3.2.24 \(\int x^{10} (a+b x)^{10} \, dx\) [124]

Optimal. Leaf size=132 \[ \frac {a^{10} x^{11}}{11}+\frac {5}{6} a^9 b x^{12}+\frac {45}{13} a^8 b^2 x^{13}+\frac {60}{7} a^7 b^3 x^{14}+14 a^6 b^4 x^{15}+\frac {63}{4} a^5 b^5 x^{16}+\frac {210}{17} a^4 b^6 x^{17}+\frac {20}{3} a^3 b^7 x^{18}+\frac {45}{19} a^2 b^8 x^{19}+\frac {1}{2} a b^9 x^{20}+\frac {b^{10} x^{21}}{21} \]

[Out]

1/11*a^10*x^11+5/6*a^9*b*x^12+45/13*a^8*b^2*x^13+60/7*a^7*b^3*x^14+14*a^6*b^4*x^15+63/4*a^5*b^5*x^16+210/17*a^
4*b^6*x^17+20/3*a^3*b^7*x^18+45/19*a^2*b^8*x^19+1/2*a*b^9*x^20+1/21*b^10*x^21

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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^{11}}{11}+\frac {5}{6} a^9 b x^{12}+\frac {45}{13} a^8 b^2 x^{13}+\frac {60}{7} a^7 b^3 x^{14}+14 a^6 b^4 x^{15}+\frac {63}{4} a^5 b^5 x^{16}+\frac {210}{17} a^4 b^6 x^{17}+\frac {20}{3} a^3 b^7 x^{18}+\frac {45}{19} a^2 b^8 x^{19}+\frac {1}{2} a b^9 x^{20}+\frac {b^{10} x^{21}}{21} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^10*(a + b*x)^10,x]

[Out]

(a^10*x^11)/11 + (5*a^9*b*x^12)/6 + (45*a^8*b^2*x^13)/13 + (60*a^7*b^3*x^14)/7 + 14*a^6*b^4*x^15 + (63*a^5*b^5
*x^16)/4 + (210*a^4*b^6*x^17)/17 + (20*a^3*b^7*x^18)/3 + (45*a^2*b^8*x^19)/19 + (a*b^9*x^20)/2 + (b^10*x^21)/2
1

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^{10} (a+b x)^{10} \, dx &=\int \left (a^{10} x^{10}+10 a^9 b x^{11}+45 a^8 b^2 x^{12}+120 a^7 b^3 x^{13}+210 a^6 b^4 x^{14}+252 a^5 b^5 x^{15}+210 a^4 b^6 x^{16}+120 a^3 b^7 x^{17}+45 a^2 b^8 x^{18}+10 a b^9 x^{19}+b^{10} x^{20}\right ) \, dx\\ &=\frac {a^{10} x^{11}}{11}+\frac {5}{6} a^9 b x^{12}+\frac {45}{13} a^8 b^2 x^{13}+\frac {60}{7} a^7 b^3 x^{14}+14 a^6 b^4 x^{15}+\frac {63}{4} a^5 b^5 x^{16}+\frac {210}{17} a^4 b^6 x^{17}+\frac {20}{3} a^3 b^7 x^{18}+\frac {45}{19} a^2 b^8 x^{19}+\frac {1}{2} a b^9 x^{20}+\frac {b^{10} x^{21}}{21}\\ \end {align*}

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Mathematica [A]
time = 0.00, size = 132, normalized size = 1.00 \begin {gather*} \frac {a^{10} x^{11}}{11}+\frac {5}{6} a^9 b x^{12}+\frac {45}{13} a^8 b^2 x^{13}+\frac {60}{7} a^7 b^3 x^{14}+14 a^6 b^4 x^{15}+\frac {63}{4} a^5 b^5 x^{16}+\frac {210}{17} a^4 b^6 x^{17}+\frac {20}{3} a^3 b^7 x^{18}+\frac {45}{19} a^2 b^8 x^{19}+\frac {1}{2} a b^9 x^{20}+\frac {b^{10} x^{21}}{21} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^10*(a + b*x)^10,x]

[Out]

(a^10*x^11)/11 + (5*a^9*b*x^12)/6 + (45*a^8*b^2*x^13)/13 + (60*a^7*b^3*x^14)/7 + 14*a^6*b^4*x^15 + (63*a^5*b^5
*x^16)/4 + (210*a^4*b^6*x^17)/17 + (20*a^3*b^7*x^18)/3 + (45*a^2*b^8*x^19)/19 + (a*b^9*x^20)/2 + (b^10*x^21)/2
1

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Mathics [A]
time = 2.15, size = 112, normalized size = 0.85 \begin {gather*} \frac {x^{11} \left (352716 a^{10}+3233230 a^9 b x+13430340 a^8 b^2 x^2+33256080 a^7 b^3 x^3+54318264 a^6 b^4 x^4+61108047 a^5 b^5 x^5+47927880 a^4 b^6 x^6+25865840 a^3 b^7 x^7+9189180 a^2 b^8 x^8+1939938 a b^9 x^9+184756 b^{10} x^{10}\right )}{3879876} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^10*(a + b*x)^10,x]')

