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

Optimal. Leaf size=25 \[ \frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{12} e (x+1)^{12} \]

[Out]

((d - e)*(1 + x)^11)/11 + (e*(1 + x)^12)/12

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Rubi [A]  time = 0.0298387, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{12} e (x+1)^{12} \]

Antiderivative was successfully verified.

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

[Out]

((d - e)*(1 + x)^11)/11 + (e*(1 + x)^12)/12

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Rubi in Sympy [A]  time = 13.6105, size = 19, normalized size = 0.76 \[ \frac{e \left (x + 1\right )^{12}}{12} + \left (\frac{d}{11} - \frac{e}{11}\right ) \left (x + 1\right )^{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(x + 1)**12/12 + (d/11 - e/11)*(x + 1)**11

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Mathematica [B]  time = 0.0427318, size = 113, normalized size = 4.52 \[ d \left (\frac{x^{11}}{11}+x^{10}+5 x^9+15 x^8+30 x^7+42 x^6+42 x^5+30 x^4+15 x^3+5 x^2+x\right )+\frac{1}{132} e \left (11 x^{10}+120 x^9+594 x^8+1760 x^7+3465 x^6+4752 x^5+4620 x^4+3168 x^3+1485 x^2+440 x+66\right ) x^2 \]

Antiderivative was successfully verified.

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

[Out]

(e*x^2*(66 + 440*x + 1485*x^2 + 3168*x^3 + 4620*x^4 + 4752*x^5 + 3465*x^6 + 1760
*x^7 + 594*x^8 + 120*x^9 + 11*x^10))/132 + d*(x + 5*x^2 + 15*x^3 + 30*x^4 + 42*x
^5 + 42*x^6 + 30*x^7 + 15*x^8 + 5*x^9 + x^10 + x^11/11)

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Maple [B]  time = 0.002, size = 127, normalized size = 5.1 \[{\frac{e{x}^{12}}{12}}+{\frac{ \left ( d+10\,e \right ){x}^{11}}{11}}+{\frac{ \left ( 10\,d+45\,e \right ){x}^{10}}{10}}+{\frac{ \left ( 45\,d+120\,e \right ){x}^{9}}{9}}+{\frac{ \left ( 120\,d+210\,e \right ){x}^{8}}{8}}+{\frac{ \left ( 210\,d+252\,e \right ){x}^{7}}{7}}+{\frac{ \left ( 252\,d+210\,e \right ){x}^{6}}{6}}+{\frac{ \left ( 210\,d+120\,e \right ){x}^{5}}{5}}+{\frac{ \left ( 120\,d+45\,e \right ){x}^{4}}{4}}+{\frac{ \left ( 45\,d+10\,e \right ){x}^{3}}{3}}+{\frac{ \left ( 10\,d+e \right ){x}^{2}}{2}}+dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(x^2+2*x+1)^5,x)

[Out]

1/12*e*x^12+1/11*(d+10*e)*x^11+1/10*(10*d+45*e)*x^10+1/9*(45*d+120*e)*x^9+1/8*(1
20*d+210*e)*x^8+1/7*(210*d+252*e)*x^7+1/6*(252*d+210*e)*x^6+1/5*(210*d+120*e)*x^
5+1/4*(120*d+45*e)*x^4+1/3*(45*d+10*e)*x^3+1/2*(10*d+e)*x^2+d*x

