3.2567 \(\int x^{-1+n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=19 \[ \frac{\left (a+b x^n\right )^9}{9 b n} \]

[Out]

(a + b*x^n)^9/(9*b*n)

_______________________________________________________________________________________

Rubi [A]  time = 0.0191555, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067 \[ \frac{\left (a+b x^n\right )^9}{9 b n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + n)*(a + b*x^n)^8,x]

[Out]

(a + b*x^n)^9/(9*b*n)

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 2.48917, size = 12, normalized size = 0.63 \[ \frac{\left (a + b x^{n}\right )^{9}}{9 b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x**n)**9/(9*b*n)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0132198, size = 19, normalized size = 1. \[ \frac{\left (a+b x^n\right )^9}{9 b n} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x^n)^9/(9*b*n)

_______________________________________________________________________________________

Maple [B]  time = 0.04, size = 132, normalized size = 7. \[{\frac{{b}^{8} \left ({x}^{n} \right ) ^{9}}{9\,n}}+{\frac{a{b}^{7} \left ({x}^{n} \right ) ^{8}}{n}}+4\,{\frac{{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{7}}{n}}+{\frac{28\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{6}}{3\,n}}+14\,{\frac{{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{5}}{n}}+14\,{\frac{{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{4}}{n}}+{\frac{28\,{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{3}}{3\,n}}+4\,{\frac{b{a}^{7} \left ({x}^{n} \right ) ^{2}}{n}}+{\frac{{a}^{8}{x}^{n}}{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+n)*(a+b*x^n)^8,x)

[Out]

1/9*b^8/n*(x^n)^9+a*b^7/n*(x^n)^8+4*a^2*b^6/n*(x^n)^7+28/3*a^3*b^5/n*(x^n)^6+14*
a^4*b^4/n*(x^n)^5+14*a^5*b^3/n*(x^n)^4+28/3*a^6*b^2/n*(x^n)^3+4*a^7*b/n*(x^n)^2+
a^8/n*x^n

_______________________________________________________________________________________

Maxima [A]  time = 1.43503, size = 23, normalized size = 1.21 \[ \frac{{\left (b x^{n} + a\right )}^{9}}{9 \, b n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*(b*x^n + a)^9/(b*n)

_______________________________________________________________________________________

Fricas [A]  time = 0.224952, size = 149, normalized size = 7.84 \[ \frac{b^{8} x^{9 \, n} + 9 \, a b^{7} x^{8 \, n} + 36 \, a^{2} b^{6} x^{7 \, n} + 84 \, a^{3} b^{5} x^{6 \, n} + 126 \, a^{4} b^{4} x^{5 \, n} + 126 \, a^{5} b^{3} x^{4 \, n} + 84 \, a^{6} b^{2} x^{3 \, n} + 36 \, a^{7} b x^{2 \, n} + 9 \, a^{8} x^{n}}{9 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*(b^8*x^(9*n) + 9*a*b^7*x^(8*n) + 36*a^2*b^6*x^(7*n) + 84*a^3*b^5*x^(6*n) + 1
26*a^4*b^4*x^(5*n) + 126*a^5*b^3*x^(4*n) + 84*a^6*b^2*x^(3*n) + 36*a^7*b*x^(2*n)
+ 9*a^8*x^n)/n

_______________________________________________________________________________________

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+n)*(a+b*x**n)**8,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.216181, size = 149, normalized size = 7.84 \[ \frac{b^{8} x^{9 \, n} + 9 \, a b^{7} x^{8 \, n} + 36 \, a^{2} b^{6} x^{7 \, n} + 84 \, a^{3} b^{5} x^{6 \, n} + 126 \, a^{4} b^{4} x^{5 \, n} + 126 \, a^{5} b^{3} x^{4 \, n} + 84 \, a^{6} b^{2} x^{3 \, n} + 36 \, a^{7} b x^{2 \, n} + 9 \, a^{8} x^{n}}{9 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/9*(b^8*x^(9*n) + 9*a*b^7*x^(8*n) + 36*a^2*b^6*x^(7*n) + 84*a^3*b^5*x^(6*n) + 1
26*a^4*b^4*x^(5*n) + 126*a^5*b^3*x^(4*n) + 84*a^6*b^2*x^(3*n) + 36*a^7*b*x^(2*n)
+ 9*a^8*x^n)/n