} \, x^{16} + 22 \, x^{15} + 33 \, x^{14} + \frac {462}{13} \, x^{13} + \frac {55}{2} \, x^{12} + 15 \, x^{11} + \frac {11}{2} \, x^{10} + \frac {11}{9} \, x^{9} + \frac {1}{8} \, x^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(1+x)*(x^2+2*x+1)^5,x, algorithm="giac")

[Out]

1/19*x^19 + 11/18*x^18 + 55/17*x^17 + 165/16*x^16 + 22*x^15 + 33*x^14 + 462/13*x^13 + 55/2*x^12 + 15*x^11 + 11
/2*x^10 + 11/9*x^9 + 1/8*x^8

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Mupad [B]
time = 0.06, size = 61, normalized size = 0.84 \begin {gather*} \frac {x^{19}}{19}+\frac {11\,x^{18}}{18}+\frac {55\,x^{17}}{17}+\frac {165\,x^{16}}{16}+22\,x^{15}+33\,x^{14}+\frac {462\,x^{13}}{13}+\frac {55\,x^{12}}{2}+15\,x^{11}+\frac {11\,x^{10}}{2}+\frac {11\,x^9}{9}+\frac {x^8}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(x + 1)*(2*x + x^2 + 1)^5,x)

[Out]

x^8/8 + (11*x^9)/9 + (11*x^10)/2 + 15*x^11 + (55*x^12)/2 + (462*x^13)/13 + 33*x^14 + 22*x^15 + (165*x^16)/16 +
(55*x^17)/17 + (11*x^18)/18 + x^19/19

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