3.5.86 \(\int x^7 (d+e x) (1+2 x+x^2)^5 \, dx\)

Optimal. Leaf size=131 \[ \frac {1}{18} (x+1)^{18} (d-8 e)-\frac {7}{17} (x+1)^{17} (d-4 e)+\frac {7}{16} (x+1)^{16} (3 d-8 e)-\frac {7}{3} (x+1)^{15} (d-2 e)+\frac {1}{2} (x+1)^{14} (5 d-8 e)-\frac {7}{13} (x+1)^{13} (3 d-4 e)+\frac {1}{12} (x+1)^{12} (7 d-8 e)-\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{19} e (x+1)^{19} \]

________________________________________________________________________________________

Rubi [A]  time = 0.13, antiderivative size = 131, 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}{18} (x+1)^{18} (d-8 e)-\frac {7}{17} (x+1)^{17} (d-4 e)+\frac {7}{16} (x+1)^{16} (3 d-8 e)-\frac {7}{3} (x+1)^{15} (d-2 e)+\frac {1}{2} (x+1)^{14} (5 d-8 e)-\frac {7}{13} (x+1)^{13} (3 d-4 e)+\frac {1}{12} (x+1)^{12} (7 d-8 e)-\frac {1}{11} (x+1)^{11} (d-e)+\frac {1}{19} e (x+1)^{19} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^7*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

-((d - e)*(1 + x)^11)/11 + ((7*d - 8*e)*(1 + x)^12)/12 - (7*(3*d - 4*e)*(1 + x)^13)/13 + ((5*d - 8*e)*(1 + x)^
14)/2 - (7*(d - 2*e)*(1 + x)^15)/3 + (7*(3*d - 8*e)*(1 + x)^16)/16 - (7*(d - 4*e)*(1 + x)^17)/17 + ((d - 8*e)*
(1 + x)^18)/18 + (e*(1 + x)^19)/19

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^7 (d+e x) \left (1+2 x+x^2\right )^5 \, dx &=\int x^7 (1+x)^{10} (d+e x) \, dx\\ &=\int \left ((-d+e) (1+x)^{10}+(7 d-8 e) (1+x)^{11}-7 (3 d-4 e) (1+x)^{12}+7 (5 d-8 e) (1+x)^{13}-35 (d-2 e) (1+x)^{14}+7 (3 d-8 e) (1+x)^{15}-7 (d-4 e) (1+x)^{16}+(d-8 e) (1+x)^{17}+e (1+x)^{18}\right ) \, dx\\ &=-\frac {1}{11} (d-e) (1+x)^{11}+\frac {1}{12} (7 d-8 e) (1+x)^{12}-\frac {7}{13} (3 d-4 e) (1+x)^{13}+\frac {1}{2} (5 d-8 e) (1+x)^{14}-\frac {7}{3} (d-2 e) (1+x)^{15}+\frac {7}{16} (3 d-8 e) (1+x)^{16}-\frac {7}{17} (d-4 e) (1+x)^{17}+\frac {1}{18} (d-8 e) (1+x)^{18}+\frac {1}{19} e (1+x)^{19}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.03, size = 149, normalized size = 1.14 \begin {gather*} \frac {1}{18} x^{18} (d+10 e)+\frac {5}{17} x^{17} (2 d+9 e)+\frac {15}{16} x^{16} (3 d+8 e)+2 x^{15} (4 d+7 e)+3 x^{14} (5 d+6 e)+\frac {42}{13} x^{13} (6 d+5 e)+\frac {5}{2} x^{12} (7 d+4 e)+\frac {15}{11} x^{11} (8 d+3 e)+\frac {1}{2} x^{10} (9 d+2 e)+\frac {1}{9} x^9 (10 d+e)+\frac {d x^8}{8}+\frac {e x^{19}}{19} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^7*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

(d*x^8)/8 + ((10*d + e)*x^9)/9 + ((9*d + 2*e)*x^10)/2 + (15*(8*d + 3*e)*x^11)/11 + (5*(7*d + 4*e)*x^12)/2 + (4
2*(6*d + 5*e)*x^13)/13 + 3*(5*d + 6*e)*x^14 + 2*(4*d + 7*e)*x^15 + (15*(3*d + 8*e)*x^16)/16 + (5*(2*d + 9*e)*x
^17)/17 + ((d + 10*e)*x^18)/18 + (e*x^19)/19

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^7 (d+e x) \left (1+2 x+x^2\right )^5 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^7*(d + e*x)*(1 + 2*x + x^2)^5,x]

[Out]

IntegrateAlgebraic[x^7*(d + e*x)*(1 + 2*x + x^2)^5, x]

