\(\int x^{11} (a+b x)^{10} \, dx\) [123]

Optimal result

Integrand size = 11, antiderivative size = 132 \[ \int x^{11} (a+b x)^{10} \, dx=\frac {a^{10} x^{12}}{12}+\frac {10}{13} a^9 b x^{13}+\frac {45}{14} a^8 b^2 x^{14}+8 a^7 b^3 x^{15}+\frac {105}{8} a^6 b^4 x^{16}+\frac {252}{17} a^5 b^5 x^{17}+\frac {35}{3} a^4 b^6 x^{18}+\frac {120}{19} a^3 b^7 x^{19}+\frac {9}{4} a^2 b^8 x^{20}+\frac {10}{21} a b^9 x^{21}+\frac {b^{10} x^{22}}{22} \]

[Out]

1/12*a^10*x^12+10/13*a^9*b*x^13+45/14*a^8*b^2*x^14+8*a^7*b^3*x^15+105/8*a^6*b^4*x^16+252/17*a^5*b^5*x^17+35/3*
a^4*b^6*x^18+120/19*a^3*b^7*x^19+9/4*a^2*b^8*x^20+10/21*a*b^9*x^21+1/22*b^10*x^22

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int x^{11} (a+b x)^{10} \, dx=\frac {a^{10} x^{12}}{12}+\frac {10}{13} a^9 b x^{13}+\frac {45}{14} a^8 b^2 x^{14}+8 a^7 b^3 x^{15}+\frac {105}{8} a^6 b^4 x^{16}+\frac {252}{17} a^5 b^5 x^{17}+\frac {35}{3} a^4 b^6 x^{18}+\frac {120}{19} a^3 b^7 x^{19}+\frac {9}{4} a^2 b^8 x^{20}+\frac {10}{21} a b^9 x^{21}+\frac {b^{10} x^{22}}{22} \]

[In]

Int[x^11*(a + b*x)^10,x]

[Out]

(a^10*x^12)/12 + (10*a^9*b*x^13)/13 + (45*a^8*b^2*x^14)/14 + 8*a^7*b^3*x^15 + (105*a^6*b^4*x^16)/8 + (252*a^5*
b^5*x^17)/17 + (35*a^4*b^6*x^18)/3 + (120*a^3*b^7*x^19)/19 + (9*a^2*b^8*x^20)/4 + (10*a*b^9*x^21)/21 + (b^10*x
^22)/22

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*} \text {integral}& = \int \left (a^{10} x^{11}+10 a^9 b x^{12}+45 a^8 b^2 x^{13}+120 a^7 b^3 x^{14}+210 a^6 b^4 x^{15}+252 a^5 b^5 x^{16}+210 a^4 b^6 x^{17}+120 a^3 b^7 x^{18}+45 a^2 b^8 x^{19}+10 a b^9 x^{20}+b^{10} x^{21}\right ) \, dx \\ & = \frac {a^{10} x^{12}}{12}+\frac {10}{13} a^9 b x^{13}+\frac {45}{14} a^8 b^2 x^{14}+8 a^7 b^3 x^{15}+\frac {105}{8} a^6 b^4 x^{16}+\frac {252}{17} a^5 b^5 x^{17}+\frac {35}{3} a^4 b^6 x^{18}+\frac {120}{19} a^3 b^7 x^{19}+\frac {9}{4} a^2 b^8 x^{20}+\frac {10}{21} a b^9 x^{21}+\frac {b^{10} x^{22}}{22} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00 \[ \int x^{11} (a+b x)^{10} \, dx=\frac {a^{10} x^{12}}{12}+\frac {10}{13} a^9 b x^{13}+\frac {45}{14} a^8 b^2 x^{14}+8 a^7 b^3 x^{15}+\frac {105}{8} a^6 b^4 x^{16}+\frac {252}{17} a^5 b^5 x^{17}+\frac {35}{3} a^4 b^6 x^{18}+\frac {120}{19} a^3 b^7 x^{19}+\frac {9}{4} a^2 b^8 x^{20}+\frac {10}{21} a b^9 x^{21}+\frac {b^{10} x^{22}}{22} \]

[In]

Integrate[x^11*(a + b*x)^10,x]

[Out]

(a^10*x^12)/12 + (10*a^9*b*x^13)/13 + (45*a^8*b^2*x^14)/14 + 8*a^7*b^3*x^15 + (105*a^6*b^4*x^16)/8 + (252*a^5*
b^5*x^17)/17 + (35*a^4*b^6*x^18)/3 + (120*a^3*b^7*x^19)/19 + (9*a^2*b^8*x^20)/4 + (10*a*b^9*x^21)/21 + (b^10*x
^22)/22

