3.563 \(\int x^3 (d+e x) (1+2 x+x^2)^5 \, dx\)

Optimal. Leaf size=69 \[ \frac{1}{14} (x+1)^{14} (d-4 e)-\frac{3}{13} (x+1)^{13} (d-2 e)+\frac{1}{12} (x+1)^{12} (3 d-4 e)-\frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{15} e (x+1)^{15} \]

[Out]

-((d - e)*(1 + x)^11)/11 + ((3*d - 4*e)*(1 + x)^12)/12 - (3*(d - 2*e)*(1 + x)^13)/13 + ((d - 4*e)*(1 + x)^14)/
14 + (e*(1 + x)^15)/15

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Rubi [A]  time = 0.0469993, antiderivative size = 69, 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} \[ \frac{1}{14} (x+1)^{14} (d-4 e)-\frac{3}{13} (x+1)^{13} (d-2 e)+\frac{1}{12} (x+1)^{12} (3 d-4 e)-\frac{1}{11} (x+1)^{11} (d-e)+\frac{1}{15} e (x+1)^{15} \]

Antiderivative was successfully verified.

[In]

Int[x^3*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

-((d - e)*(1 + x)^11)/11 + ((3*d - 4*e)*(1 + x)^12)/12 - (3*(d - 2*e)*(1 + x)^13)/13 + ((d - 4*e)*(1 + x)^14)/
14 + (e*(1 + x)^15)/15

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

Mathematica [B]  time = 0.0183849, size = 153, normalized size = 2.22 \[ \frac{1}{14} x^{14} (d+10 e)+\frac{5}{13} x^{13} (2 d+9 e)+\frac{5}{4} x^{12} (3 d+8 e)+\frac{30}{11} x^{11} (4 d+7 e)+\frac{21}{5} x^{10} (5 d+6 e)+\frac{14}{3} x^9 (6 d+5 e)+\frac{15}{4} x^8 (7 d+4 e)+\frac{15}{7} x^7 (8 d+3 e)+\frac{5}{6} x^6 (9 d+2 e)+\frac{1}{5} x^5 (10 d+e)+\frac{d x^4}{4}+\frac{e x^{15}}{15} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

(d*x^4)/4 + ((10*d + e)*x^5)/5 + (5*(9*d + 2*e)*x^6)/6 + (15*(8*d + 3*e)*x^7)/7 + (15*(7*d + 4*e)*x^8)/4 + (14
*(6*d + 5*e)*x^9)/3 + (21*(5*d + 6*e)*x^10)/5 + (30*(4*d + 7*e)*x^11)/11 + (5*(3*d + 8*e)*x^12)/4 + (5*(2*d +
9*e)*x^13)/13 + ((d + 10*e)*x^14)/14 + (e*x^15)/15

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Maple [B]  time = 0.001, size = 130, normalized size = 1.9 \begin{align*}{\frac{e{x}^{15}}{15}}+{\frac{ \left ( d+10\,e \right ){x}^{14}}{14}}+{\frac{ \left ( 10\,d+45\,e \right ){x}^{13}}{13}}+{\frac{ \left ( 45\,d+120\,e \right ){x}^{12}}{12}}+{\frac{ \left ( 120\,d+210\,e \right ){x}^{11}}{11}}+{\frac{ \left ( 210\,d+252\,e \right ){x}^{10}}{10}}+{\frac{ \left ( 252\,d+210\,e \right ){x}^{9}}{9}}+{\frac{ \left ( 210\,d+120\,e \right ){x}^{8}}{8}}+{\frac{ \left ( 120\,d+45\,e \right ){x}^{7}}{7}}+{\frac{ \left ( 45\,d+10\,e \right ){x}^{6}}{6}}+{\frac{ \left ( 10\,d+e \right ){x}^{5}}{5}}+{\frac{d{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(e*x+d)*(x^2+2*x+1)^5,x)

[Out]

1/15*e*x^15+1/14*(d+10*e)*x^14+1/13*(10*d+45*e)*x^13+1/12*(45*d+120*e)*x^12+1/11*(120*d+210*e)*x^11+1/10*(210*
d+252*e)*x^10+1/9*(252*d+210*e)*x^9+1/8*(210*d+120*e)*x^8+1/7*(120*d+45*e)*x^7+1/6*(45*d+10*e)*x^6+1/5*(10*d+e
)*x^5+1/4*d*x^4