________________________________________________________________________________________

fricas [A]  time = 0.37, size = 132, normalized size = 1.01 \begin {gather*} \frac {1}{19} x^{19} e + \frac {5}{9} x^{18} e + \frac {1}{18} x^{18} d + \frac {45}{17} x^{17} e + \frac {10}{17} x^{17} d + \frac {15}{2} x^{16} e + \frac {45}{16} x^{16} d + 14 x^{15} e + 8 x^{15} d + 18 x^{14} e + 15 x^{14} d + \frac {210}{13} x^{13} e + \frac {252}{13} x^{13} d + 10 x^{12} e + \frac {35}{2} x^{12} d + \frac {45}{11} x^{11} e + \frac {120}{11} x^{11} d + x^{10} e + \frac {9}{2} x^{10} d + \frac {1}{9} x^{9} e + \frac {10}{9} x^{9} d + \frac {1}{8} x^{8} d \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="fricas")

[Out]

1/19*x^19*e + 5/9*x^18*e + 1/18*x^18*d + 45/17*x^17*e + 10/17*x^17*d + 15/2*x^16*e + 45/16*x^16*d + 14*x^15*e
+ 8*x^15*d + 18*x^14*e + 15*x^14*d + 210/13*x^13*e + 252/13*x^13*d + 10*x^12*e + 35/2*x^12*d + 45/11*x^11*e +
120/11*x^11*d + x^10*e + 9/2*x^10*d + 1/9*x^9*e + 10/9*x^9*d + 1/8*x^8*d

________________________________________________________________________________________

giac [A]  time = 0.17, size = 143, normalized size = 1.09 \begin {gather*} \frac {1}{19} \, x^{19} e + \frac {1}{18} \, d x^{18} + \frac {5}{9} \, x^{18} e + \frac {10}{17} \, d x^{17} + \frac {45}{17} \, x^{17} e + \frac {45}{16} \, d x^{16} + \frac {15}{2} \, x^{16} e + 8 \, d x^{15} + 14 \, x^{15} e + 15 \, d x^{14} + 18 \, x^{14} e + \frac {252}{13} \, d x^{13} + \frac {210}{13} \, x^{13} e + \frac {35}{2} \, d x^{12} + 10 \, x^{12} e + \frac {120}{11} \, d x^{11} + \frac {45}{11} \, x^{11} e + \frac {9}{2} \, d x^{10} + x^{10} e + \frac {10}{9} \, d x^{9} + \frac {1}{9} \, x^{9} e + \frac {1}{8} \, d x^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="giac")

[Out]

1/19*x^19*e + 1/18*d*x^18 + 5/9*x^18*e + 10/17*d*x^17 + 45/17*x^17*e + 45/16*d*x^16 + 15/2*x^16*e + 8*d*x^15 +
14*x^15*e + 15*d*x^14 + 18*x^14*e + 252/13*d*x^13 + 210/13*x^13*e + 35/2*d*x^12 + 10*x^12*e + 120/11*d*x^11 +
45/11*x^11*e + 9/2*d*x^10 + x^10*e + 10/9*d*x^9 + 1/9*x^9*e + 1/8*d*x^8

________________________________________________________________________________________

maple [A]  time = 0.05, size = 130, normalized size = 0.99 \begin {gather*} \frac {e \,x^{19}}{19}+\frac {\left (d +10 e \right ) x^{18}}{18}+\frac {\left (10 d +45 e \right ) x^{17}}{17}+\frac {\left (45 d +120 e \right ) x^{16}}{16}+\frac {\left (120 d +210 e \right ) x^{15}}{15}+\frac {\left (210 d +252 e \right ) x^{14}}{14}+\frac {\left (252 d +210 e \right ) x^{13}}{13}+\frac {\left (210 d +120 e \right ) x^{12}}{12}+\frac {\left (120 d +45 e \right ) x^{11}}{11}+\frac {\left (45 d +10 e \right ) x^{10}}{10}+\frac {d \,x^{8}}{8}+\frac {\left (10 d +e \right ) x^{9}}{9} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(e*x+d)*(x^2+2*x+1)^5,x)

[Out]

1/19*e*x^19+1/18*(d+10*e)*x^18+1/17*(10*d+45*e)*x^17+1/16*(45*d+120*e)*x^16+1/15*(120*d+210*e)*x^15+1/14*(210*
d+252*e)*x^14+1/13*(252*d+210*e)*x^13+1/12*(210*d+120*e)*x^12+1/11*(120*d+45*e)*x^11+1/10*(45*d+10*e)*x^10+1/9
*(10*d+e)*x^9+1/8*d*x^8

