3.127 \(\int x^7 (a+b x)^{10} \, dx\)

Optimal. Leaf size=132 \[ -\frac{a^7 (a+b x)^{11}}{11 b^8}+\frac{7 a^6 (a+b x)^{12}}{12 b^8}-\frac{21 a^5 (a+b x)^{13}}{13 b^8}+\frac{5 a^4 (a+b x)^{14}}{2 b^8}-\frac{7 a^3 (a+b x)^{15}}{3 b^8}+\frac{21 a^2 (a+b x)^{16}}{16 b^8}+\frac{(a+b x)^{18}}{18 b^8}-\frac{7 a (a+b x)^{17}}{17 b^8} \]

[Out]

-(a^7*(a + b*x)^11)/(11*b^8) + (7*a^6*(a + b*x)^12)/(12*b^8) - (21*a^5*(a + b*x)
^13)/(13*b^8) + (5*a^4*(a + b*x)^14)/(2*b^8) - (7*a^3*(a + b*x)^15)/(3*b^8) + (2
1*a^2*(a + b*x)^16)/(16*b^8) - (7*a*(a + b*x)^17)/(17*b^8) + (a + b*x)^18/(18*b^
8)

_______________________________________________________________________________________

Rubi [A]  time = 0.139521, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a^7 (a+b x)^{11}}{11 b^8}+\frac{7 a^6 (a+b x)^{12}}{12 b^8}-\frac{21 a^5 (a+b x)^{13}}{13 b^8}+\frac{5 a^4 (a+b x)^{14}}{2 b^8}-\frac{7 a^3 (a+b x)^{15}}{3 b^8}+\frac{21 a^2 (a+b x)^{16}}{16 b^8}+\frac{(a+b x)^{18}}{18 b^8}-\frac{7 a (a+b x)^{17}}{17 b^8} \]

Antiderivative was successfully verified.

[In]  Int[x^7*(a + b*x)^10,x]

[Out]

-(a^7*(a + b*x)^11)/(11*b^8) + (7*a^6*(a + b*x)^12)/(12*b^8) - (21*a^5*(a + b*x)
^13)/(13*b^8) + (5*a^4*(a + b*x)^14)/(2*b^8) - (7*a^3*(a + b*x)^15)/(3*b^8) + (2
1*a^2*(a + b*x)^16)/(16*b^8) - (7*a*(a + b*x)^17)/(17*b^8) + (a + b*x)^18/(18*b^
8)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 26.2368, size = 131, normalized size = 0.99 \[ \frac{a^{10} x^{8}}{8} + \frac{10 a^{9} b x^{9}}{9} + \frac{9 a^{8} b^{2} x^{10}}{2} + \frac{120 a^{7} b^{3} x^{11}}{11} + \frac{35 a^{6} b^{4} x^{12}}{2} + \frac{252 a^{5} b^{5} x^{13}}{13} + 15 a^{4} b^{6} x^{14} + 8 a^{3} b^{7} x^{15} + \frac{45 a^{2} b^{8} x^{16}}{16} + \frac{10 a b^{9} x^{17}}{17} + \frac{b^{10} x^{18}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7*(b*x+a)**10,x)

[Out]

a**10*x**8/8 + 10*a**9*b*x**9/9 + 9*a**8*b**2*x**10/2 + 120*a**7*b**3*x**11/11 +
35*a**6*b**4*x**12/2 + 252*a**5*b**5*x**13/13 + 15*a**4*b**6*x**14 + 8*a**3*b**
7*x**15 + 45*a**2*b**8*x**16/16 + 10*a*b**9*x**17/17 + b**10*x**18/18

_______________________________________________________________________________________

Mathematica [A]  time = 0.00421802, size = 130, normalized size = 0.98 \[ \frac{a^{10} x^8}{8}+\frac{10}{9} a^9 b x^9+\frac{9}{2} a^8 b^2 x^{10}+\frac{120}{11} a^7 b^3 x^{11}+\frac{35}{2} a^6 b^4 x^{12}+\frac{252}{13} a^5 b^5 x^{13}+15 a^4 b^6 x^{14}+8 a^3 b^7 x^{15}+\frac{45}{16} a^2 b^8 x^{16}+\frac{10}{17} a b^9 x^{17}+\frac{b^{10} x^{18}}{18} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7*(a + b*x)^10,x]

[Out]

(a^10*x^8)/8 + (10*a^9*b*x^9)/9 + (9*a^8*b^2*x^10)/2 + (120*a^7*b^3*x^11)/11 + (
35*a^6*b^4*x^12)/2 + (252*a^5*b^5*x^13)/13 + 15*a^4*b^6*x^14 + 8*a^3*b^7*x^15 +
(45*a^2*b^8*x^16)/16 + (10*a*b^9*x^17)/17 + (b^10*x^18)/18

_______________________________________________________________________________________

Maple [A]  time = 0.001, size = 113, normalized size = 0.9 \[{\frac{{b}^{10}{x}^{18}}{18}}+{\frac{10\,a{b}^{9}{x}^{17}}{17}}+{\frac{45\,{a}^{2}{b}^{8}{x}^{16}}{16}}+8\,{a}^{3}{b}^{7}{x}^{15}+15\,{a}^{4}{b}^{6}{x}^{14}+{\frac{252\,{a}^{5}{b}^{5}{x}^{13}}{13}}+{\frac{35\,{a}^{6}{b}^{4}{x}^{12}}{2}}+{\frac{120\,{a}^{7}{b}^{3}{x}^{11}}{11}}+{\frac{9\,{a}^{8}{b}^{2}{x}^{10}}{2}}+{\frac{10\,{a}^{9}b{x}^{9}}{9}}+{\frac{{a}^{10}{x}^{8}}{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7*(b*x+a)^10,x)

