3.559 \(\int x^7 (d+e x) \left (1+2 x+x^2\right )^5 \, dx\)

Optimal. Leaf size=131 \[ \frac{1}{18} (x+1)^{18} (d-8 e)-\frac{7}{17} (x+1)^{17} (d-4 e)+\frac{7}{16} (x+1)^{16} (3 d-8 e)-\frac{7}{3} (x+1)^{15} (d-2 e)+\frac{1}{2} (x+1)^{14} (5 d-8 e)-\frac{7}{13} (x+1)^{13} (3 d-4 e)+\frac{1}{12} (x+1)^{12} (7 d-8 e)-\frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{19} e (x+1)^{19} \]

[Out]

-((d - e)*(1 + x)^11)/11 + ((7*d - 8*e)*(1 + x)^12)/12 - (7*(3*d - 4*e)*(1 + x)^
13)/13 + ((5*d - 8*e)*(1 + x)^14)/2 - (7*(d - 2*e)*(1 + x)^15)/3 + (7*(3*d - 8*e
)*(1 + x)^16)/16 - (7*(d - 4*e)*(1 + x)^17)/17 + ((d - 8*e)*(1 + x)^18)/18 + (e*
(1 + x)^19)/19

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Rubi [A]  time = 0.262791, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{1}{18} (x+1)^{18} (d-8 e)-\frac{7}{17} (x+1)^{17} (d-4 e)+\frac{7}{16} (x+1)^{16} (3 d-8 e)-\frac{7}{3} (x+1)^{15} (d-2 e)+\frac{1}{2} (x+1)^{14} (5 d-8 e)-\frac{7}{13} (x+1)^{13} (3 d-4 e)+\frac{1}{12} (x+1)^{12} (7 d-8 e)-\frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{19} e (x+1)^{19} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

-((d - e)*(1 + x)^11)/11 + ((7*d - 8*e)*(1 + x)^12)/12 - (7*(3*d - 4*e)*(1 + x)^
13)/13 + ((5*d - 8*e)*(1 + x)^14)/2 - (7*(d - 2*e)*(1 + x)^15)/3 + (7*(3*d - 8*e
)*(1 + x)^16)/16 - (7*(d - 4*e)*(1 + x)^17)/17 + ((d - 8*e)*(1 + x)^18)/18 + (e*
(1 + x)^19)/19

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Rubi in Sympy [A]  time = 31.888, size = 122, normalized size = 0.93 \[ \frac{e \left (x + 1\right )^{19}}{19} + \left (\frac{d}{18} - \frac{4 e}{9}\right ) \left (x + 1\right )^{18} - \left (\frac{d}{11} - \frac{e}{11}\right ) \left (x + 1\right )^{11} - \left (\frac{7 d}{17} - \frac{28 e}{17}\right ) \left (x + 1\right )^{17} + \left (\frac{7 d}{12} - \frac{2 e}{3}\right ) \left (x + 1\right )^{12} + \left (\frac{21 d}{16} - \frac{7 e}{2}\right ) \left (x + 1\right )^{16} - \left (\frac{21 d}{13} - \frac{28 e}{13}\right ) \left (x + 1\right )^{13} - \left (\frac{7 d}{3} - \frac{14 e}{3}\right ) \left (x + 1\right )^{15} + \left (\frac{5 d}{2} - 4 e\right ) \left (x + 1\right )^{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(e*x+d)*(x**2+2*x+1)**5,x)

[Out]

e*(x + 1)**19/19 + (d/18 - 4*e/9)*(x + 1)**18 - (d/11 - e/11)*(x + 1)**11 - (7*d
/17 - 28*e/17)*(x + 1)**17 + (7*d/12 - 2*e/3)*(x + 1)**12 + (21*d/16 - 7*e/2)*(x
+ 1)**16 - (21*d/13 - 28*e/13)*(x + 1)**13 - (7*d/3 - 14*e/3)*(x + 1)**15 + (5*
d/2 - 4*e)*(x + 1)**14

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Mathematica [A]  time = 0.0443384, size = 149, normalized size = 1.14 \[ \frac{1}{18} x^{18} (d+10 e)+\frac{5}{17} x^{17} (2 d+9 e)+\frac{15}{16} x^{16} (3 d+8 e)+2 x^{15} (4 d+7 e)+3 x^{14} (5 d+6 e)+\frac{42}{13} x^{13} (6 d+5 e)+\frac{5}{2} x^{12} (7 d+4 e)+\frac{15}{11} x^{11} (8 d+3 e)+\frac{1}{2} x^{10} (9 d+2 e)+\frac{1}{9} x^9 (10 d+e)+\frac{d x^8}{8}+\frac{e x^{19}}{19} \]

Antiderivative was successfully verified.

