3.5.87 \(\int x^6 (d+e x) (1+2 x+x^2)^5 \, dx\)

Optimal. Leaf size=119 \[ \frac {1}{17} (x+1)^{17} (d-7 e)-\frac {3}{16} (x+1)^{16} (2 d-7 e)+\frac {1}{3} (x+1)^{15} (3 d-7 e)-\frac {5}{14} (x+1)^{14} (4 d-7 e)+\frac {3}{13} (x+1)^{13} (5 d-7 e)-\frac {1}{12} (x+1)^{12} (6 d-7 e)+\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{18} e (x+1)^{18} \]

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Rubi [A]  time = 0.07, antiderivative size = 119, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {1}{17} (x+1)^{17} (d-7 e)-\frac {3}{16} (x+1)^{16} (2 d-7 e)+\frac {1}{3} (x+1)^{15} (3 d-7 e)-\frac {5}{14} (x+1)^{14} (4 d-7 e)+\frac {3}{13} (x+1)^{13} (5 d-7 e)-\frac {1}{12} (x+1)^{12} (6 d-7 e)+\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{18} e (x+1)^{18} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^6*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

((d - e)*(1 + x)^11)/11 - ((6*d - 7*e)*(1 + x)^12)/12 + (3*(5*d - 7*e)*(1 + x)^13)/13 - (5*(4*d - 7*e)*(1 + x)
^14)/14 + ((3*d - 7*e)*(1 + x)^15)/3 - (3*(2*d - 7*e)*(1 + x)^16)/16 + ((d - 7*e)*(1 + x)^17)/17 + (e*(1 + x)^
18)/18

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^6 (d+e x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^6 (1+x)^{10} (d+e x) \, dx\\ &=\int \left ((d-e) (1+x)^{10}+(-6 d+7 e) (1+x)^{11}+3 (5 d-7 e) (1+x)^{12}-5 (4 d-7 e) (1+x)^{13}+5 (3 d-7 e) (1+x)^{14}-3 (2 d-7 e) (1+x)^{15}+(d-7 e) (1+x)^{16}+e (1+x)^{17}\right ) \, dx\\ &=\frac {1}{11} (d-e) (1+x)^{11}-\frac {1}{12} (6 d-7 e) (1+x)^{12}+\frac {3}{13} (5 d-7 e) (1+x)^{13}-\frac {5}{14} (4 d-7 e) (1+x)^{14}+\frac {1}{3} (3 d-7 e) (1+x)^{15}-\frac {3}{16} (2 d-7 e) (1+x)^{16}+\frac {1}{17} (d-7 e) (1+x)^{17}+\frac {1}{18} e (1+x)^{18}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 150, normalized size = 1.26 \begin {gather*} \frac {1}{17} x^{17} (d+10 e)+\frac {5}{16} x^{16} (2 d+9 e)+x^{15} (3 d+8 e)+\frac {15}{7} x^{14} (4 d+7 e)+\frac {42}{13} x^{13} (5 d+6 e)+\frac {7}{2} x^{12} (6 d+5 e)+\frac {30}{11} x^{11} (7 d+4 e)+\frac {3}{2} x^{10} (8 d+3 e)+\frac {5}{9} x^9 (9 d+2 e)+\frac {1}{8} x^8 (10 d+e)+\frac {d x^7}{7}+\frac {e x^{18}}{18} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^6*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

(d*x^7)/7 + ((10*d + e)*x^8)/8 + (5*(9*d + 2*e)*x^9)/9 + (3*(8*d + 3*e)*x^10)/2 + (30*(7*d + 4*e)*x^11)/11 + (
7*(6*d + 5*e)*x^12)/2 + (42*(5*d + 6*e)*x^13)/13 + (15*(4*d + 7*e)*x^14)/7 + (3*d + 8*e)*x^15 + (5*(2*d + 9*e)
*x^16)/16 + ((d + 10*e)*x^17)/17 + (e*x^18)/18

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^6 (d+e x) \left (1+2 x+x^2\right )^5 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^6*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

IntegrateAlgebraic[x^6*(d + e*x)*(1 + 2*x + x^2)^5, x]