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Maxima [B]  time = 1.01501, size = 174, normalized size = 2.52 \begin{align*} \frac{1}{15} \, e x^{15} + \frac{1}{14} \,{\left (d + 10 \, e\right )} x^{14} + \frac{5}{13} \,{\left (2 \, d + 9 \, e\right )} x^{13} + \frac{5}{4} \,{\left (3 \, d + 8 \, e\right )} x^{12} + \frac{30}{11} \,{\left (4 \, d + 7 \, e\right )} x^{11} + \frac{21}{5} \,{\left (5 \, d + 6 \, e\right )} x^{10} + \frac{14}{3} \,{\left (6 \, d + 5 \, e\right )} x^{9} + \frac{15}{4} \,{\left (7 \, d + 4 \, e\right )} x^{8} + \frac{15}{7} \,{\left (8 \, d + 3 \, e\right )} x^{7} + \frac{5}{6} \,{\left (9 \, d + 2 \, e\right )} x^{6} + \frac{1}{5} \,{\left (10 \, d + e\right )} x^{5} + \frac{1}{4} \, d x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.22425, size = 392, normalized size = 5.68 \begin{align*} \frac{1}{15} x^{15} e + \frac{5}{7} x^{14} e + \frac{1}{14} x^{14} d + \frac{45}{13} x^{13} e + \frac{10}{13} x^{13} d + 10 x^{12} e + \frac{15}{4} x^{12} d + \frac{210}{11} x^{11} e + \frac{120}{11} x^{11} d + \frac{126}{5} x^{10} e + 21 x^{10} d + \frac{70}{3} x^{9} e + 28 x^{9} d + 15 x^{8} e + \frac{105}{4} x^{8} d + \frac{45}{7} x^{7} e + \frac{120}{7} x^{7} d + \frac{5}{3} x^{6} e + \frac{15}{2} x^{6} d + \frac{1}{5} x^{5} e + 2 x^{5} d + \frac{1}{4} x^{4} d \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")

[Out]

1/15*x^15*e + 5/7*x^14*e + 1/14*x^14*d + 45/13*x^13*e + 10/13*x^13*d + 10*x^12*e + 15/4*x^12*d + 210/11*x^11*e
+ 120/11*x^11*d + 126/5*x^10*e + 21*x^10*d + 70/3*x^9*e + 28*x^9*d + 15*x^8*e + 105/4*x^8*d + 45/7*x^7*e + 12
0/7*x^7*d + 5/3*x^6*e + 15/2*x^6*d + 1/5*x^5*e + 2*x^5*d + 1/4*x^4*d

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Sympy [B]  time = 0.126569, size = 136, normalized size = 1.97 \begin{align*} \frac{d x^{4}}{4} + \frac{e x^{15}}{15} + x^{14} \left (\frac{d}{14} + \frac{5 e}{7}\right ) + x^{13} \left (\frac{10 d}{13} + \frac{45 e}{13}\right ) + x^{12} \left (\frac{15 d}{4} + 10 e\right ) + x^{11} \left (\frac{120 d}{11} + \frac{210 e}{11}\right ) + x^{10} \left (21 d + \frac{126 e}{5}\right ) + x^{9} \left (28 d + \frac{70 e}{3}\right ) + x^{8} \left (\frac{105 d}{4} + 15 e\right ) + x^{7} \left (\frac{120 d}{7} + \frac{45 e}{7}\right ) + x^{6} \left (\frac{15 d}{2} + \frac{5 e}{3}\right ) + x^{5} \left (2 d + \frac{e}{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(e*x+d)*(x**2+2*x+1)**5,x)

[Out]

d*x**4/4 + e*x**15/15 + x**14*(d/14 + 5*e/7) + x**13*(10*d/13 + 45*e/13) + x**12*(15*d/4 + 10*e) + x**11*(120*
d/11 + 210*e/11) + x**10*(21*d + 126*e/5) + x**9*(28*d + 70*e/3) + x**8*(105*d/4 + 15*e) + x**7*(120*d/7 + 45*
e/7) + x**6*(15*d/2 + 5*e/3) + x**5*(2*d + e/5)

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Giac [B]  time = 1.17509, size = 194, normalized size = 2.81 \begin{align*} \frac{1}{15} \, x^{15} e + \frac{1}{14} \, d x^{14} + \frac{5}{7} \, x^{14} e + \frac{10}{13} \, d x^{13} + \frac{45}{13} \, x^{13} e + \frac{15}{4} \, d x^{12} + 10 \, x^{12} e + \frac{120}{11} \, d x^{11} + \frac{210}{11} \, x^{11} e + 21 \, d x^{10} + \frac{126}{5} \, x^{10} e + 28 \, d x^{9} + \frac{70}{3} \, x^{9} e + \frac{105}{4} \, d x^{8} + 15 \, x^{8} e + \frac{120}{7} \, d x^{7} + \frac{45}{7} \, x^{7} e + \frac{15}{2} \, d x^{6} + \frac{5}{3} \, x^{6} e + 2 \, d x^{5} + \frac{1}{5} \, x^{5} e + \frac{1}{4} \, d x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")

[Out]

1/15*x^15*e + 1/14*d*x^14 + 5/7*x^14*e + 10/13*d*x^13 + 45/13*x^13*e + 15/4*d*x^12 + 10*x^12*e + 120/11*d*x^11
+ 210/11*x^11*e + 21*d*x^10 + 126/5*x^10*e + 28*d*x^9 + 70/3*x^9*e + 105/4*d*x^8 + 15*x^8*e + 120/7*d*x^7 + 4
5/7*x^7*e + 15/2*d*x^6 + 5/3*x^6*e + 2*d*x^5 + 1/5*x^5*e + 1/4*d*x^4