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

[Out]

(d*x^8)/8 + ((10*d + e)*x^9)/9 + ((9*d + 2*e)*x^10)/2 + (15*(8*d + 3*e)*x^11)/11
+ (5*(7*d + 4*e)*x^12)/2 + (42*(6*d + 5*e)*x^13)/13 + 3*(5*d + 6*e)*x^14 + 2*(4
*d + 7*e)*x^15 + (15*(3*d + 8*e)*x^16)/16 + (5*(2*d + 9*e)*x^17)/17 + ((d + 10*e
)*x^18)/18 + (e*x^19)/19

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Maple [A]  time = 0.002, size = 130, normalized size = 1. \[{\frac{e{x}^{19}}{19}}+{\frac{ \left ( d+10\,e \right ){x}^{18}}{18}}+{\frac{ \left ( 10\,d+45\,e \right ){x}^{17}}{17}}+{\frac{ \left ( 45\,d+120\,e \right ){x}^{16}}{16}}+{\frac{ \left ( 120\,d+210\,e \right ){x}^{15}}{15}}+{\frac{ \left ( 210\,d+252\,e \right ){x}^{14}}{14}}+{\frac{ \left ( 252\,d+210\,e \right ){x}^{13}}{13}}+{\frac{ \left ( 210\,d+120\,e \right ){x}^{12}}{12}}+{\frac{ \left ( 120\,d+45\,e \right ){x}^{11}}{11}}+{\frac{ \left ( 45\,d+10\,e \right ){x}^{10}}{10}}+{\frac{ \left ( 10\,d+e \right ){x}^{9}}{9}}+{\frac{d{x}^{8}}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(e*x+d)*(x^2+2*x+1)^5,x)

[Out]

1/19*e*x^19+1/18*(d+10*e)*x^18+1/17*(10*d+45*e)*x^17+1/16*(45*d+120*e)*x^16+1/15
*(120*d+210*e)*x^15+1/14*(210*d+252*e)*x^14+1/13*(252*d+210*e)*x^13+1/12*(210*d+
120*e)*x^12+1/11*(120*d+45*e)*x^11+1/10*(45*d+10*e)*x^10+1/9*(10*d+e)*x^9+1/8*d*
x^8

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Maxima [A]  time = 0.682497, size = 174, normalized size = 1.33 \[ \frac{1}{19} \, e x^{19} + \frac{1}{18} \,{\left (d + 10 \, e\right )} x^{18} + \frac{5}{17} \,{\left (2 \, d + 9 \, e\right )} x^{17} + \frac{15}{16} \,{\left (3 \, d + 8 \, e\right )} x^{16} + 2 \,{\left (4 \, d + 7 \, e\right )} x^{15} + 3 \,{\left (5 \, d + 6 \, e\right )} x^{14} + \frac{42}{13} \,{\left (6 \, d + 5 \, e\right )} x^{13} + \frac{5}{2} \,{\left (7 \, d + 4 \, e\right )} x^{12} + \frac{15}{11} \,{\left (8 \, d + 3 \, e\right )} x^{11} + \frac{1}{2} \,{\left (9 \, d + 2 \, e\right )} x^{10} + \frac{1}{9} \,{\left (10 \, d + e\right )} x^{9} + \frac{1}{8} \, d x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 + 2*x + 1)^5*x^7,x, algorithm="maxima")

[Out]

1/19*e*x^19 + 1/18*(d + 10*e)*x^18 + 5/17*(2*d + 9*e)*x^17 + 15/16*(3*d + 8*e)*x
^16 + 2*(4*d + 7*e)*x^15 + 3*(5*d + 6*e)*x^14 + 42/13*(6*d + 5*e)*x^13 + 5/2*(7*
d + 4*e)*x^12 + 15/11*(8*d + 3*e)*x^11 + 1/2*(9*d + 2*e)*x^10 + 1/9*(10*d + e)*x
^9 + 1/8*d*x^8

