3.560 \(\int x^6 (d+e x) (1+2 x+x^2)^5 \, dx\)

Optimal. Leaf size=119 \[ \frac{1}{17} (x+1)^{17} (d-7 e)-\frac{3}{16} (x+1)^{16} (2 d-7 e)+\frac{1}{3} (x+1)^{15} (3 d-7 e)-\frac{5}{14} (x+1)^{14} (4 d-7 e)+\frac{3}{13} (x+1)^{13} (5 d-7 e)-\frac{1}{12} (x+1)^{12} (6 d-7 e)+\frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{18} e (x+1)^{18} \]

[Out]

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

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Rubi [A]  time = 0.074182, antiderivative size = 119, 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, Rules used = {27, 76} \[ \frac{1}{17} (x+1)^{17} (d-7 e)-\frac{3}{16} (x+1)^{16} (2 d-7 e)+\frac{1}{3} (x+1)^{15} (3 d-7 e)-\frac{5}{14} (x+1)^{14} (4 d-7 e)+\frac{3}{13} (x+1)^{13} (5 d-7 e)-\frac{1}{12} (x+1)^{12} (6 d-7 e)+\frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{18} e (x+1)^{18} \]

Antiderivative was successfully verified.

[In]

Int[x^6*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

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

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 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int x^6 (d+e x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^6 (1+x)^{10} (d+e x) \, dx\\ &=\int \left ((d-e) (1+x)^{10}+(-6 d+7 e) (1+x)^{11}+3 (5 d-7 e) (1+x)^{12}-5 (4 d-7 e) (1+x)^{13}+5 (3 d-7 e) (1+x)^{14}-3 (2 d-7 e) (1+x)^{15}+(d-7 e) (1+x)^{16}+e (1+x)^{17}\right ) \, dx\\ &=\frac{1}{11} (d-e) (1+x)^{11}-\frac{1}{12} (6 d-7 e) (1+x)^{12}+\frac{3}{13} (5 d-7 e) (1+x)^{13}-\frac{5}{14} (4 d-7 e) (1+x)^{14}+\frac{1}{3} (3 d-7 e) (1+x)^{15}-\frac{3}{16} (2 d-7 e) (1+x)^{16}+\frac{1}{17} (d-7 e) (1+x)^{17}+\frac{1}{18} e (1+x)^{18}\\ \end{align*}

Mathematica [A]  time = 0.0182863, size = 150, normalized size = 1.26 \[ \frac{1}{17} x^{17} (d+10 e)+\frac{5}{16} x^{16} (2 d+9 e)+x^{15} (3 d+8 e)+\frac{15}{7} x^{14} (4 d+7 e)+\frac{42}{13} x^{13} (5 d+6 e)+\frac{7}{2} x^{12} (6 d+5 e)+\frac{30}{11} x^{11} (7 d+4 e)+\frac{3}{2} x^{10} (8 d+3 e)+\frac{5}{9} x^9 (9 d+2 e)+\frac{1}{8} x^8 (10 d+e)+\frac{d x^7}{7}+\frac{e x^{18}}{18} \]

Antiderivative was successfully verified.

[In]

Integrate[x^6*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

(d*x^7)/7 + ((10*d + e)*x^8)/8 + (5*(9*d + 2*e)*x^9)/9 + (3*(8*d + 3*e)*x^10)/2 + (30*(7*d + 4*e)*x^11)/11 + (
7*(6*d + 5*e)*x^12)/2 + (42*(5*d + 6*e)*x^13)/13 + (15*(4*d + 7*e)*x^14)/7 + (3*d + 8*e)*x^15 + (5*(2*d + 9*e)
*x^16)/16 + ((d + 10*e)*x^17)/17 + (e*x^18)/18

