3.2560 \(\int x^{-1+8 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=151 \[ \frac{a^8 x^{8 n}}{8 n}+\frac{8 a^7 b x^{9 n}}{9 n}+\frac{14 a^6 b^2 x^{10 n}}{5 n}+\frac{56 a^5 b^3 x^{11 n}}{11 n}+\frac{35 a^4 b^4 x^{12 n}}{6 n}+\frac{56 a^3 b^5 x^{13 n}}{13 n}+\frac{2 a^2 b^6 x^{14 n}}{n}+\frac{8 a b^7 x^{15 n}}{15 n}+\frac{b^8 x^{16 n}}{16 n} \]

[Out]

(a^8*x^(8*n))/(8*n) + (8*a^7*b*x^(9*n))/(9*n) + (14*a^6*b^2*x^(10*n))/(5*n) + (5
6*a^5*b^3*x^(11*n))/(11*n) + (35*a^4*b^4*x^(12*n))/(6*n) + (56*a^3*b^5*x^(13*n))
/(13*n) + (2*a^2*b^6*x^(14*n))/n + (8*a*b^7*x^(15*n))/(15*n) + (b^8*x^(16*n))/(1
6*n)

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Rubi [A]  time = 0.185411, 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 \[ \frac{a^8 x^{8 n}}{8 n}+\frac{8 a^7 b x^{9 n}}{9 n}+\frac{14 a^6 b^2 x^{10 n}}{5 n}+\frac{56 a^5 b^3 x^{11 n}}{11 n}+\frac{35 a^4 b^4 x^{12 n}}{6 n}+\frac{56 a^3 b^5 x^{13 n}}{13 n}+\frac{2 a^2 b^6 x^{14 n}}{n}+\frac{8 a b^7 x^{15 n}}{15 n}+\frac{b^8 x^{16 n}}{16 n} \]

Antiderivative was successfully verified.

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

[Out]

(a^8*x^(8*n))/(8*n) + (8*a^7*b*x^(9*n))/(9*n) + (14*a^6*b^2*x^(10*n))/(5*n) + (5
6*a^5*b^3*x^(11*n))/(11*n) + (35*a^4*b^4*x^(12*n))/(6*n) + (56*a^3*b^5*x^(13*n))
/(13*n) + (2*a^2*b^6*x^(14*n))/n + (8*a*b^7*x^(15*n))/(15*n) + (b^8*x^(16*n))/(1
6*n)

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Rubi in Sympy [A]  time = 30.515, size = 136, normalized size = 0.9 \[ \frac{a^{8} x^{8 n}}{8 n} + \frac{8 a^{7} b x^{9 n}}{9 n} + \frac{14 a^{6} b^{2} x^{10 n}}{5 n} + \frac{56 a^{5} b^{3} x^{11 n}}{11 n} + \frac{35 a^{4} b^{4} x^{12 n}}{6 n} + \frac{56 a^{3} b^{5} x^{13 n}}{13 n} + \frac{2 a^{2} b^{6} x^{14 n}}{n} + \frac{8 a b^{7} x^{15 n}}{15 n} + \frac{b^{8} x^{16 n}}{16 n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**8*x**(8*n)/(8*n) + 8*a**7*b*x**(9*n)/(9*n) + 14*a**6*b**2*x**(10*n)/(5*n) + 5
6*a**5*b**3*x**(11*n)/(11*n) + 35*a**4*b**4*x**(12*n)/(6*n) + 56*a**3*b**5*x**(1
3*n)/(13*n) + 2*a**2*b**6*x**(14*n)/n + 8*a*b**7*x**(15*n)/(15*n) + b**8*x**(16*
n)/(16*n)

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Mathematica [A]  time = 0.043549, size = 113, normalized size = 0.75 \[ \frac{x^{8 n} \left (12870 a^8+91520 a^7 b x^n+288288 a^6 b^2 x^{2 n}+524160 a^5 b^3 x^{3 n}+600600 a^4 b^4 x^{4 n}+443520 a^3 b^5 x^{5 n}+205920 a^2 b^6 x^{6 n}+54912 a b^7 x^{7 n}+6435 b^8 x^{8 n}\right )}{102960 n} \]

Antiderivative was successfully verified.

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

[Out]

(x^(8*n)*(12870*a^8 + 91520*a^7*b*x^n + 288288*a^6*b^2*x^(2*n) + 524160*a^5*b^3*
x^(3*n) + 600600*a^4*b^4*x^(4*n) + 443520*a^3*b^5*x^(5*n) + 205920*a^2*b^6*x^(6*
n) + 54912*a*b^7*x^(7*n) + 6435*b^8*x^(8*n)))/(102960*n)

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Maple [A]  time = 0.04, size = 136, normalized size = 0.9 \[{\frac{{b}^{8} \left ({x}^{n} \right ) ^{16}}{16\,n}}+{\frac{8\,a{b}^{7} \left ({x}^{n} \right ) ^{15}}{15\,n}}+2\,{\frac{{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{14}}{n}}+{\frac{56\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{13}}{13\,n}}+{\frac{35\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{12}}{6\,n}}+{\frac{56\,{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{11}}{11\,n}}+{\frac{14\,{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{10}}{5\,n}}+{\frac{8\,b{a}^{7} \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{8}}{8\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+8*n)*(a+b*x^n)^8,x)

[Out]

1/16*b^8/n*(x^n)^16+8/15*a*b^7/n*(x^n)^15+2*a^2*b^6/n*(x^n)^14+56/13*a^3*b^5/n*(
x^n)^13+35/6*a^4*b^4/n*(x^n)^12+56/11*a^5*b^3/n*(x^n)^11+14/5*a^6*b^2/n*(x^n)^10
+8/9*a^7*b/n*(x^n)^9+1/8*a^8/n*(x^n)^8

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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^(8*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225603, size = 153, normalized size = 1.01 \[ \frac{6435 \, b^{8} x^{16 \, n} + 54912 \, a b^{7} x^{15 \, n} + 205920 \, a^{2} b^{6} x^{14 \, n} + 443520 \, a^{3} b^{5} x^{13 \, n} + 600600 \, a^{4} b^{4} x^{12 \, n} + 524160 \, a^{5} b^{3} x^{11 \, n} + 288288 \, a^{6} b^{2} x^{10 \, n} + 91520 \, a^{7} b x^{9 \, n} + 12870 \, a^{8} x^{8 \, n}}{102960 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/102960*(6435*b^8*x^(16*n) + 54912*a*b^7*x^(15*n) + 205920*a^2*b^6*x^(14*n) + 4
43520*a^3*b^5*x^(13*n) + 600600*a^4*b^4*x^(12*n) + 524160*a^5*b^3*x^(11*n) + 288
288*a^6*b^2*x^(10*n) + 91520*a^7*b*x^(9*n) + 12870*a^8*x^(8*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+8*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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