3.2.25 \(\int x^9 (a+b x)^{10} \, dx\) [125]

Optimal. Leaf size=132 \[ \frac {a^{10} x^{10}}{10}+\frac {10}{11} a^9 b x^{11}+\frac {15}{4} a^8 b^2 x^{12}+\frac {120}{13} a^7 b^3 x^{13}+15 a^6 b^4 x^{14}+\frac {84}{5} a^5 b^5 x^{15}+\frac {105}{8} a^4 b^6 x^{16}+\frac {120}{17} a^3 b^7 x^{17}+\frac {5}{2} a^2 b^8 x^{18}+\frac {10}{19} a b^9 x^{19}+\frac {b^{10} x^{20}}{20} \]

[Out]

1/10*a^10*x^10+10/11*a^9*b*x^11+15/4*a^8*b^2*x^12+120/13*a^7*b^3*x^13+15*a^6*b^4*x^14+84/5*a^5*b^5*x^15+105/8*
a^4*b^6*x^16+120/17*a^3*b^7*x^17+5/2*a^2*b^8*x^18+10/19*a*b^9*x^19+1/20*b^10*x^20

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Rubi [A]
time = 0.04, 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 = {45} \begin {gather*} \frac {a^{10} x^{10}}{10}+\frac {10}{11} a^9 b x^{11}+\frac {15}{4} a^8 b^2 x^{12}+\frac {120}{13} a^7 b^3 x^{13}+15 a^6 b^4 x^{14}+\frac {84}{5} a^5 b^5 x^{15}+\frac {105}{8} a^4 b^6 x^{16}+\frac {120}{17} a^3 b^7 x^{17}+\frac {5}{2} a^2 b^8 x^{18}+\frac {10}{19} a b^9 x^{19}+\frac {b^{10} x^{20}}{20} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^9*(a + b*x)^10,x]

[Out]

(a^10*x^10)/10 + (10*a^9*b*x^11)/11 + (15*a^8*b^2*x^12)/4 + (120*a^7*b^3*x^13)/13 + 15*a^6*b^4*x^14 + (84*a^5*
b^5*x^15)/5 + (105*a^4*b^6*x^16)/8 + (120*a^3*b^7*x^17)/17 + (5*a^2*b^8*x^18)/2 + (10*a*b^9*x^19)/19 + (b^10*x
^20)/20

Rule 45

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^9 (a+b x)^{10} \, dx &=\int \left (a^{10} x^9+10 a^9 b x^{10}+45 a^8 b^2 x^{11}+120 a^7 b^3 x^{12}+210 a^6 b^4 x^{13}+252 a^5 b^5 x^{14}+210 a^4 b^6 x^{15}+120 a^3 b^7 x^{16}+45 a^2 b^8 x^{17}+10 a b^9 x^{18}+b^{10} x^{19}\right ) \, dx\\ &=\frac {a^{10} x^{10}}{10}+\frac {10}{11} a^9 b x^{11}+\frac {15}{4} a^8 b^2 x^{12}+\frac {120}{13} a^7 b^3 x^{13}+15 a^6 b^4 x^{14}+\frac {84}{5} a^5 b^5 x^{15}+\frac {105}{8} a^4 b^6 x^{16}+\frac {120}{17} a^3 b^7 x^{17}+\frac {5}{2} a^2 b^8 x^{18}+\frac {10}{19} a b^9 x^{19}+\frac {b^{10} x^{20}}{20}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 132, normalized size = 1.00 \begin {gather*} \frac {a^{10} x^{10}}{10}+\frac {10}{11} a^9 b x^{11}+\frac {15}{4} a^8 b^2 x^{12}+\frac {120}{13} a^7 b^3 x^{13}+15 a^6 b^4 x^{14}+\frac {84}{5} a^5 b^5 x^{15}+\frac {105}{8} a^4 b^6 x^{16}+\frac {120}{17} a^3 b^7 x^{17}+\frac {5}{2} a^2 b^8 x^{18}+\frac {10}{19} a b^9 x^{19}+\frac {b^{10} x^{20}}{20} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^9*(a + b*x)^10,x]

[Out]

(a^10*x^10)/10 + (10*a^9*b*x^11)/11 + (15*a^8*b^2*x^12)/4 + (120*a^7*b^3*x^13)/13 + 15*a^6*b^4*x^14 + (84*a^5*
b^5*x^15)/5 + (105*a^4*b^6*x^16)/8 + (120*a^3*b^7*x^17)/17 + (5*a^2*b^8*x^18)/2 + (10*a*b^9*x^19)/19 + (b^10*x
^20)/20