[Out]

x ^ 11 (352716 a ^ 10 + 3233230 a ^ 9 b x + 13430340 a ^ 8 b ^ 2 x ^ 2 + 33256080 a ^ 7 b ^ 3 x ^ 3 + 54318264
a ^ 6 b ^ 4 x ^ 4 + 61108047 a ^ 5 b ^ 5 x ^ 5 + 47927880 a ^ 4 b ^ 6 x ^ 6 + 25865840 a ^ 3 b ^ 7 x ^ 7 + 91
89180 a ^ 2 b ^ 8 x ^ 8 + 1939938 a b ^ 9 x ^ 9 + 184756 b ^ 10 x ^ 10) / 3879876

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Maple [A]
time = 0.07, size = 113, normalized size = 0.86

method	result	size
gosper	\(\frac {1}{11} a^{10} x^{11}+\frac {5}{6} a^{9} b \,x^{12}+\frac {45}{13} a^{8} b^{2} x^{13}+\frac {60}{7} a^{7} b^{3} x^{14}+14 a^{6} b^{4} x^{15}+\frac {63}{4} a^{5} b^{5} x^{16}+\frac {210}{17} a^{4} b^{6} x^{17}+\frac {20}{3} a^{3} b^{7} x^{18}+\frac {45}{19} a^{2} b^{8} x^{19}+\frac {1}{2} a \,b^{9} x^{20}+\frac {1}{21} b^{10} x^{21}\)	\(113\)
default	\(\frac {1}{11} a^{10} x^{11}+\frac {5}{6} a^{9} b \,x^{12}+\frac {45}{13} a^{8} b^{2} x^{13}+\frac {60}{7} a^{7} b^{3} x^{14}+14 a^{6} b^{4} x^{15}+\frac {63}{4} a^{5} b^{5} x^{16}+\frac {210}{17} a^{4} b^{6} x^{17}+\frac {20}{3} a^{3} b^{7} x^{18}+\frac {45}{19} a^{2} b^{8} x^{19}+\frac {1}{2} a \,b^{9} x^{20}+\frac {1}{21} b^{10} x^{21}\)	\(113\)
norman	\(\frac {1}{11} a^{10} x^{11}+\frac {5}{6} a^{9} b \,x^{12}+\frac {45}{13} a^{8} b^{2} x^{13}+\frac {60}{7} a^{7} b^{3} x^{14}+14 a^{6} b^{4} x^{15}+\frac {63}{4} a^{5} b^{5} x^{16}+\frac {210}{17} a^{4} b^{6} x^{17}+\frac {20}{3} a^{3} b^{7} x^{18}+\frac {45}{19} a^{2} b^{8} x^{19}+\frac {1}{2} a \,b^{9} x^{20}+\frac {1}{21} b^{10} x^{21}\)	\(113\)
risch	\(\frac {1}{11} a^{10} x^{11}+\frac {5}{6} a^{9} b \,x^{12}+\frac {45}{13} a^{8} b^{2} x^{13}+\frac {60}{7} a^{7} b^{3} x^{14}+14 a^{6} b^{4} x^{15}+\frac {63}{4} a^{5} b^{5} x^{16}+\frac {210}{17} a^{4} b^{6} x^{17}+\frac {20}{3} a^{3} b^{7} x^{18}+\frac {45}{19} a^{2} b^{8} x^{19}+\frac {1}{2} a \,b^{9} x^{20}+\frac {1}{21} b^{10} x^{21}\)	\(113\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*a^10*x^11+5/6*a^9*b*x^12+45/13*a^8*b^2*x^13+60/7*a^7*b^3*x^14+14*a^6*b^4*x^15+63/4*a^5*b^5*x^16+210/17*a^
4*b^6*x^17+20/3*a^3*b^7*x^18+45/19*a^2*b^8*x^19+1/2*a*b^9*x^20+1/21*b^10*x^21

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Maxima [A]
time = 0.25, size = 112, normalized size = 0.85 \begin {gather*} \frac {1}{21} \, b^{10} x^{21} + \frac {1}{2} \, a b^{9} x^{20} + \frac {45}{19} \, a^{2} b^{8} x^{19} + \frac {20}{3} \, a^{3} b^{7} x^{18} + \frac {210}{17} \, a^{4} b^{6} x^{17} + \frac {63}{4} \, a^{5} b^{5} x^{16} + 14 \, a^{6} b^{4} x^{15} + \frac {60}{7} \, a^{7} b^{3} x^{14} + \frac {45}{13} \, a^{8} b^{2} x^{13} + \frac {5}{6} \, a^{9} b x^{12} + \frac {1}{11} \, a^{10} x^{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*b^10*x^21 + 1/2*a*b^9*x^20 + 45/19*a^2*b^8*x^19 + 20/3*a^3*b^7*x^18 + 210/17*a^4*b^6*x^17 + 63/4*a^5*b^5*
x^16 + 14*a^6*b^4*x^15 + 60/7*a^7*b^3*x^14 + 45/13*a^8*b^2*x^13 + 5/6*a^9*b*x^12 + 1/11*a^10*x^11