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fricas [A]  time = 0.35, size = 133, normalized size = 1.12 \begin {gather*} \frac {1}{18} x^{18} e + \frac {10}{17} x^{17} e + \frac {1}{17} x^{17} d + \frac {45}{16} x^{16} e + \frac {5}{8} x^{16} d + 8 x^{15} e + 3 x^{15} d + 15 x^{14} e + \frac {60}{7} x^{14} d + \frac {252}{13} x^{13} e + \frac {210}{13} x^{13} d + \frac {35}{2} x^{12} e + 21 x^{12} d + \frac {120}{11} x^{11} e + \frac {210}{11} x^{11} d + \frac {9}{2} x^{10} e + 12 x^{10} d + \frac {10}{9} x^{9} e + 5 x^{9} d + \frac {1}{8} x^{8} e + \frac {5}{4} x^{8} d + \frac {1}{7} x^{7} d \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")

[Out]

1/18*x^18*e + 10/17*x^17*e + 1/17*x^17*d + 45/16*x^16*e + 5/8*x^16*d + 8*x^15*e + 3*x^15*d + 15*x^14*e + 60/7*
x^14*d + 252/13*x^13*e + 210/13*x^13*d + 35/2*x^12*e + 21*x^12*d + 120/11*x^11*e + 210/11*x^11*d + 9/2*x^10*e
+ 12*x^10*d + 10/9*x^9*e + 5*x^9*d + 1/8*x^8*e + 5/4*x^8*d + 1/7*x^7*d

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giac [A]  time = 0.15, size = 144, normalized size = 1.21 \begin {gather*} \frac {1}{18} \, x^{18} e + \frac {1}{17} \, d x^{17} + \frac {10}{17} \, x^{17} e + \frac {5}{8} \, d x^{16} + \frac {45}{16} \, x^{16} e + 3 \, d x^{15} + 8 \, x^{15} e + \frac {60}{7} \, d x^{14} + 15 \, x^{14} e + \frac {210}{13} \, d x^{13} + \frac {252}{13} \, x^{13} e + 21 \, d x^{12} + \frac {35}{2} \, x^{12} e + \frac {210}{11} \, d x^{11} + \frac {120}{11} \, x^{11} e + 12 \, d x^{10} + \frac {9}{2} \, x^{10} e + 5 \, d x^{9} + \frac {10}{9} \, x^{9} e + \frac {5}{4} \, d x^{8} + \frac {1}{8} \, x^{8} e + \frac {1}{7} \, d x^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")

[Out]

1/18*x^18*e + 1/17*d*x^17 + 10/17*x^17*e + 5/8*d*x^16 + 45/16*x^16*e + 3*d*x^15 + 8*x^15*e + 60/7*d*x^14 + 15*
x^14*e + 210/13*d*x^13 + 252/13*x^13*e + 21*d*x^12 + 35/2*x^12*e + 210/11*d*x^11 + 120/11*x^11*e + 12*d*x^10 +
9/2*x^10*e + 5*d*x^9 + 10/9*x^9*e + 5/4*d*x^8 + 1/8*x^8*e + 1/7*d*x^7

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maple [A]  time = 0.04, size = 130, normalized size = 1.09 \begin {gather*} \frac {e \,x^{18}}{18}+\frac {\left (d +10 e \right ) x^{17}}{17}+\frac {\left (10 d +45 e \right ) x^{16}}{16}+\frac {\left (45 d +120 e \right ) x^{15}}{15}+\frac {\left (120 d +210 e \right ) x^{14}}{14}+\frac {\left (210 d +252 e \right ) x^{13}}{13}+\frac {\left (252 d +210 e \right ) x^{12}}{12}+\frac {\left (210 d +120 e \right ) x^{11}}{11}+\frac {\left (120 d +45 e \right ) x^{10}}{10}+\frac {\left (45 d +10 e \right ) x^{9}}{9}+\frac {d \,x^{7}}{7}+\frac {\left (10 d +e \right ) x^{8}}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(e*x+d)*(x^2+2*x+1)^5,x)

[Out]

1/18*e*x^18+1/17*(d+10*e)*x^17+1/16*(10*d+45*e)*x^16+1/15*(45*d+120*e)*x^15+1/14*(120*d+210*e)*x^14+1/13*(210*
d+252*e)*x^13+1/12*(252*d+210*e)*x^12+1/11*(210*d+120*e)*x^11+1/10*(120*d+45*e)*x^10+1/9*(45*d+10*e)*x^9+1/8*(
10*d+e)*x^8+1/7*d*x^7

