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

Optimal. Leaf size=99 \[ \frac{1}{16} (x+1)^{16} (d-6 e)-\frac{1}{3} (x+1)^{15} (d-3 e)+\frac{5}{7} (x+1)^{14} (d-2 e)-\frac{5}{13} (x+1)^{13} (2 d-3 e)+\frac{1}{12} (x+1)^{12} (5 d-6 e)-\frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{17} e (x+1)^{17} \]

[Out]

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

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Rubi [A]  time = 0.0636043, antiderivative size = 99, 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} \[ \frac{1}{16} (x+1)^{16} (d-6 e)-\frac{1}{3} (x+1)^{15} (d-3 e)+\frac{5}{7} (x+1)^{14} (d-2 e)-\frac{5}{13} (x+1)^{13} (2 d-3 e)+\frac{1}{12} (x+1)^{12} (5 d-6 e)-\frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{17} e (x+1)^{17} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0188952, size = 151, normalized size = 1.53 \[ \frac{1}{16} x^{16} (d+10 e)+\frac{1}{3} x^{15} (2 d+9 e)+\frac{15}{14} x^{14} (3 d+8 e)+\frac{30}{13} x^{13} (4 d+7 e)+\frac{7}{2} x^{12} (5 d+6 e)+\frac{42}{11} x^{11} (6 d+5 e)+3 x^{10} (7 d+4 e)+\frac{5}{3} x^9 (8 d+3 e)+\frac{5}{8} x^8 (9 d+2 e)+\frac{1}{7} x^7 (10 d+e)+\frac{d x^6}{6}+\frac{e x^{17}}{17} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 130, normalized size = 1.3 \begin{align*}{\frac{e{x}^{17}}{17}}+{\frac{ \left ( d+10\,e \right ){x}^{16}}{16}}+{\frac{ \left ( 10\,d+45\,e \right ){x}^{15}}{15}}+{\frac{ \left ( 45\,d+120\,e \right ){x}^{14}}{14}}+{\frac{ \left ( 120\,d+210\,e \right ){x}^{13}}{13}}+{\frac{ \left ( 210\,d+252\,e \right ){x}^{12}}{12}}+{\frac{ \left ( 252\,d+210\,e \right ){x}^{11}}{11}}+{\frac{ \left ( 210\,d+120\,e \right ){x}^{10}}{10}}+{\frac{ \left ( 120\,d+45\,e \right ){x}^{9}}{9}}+{\frac{ \left ( 45\,d+10\,e \right ){x}^{8}}{8}}+{\frac{ \left ( 10\,d+e \right ){x}^{7}}{7}}+{\frac{d{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17*e*x^17+1/16*(d+10*e)*x^16+1/15*(10*d+45*e)*x^15+1/14*(45*d+120*e)*x^14+1/13*(120*d+210*e)*x^13+1/12*(210*
d+252*e)*x^12+1/11*(252*d+210*e)*x^11+1/10*(210*d+120*e)*x^10+1/9*(120*d+45*e)*x^9+1/8*(45*d+10*e)*x^8+1/7*(10
*d+e)*x^7+1/6*d*x^6