________________________________________________________________________________________

maxima [A]  time = 0.48, size = 129, normalized size = 0.98 \begin {gather*} \frac {1}{19} \, e x^{19} + \frac {1}{18} \, {\left (d + 10 \, e\right )} x^{18} + \frac {5}{17} \, {\left (2 \, d + 9 \, e\right )} x^{17} + \frac {15}{16} \, {\left (3 \, d + 8 \, e\right )} x^{16} + 2 \, {\left (4 \, d + 7 \, e\right )} x^{15} + 3 \, {\left (5 \, d + 6 \, e\right )} x^{14} + \frac {42}{13} \, {\left (6 \, d + 5 \, e\right )} x^{13} + \frac {5}{2} \, {\left (7 \, d + 4 \, e\right )} x^{12} + \frac {15}{11} \, {\left (8 \, d + 3 \, e\right )} x^{11} + \frac {1}{2} \, {\left (9 \, d + 2 \, e\right )} x^{10} + \frac {1}{9} \, {\left (10 \, d + e\right )} x^{9} + \frac {1}{8} \, d x^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(e*x+d)*(x^2+2*x+1)^5,x, algorithm="maxima")

[Out]

1/19*e*x^19 + 1/18*(d + 10*e)*x^18 + 5/17*(2*d + 9*e)*x^17 + 15/16*(3*d + 8*e)*x^16 + 2*(4*d + 7*e)*x^15 + 3*(
5*d + 6*e)*x^14 + 42/13*(6*d + 5*e)*x^13 + 5/2*(7*d + 4*e)*x^12 + 15/11*(8*d + 3*e)*x^11 + 1/2*(9*d + 2*e)*x^1
0 + 1/9*(10*d + e)*x^9 + 1/8*d*x^8

________________________________________________________________________________________

mupad [B]  time = 1.12, size = 121, normalized size = 0.92 \begin {gather*} \frac {e\,x^{19}}{19}+\left (\frac {d}{18}+\frac {5\,e}{9}\right )\,x^{18}+\left (\frac {10\,d}{17}+\frac {45\,e}{17}\right )\,x^{17}+\left (\frac {45\,d}{16}+\frac {15\,e}{2}\right )\,x^{16}+\left (8\,d+14\,e\right )\,x^{15}+\left (15\,d+18\,e\right )\,x^{14}+\left (\frac {252\,d}{13}+\frac {210\,e}{13}\right )\,x^{13}+\left (\frac {35\,d}{2}+10\,e\right )\,x^{12}+\left (\frac {120\,d}{11}+\frac {45\,e}{11}\right )\,x^{11}+\left (\frac {9\,d}{2}+e\right )\,x^{10}+\left (\frac {10\,d}{9}+\frac {e}{9}\right )\,x^9+\frac {d\,x^8}{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(d + e*x)*(2*x + x^2 + 1)^5,x)

[Out]

x^15*(8*d + 14*e) + x^9*((10*d)/9 + e/9) + x^14*(15*d + 18*e) + x^18*(d/18 + (5*e)/9) + x^12*((35*d)/2 + 10*e)
+ x^16*((45*d)/16 + (15*e)/2) + x^17*((10*d)/17 + (45*e)/17) + x^11*((120*d)/11 + (45*e)/11) + x^13*((252*d)/
13 + (210*e)/13) + (d*x^8)/8 + (e*x^19)/19 + x^10*((9*d)/2 + e)

________________________________________________________________________________________

sympy [A]  time = 0.10, size = 133, normalized size = 1.02 \begin {gather*} \frac {d x^{8}}{8} + \frac {e x^{19}}{19} + x^{18} \left (\frac {d}{18} + \frac {5 e}{9}\right ) + x^{17} \left (\frac {10 d}{17} + \frac {45 e}{17}\right ) + x^{16} \left (\frac {45 d}{16} + \frac {15 e}{2}\right ) + x^{15} \left (8 d + 14 e\right ) + x^{14} \left (15 d + 18 e\right ) + x^{13} \left (\frac {252 d}{13} + \frac {210 e}{13}\right ) + x^{12} \left (\frac {35 d}{2} + 10 e\right ) + x^{11} \left (\frac {120 d}{11} + \frac {45 e}{11}\right ) + x^{10} \left (\frac {9 d}{2} + e\right ) + x^{9} \left (\frac {10 d}{9} + \frac {e}{9}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**7*(e*x+d)*(x**2+2*x+1)**5,x)

[Out]

d*x**8/8 + e*x**19/19 + x**18*(d/18 + 5*e/9) + x**17*(10*d/17 + 45*e/17) + x**16*(45*d/16 + 15*e/2) + x**15*(8
*d + 14*e) + x**14*(15*d + 18*e) + x**13*(252*d/13 + 210*e/13) + x**12*(35*d/2 + 10*e) + x**11*(120*d/11 + 45*
e/11) + x**10*(9*d/2 + e) + x**9*(10*d/9 + e/9)

________________________________________________________________________________________