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Maple [A]  time = 0.001, size = 130, normalized size = 1.1 \begin{align*}{\frac{e{x}^{18}}{18}}+{\frac{ \left ( d+10\,e \right ){x}^{17}}{17}}+{\frac{ \left ( 10\,d+45\,e \right ){x}^{16}}{16}}+{\frac{ \left ( 45\,d+120\,e \right ){x}^{15}}{15}}+{\frac{ \left ( 120\,d+210\,e \right ){x}^{14}}{14}}+{\frac{ \left ( 210\,d+252\,e \right ){x}^{13}}{13}}+{\frac{ \left ( 252\,d+210\,e \right ){x}^{12}}{12}}+{\frac{ \left ( 210\,d+120\,e \right ){x}^{11}}{11}}+{\frac{ \left ( 120\,d+45\,e \right ){x}^{10}}{10}}+{\frac{ \left ( 45\,d+10\,e \right ){x}^{9}}{9}}+{\frac{ \left ( 10\,d+e \right ){x}^{8}}{8}}+{\frac{d{x}^{7}}{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(e*x+d)*(x^2+2*x+1)^5,x)

[Out]

1/18*e*x^18+1/17*(d+10*e)*x^17+1/16*(10*d+45*e)*x^16+1/15*(45*d+120*e)*x^15+1/14*(120*d+210*e)*x^14+1/13*(210*
d+252*e)*x^13+1/12*(252*d+210*e)*x^12+1/11*(210*d+120*e)*x^11+1/10*(120*d+45*e)*x^10+1/9*(45*d+10*e)*x^9+1/8*(
10*d+e)*x^8+1/7*d*x^7

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Maxima [A]  time = 0.985174, size = 173, normalized size = 1.45 \begin{align*} \frac{1}{18} \, e x^{18} + \frac{1}{17} \,{\left (d + 10 \, e\right )} x^{17} + \frac{5}{16} \,{\left (2 \, d + 9 \, e\right )} x^{16} +{\left (3 \, d + 8 \, e\right )} x^{15} + \frac{15}{7} \,{\left (4 \, d + 7 \, e\right )} x^{14} + \frac{42}{13} \,{\left (5 \, d + 6 \, e\right )} x^{13} + \frac{7}{2} \,{\left (6 \, d + 5 \, e\right )} x^{12} + \frac{30}{11} \,{\left (7 \, d + 4 \, e\right )} x^{11} + \frac{3}{2} \,{\left (8 \, d + 3 \, e\right )} x^{10} + \frac{5}{9} \,{\left (9 \, d + 2 \, e\right )} x^{9} + \frac{1}{8} \,{\left (10 \, d + e\right )} x^{8} + \frac{1}{7} \, d x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")

[Out]

1/18*e*x^18 + 1/17*(d + 10*e)*x^17 + 5/16*(2*d + 9*e)*x^16 + (3*d + 8*e)*x^15 + 15/7*(4*d + 7*e)*x^14 + 42/13*
(5*d + 6*e)*x^13 + 7/2*(6*d + 5*e)*x^12 + 30/11*(7*d + 4*e)*x^11 + 3/2*(8*d + 3*e)*x^10 + 5/9*(9*d + 2*e)*x^9
+ 1/8*(10*d + e)*x^8 + 1/7*d*x^7

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Fricas [A]  time = 1.2034, size = 394, normalized size = 3.31 \begin{align*} \frac{1}{18} x^{18} e + \frac{10}{17} x^{17} e + \frac{1}{17} x^{17} d + \frac{45}{16} x^{16} e + \frac{5}{8} x^{16} d + 8 x^{15} e + 3 x^{15} d + 15 x^{14} e + \frac{60}{7} x^{14} d + \frac{252}{13} x^{13} e + \frac{210}{13} x^{13} d + \frac{35}{2} x^{12} e + 21 x^{12} d + \frac{120}{11} x^{11} e + \frac{210}{11} x^{11} d + \frac{9}{2} x^{10} e + 12 x^{10} d + \frac{10}{9} x^{9} e + 5 x^{9} d + \frac{1}{8} x^{8} e + \frac{5}{4} x^{8} d + \frac{1}{7} x^{7} d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")