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Maxima [A]  time = 1.00989, size = 174, normalized size = 1.76 \begin{align*} \frac{1}{17} \, e x^{17} + \frac{1}{16} \,{\left (d + 10 \, e\right )} x^{16} + \frac{1}{3} \,{\left (2 \, d + 9 \, e\right )} x^{15} + \frac{15}{14} \,{\left (3 \, d + 8 \, e\right )} x^{14} + \frac{30}{13} \,{\left (4 \, d + 7 \, e\right )} x^{13} + \frac{7}{2} \,{\left (5 \, d + 6 \, e\right )} x^{12} + \frac{42}{11} \,{\left (6 \, d + 5 \, e\right )} x^{11} + 3 \,{\left (7 \, d + 4 \, e\right )} x^{10} + \frac{5}{3} \,{\left (8 \, d + 3 \, e\right )} x^{9} + \frac{5}{8} \,{\left (9 \, d + 2 \, e\right )} x^{8} + \frac{1}{7} \,{\left (10 \, d + e\right )} x^{7} + \frac{1}{6} \, d x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.0647, size = 394, normalized size = 3.98 \begin{align*} \frac{1}{17} x^{17} e + \frac{5}{8} x^{16} e + \frac{1}{16} x^{16} d + 3 x^{15} e + \frac{2}{3} x^{15} d + \frac{60}{7} x^{14} e + \frac{45}{14} x^{14} d + \frac{210}{13} x^{13} e + \frac{120}{13} x^{13} d + 21 x^{12} e + \frac{35}{2} x^{12} d + \frac{210}{11} x^{11} e + \frac{252}{11} x^{11} d + 12 x^{10} e + 21 x^{10} d + 5 x^{9} e + \frac{40}{3} x^{9} d + \frac{5}{4} x^{8} e + \frac{45}{8} x^{8} d + \frac{1}{7} x^{7} e + \frac{10}{7} x^{7} d + \frac{1}{6} x^{6} d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17*x^17*e + 5/8*x^16*e + 1/16*x^16*d + 3*x^15*e + 2/3*x^15*d + 60/7*x^14*e + 45/14*x^14*d + 210/13*x^13*e +
120/13*x^13*d + 21*x^12*e + 35/2*x^12*d + 210/11*x^11*e + 252/11*x^11*d + 12*x^10*e + 21*x^10*d + 5*x^9*e + 40
/3*x^9*d + 5/4*x^8*e + 45/8*x^8*d + 1/7*x^7*e + 10/7*x^7*d + 1/6*x^6*d

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Sympy [A]  time = 0.232054, size = 136, normalized size = 1.37 \begin{align*} \frac{d x^{6}}{6} + \frac{e x^{17}}{17} + x^{16} \left (\frac{d}{16} + \frac{5 e}{8}\right ) + x^{15} \left (\frac{2 d}{3} + 3 e\right ) + x^{14} \left (\frac{45 d}{14} + \frac{60 e}{7}\right ) + x^{13} \left (\frac{120 d}{13} + \frac{210 e}{13}\right ) + x^{12} \left (\frac{35 d}{2} + 21 e\right ) + x^{11} \left (\frac{252 d}{11} + \frac{210 e}{11}\right ) + x^{10} \left (21 d + 12 e\right ) + x^{9} \left (\frac{40 d}{3} + 5 e\right ) + x^{8} \left (\frac{45 d}{8} + \frac{5 e}{4}\right ) + x^{7} \left (\frac{10 d}{7} + \frac{e}{7}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

d*x**6/6 + e*x**17/17 + x**16*(d/16 + 5*e/8) + x**15*(2*d/3 + 3*e) + x**14*(45*d/14 + 60*e/7) + x**13*(120*d/1
3 + 210*e/13) + x**12*(35*d/2 + 21*e) + x**11*(252*d/11 + 210*e/11) + x**10*(21*d + 12*e) + x**9*(40*d/3 + 5*e
) + x**8*(45*d/8 + 5*e/4) + x**7*(10*d/7 + e/7)

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Giac [A]  time = 1.15148, size = 194, normalized size = 1.96 \begin{align*} \frac{1}{17} \, x^{17} e + \frac{1}{16} \, d x^{16} + \frac{5}{8} \, x^{16} e + \frac{2}{3} \, d x^{15} + 3 \, x^{15} e + \frac{45}{14} \, d x^{14} + \frac{60}{7} \, x^{14} e + \frac{120}{13} \, d x^{13} + \frac{210}{13} \, x^{13} e + \frac{35}{2} \, d x^{12} + 21 \, x^{12} e + \frac{252}{11} \, d x^{11} + \frac{210}{11} \, x^{11} e + 21 \, d x^{10} + 12 \, x^{10} e + \frac{40}{3} \, d x^{9} + 5 \, x^{9} e + \frac{45}{8} \, d x^{8} + \frac{5}{4} \, x^{8} e + \frac{10}{7} \, d x^{7} + \frac{1}{7} \, x^{7} e + \frac{1}{6} \, d x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/17*x^17*e + 1/16*d*x^16 + 5/8*x^16*e + 2/3*d*x^15 + 3*x^15*e + 45/14*d*x^14 + 60/7*x^14*e + 120/13*d*x^13 +
210/13*x^13*e + 35/2*d*x^12 + 21*x^12*e + 252/11*d*x^11 + 210/11*x^11*e + 21*d*x^10 + 12*x^10*e + 40/3*d*x^9 +
5*x^9*e + 45/8*d*x^8 + 5/4*x^8*e + 10/7*d*x^7 + 1/7*x^7*e + 1/6*d*x^6