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]

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

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Rubi [A]  time = 0.06, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {43} \[ \frac {21 a^2 (a+b x)^{16}}{16 b^8}-\frac {7 a^3 (a+b x)^{15}}{3 b^8}+\frac {5 a^4 (a+b x)^{14}}{2 b^8}-\frac {21 a^5 (a+b x)^{13}}{13 b^8}+\frac {7 a^6 (a+b x)^{12}}{12 b^8}-\frac {a^7 (a+b x)^{11}}{11 b^8}+\frac {(a+b x)^{18}}{18 b^8}-\frac {7 a (a+b x)^{17}}{17 b^8} \]

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) + (21*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)

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 x^7 (a+b x)^{10} \, dx &=\int \left (-\frac {a^7 (a+b x)^{10}}{b^7}+\frac {7 a^6 (a+b x)^{11}}{b^7}-\frac {21 a^5 (a+b x)^{12}}{b^7}+\frac {35 a^4 (a+b x)^{13}}{b^7}-\frac {35 a^3 (a+b x)^{14}}{b^7}+\frac {21 a^2 (a+b x)^{15}}{b^7}-\frac {7 a (a+b x)^{16}}{b^7}+\frac {(a+b x)^{17}}{b^7}\right ) \, dx\\ &=-\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 {7 a (a+b x)^{17}}{17 b^8}+\frac {(a+b x)^{18}}{18 b^8}\\ \end {align*}

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Mathematica [A]  time = 0.00, 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

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fricas [A]  time = 0.65, size = 112, normalized size = 0.85 \[ \frac {1}{18} x^{18} b^{10} + \frac {10}{17} x^{17} b^{9} a + \frac {45}{16} x^{16} b^{8} a^{2} + 8 x^{15} b^{7} a^{3} + 15 x^{14} b^{6} a^{4} + \frac {252}{13} x^{13} b^{5} a^{5} + \frac {35}{2} x^{12} b^{4} a^{6} + \frac {120}{11} x^{11} b^{3} a^{7} + \frac {9}{2} x^{10} b^{2} a^{8} + \frac {10}{9} x^{9} b a^{9} + \frac {1}{8} x^{8} a^{10} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x+a)^10,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^14*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

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giac [A]  time = 0.84, size = 112, normalized size = 0.85 \[ \frac {1}{18} \, b^{10} x^{18} + \frac {10}{17} \, a b^{9} x^{17} + \frac {45}{16} \, a^{2} b^{8} x^{16} + 8 \, a^{3} b^{7} x^{15} + 15 \, a^{4} b^{6} x^{14} + \frac {252}{13} \, a^{5} b^{5} x^{13} + \frac {35}{2} \, a^{6} b^{4} x^{12} + \frac {120}{11} \, a^{7} b^{3} x^{11} + \frac {9}{2} \, a^{8} b^{2} x^{10} + \frac {10}{9} \, a^{9} b x^{9} + \frac {1}{8} \, a^{10} x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x+a)^10,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^1
3 + 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

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maple [A]  time = 0.00, size = 113, normalized size = 0.86 \[ \frac {1}{18} b^{10} x^{18}+\frac {10}{17} a \,b^{9} x^{17}+\frac {45}{16} a^{2} b^{8} x^{16}+8 a^{3} b^{7} x^{15}+15 a^{4} b^{6} x^{14}+\frac {252}{13} a^{5} b^{5} x^{13}+\frac {35}{2} a^{6} b^{4} x^{12}+\frac {120}{11} a^{7} b^{3} x^{11}+\frac {9}{2} a^{8} b^{2} x^{10}+\frac {10}{9} a^{9} b \,x^{9}+\frac {1}{8} a^{10} x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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^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

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maxima [A]  time = 1.34, size = 112, normalized size = 0.85 \[ \frac {1}{18} \, b^{10} x^{18} + \frac {10}{17} \, a b^{9} x^{17} + \frac {45}{16} \, a^{2} b^{8} x^{16} + 8 \, a^{3} b^{7} x^{15} + 15 \, a^{4} b^{6} x^{14} + \frac {252}{13} \, a^{5} b^{5} x^{13} + \frac {35}{2} \, a^{6} b^{4} x^{12} + \frac {120}{11} \, a^{7} b^{3} x^{11} + \frac {9}{2} \, a^{8} b^{2} x^{10} + \frac {10}{9} \, a^{9} b x^{9} + \frac {1}{8} \, a^{10} x^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^7*(b*x+a)^10,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^1
3 + 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

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mupad [B]  time = 0.08, size = 112, normalized size = 0.85 \[ \frac {a^{10}\,x^8}{8}+\frac {10\,a^9\,b\,x^9}{9}+\frac {9\,a^8\,b^2\,x^{10}}{2}+\frac {120\,a^7\,b^3\,x^{11}}{11}+\frac {35\,a^6\,b^4\,x^{12}}{2}+\frac {252\,a^5\,b^5\,x^{13}}{13}+15\,a^4\,b^6\,x^{14}+8\,a^3\,b^7\,x^{15}+\frac {45\,a^2\,b^8\,x^{16}}{16}+\frac {10\,a\,b^9\,x^{17}}{17}+\frac {b^{10}\,x^{18}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^7*(a + b*x)^10,x)

[Out]

(a^10*x^8)/8 + (b^10*x^18)/18 + (10*a^9*b*x^9)/9 + (10*a*b^9*x^17)/17 + (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

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sympy [A]  time = 0.11, 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

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