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Fricas [A]
time = 0.30, size = 112, normalized size = 0.85 \begin {gather*} \frac {1}{21} \, b^{10} x^{21} + \frac {1}{2} \, a b^{9} x^{20} + \frac {45}{19} \, a^{2} b^{8} x^{19} + \frac {20}{3} \, a^{3} b^{7} x^{18} + \frac {210}{17} \, a^{4} b^{6} x^{17} + \frac {63}{4} \, a^{5} b^{5} x^{16} + 14 \, a^{6} b^{4} x^{15} + \frac {60}{7} \, a^{7} b^{3} x^{14} + \frac {45}{13} \, a^{8} b^{2} x^{13} + \frac {5}{6} \, a^{9} b x^{12} + \frac {1}{11} \, a^{10} x^{11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*b^10*x^21 + 1/2*a*b^9*x^20 + 45/19*a^2*b^8*x^19 + 20/3*a^3*b^7*x^18 + 210/17*a^4*b^6*x^17 + 63/4*a^5*b^5*
x^16 + 14*a^6*b^4*x^15 + 60/7*a^7*b^3*x^14 + 45/13*a^8*b^2*x^13 + 5/6*a^9*b*x^12 + 1/11*a^10*x^11

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Sympy [A]
time = 0.05, size = 131, normalized size = 0.99 \begin {gather*} \frac {a^{10} x^{11}}{11} + \frac {5 a^{9} b x^{12}}{6} + \frac {45 a^{8} b^{2} x^{13}}{13} + \frac {60 a^{7} b^{3} x^{14}}{7} + 14 a^{6} b^{4} x^{15} + \frac {63 a^{5} b^{5} x^{16}}{4} + \frac {210 a^{4} b^{6} x^{17}}{17} + \frac {20 a^{3} b^{7} x^{18}}{3} + \frac {45 a^{2} b^{8} x^{19}}{19} + \frac {a b^{9} x^{20}}{2} + \frac {b^{10} x^{21}}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**10*x**11/11 + 5*a**9*b*x**12/6 + 45*a**8*b**2*x**13/13 + 60*a**7*b**3*x**14/7 + 14*a**6*b**4*x**15 + 63*a**
5*b**5*x**16/4 + 210*a**4*b**6*x**17/17 + 20*a**3*b**7*x**18/3 + 45*a**2*b**8*x**19/19 + a*b**9*x**20/2 + b**1
0*x**21/21

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Giac [A]
time = 0.00, size = 132, normalized size = 1.00 \begin {gather*} \frac {1}{21} x^{21} b^{10}+\frac {1}{2} x^{20} b^{9} a+\frac {45}{19} x^{19} b^{8} a^{2}+\frac {20}{3} x^{18} b^{7} a^{3}+\frac {210}{17} x^{17} b^{6} a^{4}+\frac {63}{4} x^{16} b^{5} a^{5}+14 x^{15} b^{4} a^{6}+\frac {60}{7} x^{14} b^{3} a^{7}+\frac {45}{13} x^{13} b^{2} a^{8}+\frac {5}{6} x^{12} b a^{9}+\frac {1}{11} x^{11} a^{10} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*b^10*x^21 + 1/2*a*b^9*x^20 + 45/19*a^2*b^8*x^19 + 20/3*a^3*b^7*x^18 + 210/17*a^4*b^6*x^17 + 63/4*a^5*b^5*
x^16 + 14*a^6*b^4*x^15 + 60/7*a^7*b^3*x^14 + 45/13*a^8*b^2*x^13 + 5/6*a^9*b*x^12 + 1/11*a^10*x^11

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Mupad [B]
time = 0.08, size = 112, normalized size = 0.85 \begin {gather*} \frac {a^{10}\,x^{11}}{11}+\frac {5\,a^9\,b\,x^{12}}{6}+\frac {45\,a^8\,b^2\,x^{13}}{13}+\frac {60\,a^7\,b^3\,x^{14}}{7}+14\,a^6\,b^4\,x^{15}+\frac {63\,a^5\,b^5\,x^{16}}{4}+\frac {210\,a^4\,b^6\,x^{17}}{17}+\frac {20\,a^3\,b^7\,x^{18}}{3}+\frac {45\,a^2\,b^8\,x^{19}}{19}+\frac {a\,b^9\,x^{20}}{2}+\frac {b^{10}\,x^{21}}{21} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^10*(a + b*x)^10,x)

[Out]

(a^10*x^11)/11 + (b^10*x^21)/21 + (5*a^9*b*x^12)/6 + (a*b^9*x^20)/2 + (45*a^8*b^2*x^13)/13 + (60*a^7*b^3*x^14)
/7 + 14*a^6*b^4*x^15 + (63*a^5*b^5*x^16)/4 + (210*a^4*b^6*x^17)/17 + (20*a^3*b^7*x^18)/3 + (45*a^2*b^8*x^19)/1
9

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