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Maple [A]
time = 0.08, size = 113, normalized size = 0.86

method	result	size
gosper	\(\frac {1}{10} a^{10} x^{10}+\frac {10}{11} a^{9} b \,x^{11}+\frac {15}{4} a^{8} b^{2} x^{12}+\frac {120}{13} a^{7} b^{3} x^{13}+15 a^{6} b^{4} x^{14}+\frac {84}{5} a^{5} b^{5} x^{15}+\frac {105}{8} a^{4} b^{6} x^{16}+\frac {120}{17} a^{3} b^{7} x^{17}+\frac {5}{2} a^{2} b^{8} x^{18}+\frac {10}{19} a \,b^{9} x^{19}+\frac {1}{20} b^{10} x^{20}\)	\(113\)
default	\(\frac {1}{10} a^{10} x^{10}+\frac {10}{11} a^{9} b \,x^{11}+\frac {15}{4} a^{8} b^{2} x^{12}+\frac {120}{13} a^{7} b^{3} x^{13}+15 a^{6} b^{4} x^{14}+\frac {84}{5} a^{5} b^{5} x^{15}+\frac {105}{8} a^{4} b^{6} x^{16}+\frac {120}{17} a^{3} b^{7} x^{17}+\frac {5}{2} a^{2} b^{8} x^{18}+\frac {10}{19} a \,b^{9} x^{19}+\frac {1}{20} b^{10} x^{20}\)	\(113\)
norman	\(\frac {1}{10} a^{10} x^{10}+\frac {10}{11} a^{9} b \,x^{11}+\frac {15}{4} a^{8} b^{2} x^{12}+\frac {120}{13} a^{7} b^{3} x^{13}+15 a^{6} b^{4} x^{14}+\frac {84}{5} a^{5} b^{5} x^{15}+\frac {105}{8} a^{4} b^{6} x^{16}+\frac {120}{17} a^{3} b^{7} x^{17}+\frac {5}{2} a^{2} b^{8} x^{18}+\frac {10}{19} a \,b^{9} x^{19}+\frac {1}{20} b^{10} x^{20}\)	\(113\)
risch	\(\frac {1}{10} a^{10} x^{10}+\frac {10}{11} a^{9} b \,x^{11}+\frac {15}{4} a^{8} b^{2} x^{12}+\frac {120}{13} a^{7} b^{3} x^{13}+15 a^{6} b^{4} x^{14}+\frac {84}{5} a^{5} b^{5} x^{15}+\frac {105}{8} a^{4} b^{6} x^{16}+\frac {120}{17} a^{3} b^{7} x^{17}+\frac {5}{2} a^{2} b^{8} x^{18}+\frac {10}{19} a \,b^{9} x^{19}+\frac {1}{20} b^{10} x^{20}\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*(b*x+a)^10,x,method=_RETURNVERBOSE)

[Out]

1/10*a^10*x^10+10/11*a^9*b*x^11+15/4*a^8*b^2*x^12+120/13*a^7*b^3*x^13+15*a^6*b^4*x^14+84/5*a^5*b^5*x^15+105/8*
a^4*b^6*x^16+120/17*a^3*b^7*x^17+5/2*a^2*b^8*x^18+10/19*a*b^9*x^19+1/20*b^10*x^20

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Maxima [A]
time = 0.28, size = 112, normalized size = 0.85 \begin {gather*} \frac {1}{20} \, b^{10} x^{20} + \frac {10}{19} \, a b^{9} x^{19} + \frac {5}{2} \, a^{2} b^{8} x^{18} + \frac {120}{17} \, a^{3} b^{7} x^{17} + \frac {105}{8} \, a^{4} b^{6} x^{16} + \frac {84}{5} \, a^{5} b^{5} x^{15} + 15 \, a^{6} b^{4} x^{14} + \frac {120}{13} \, a^{7} b^{3} x^{13} + \frac {15}{4} \, a^{8} b^{2} x^{12} + \frac {10}{11} \, a^{9} b x^{11} + \frac {1}{10} \, a^{10} x^{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(b*x+a)^10,x, algorithm="maxima")

[Out]

1/20*b^10*x^20 + 10/19*a*b^9*x^19 + 5/2*a^2*b^8*x^18 + 120/17*a^3*b^7*x^17 + 105/8*a^4*b^6*x^16 + 84/5*a^5*b^5
*x^15 + 15*a^6*b^4*x^14 + 120/13*a^7*b^3*x^13 + 15/4*a^8*b^2*x^12 + 10/11*a^9*b*x^11 + 1/10*a^10*x^10