[Out]

1/18*b^10*x^18+10/17*a*b^9*x^17+45/16*a^2*b^8*x^16+8*a^3*b^7*x^15+15*a^4*b^6*x^1
4+252/13*a^5*b^5*x^13+35/2*a^6*b^4*x^12+120/11*a^7*b^3*x^11+9/2*a^8*b^2*x^10+10/
9*a^9*b*x^9+1/8*a^10*x^8

_______________________________________________________________________________________

Maxima [A]  time = 1.33982, size = 151, normalized size = 1.14 \[ \frac{1}{18} \, b^{10} x^{18} + \frac{10}{17} \, a b^{9} x^{17} + \frac{45}{16} \, a^{2} b^{8} x^{16} + 8 \, a^{3} b^{7} x^{15} + 15 \, a^{4} b^{6} x^{14} + \frac{252}{13} \, a^{5} b^{5} x^{13} + \frac{35}{2} \, a^{6} b^{4} x^{12} + \frac{120}{11} \, a^{7} b^{3} x^{11} + \frac{9}{2} \, a^{8} b^{2} x^{10} + \frac{10}{9} \, a^{9} b x^{9} + \frac{1}{8} \, a^{10} x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10*x^7,x, algorithm="maxima")

[Out]

1/18*b^10*x^18 + 10/17*a*b^9*x^17 + 45/16*a^2*b^8*x^16 + 8*a^3*b^7*x^15 + 15*a^4
*b^6*x^14 + 252/13*a^5*b^5*x^13 + 35/2*a^6*b^4*x^12 + 120/11*a^7*b^3*x^11 + 9/2*
a^8*b^2*x^10 + 10/9*a^9*b*x^9 + 1/8*a^10*x^8

_______________________________________________________________________________________

Fricas [A]  time = 0.176141, size = 1, normalized size = 0.01 \[ \frac{1}{18} x^{18} b^{10} + \frac{10}{17} x^{17} b^{9} a + \frac{45}{16} x^{16} b^{8} a^{2} + 8 x^{15} b^{7} a^{3} + 15 x^{14} b^{6} a^{4} + \frac{252}{13} x^{13} b^{5} a^{5} + \frac{35}{2} x^{12} b^{4} a^{6} + \frac{120}{11} x^{11} b^{3} a^{7} + \frac{9}{2} x^{10} b^{2} a^{8} + \frac{10}{9} x^{9} b a^{9} + \frac{1}{8} x^{8} a^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10*x^7,x, algorithm="fricas")

[Out]

1/18*x^18*b^10 + 10/17*x^17*b^9*a + 45/16*x^16*b^8*a^2 + 8*x^15*b^7*a^3 + 15*x^1
4*b^6*a^4 + 252/13*x^13*b^5*a^5 + 35/2*x^12*b^4*a^6 + 120/11*x^11*b^3*a^7 + 9/2*
x^10*b^2*a^8 + 10/9*x^9*b*a^9 + 1/8*x^8*a^10

_______________________________________________________________________________________

Sympy [A]  time = 0.173615, size = 131, normalized size = 0.99 \[ \frac{a^{10} x^{8}}{8} + \frac{10 a^{9} b x^{9}}{9} + \frac{9 a^{8} b^{2} x^{10}}{2} + \frac{120 a^{7} b^{3} x^{11}}{11} + \frac{35 a^{6} b^{4} x^{12}}{2} + \frac{252 a^{5} b^{5} x^{13}}{13} + 15 a^{4} b^{6} x^{14} + 8 a^{3} b^{7} x^{15} + \frac{45 a^{2} b^{8} x^{16}}{16} + \frac{10 a b^{9} x^{17}}{17} + \frac{b^{10} x^{18}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7*(b*x+a)**10,x)

[Out]

a**10*x**8/8 + 10*a**9*b*x**9/9 + 9*a**8*b**2*x**10/2 + 120*a**7*b**3*x**11/11 +
35*a**6*b**4*x**12/2 + 252*a**5*b**5*x**13/13 + 15*a**4*b**6*x**14 + 8*a**3*b**
7*x**15 + 45*a**2*b**8*x**16/16 + 10*a*b**9*x**17/17 + b**10*x**18/18

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.200192, size = 151, normalized size = 1.14 \[ \frac{1}{18} \, b^{10} x^{18} + \frac{10}{17} \, a b^{9} x^{17} + \frac{45}{16} \, a^{2} b^{8} x^{16} + 8 \, a^{3} b^{7} x^{15} + 15 \, a^{4} b^{6} x^{14} + \frac{252}{13} \, a^{5} b^{5} x^{13} + \frac{35}{2} \, a^{6} b^{4} x^{12} + \frac{120}{11} \, a^{7} b^{3} x^{11} + \frac{9}{2} \, a^{8} b^{2} x^{10} + \frac{10}{9} \, a^{9} b x^{9} + \frac{1}{8} \, a^{10} x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^10*x^7,x, algorithm="giac")

[Out]

1/18*b^10*x^18 + 10/17*a*b^9*x^17 + 45/16*a^2*b^8*x^16 + 8*a^3*b^7*x^15 + 15*a^4
*b^6*x^14 + 252/13*a^5*b^5*x^13 + 35/2*a^6*b^4*x^12 + 120/11*a^7*b^3*x^11 + 9/2*
a^8*b^2*x^10 + 10/9*a^9*b*x^9 + 1/8*a^10*x^8