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Maxima [A]  time = 0.678574, size = 170, normalized size = 6.8 \[ \frac{1}{12} \, e x^{12} + \frac{1}{11} \,{\left (d + 10 \, e\right )} x^{11} + \frac{1}{2} \,{\left (2 \, d + 9 \, e\right )} x^{10} + \frac{5}{3} \,{\left (3 \, d + 8 \, e\right )} x^{9} + \frac{15}{4} \,{\left (4 \, d + 7 \, e\right )} x^{8} + 6 \,{\left (5 \, d + 6 \, e\right )} x^{7} + 7 \,{\left (6 \, d + 5 \, e\right )} x^{6} + 6 \,{\left (7 \, d + 4 \, e\right )} x^{5} + \frac{15}{4} \,{\left (8 \, d + 3 \, e\right )} x^{4} + \frac{5}{3} \,{\left (9 \, d + 2 \, e\right )} x^{3} + \frac{1}{2} \,{\left (10 \, d + e\right )} x^{2} + d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.249421, size = 1, normalized size = 0.04 \[ \frac{1}{12} x^{12} e + \frac{10}{11} x^{11} e + \frac{1}{11} x^{11} d + \frac{9}{2} x^{10} e + x^{10} d + \frac{40}{3} x^{9} e + 5 x^{9} d + \frac{105}{4} x^{8} e + 15 x^{8} d + 36 x^{7} e + 30 x^{7} d + 35 x^{6} e + 42 x^{6} d + 24 x^{5} e + 42 x^{5} d + \frac{45}{4} x^{4} e + 30 x^{4} d + \frac{10}{3} x^{3} e + 15 x^{3} d + \frac{1}{2} x^{2} e + 5 x^{2} d + x d \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*x^12*e + 10/11*x^11*e + 1/11*x^11*d + 9/2*x^10*e + x^10*d + 40/3*x^9*e + 5*
x^9*d + 105/4*x^8*e + 15*x^8*d + 36*x^7*e + 30*x^7*d + 35*x^6*e + 42*x^6*d + 24*
x^5*e + 42*x^5*d + 45/4*x^4*e + 30*x^4*d + 10/3*x^3*e + 15*x^3*d + 1/2*x^2*e + 5
*x^2*d + x*d

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Sympy [A]  time = 0.189618, size = 119, normalized size = 4.76 \[ d x + \frac{e x^{12}}{12} + x^{11} \left (\frac{d}{11} + \frac{10 e}{11}\right ) + x^{10} \left (d + \frac{9 e}{2}\right ) + x^{9} \left (5 d + \frac{40 e}{3}\right ) + x^{8} \left (15 d + \frac{105 e}{4}\right ) + x^{7} \left (30 d + 36 e\right ) + x^{6} \left (42 d + 35 e\right ) + x^{5} \left (42 d + 24 e\right ) + x^{4} \left (30 d + \frac{45 e}{4}\right ) + x^{3} \left (15 d + \frac{10 e}{3}\right ) + x^{2} \left (5 d + \frac{e}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

d*x + e*x**12/12 + x**11*(d/11 + 10*e/11) + x**10*(d + 9*e/2) + x**9*(5*d + 40*e
/3) + x**8*(15*d + 105*e/4) + x**7*(30*d + 36*e) + x**6*(42*d + 35*e) + x**5*(42
*d + 24*e) + x**4*(30*d + 45*e/4) + x**3*(15*d + 10*e/3) + x**2*(5*d + e/2)

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GIAC/XCAS [A]  time = 0.266624, size = 189, normalized size = 7.56 \[ \frac{1}{12} \, x^{12} e + \frac{1}{11} \, d x^{11} + \frac{10}{11} \, x^{11} e + d x^{10} + \frac{9}{2} \, x^{10} e + 5 \, d x^{9} + \frac{40}{3} \, x^{9} e + 15 \, d x^{8} + \frac{105}{4} \, x^{8} e + 30 \, d x^{7} + 36 \, x^{7} e + 42 \, d x^{6} + 35 \, x^{6} e + 42 \, d x^{5} + 24 \, x^{5} e + 30 \, d x^{4} + \frac{45}{4} \, x^{4} e + 15 \, d x^{3} + \frac{10}{3} \, x^{3} e + 5 \, d x^{2} + \frac{1}{2} \, x^{2} e + d x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/12*x^12*e + 1/11*d*x^11 + 10/11*x^11*e + d*x^10 + 9/2*x^10*e + 5*d*x^9 + 40/3*
x^9*e + 15*d*x^8 + 105/4*x^8*e + 30*d*x^7 + 36*x^7*e + 42*d*x^6 + 35*x^6*e + 42*
d*x^5 + 24*x^5*e + 30*d*x^4 + 45/4*x^4*e + 15*d*x^3 + 10/3*x^3*e + 5*d*x^2 + 1/2
*x^2*e + d*x