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.86

method	result	size
gosper	\(\frac {1}{12} a^{10} x^{12}+\frac {10}{13} a^{9} b \,x^{13}+\frac {45}{14} a^{8} b^{2} x^{14}+8 a^{7} b^{3} x^{15}+\frac {105}{8} a^{6} b^{4} x^{16}+\frac {252}{17} a^{5} b^{5} x^{17}+\frac {35}{3} a^{4} b^{6} x^{18}+\frac {120}{19} a^{3} b^{7} x^{19}+\frac {9}{4} a^{2} b^{8} x^{20}+\frac {10}{21} a \,b^{9} x^{21}+\frac {1}{22} b^{10} x^{22}\)	\(113\)
default	\(\frac {1}{12} a^{10} x^{12}+\frac {10}{13} a^{9} b \,x^{13}+\frac {45}{14} a^{8} b^{2} x^{14}+8 a^{7} b^{3} x^{15}+\frac {105}{8} a^{6} b^{4} x^{16}+\frac {252}{17} a^{5} b^{5} x^{17}+\frac {35}{3} a^{4} b^{6} x^{18}+\frac {120}{19} a^{3} b^{7} x^{19}+\frac {9}{4} a^{2} b^{8} x^{20}+\frac {10}{21} a \,b^{9} x^{21}+\frac {1}{22} b^{10} x^{22}\)	\(113\)
norman	\(\frac {1}{12} a^{10} x^{12}+\frac {10}{13} a^{9} b \,x^{13}+\frac {45}{14} a^{8} b^{2} x^{14}+8 a^{7} b^{3} x^{15}+\frac {105}{8} a^{6} b^{4} x^{16}+\frac {252}{17} a^{5} b^{5} x^{17}+\frac {35}{3} a^{4} b^{6} x^{18}+\frac {120}{19} a^{3} b^{7} x^{19}+\frac {9}{4} a^{2} b^{8} x^{20}+\frac {10}{21} a \,b^{9} x^{21}+\frac {1}{22} b^{10} x^{22}\)	\(113\)
risch	\(\frac {1}{12} a^{10} x^{12}+\frac {10}{13} a^{9} b \,x^{13}+\frac {45}{14} a^{8} b^{2} x^{14}+8 a^{7} b^{3} x^{15}+\frac {105}{8} a^{6} b^{4} x^{16}+\frac {252}{17} a^{5} b^{5} x^{17}+\frac {35}{3} a^{4} b^{6} x^{18}+\frac {120}{19} a^{3} b^{7} x^{19}+\frac {9}{4} a^{2} b^{8} x^{20}+\frac {10}{21} a \,b^{9} x^{21}+\frac {1}{22} b^{10} x^{22}\)	\(113\)
parallelrisch	\(\frac {1}{12} a^{10} x^{12}+\frac {10}{13} a^{9} b \,x^{13}+\frac {45}{14} a^{8} b^{2} x^{14}+8 a^{7} b^{3} x^{15}+\frac {105}{8} a^{6} b^{4} x^{16}+\frac {252}{17} a^{5} b^{5} x^{17}+\frac {35}{3} a^{4} b^{6} x^{18}+\frac {120}{19} a^{3} b^{7} x^{19}+\frac {9}{4} a^{2} b^{8} x^{20}+\frac {10}{21} a \,b^{9} x^{21}+\frac {1}{22} b^{10} x^{22}\)	\(113\)

[In]

int(x^11*(b*x+a)^10,x,method=_RETURNVERBOSE)

[Out]

1/12*a^10*x^12+10/13*a^9*b*x^13+45/14*a^8*b^2*x^14+8*a^7*b^3*x^15+105/8*a^6*b^4*x^16+252/17*a^5*b^5*x^17+35/3*
a^4*b^6*x^18+120/19*a^3*b^7*x^19+9/4*a^2*b^8*x^20+10/21*a*b^9*x^21+1/22*b^10*x^22

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^{11} (a+b x)^{10} \, dx=\frac {1}{22} \, b^{10} x^{22} + \frac {10}{21} \, a b^{9} x^{21} + \frac {9}{4} \, a^{2} b^{8} x^{20} + \frac {120}{19} \, a^{3} b^{7} x^{19} + \frac {35}{3} \, a^{4} b^{6} x^{18} + \frac {252}{17} \, a^{5} b^{5} x^{17} + \frac {105}{8} \, a^{6} b^{4} x^{16} + 8 \, a^{7} b^{3} x^{15} + \frac {45}{14} \, a^{8} b^{2} x^{14} + \frac {10}{13} \, a^{9} b x^{13} + \frac {1}{12} \, a^{10} x^{12} \]

[In]

integrate(x^11*(b*x+a)^10,x, algorithm="fricas")

[Out]

1/22*b^10*x^22 + 10/21*a*b^9*x^21 + 9/4*a^2*b^8*x^20 + 120/19*a^3*b^7*x^19 + 35/3*a^4*b^6*x^18 + 252/17*a^5*b^
5*x^17 + 105/8*a^6*b^4*x^16 + 8*a^7*b^3*x^15 + 45/14*a^8*b^2*x^14 + 10/13*a^9*b*x^13 + 1/12*a^10*x^12

