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

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

[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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Rubi [A]  time = 0.0592057, 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, Rules used = {43} \[ \frac{45}{19} a^2 b^8 x^{19}+\frac{20}{3} a^3 b^7 x^{18}+\frac{210}{17} a^4 b^6 x^{17}+\frac{63}{4} a^5 b^5 x^{16}+14 a^6 b^4 x^{15}+\frac{60}{7} a^7 b^3 x^{14}+\frac{45}{13} a^8 b^2 x^{13}+\frac{5}{6} a^9 b x^{12}+\frac{a^{10} x^{11}}{11}+\frac{1}{2} a b^9 x^{20}+\frac{b^{10} x^{21}}{21} \]

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 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^{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*}

Mathematica [A]  time = 0.0045776, size = 132, normalized size = 1. \[ \frac{45}{19} a^2 b^8 x^{19}+\frac{20}{3} a^3 b^7 x^{18}+\frac{210}{17} a^4 b^6 x^{17}+\frac{63}{4} a^5 b^5 x^{16}+14 a^6 b^4 x^{15}+\frac{60}{7} a^7 b^3 x^{14}+\frac{45}{13} a^8 b^2 x^{13}+\frac{5}{6} a^9 b x^{12}+\frac{a^{10} x^{11}}{11}+\frac{1}{2} a b^9 x^{20}+\frac{b^{10} x^{21}}{21} \]

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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Maple [A]  time = 0., size = 113, normalized size = 0.9 \begin{align*}{\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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 1.05995, size = 151, normalized size = 1.14 \begin{align*} \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{align*}

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 = 1.50767, size = 282, normalized size = 2.14 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 0.12072, size = 131, normalized size = 0.99 \begin{align*} \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{align*}

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 = 1.17783, size = 151, normalized size = 1.14 \begin{align*} \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{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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