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

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 = 0.0085815, 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, Rules used = {27, 43} \[ \frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{12} e (x+1)^{12} \]

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

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int (d+e x) \left (1+2 x+x^2\right )^5 \, dx &=\int (1+x)^{10} (d+e x) \, dx\\ &=\int \left ((d-e) (1+x)^{10}+e (1+x)^{11}\right ) \, dx\\ &=\frac{1}{11} (d-e) (1+x)^{11}+\frac{1}{12} e (1+x)^{12}\\ \end{align*}

Mathematica [B]  time = 0.0203552, 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 \begin{align*}{\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 \end{align*}

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*(120*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 [B]  time = 1.00162, size = 170, normalized size = 6.8 \begin{align*} \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 \end{align*}

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 [B]  time = 1.14679, size = 343, normalized size = 13.72 \begin{align*} \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 \end{align*}

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 [B]  time = 0.134725, size = 119, normalized size = 4.76 \begin{align*} 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 ) \end{align*}

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 [B]  time = 1.14152, size = 189, normalized size = 7.56 \begin{align*} \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 \end{align*}

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