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Fricas [A]  time = 0.271313, size = 1, normalized size = 0.01 \[ \frac{1}{19} x^{19} e + \frac{5}{9} x^{18} e + \frac{1}{18} x^{18} d + \frac{45}{17} x^{17} e + \frac{10}{17} x^{17} d + \frac{15}{2} x^{16} e + \frac{45}{16} x^{16} d + 14 x^{15} e + 8 x^{15} d + 18 x^{14} e + 15 x^{14} d + \frac{210}{13} x^{13} e + \frac{252}{13} x^{13} d + 10 x^{12} e + \frac{35}{2} x^{12} d + \frac{45}{11} x^{11} e + \frac{120}{11} x^{11} d + x^{10} e + \frac{9}{2} x^{10} d + \frac{1}{9} x^{9} e + \frac{10}{9} x^{9} d + \frac{1}{8} x^{8} d \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 + 2*x + 1)^5*x^7,x, algorithm="fricas")

[Out]

1/19*x^19*e + 5/9*x^18*e + 1/18*x^18*d + 45/17*x^17*e + 10/17*x^17*d + 15/2*x^16
*e + 45/16*x^16*d + 14*x^15*e + 8*x^15*d + 18*x^14*e + 15*x^14*d + 210/13*x^13*e
+ 252/13*x^13*d + 10*x^12*e + 35/2*x^12*d + 45/11*x^11*e + 120/11*x^11*d + x^10
*e + 9/2*x^10*d + 1/9*x^9*e + 10/9*x^9*d + 1/8*x^8*d

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Sympy [A]  time = 0.182989, size = 133, normalized size = 1.02 \[ \frac{d x^{8}}{8} + \frac{e x^{19}}{19} + x^{18} \left (\frac{d}{18} + \frac{5 e}{9}\right ) + x^{17} \left (\frac{10 d}{17} + \frac{45 e}{17}\right ) + x^{16} \left (\frac{45 d}{16} + \frac{15 e}{2}\right ) + x^{15} \left (8 d + 14 e\right ) + x^{14} \left (15 d + 18 e\right ) + x^{13} \left (\frac{252 d}{13} + \frac{210 e}{13}\right ) + x^{12} \left (\frac{35 d}{2} + 10 e\right ) + x^{11} \left (\frac{120 d}{11} + \frac{45 e}{11}\right ) + x^{10} \left (\frac{9 d}{2} + e\right ) + x^{9} \left (\frac{10 d}{9} + \frac{e}{9}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(e*x+d)*(x**2+2*x+1)**5,x)

[Out]

d*x**8/8 + e*x**19/19 + x**18*(d/18 + 5*e/9) + x**17*(10*d/17 + 45*e/17) + x**16
*(45*d/16 + 15*e/2) + x**15*(8*d + 14*e) + x**14*(15*d + 18*e) + x**13*(252*d/13
+ 210*e/13) + x**12*(35*d/2 + 10*e) + x**11*(120*d/11 + 45*e/11) + x**10*(9*d/2
+ e) + x**9*(10*d/9 + e/9)

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GIAC/XCAS [A]  time = 0.268568, size = 193, normalized size = 1.47 \[ \frac{1}{19} \, x^{19} e + \frac{1}{18} \, d x^{18} + \frac{5}{9} \, x^{18} e + \frac{10}{17} \, d x^{17} + \frac{45}{17} \, x^{17} e + \frac{45}{16} \, d x^{16} + \frac{15}{2} \, x^{16} e + 8 \, d x^{15} + 14 \, x^{15} e + 15 \, d x^{14} + 18 \, x^{14} e + \frac{252}{13} \, d x^{13} + \frac{210}{13} \, x^{13} e + \frac{35}{2} \, d x^{12} + 10 \, x^{12} e + \frac{120}{11} \, d x^{11} + \frac{45}{11} \, x^{11} e + \frac{9}{2} \, d x^{10} + x^{10} e + \frac{10}{9} \, d x^{9} + \frac{1}{9} \, x^{9} e + \frac{1}{8} \, d x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 + 2*x + 1)^5*x^7,x, algorithm="giac")

[Out]

1/19*x^19*e + 1/18*d*x^18 + 5/9*x^18*e + 10/17*d*x^17 + 45/17*x^17*e + 45/16*d*x
^16 + 15/2*x^16*e + 8*d*x^15 + 14*x^15*e + 15*d*x^14 + 18*x^14*e + 252/13*d*x^13
+ 210/13*x^13*e + 35/2*d*x^12 + 10*x^12*e + 120/11*d*x^11 + 45/11*x^11*e + 9/2*
d*x^10 + x^10*e + 10/9*d*x^9 + 1/9*x^9*e + 1/8*d*x^8