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maxima [A]  time = 0.63, size = 128, normalized size = 1.08 \begin {gather*} \frac {1}{18} \, e x^{18} + \frac {1}{17} \, {\left (d + 10 \, e\right )} x^{17} + \frac {5}{16} \, {\left (2 \, d + 9 \, e\right )} x^{16} + {\left (3 \, d + 8 \, e\right )} x^{15} + \frac {15}{7} \, {\left (4 \, d + 7 \, e\right )} x^{14} + \frac {42}{13} \, {\left (5 \, d + 6 \, e\right )} x^{13} + \frac {7}{2} \, {\left (6 \, d + 5 \, e\right )} x^{12} + \frac {30}{11} \, {\left (7 \, d + 4 \, e\right )} x^{11} + \frac {3}{2} \, {\left (8 \, d + 3 \, e\right )} x^{10} + \frac {5}{9} \, {\left (9 \, d + 2 \, e\right )} x^{9} + \frac {1}{8} \, {\left (10 \, d + e\right )} x^{8} + \frac {1}{7} \, d x^{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^6*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")

[Out]

1/18*e*x^18 + 1/17*(d + 10*e)*x^17 + 5/16*(2*d + 9*e)*x^16 + (3*d + 8*e)*x^15 + 15/7*(4*d + 7*e)*x^14 + 42/13*
(5*d + 6*e)*x^13 + 7/2*(6*d + 5*e)*x^12 + 30/11*(7*d + 4*e)*x^11 + 3/2*(8*d + 3*e)*x^10 + 5/9*(9*d + 2*e)*x^9
+ 1/8*(10*d + e)*x^8 + 1/7*d*x^7

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mupad [B]  time = 0.08, size = 123, normalized size = 1.03 \begin {gather*} \frac {e\,x^{18}}{18}+\left (\frac {d}{17}+\frac {10\,e}{17}\right )\,x^{17}+\left (\frac {5\,d}{8}+\frac {45\,e}{16}\right )\,x^{16}+\left (3\,d+8\,e\right )\,x^{15}+\left (\frac {60\,d}{7}+15\,e\right )\,x^{14}+\left (\frac {210\,d}{13}+\frac {252\,e}{13}\right )\,x^{13}+\left (21\,d+\frac {35\,e}{2}\right )\,x^{12}+\left (\frac {210\,d}{11}+\frac {120\,e}{11}\right )\,x^{11}+\left (12\,d+\frac {9\,e}{2}\right )\,x^{10}+\left (5\,d+\frac {10\,e}{9}\right )\,x^9+\left (\frac {5\,d}{4}+\frac {e}{8}\right )\,x^8+\frac {d\,x^7}{7} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^6*(d + e*x)*(2*x + x^2 + 1)^5,x)

[Out]

x^8*((5*d)/4 + e/8) + x^15*(3*d + 8*e) + x^9*(5*d + (10*e)/9) + x^10*(12*d + (9*e)/2) + x^17*(d/17 + (10*e)/17
) + x^12*(21*d + (35*e)/2) + x^16*((5*d)/8 + (45*e)/16) + x^14*((60*d)/7 + 15*e) + x^11*((210*d)/11 + (120*e)/
11) + x^13*((210*d)/13 + (252*e)/13) + (d*x^7)/7 + (e*x^18)/18

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sympy [A]  time = 0.10, size = 134, normalized size = 1.13 \begin {gather*} \frac {d x^{7}}{7} + \frac {e x^{18}}{18} + x^{17} \left (\frac {d}{17} + \frac {10 e}{17}\right ) + x^{16} \left (\frac {5 d}{8} + \frac {45 e}{16}\right ) + x^{15} \left (3 d + 8 e\right ) + x^{14} \left (\frac {60 d}{7} + 15 e\right ) + x^{13} \left (\frac {210 d}{13} + \frac {252 e}{13}\right ) + x^{12} \left (21 d + \frac {35 e}{2}\right ) + x^{11} \left (\frac {210 d}{11} + \frac {120 e}{11}\right ) + x^{10} \left (12 d + \frac {9 e}{2}\right ) + x^{9} \left (5 d + \frac {10 e}{9}\right ) + x^{8} \left (\frac {5 d}{4} + \frac {e}{8}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**6*(e*x+d)*(x**2+2*x+1)**5,x)

[Out]

d*x**7/7 + e*x**18/18 + x**17*(d/17 + 10*e/17) + x**16*(5*d/8 + 45*e/16) + x**15*(3*d + 8*e) + x**14*(60*d/7 +
15*e) + x**13*(210*d/13 + 252*e/13) + x**12*(21*d + 35*e/2) + x**11*(210*d/11 + 120*e/11) + x**10*(12*d + 9*e
/2) + x**9*(5*d + 10*e/9) + x**8*(5*d/4 + e/8)

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