3.562 \(\int x^4 (d+e x) (1+2 x+x^2)^5 \, dx\)

Optimal. Leaf size=87 \[ \frac{1}{15} (x+1)^{15} (d-5 e)-\frac{1}{7} (x+1)^{14} (2 d-5 e)+\frac{2}{13} (x+1)^{13} (3 d-5 e)-\frac{1}{12} (x+1)^{12} (4 d-5 e)+\frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{16} e (x+1)^{16} \]

[Out]

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

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Rubi [A]  time = 0.0567202, antiderivative size = 87, 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} \[ \frac{1}{15} (x+1)^{15} (d-5 e)-\frac{1}{7} (x+1)^{14} (2 d-5 e)+\frac{2}{13} (x+1)^{13} (3 d-5 e)-\frac{1}{12} (x+1)^{12} (4 d-5 e)+\frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{16} e (x+1)^{16} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

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

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

Mathematica [A]  time = 0.0182847, size = 153, normalized size = 1.76 \[ \frac{1}{15} x^{15} (d+10 e)+\frac{5}{14} x^{14} (2 d+9 e)+\frac{15}{13} x^{13} (3 d+8 e)+\frac{5}{2} x^{12} (4 d+7 e)+\frac{42}{11} x^{11} (5 d+6 e)+\frac{21}{5} x^{10} (6 d+5 e)+\frac{10}{3} x^9 (7 d+4 e)+\frac{15}{8} x^8 (8 d+3 e)+\frac{5}{7} x^7 (9 d+2 e)+\frac{1}{6} x^6 (10 d+e)+\frac{d x^5}{5}+\frac{e x^{16}}{16} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

(d*x^5)/5 + ((10*d + e)*x^6)/6 + (5*(9*d + 2*e)*x^7)/7 + (15*(8*d + 3*e)*x^8)/8 + (10*(7*d + 4*e)*x^9)/3 + (21
*(6*d + 5*e)*x^10)/5 + (42*(5*d + 6*e)*x^11)/11 + (5*(4*d + 7*e)*x^12)/2 + (15*(3*d + 8*e)*x^13)/13 + (5*(2*d
+ 9*e)*x^14)/14 + ((d + 10*e)*x^15)/15 + (e*x^16)/16

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Maple [A]  time = 0., size = 130, normalized size = 1.5 \begin{align*}{\frac{e{x}^{16}}{16}}+{\frac{ \left ( d+10\,e \right ){x}^{15}}{15}}+{\frac{ \left ( 10\,d+45\,e \right ){x}^{14}}{14}}+{\frac{ \left ( 45\,d+120\,e \right ){x}^{13}}{13}}+{\frac{ \left ( 120\,d+210\,e \right ){x}^{12}}{12}}+{\frac{ \left ( 210\,d+252\,e \right ){x}^{11}}{11}}+{\frac{ \left ( 252\,d+210\,e \right ){x}^{10}}{10}}+{\frac{ \left ( 210\,d+120\,e \right ){x}^{9}}{9}}+{\frac{ \left ( 120\,d+45\,e \right ){x}^{8}}{8}}+{\frac{ \left ( 45\,d+10\,e \right ){x}^{7}}{7}}+{\frac{ \left ( 10\,d+e \right ){x}^{6}}{6}}+{\frac{d{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(e*x+d)*(x^2+2*x+1)^5,x)

[Out]

1/16*e*x^16+1/15*(d+10*e)*x^15+1/14*(10*d+45*e)*x^14+1/13*(45*d+120*e)*x^13+1/12*(120*d+210*e)*x^12+1/11*(210*
d+252*e)*x^11+1/10*(252*d+210*e)*x^10+1/9*(210*d+120*e)*x^9+1/8*(120*d+45*e)*x^8+1/7*(45*d+10*e)*x^7+1/6*(10*d
+e)*x^6+1/5*d*x^5

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Maxima [A]  time = 0.966038, size = 174, normalized size = 2. \begin{align*} \frac{1}{16} \, e x^{16} + \frac{1}{15} \,{\left (d + 10 \, e\right )} x^{15} + \frac{5}{14} \,{\left (2 \, d + 9 \, e\right )} x^{14} + \frac{15}{13} \,{\left (3 \, d + 8 \, e\right )} x^{13} + \frac{5}{2} \,{\left (4 \, d + 7 \, e\right )} x^{12} + \frac{42}{11} \,{\left (5 \, d + 6 \, e\right )} x^{11} + \frac{21}{5} \,{\left (6 \, d + 5 \, e\right )} x^{10} + \frac{10}{3} \,{\left (7 \, d + 4 \, e\right )} x^{9} + \frac{15}{8} \,{\left (8 \, d + 3 \, e\right )} x^{8} + \frac{5}{7} \,{\left (9 \, d + 2 \, e\right )} x^{7} + \frac{1}{6} \,{\left (10 \, d + e\right )} x^{6} + \frac{1}{5} \, d x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")

[Out]