[Out]

1/18*x^18*e + 10/17*x^17*e + 1/17*x^17*d + 45/16*x^16*e + 5/8*x^16*d + 8*x^15*e + 3*x^15*d + 15*x^14*e + 60/7*
x^14*d + 252/13*x^13*e + 210/13*x^13*d + 35/2*x^12*e + 21*x^12*d + 120/11*x^11*e + 210/11*x^11*d + 9/2*x^10*e
+ 12*x^10*d + 10/9*x^9*e + 5*x^9*d + 1/8*x^8*e + 5/4*x^8*d + 1/7*x^7*d

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Sympy [A]  time = 0.309396, size = 134, normalized size = 1.13 \begin{align*} \frac{d x^{7}}{7} + \frac{e x^{18}}{18} + x^{17} \left (\frac{d}{17} + \frac{10 e}{17}\right ) + x^{16} \left (\frac{5 d}{8} + \frac{45 e}{16}\right ) + x^{15} \left (3 d + 8 e\right ) + x^{14} \left (\frac{60 d}{7} + 15 e\right ) + x^{13} \left (\frac{210 d}{13} + \frac{252 e}{13}\right ) + x^{12} \left (21 d + \frac{35 e}{2}\right ) + x^{11} \left (\frac{210 d}{11} + \frac{120 e}{11}\right ) + x^{10} \left (12 d + \frac{9 e}{2}\right ) + x^{9} \left (5 d + \frac{10 e}{9}\right ) + x^{8} \left (\frac{5 d}{4} + \frac{e}{8}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(e*x+d)*(x**2+2*x+1)**5,x)

[Out]

d*x**7/7 + e*x**18/18 + x**17*(d/17 + 10*e/17) + x**16*(5*d/8 + 45*e/16) + x**15*(3*d + 8*e) + x**14*(60*d/7 +
15*e) + x**13*(210*d/13 + 252*e/13) + x**12*(21*d + 35*e/2) + x**11*(210*d/11 + 120*e/11) + x**10*(12*d + 9*e
/2) + x**9*(5*d + 10*e/9) + x**8*(5*d/4 + e/8)

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Giac [A]  time = 1.12444, size = 194, normalized size = 1.63 \begin{align*} \frac{1}{18} \, x^{18} e + \frac{1}{17} \, d x^{17} + \frac{10}{17} \, x^{17} e + \frac{5}{8} \, d x^{16} + \frac{45}{16} \, x^{16} e + 3 \, d x^{15} + 8 \, x^{15} e + \frac{60}{7} \, d x^{14} + 15 \, x^{14} e + \frac{210}{13} \, d x^{13} + \frac{252}{13} \, x^{13} e + 21 \, d x^{12} + \frac{35}{2} \, x^{12} e + \frac{210}{11} \, d x^{11} + \frac{120}{11} \, x^{11} e + 12 \, d x^{10} + \frac{9}{2} \, x^{10} e + 5 \, d x^{9} + \frac{10}{9} \, x^{9} e + \frac{5}{4} \, d x^{8} + \frac{1}{8} \, x^{8} e + \frac{1}{7} \, d x^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")

[Out]

1/18*x^18*e + 1/17*d*x^17 + 10/17*x^17*e + 5/8*d*x^16 + 45/16*x^16*e + 3*d*x^15 + 8*x^15*e + 60/7*d*x^14 + 15*
x^14*e + 210/13*d*x^13 + 252/13*x^13*e + 21*d*x^12 + 35/2*x^12*e + 210/11*d*x^11 + 120/11*x^11*e + 12*d*x^10 +
9/2*x^10*e + 5*d*x^9 + 10/9*x^9*e + 5/4*d*x^8 + 1/8*x^8*e + 1/7*d*x^7