3.126 \(\int x^8 (a+b x)^{10} \, dx\)

Optimal. Leaf size=147 \[ \frac{28 a^2 (a+b x)^{17}}{17 b^9}-\frac{7 a^3 (a+b x)^{16}}{2 b^9}+\frac{14 a^4 (a+b x)^{15}}{3 b^9}-\frac{4 a^5 (a+b x)^{14}}{b^9}+\frac{28 a^6 (a+b x)^{13}}{13 b^9}-\frac{2 a^7 (a+b x)^{12}}{3 b^9}+\frac{a^8 (a+b x)^{11}}{11 b^9}+\frac{(a+b x)^{19}}{19 b^9}-\frac{4 a (a+b x)^{18}}{9 b^9} \]

[Out]

(a^8*(a + b*x)^11)/(11*b^9) - (2*a^7*(a + b*x)^12)/(3*b^9) + (28*a^6*(a + b*x)^13)/(13*b^9) - (4*a^5*(a + b*x)
^14)/b^9 + (14*a^4*(a + b*x)^15)/(3*b^9) - (7*a^3*(a + b*x)^16)/(2*b^9) + (28*a^2*(a + b*x)^17)/(17*b^9) - (4*
a*(a + b*x)^18)/(9*b^9) + (a + b*x)^19/(19*b^9)

________________________________________________________________________________________

Rubi [A]  time = 0.0642461, antiderivative size = 147, 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} \[ \frac{28 a^2 (a+b x)^{17}}{17 b^9}-\frac{7 a^3 (a+b x)^{16}}{2 b^9}+\frac{14 a^4 (a+b x)^{15}}{3 b^9}-\frac{4 a^5 (a+b x)^{14}}{b^9}+\frac{28 a^6 (a+b x)^{13}}{13 b^9}-\frac{2 a^7 (a+b x)^{12}}{3 b^9}+\frac{a^8 (a+b x)^{11}}{11 b^9}+\frac{(a+b x)^{19}}{19 b^9}-\frac{4 a (a+b x)^{18}}{9 b^9} \]

Antiderivative was successfully verified.

[In]

Int[x^8*(a + b*x)^10,x]

[Out]

(a^8*(a + b*x)^11)/(11*b^9) - (2*a^7*(a + b*x)^12)/(3*b^9) + (28*a^6*(a + b*x)^13)/(13*b^9) - (4*a^5*(a + b*x)
^14)/b^9 + (14*a^4*(a + b*x)^15)/(3*b^9) - (7*a^3*(a + b*x)^16)/(2*b^9) + (28*a^2*(a + b*x)^17)/(17*b^9) - (4*
a*(a + b*x)^18)/(9*b^9) + (a + b*x)^19/(19*b^9)

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^8 (a+b x)^{10} \, dx &=\int \left (\frac{a^8 (a+b x)^{10}}{b^8}-\frac{8 a^7 (a+b x)^{11}}{b^8}+\frac{28 a^6 (a+b x)^{12}}{b^8}-\frac{56 a^5 (a+b x)^{13}}{b^8}+\frac{70 a^4 (a+b x)^{14}}{b^8}-\frac{56 a^3 (a+b x)^{15}}{b^8}+\frac{28 a^2 (a+b x)^{16}}{b^8}-\frac{8 a (a+b x)^{17}}{b^8}+\frac{(a+b x)^{18}}{b^8}\right ) \, dx\\ &=\frac{a^8 (a+b x)^{11}}{11 b^9}-\frac{2 a^7 (a+b x)^{12}}{3 b^9}+\frac{28 a^6 (a+b x)^{13}}{13 b^9}-\frac{4 a^5 (a+b x)^{14}}{b^9}+\frac{14 a^4 (a+b x)^{15}}{3 b^9}-\frac{7 a^3 (a+b x)^{16}}{2 b^9}+\frac{28 a^2 (a+b x)^{17}}{17 b^9}-\frac{4 a (a+b x)^{18}}{9 b^9}+\frac{(a+b x)^{19}}{19 b^9}\\ \end{align*}

Mathematica [A]  time = 0.0035812, size = 125, normalized size = 0.85 \[ \frac{45}{17} a^2 b^8 x^{17}+\frac{15}{2} a^3 b^7 x^{16}+14 a^4 b^6 x^{15}+18 a^5 b^5 x^{14}+\frac{210}{13} a^6 b^4 x^{13}+10 a^7 b^3 x^{12}+\frac{45}{11} a^8 b^2 x^{11}+a^9 b x^{10}+\frac{a^{10} x^9}{9}+\frac{5}{9} a b^9 x^{18}+\frac{b^{10} x^{19}}{19} \]

Antiderivative was successfully verified.

