3.2559 \(\int x^{-1+9 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=151 \[ \frac{a^8 x^{9 n}}{9 n}+\frac{4 a^7 b x^{10 n}}{5 n}+\frac{28 a^6 b^2 x^{11 n}}{11 n}+\frac{14 a^5 b^3 x^{12 n}}{3 n}+\frac{70 a^4 b^4 x^{13 n}}{13 n}+\frac{4 a^3 b^5 x^{14 n}}{n}+\frac{28 a^2 b^6 x^{15 n}}{15 n}+\frac{a b^7 x^{16 n}}{2 n}+\frac{b^8 x^{17 n}}{17 n} \]

[Out]

(a^8*x^(9*n))/(9*n) + (4*a^7*b*x^(10*n))/(5*n) + (28*a^6*b^2*x^(11*n))/(11*n) +
(14*a^5*b^3*x^(12*n))/(3*n) + (70*a^4*b^4*x^(13*n))/(13*n) + (4*a^3*b^5*x^(14*n)
)/n + (28*a^2*b^6*x^(15*n))/(15*n) + (a*b^7*x^(16*n))/(2*n) + (b^8*x^(17*n))/(17
*n)

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Rubi [A]  time = 0.192947, antiderivative size = 151, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{a^8 x^{9 n}}{9 n}+\frac{4 a^7 b x^{10 n}}{5 n}+\frac{28 a^6 b^2 x^{11 n}}{11 n}+\frac{14 a^5 b^3 x^{12 n}}{3 n}+\frac{70 a^4 b^4 x^{13 n}}{13 n}+\frac{4 a^3 b^5 x^{14 n}}{n}+\frac{28 a^2 b^6 x^{15 n}}{15 n}+\frac{a b^7 x^{16 n}}{2 n}+\frac{b^8 x^{17 n}}{17 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 9*n)*(a + b*x^n)^8,x]

[Out]

(a^8*x^(9*n))/(9*n) + (4*a^7*b*x^(10*n))/(5*n) + (28*a^6*b^2*x^(11*n))/(11*n) +
(14*a^5*b^3*x^(12*n))/(3*n) + (70*a^4*b^4*x^(13*n))/(13*n) + (4*a^3*b^5*x^(14*n)
)/n + (28*a^2*b^6*x^(15*n))/(15*n) + (a*b^7*x^(16*n))/(2*n) + (b^8*x^(17*n))/(17
*n)

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Rubi in Sympy [A]  time = 31.1735, size = 134, normalized size = 0.89 \[ \frac{a^{8} x^{9 n}}{9 n} + \frac{4 a^{7} b x^{10 n}}{5 n} + \frac{28 a^{6} b^{2} x^{11 n}}{11 n} + \frac{14 a^{5} b^{3} x^{12 n}}{3 n} + \frac{70 a^{4} b^{4} x^{13 n}}{13 n} + \frac{4 a^{3} b^{5} x^{14 n}}{n} + \frac{28 a^{2} b^{6} x^{15 n}}{15 n} + \frac{a b^{7} x^{16 n}}{2 n} + \frac{b^{8} x^{17 n}}{17 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+9*n)*(a+b*x**n)**8,x)

[Out]

a**8*x**(9*n)/(9*n) + 4*a**7*b*x**(10*n)/(5*n) + 28*a**6*b**2*x**(11*n)/(11*n) +
14*a**5*b**3*x**(12*n)/(3*n) + 70*a**4*b**4*x**(13*n)/(13*n) + 4*a**3*b**5*x**(
14*n)/n + 28*a**2*b**6*x**(15*n)/(15*n) + a*b**7*x**(16*n)/(2*n) + b**8*x**(17*n
)/(17*n)

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Mathematica [A]  time = 0.0448421, size = 113, normalized size = 0.75 \[ \frac{x^{9 n} \left (24310 a^8+175032 a^7 b x^n+556920 a^6 b^2 x^{2 n}+1021020 a^5 b^3 x^{3 n}+1178100 a^4 b^4 x^{4 n}+875160 a^3 b^5 x^{5 n}+408408 a^2 b^6 x^{6 n}+109395 a b^7 x^{7 n}+12870 b^8 x^{8 n}\right )}{218790 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 9*n)*(a + b*x^n)^8,x]

[Out]

(x^(9*n)*(24310*a^8 + 175032*a^7*b*x^n + 556920*a^6*b^2*x^(2*n) + 1021020*a^5*b^
3*x^(3*n) + 1178100*a^4*b^4*x^(4*n) + 875160*a^3*b^5*x^(5*n) + 408408*a^2*b^6*x^
(6*n) + 109395*a*b^7*x^(7*n) + 12870*b^8*x^(8*n)))/(218790*n)

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Maple [A]  time = 0.041, size = 136, normalized size = 0.9 \[{\frac{{b}^{8} \left ({x}^{n} \right ) ^{17}}{17\,n}}+{\frac{a{b}^{7} \left ({x}^{n} \right ) ^{16}}{2\,n}}+{\frac{28\,{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{15}}{15\,n}}+4\,{\frac{{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{14}}{n}}+{\frac{70\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{13}}{13\,n}}+{\frac{14\,{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{12}}{3\,n}}+{\frac{28\,{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{11}}{11\,n}}+{\frac{4\,b{a}^{7} \left ({x}^{n} \right ) ^{10}}{5\,n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{9}}{9\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+9*n)*(a+b*x^n)^8,x)

[Out]

1/17*b^8/n*(x^n)^17+1/2*a*b^7/n*(x^n)^16+28/15*a^2*b^6/n*(x^n)^15+4*a^3*b^5/n*(x
^n)^14+70/13*a^4*b^4/n*(x^n)^13+14/3*a^5*b^3/n*(x^n)^12+28/11*a^6*b^2/n*(x^n)^11
+4/5*a^7*b/n*(x^n)^10+1/9*a^8/n*(x^n)^9

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(9*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227268, size = 153, normalized size = 1.01 \[ \frac{12870 \, b^{8} x^{17 \, n} + 109395 \, a b^{7} x^{16 \, n} + 408408 \, a^{2} b^{6} x^{15 \, n} + 875160 \, a^{3} b^{5} x^{14 \, n} + 1178100 \, a^{4} b^{4} x^{13 \, n} + 1021020 \, a^{5} b^{3} x^{12 \, n} + 556920 \, a^{6} b^{2} x^{11 \, n} + 175032 \, a^{7} b x^{10 \, n} + 24310 \, a^{8} x^{9 \, n}}{218790 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(9*n - 1),x, algorithm="fricas")

[Out]

1/218790*(12870*b^8*x^(17*n) + 109395*a*b^7*x^(16*n) + 408408*a^2*b^6*x^(15*n) +
875160*a^3*b^5*x^(14*n) + 1178100*a^4*b^4*x^(13*n) + 1021020*a^5*b^3*x^(12*n) +
556920*a^6*b^2*x^(11*n) + 175032*a^7*b*x^(10*n) + 24310*a^8*x^(9*n))/n

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1+9*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{8} x^{9 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(9*n - 1),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^8*x^(9*n - 1), x)