Sympy [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.01 \[ \int x^{11} (a+b x)^{10} \, dx=\frac {a^{10} x^{12}}{12} + \frac {10 a^{9} b x^{13}}{13} + \frac {45 a^{8} b^{2} x^{14}}{14} + 8 a^{7} b^{3} x^{15} + \frac {105 a^{6} b^{4} x^{16}}{8} + \frac {252 a^{5} b^{5} x^{17}}{17} + \frac {35 a^{4} b^{6} x^{18}}{3} + \frac {120 a^{3} b^{7} x^{19}}{19} + \frac {9 a^{2} b^{8} x^{20}}{4} + \frac {10 a b^{9} x^{21}}{21} + \frac {b^{10} x^{22}}{22} \]

[In]

integrate(x**11*(b*x+a)**10,x)

[Out]

a**10*x**12/12 + 10*a**9*b*x**13/13 + 45*a**8*b**2*x**14/14 + 8*a**7*b**3*x**15 + 105*a**6*b**4*x**16/8 + 252*
a**5*b**5*x**17/17 + 35*a**4*b**6*x**18/3 + 120*a**3*b**7*x**19/19 + 9*a**2*b**8*x**20/4 + 10*a*b**9*x**21/21
+ b**10*x**22/22

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^{11} (a+b x)^{10} \, dx=\frac {1}{22} \, b^{10} x^{22} + \frac {10}{21} \, a b^{9} x^{21} + \frac {9}{4} \, a^{2} b^{8} x^{20} + \frac {120}{19} \, a^{3} b^{7} x^{19} + \frac {35}{3} \, a^{4} b^{6} x^{18} + \frac {252}{17} \, a^{5} b^{5} x^{17} + \frac {105}{8} \, a^{6} b^{4} x^{16} + 8 \, a^{7} b^{3} x^{15} + \frac {45}{14} \, a^{8} b^{2} x^{14} + \frac {10}{13} \, a^{9} b x^{13} + \frac {1}{12} \, a^{10} x^{12} \]

[In]

integrate(x^11*(b*x+a)^10,x, algorithm="maxima")

[Out]

1/22*b^10*x^22 + 10/21*a*b^9*x^21 + 9/4*a^2*b^8*x^20 + 120/19*a^3*b^7*x^19 + 35/3*a^4*b^6*x^18 + 252/17*a^5*b^
5*x^17 + 105/8*a^6*b^4*x^16 + 8*a^7*b^3*x^15 + 45/14*a^8*b^2*x^14 + 10/13*a^9*b*x^13 + 1/12*a^10*x^12

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^{11} (a+b x)^{10} \, dx=\frac {1}{22} \, b^{10} x^{22} + \frac {10}{21} \, a b^{9} x^{21} + \frac {9}{4} \, a^{2} b^{8} x^{20} + \frac {120}{19} \, a^{3} b^{7} x^{19} + \frac {35}{3} \, a^{4} b^{6} x^{18} + \frac {252}{17} \, a^{5} b^{5} x^{17} + \frac {105}{8} \, a^{6} b^{4} x^{16} + 8 \, a^{7} b^{3} x^{15} + \frac {45}{14} \, a^{8} b^{2} x^{14} + \frac {10}{13} \, a^{9} b x^{13} + \frac {1}{12} \, a^{10} x^{12} \]

[In]

integrate(x^11*(b*x+a)^10,x, algorithm="giac")

[Out]

1/22*b^10*x^22 + 10/21*a*b^9*x^21 + 9/4*a^2*b^8*x^20 + 120/19*a^3*b^7*x^19 + 35/3*a^4*b^6*x^18 + 252/17*a^5*b^
5*x^17 + 105/8*a^6*b^4*x^16 + 8*a^7*b^3*x^15 + 45/14*a^8*b^2*x^14 + 10/13*a^9*b*x^13 + 1/12*a^10*x^12

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.85 \[ \int x^{11} (a+b x)^{10} \, dx=\frac {a^{10}\,x^{12}}{12}+\frac {10\,a^9\,b\,x^{13}}{13}+\frac {45\,a^8\,b^2\,x^{14}}{14}+8\,a^7\,b^3\,x^{15}+\frac {105\,a^6\,b^4\,x^{16}}{8}+\frac {252\,a^5\,b^5\,x^{17}}{17}+\frac {35\,a^4\,b^6\,x^{18}}{3}+\frac {120\,a^3\,b^7\,x^{19}}{19}+\frac {9\,a^2\,b^8\,x^{20}}{4}+\frac {10\,a\,b^9\,x^{21}}{21}+\frac {b^{10}\,x^{22}}{22} \]

[In]

int(x^11*(a + b*x)^10,x)

[Out]

(a^10*x^12)/12 + (b^10*x^22)/22 + (10*a^9*b*x^13)/13 + (10*a*b^9*x^21)/21 + (45*a^8*b^2*x^14)/14 + 8*a^7*b^3*x
^15 + (105*a^6*b^4*x^16)/8 + (252*a^5*b^5*x^17)/17 + (35*a^4*b^6*x^18)/3 + (120*a^3*b^7*x^19)/19 + (9*a^2*b^8*
x^20)/4