[In]

Integrate[x^8*(a + b*x)^10,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.001, size = 112, normalized size = 0.8 \begin{align*}{\frac{{b}^{10}{x}^{19}}{19}}+{\frac{5\,a{b}^{9}{x}^{18}}{9}}+{\frac{45\,{a}^{2}{b}^{8}{x}^{17}}{17}}+{\frac{15\,{a}^{3}{b}^{7}{x}^{16}}{2}}+14\,{a}^{4}{b}^{6}{x}^{15}+18\,{a}^{5}{b}^{5}{x}^{14}+{\frac{210\,{a}^{6}{b}^{4}{x}^{13}}{13}}+10\,{a}^{7}{b}^{3}{x}^{12}+{\frac{45\,{a}^{8}{b}^{2}{x}^{11}}{11}}+{a}^{9}b{x}^{10}+{\frac{{a}^{10}{x}^{9}}{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^8*(b*x+a)^10,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.03017, size = 150, normalized size = 1.02 \begin{align*} \frac{1}{19} \, b^{10} x^{19} + \frac{5}{9} \, a b^{9} x^{18} + \frac{45}{17} \, a^{2} b^{8} x^{17} + \frac{15}{2} \, a^{3} b^{7} x^{16} + 14 \, a^{4} b^{6} x^{15} + 18 \, a^{5} b^{5} x^{14} + \frac{210}{13} \, a^{6} b^{4} x^{13} + 10 \, a^{7} b^{3} x^{12} + \frac{45}{11} \, a^{8} b^{2} x^{11} + a^{9} b x^{10} + \frac{1}{9} \, a^{10} x^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x+a)^10,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.57643, size = 269, normalized size = 1.83 \begin{align*} \frac{1}{19} x^{19} b^{10} + \frac{5}{9} x^{18} b^{9} a + \frac{45}{17} x^{17} b^{8} a^{2} + \frac{15}{2} x^{16} b^{7} a^{3} + 14 x^{15} b^{6} a^{4} + 18 x^{14} b^{5} a^{5} + \frac{210}{13} x^{13} b^{4} a^{6} + 10 x^{12} b^{3} a^{7} + \frac{45}{11} x^{11} b^{2} a^{8} + x^{10} b a^{9} + \frac{1}{9} x^{9} a^{10} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x+a)^10,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.111013, size = 126, normalized size = 0.86 \begin{align*} \frac{a^{10} x^{9}}{9} + a^{9} b x^{10} + \frac{45 a^{8} b^{2} x^{11}}{11} + 10 a^{7} b^{3} x^{12} + \frac{210 a^{6} b^{4} x^{13}}{13} + 18 a^{5} b^{5} x^{14} + 14 a^{4} b^{6} x^{15} + \frac{15 a^{3} b^{7} x^{16}}{2} + \frac{45 a^{2} b^{8} x^{17}}{17} + \frac{5 a b^{9} x^{18}}{9} + \frac{b^{10} x^{19}}{19} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**8*(b*x+a)**10,x)

[Out]

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

________________________________________________________________________________________

Giac [A]  time = 1.21193, size = 150, normalized size = 1.02 \begin{align*} \frac{1}{19} \, b^{10} x^{19} + \frac{5}{9} \, a b^{9} x^{18} + \frac{45}{17} \, a^{2} b^{8} x^{17} + \frac{15}{2} \, a^{3} b^{7} x^{16} + 14 \, a^{4} b^{6} x^{15} + 18 \, a^{5} b^{5} x^{14} + \frac{210}{13} \, a^{6} b^{4} x^{13} + 10 \, a^{7} b^{3} x^{12} + \frac{45}{11} \, a^{8} b^{2} x^{11} + a^{9} b x^{10} + \frac{1}{9} \, a^{10} x^{9} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^8*(b*x+a)^10,x, algorithm="giac")

[Out]

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