1/16*e*x^16 + 1/15*(d + 10*e)*x^15 + 5/14*(2*d + 9*e)*x^14 + 15/13*(3*d + 8*e)*x^13 + 5/2*(4*d + 7*e)*x^12 + 4
2/11*(5*d + 6*e)*x^11 + 21/5*(6*d + 5*e)*x^10 + 10/3*(7*d + 4*e)*x^9 + 15/8*(8*d + 3*e)*x^8 + 5/7*(9*d + 2*e)*
x^7 + 1/6*(10*d + e)*x^6 + 1/5*d*x^5

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Fricas [A]  time = 1.10661, size = 400, normalized size = 4.6 \begin{align*} \frac{1}{16} x^{16} e + \frac{2}{3} x^{15} e + \frac{1}{15} x^{15} d + \frac{45}{14} x^{14} e + \frac{5}{7} x^{14} d + \frac{120}{13} x^{13} e + \frac{45}{13} x^{13} d + \frac{35}{2} x^{12} e + 10 x^{12} d + \frac{252}{11} x^{11} e + \frac{210}{11} x^{11} d + 21 x^{10} e + \frac{126}{5} x^{10} d + \frac{40}{3} x^{9} e + \frac{70}{3} x^{9} d + \frac{45}{8} x^{8} e + 15 x^{8} d + \frac{10}{7} x^{7} e + \frac{45}{7} x^{7} d + \frac{1}{6} x^{6} e + \frac{5}{3} x^{6} d + \frac{1}{5} x^{5} d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")

[Out]

1/16*x^16*e + 2/3*x^15*e + 1/15*x^15*d + 45/14*x^14*e + 5/7*x^14*d + 120/13*x^13*e + 45/13*x^13*d + 35/2*x^12*
e + 10*x^12*d + 252/11*x^11*e + 210/11*x^11*d + 21*x^10*e + 126/5*x^10*d + 40/3*x^9*e + 70/3*x^9*d + 45/8*x^8*
e + 15*x^8*d + 10/7*x^7*e + 45/7*x^7*d + 1/6*x^6*e + 5/3*x^6*d + 1/5*x^5*d

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Sympy [A]  time = 0.137223, size = 139, normalized size = 1.6 \begin{align*} \frac{d x^{5}}{5} + \frac{e x^{16}}{16} + x^{15} \left (\frac{d}{15} + \frac{2 e}{3}\right ) + x^{14} \left (\frac{5 d}{7} + \frac{45 e}{14}\right ) + x^{13} \left (\frac{45 d}{13} + \frac{120 e}{13}\right ) + x^{12} \left (10 d + \frac{35 e}{2}\right ) + x^{11} \left (\frac{210 d}{11} + \frac{252 e}{11}\right ) + x^{10} \left (\frac{126 d}{5} + 21 e\right ) + x^{9} \left (\frac{70 d}{3} + \frac{40 e}{3}\right ) + x^{8} \left (15 d + \frac{45 e}{8}\right ) + x^{7} \left (\frac{45 d}{7} + \frac{10 e}{7}\right ) + x^{6} \left (\frac{5 d}{3} + \frac{e}{6}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(e*x+d)*(x**2+2*x+1)**5,x)

[Out]

d*x**5/5 + e*x**16/16 + x**15*(d/15 + 2*e/3) + x**14*(5*d/7 + 45*e/14) + x**13*(45*d/13 + 120*e/13) + x**12*(1
0*d + 35*e/2) + x**11*(210*d/11 + 252*e/11) + x**10*(126*d/5 + 21*e) + x**9*(70*d/3 + 40*e/3) + x**8*(15*d + 4
5*e/8) + x**7*(45*d/7 + 10*e/7) + x**6*(5*d/3 + e/6)

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Giac [A]  time = 1.10664, size = 194, normalized size = 2.23 \begin{align*} \frac{1}{16} \, x^{16} e + \frac{1}{15} \, d x^{15} + \frac{2}{3} \, x^{15} e + \frac{5}{7} \, d x^{14} + \frac{45}{14} \, x^{14} e + \frac{45}{13} \, d x^{13} + \frac{120}{13} \, x^{13} e + 10 \, d x^{12} + \frac{35}{2} \, x^{12} e + \frac{210}{11} \, d x^{11} + \frac{252}{11} \, x^{11} e + \frac{126}{5} \, d x^{10} + 21 \, x^{10} e + \frac{70}{3} \, d x^{9} + \frac{40}{3} \, x^{9} e + 15 \, d x^{8} + \frac{45}{8} \, x^{8} e + \frac{45}{7} \, d x^{7} + \frac{10}{7} \, x^{7} e + \frac{5}{3} \, d x^{6} + \frac{1}{6} \, x^{6} e + \frac{1}{5} \, d x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")

[Out]

1/16*x^16*e + 1/15*d*x^15 + 2/3*x^15*e + 5/7*d*x^14 + 45/14*x^14*e + 45/13*d*x^13 + 120/13*x^13*e + 10*d*x^12
+ 35/2*x^12*e + 210/11*d*x^11 + 252/11*x^11*e + 126/5*d*x^10 + 21*x^10*e + 70/3*d*x^9 + 40/3*x^9*e + 15*d*x^8
+ 45/8*x^8*e + 45/7*d*x^7 + 10/7*x^7*e + 5/3*d*x^6 + 1/6*x^6*e + 1/5*d*x^5