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Fricas [A]
time = 1.23, size = 112, normalized size = 0.85 \begin {gather*} \frac {1}{20} \, b^{10} x^{20} + \frac {10}{19} \, a b^{9} x^{19} + \frac {5}{2} \, a^{2} b^{8} x^{18} + \frac {120}{17} \, a^{3} b^{7} x^{17} + \frac {105}{8} \, a^{4} b^{6} x^{16} + \frac {84}{5} \, a^{5} b^{5} x^{15} + 15 \, a^{6} b^{4} x^{14} + \frac {120}{13} \, a^{7} b^{3} x^{13} + \frac {15}{4} \, a^{8} b^{2} x^{12} + \frac {10}{11} \, a^{9} b x^{11} + \frac {1}{10} \, a^{10} x^{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(b*x+a)^10,x, algorithm="fricas")

[Out]

1/20*b^10*x^20 + 10/19*a*b^9*x^19 + 5/2*a^2*b^8*x^18 + 120/17*a^3*b^7*x^17 + 105/8*a^4*b^6*x^16 + 84/5*a^5*b^5
*x^15 + 15*a^6*b^4*x^14 + 120/13*a^7*b^3*x^13 + 15/4*a^8*b^2*x^12 + 10/11*a^9*b*x^11 + 1/10*a^10*x^10

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Sympy [A]
time = 0.01, size = 133, normalized size = 1.01 \begin {gather*} \frac {a^{10} x^{10}}{10} + \frac {10 a^{9} b x^{11}}{11} + \frac {15 a^{8} b^{2} x^{12}}{4} + \frac {120 a^{7} b^{3} x^{13}}{13} + 15 a^{6} b^{4} x^{14} + \frac {84 a^{5} b^{5} x^{15}}{5} + \frac {105 a^{4} b^{6} x^{16}}{8} + \frac {120 a^{3} b^{7} x^{17}}{17} + \frac {5 a^{2} b^{8} x^{18}}{2} + \frac {10 a b^{9} x^{19}}{19} + \frac {b^{10} x^{20}}{20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9*(b*x+a)**10,x)

[Out]

a**10*x**10/10 + 10*a**9*b*x**11/11 + 15*a**8*b**2*x**12/4 + 120*a**7*b**3*x**13/13 + 15*a**6*b**4*x**14 + 84*
a**5*b**5*x**15/5 + 105*a**4*b**6*x**16/8 + 120*a**3*b**7*x**17/17 + 5*a**2*b**8*x**18/2 + 10*a*b**9*x**19/19
+ b**10*x**20/20

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Giac [A]
time = 1.62, size = 112, normalized size = 0.85 \begin {gather*} \frac {1}{20} \, b^{10} x^{20} + \frac {10}{19} \, a b^{9} x^{19} + \frac {5}{2} \, a^{2} b^{8} x^{18} + \frac {120}{17} \, a^{3} b^{7} x^{17} + \frac {105}{8} \, a^{4} b^{6} x^{16} + \frac {84}{5} \, a^{5} b^{5} x^{15} + 15 \, a^{6} b^{4} x^{14} + \frac {120}{13} \, a^{7} b^{3} x^{13} + \frac {15}{4} \, a^{8} b^{2} x^{12} + \frac {10}{11} \, a^{9} b x^{11} + \frac {1}{10} \, a^{10} x^{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9*(b*x+a)^10,x, algorithm="giac")

[Out]

1/20*b^10*x^20 + 10/19*a*b^9*x^19 + 5/2*a^2*b^8*x^18 + 120/17*a^3*b^7*x^17 + 105/8*a^4*b^6*x^16 + 84/5*a^5*b^5
*x^15 + 15*a^6*b^4*x^14 + 120/13*a^7*b^3*x^13 + 15/4*a^8*b^2*x^12 + 10/11*a^9*b*x^11 + 1/10*a^10*x^10

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Mupad [B]
time = 0.12, size = 112, normalized size = 0.85 \begin {gather*} \frac {a^{10}\,x^{10}}{10}+\frac {10\,a^9\,b\,x^{11}}{11}+\frac {15\,a^8\,b^2\,x^{12}}{4}+\frac {120\,a^7\,b^3\,x^{13}}{13}+15\,a^6\,b^4\,x^{14}+\frac {84\,a^5\,b^5\,x^{15}}{5}+\frac {105\,a^4\,b^6\,x^{16}}{8}+\frac {120\,a^3\,b^7\,x^{17}}{17}+\frac {5\,a^2\,b^8\,x^{18}}{2}+\frac {10\,a\,b^9\,x^{19}}{19}+\frac {b^{10}\,x^{20}}{20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*(a + b*x)^10,x)

[Out]

(a^10*x^10)/10 + (b^10*x^20)/20 + (10*a^9*b*x^11)/11 + (10*a*b^9*x^19)/19 + (15*a^8*b^2*x^12)/4 + (120*a^7*b^3
*x^13)/13 + 15*a^6*b^4*x^14 + (84*a^5*b^5*x^15)/5 + (105*a^4*b^6*x^16)/8 + (120*a^3*b^7*x^17)/17 + (5*a^2*b^8*
x^18)/2

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