3.2562 \(\int x^{-1+6 n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=128 \[ -\frac{a^5 \left (a+b x^n\right )^9}{9 b^6 n}+\frac{a^4 \left (a+b x^n\right )^{10}}{2 b^6 n}-\frac{10 a^3 \left (a+b x^n\right )^{11}}{11 b^6 n}+\frac{5 a^2 \left (a+b x^n\right )^{12}}{6 b^6 n}+\frac{\left (a+b x^n\right )^{14}}{14 b^6 n}-\frac{5 a \left (a+b x^n\right )^{13}}{13 b^6 n} \]

[Out]

-(a^5*(a + b*x^n)^9)/(9*b^6*n) + (a^4*(a + b*x^n)^10)/(2*b^6*n) - (10*a^3*(a + b
*x^n)^11)/(11*b^6*n) + (5*a^2*(a + b*x^n)^12)/(6*b^6*n) - (5*a*(a + b*x^n)^13)/(
13*b^6*n) + (a + b*x^n)^14/(14*b^6*n)

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Rubi [A]  time = 0.170739, antiderivative size = 128, 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 \[ -\frac{a^5 \left (a+b x^n\right )^9}{9 b^6 n}+\frac{a^4 \left (a+b x^n\right )^{10}}{2 b^6 n}-\frac{10 a^3 \left (a+b x^n\right )^{11}}{11 b^6 n}+\frac{5 a^2 \left (a+b x^n\right )^{12}}{6 b^6 n}+\frac{\left (a+b x^n\right )^{14}}{14 b^6 n}-\frac{5 a \left (a+b x^n\right )^{13}}{13 b^6 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 6*n)*(a + b*x^n)^8,x]

[Out]

-(a^5*(a + b*x^n)^9)/(9*b^6*n) + (a^4*(a + b*x^n)^10)/(2*b^6*n) - (10*a^3*(a + b
*x^n)^11)/(11*b^6*n) + (5*a^2*(a + b*x^n)^12)/(6*b^6*n) - (5*a*(a + b*x^n)^13)/(
13*b^6*n) + (a + b*x^n)^14/(14*b^6*n)

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Rubi in Sympy [A]  time = 30.619, size = 110, normalized size = 0.86 \[ - \frac{a^{5} \left (a + b x^{n}\right )^{9}}{9 b^{6} n} + \frac{a^{4} \left (a + b x^{n}\right )^{10}}{2 b^{6} n} - \frac{10 a^{3} \left (a + b x^{n}\right )^{11}}{11 b^{6} n} + \frac{5 a^{2} \left (a + b x^{n}\right )^{12}}{6 b^{6} n} - \frac{5 a \left (a + b x^{n}\right )^{13}}{13 b^{6} n} + \frac{\left (a + b x^{n}\right )^{14}}{14 b^{6} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+6*n)*(a+b*x**n)**8,x)

[Out]

-a**5*(a + b*x**n)**9/(9*b**6*n) + a**4*(a + b*x**n)**10/(2*b**6*n) - 10*a**3*(a
+ b*x**n)**11/(11*b**6*n) + 5*a**2*(a + b*x**n)**12/(6*b**6*n) - 5*a*(a + b*x**
n)**13/(13*b**6*n) + (a + b*x**n)**14/(14*b**6*n)

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Mathematica [A]  time = 0.0432265, size = 113, normalized size = 0.88 \[ \frac{x^{6 n} \left (3003 a^8+20592 a^7 b x^n+63063 a^6 b^2 x^{2 n}+112112 a^5 b^3 x^{3 n}+126126 a^4 b^4 x^{4 n}+91728 a^3 b^5 x^{5 n}+42042 a^2 b^6 x^{6 n}+11088 a b^7 x^{7 n}+1287 b^8 x^{8 n}\right )}{18018 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 6*n)*(a + b*x^n)^8,x]

[Out]

(x^(6*n)*(3003*a^8 + 20592*a^7*b*x^n + 63063*a^6*b^2*x^(2*n) + 112112*a^5*b^3*x^
(3*n) + 126126*a^4*b^4*x^(4*n) + 91728*a^3*b^5*x^(5*n) + 42042*a^2*b^6*x^(6*n) +
11088*a*b^7*x^(7*n) + 1287*b^8*x^(8*n)))/(18018*n)

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Maple [A]  time = 0.041, size = 136, normalized size = 1.1 \[{\frac{{b}^{8} \left ({x}^{n} \right ) ^{14}}{14\,n}}+{\frac{8\,a{b}^{7} \left ({x}^{n} \right ) ^{13}}{13\,n}}+{\frac{7\,{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{12}}{3\,n}}+{\frac{56\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{11}}{11\,n}}+7\,{\frac{{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{10}}{n}}+{\frac{56\,{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{7\,{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{8}}{2\,n}}+{\frac{8\,b{a}^{7} \left ({x}^{n} \right ) ^{7}}{7\,n}}+{\frac{{a}^{8} \left ({x}^{n} \right ) ^{6}}{6\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+6*n)*(a+b*x^n)^8,x)

[Out]

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

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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^(6*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.227374, size = 153, normalized size = 1.2 \[ \frac{1287 \, b^{8} x^{14 \, n} + 11088 \, a b^{7} x^{13 \, n} + 42042 \, a^{2} b^{6} x^{12 \, n} + 91728 \, a^{3} b^{5} x^{11 \, n} + 126126 \, a^{4} b^{4} x^{10 \, n} + 112112 \, a^{5} b^{3} x^{9 \, n} + 63063 \, a^{6} b^{2} x^{8 \, n} + 20592 \, a^{7} b x^{7 \, n} + 3003 \, a^{8} x^{6 \, n}}{18018 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(6*n - 1),x, algorithm="fricas")

[Out]

1/18018*(1287*b^8*x^(14*n) + 11088*a*b^7*x^(13*n) + 42042*a^2*b^6*x^(12*n) + 917
28*a^3*b^5*x^(11*n) + 126126*a^4*b^4*x^(10*n) + 112112*a^5*b^3*x^(9*n) + 63063*a
^6*b^2*x^(8*n) + 20592*a^7*b*x^(7*n) + 3003*a^8*x^(6*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+6*n)*(a+b*x**n)**8,x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{n} + a\right )}^{8} x^{6 \, n - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^n + a)^8*x^(6*n - 1),x, algorithm="giac")

[Out]

integrate((b*x^n + a)^8*x^(6*n - 1), x)