3.2569 \(\int x^{-1-n} \left (a+b x^n\right )^8 \, dx\)

Optimal. Leaf size=135 \[ -\frac{a^8 x^{-n}}{n}+8 a^7 b \log (x)+\frac{28 a^6 b^2 x^n}{n}+\frac{28 a^5 b^3 x^{2 n}}{n}+\frac{70 a^4 b^4 x^{3 n}}{3 n}+\frac{14 a^3 b^5 x^{4 n}}{n}+\frac{28 a^2 b^6 x^{5 n}}{5 n}+\frac{4 a b^7 x^{6 n}}{3 n}+\frac{b^8 x^{7 n}}{7 n} \]

[Out]

-(a^8/(n*x^n)) + (28*a^6*b^2*x^n)/n + (28*a^5*b^3*x^(2*n))/n + (70*a^4*b^4*x^(3*
n))/(3*n) + (14*a^3*b^5*x^(4*n))/n + (28*a^2*b^6*x^(5*n))/(5*n) + (4*a*b^7*x^(6*
n))/(3*n) + (b^8*x^(7*n))/(7*n) + 8*a^7*b*Log[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.15267, antiderivative size = 135, 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 \[ -\frac{a^8 x^{-n}}{n}+8 a^7 b \log (x)+\frac{28 a^6 b^2 x^n}{n}+\frac{28 a^5 b^3 x^{2 n}}{n}+\frac{70 a^4 b^4 x^{3 n}}{3 n}+\frac{14 a^3 b^5 x^{4 n}}{n}+\frac{28 a^2 b^6 x^{5 n}}{5 n}+\frac{4 a b^7 x^{6 n}}{3 n}+\frac{b^8 x^{7 n}}{7 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 - n)*(a + b*x^n)^8,x]

[Out]

-(a^8/(n*x^n)) + (28*a^6*b^2*x^n)/n + (28*a^5*b^3*x^(2*n))/n + (70*a^4*b^4*x^(3*
n))/(3*n) + (14*a^3*b^5*x^(4*n))/n + (28*a^2*b^6*x^(5*n))/(5*n) + (4*a*b^7*x^(6*
n))/(3*n) + (b^8*x^(7*n))/(7*n) + 8*a^7*b*Log[x]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{8} x^{- n}}{n} + \frac{8 a^{7} b \log{\left (x^{n} \right )}}{n} + \frac{28 a^{6} b^{2} x^{n}}{n} + \frac{56 a^{5} b^{3} \int ^{x^{n}} x\, dx}{n} + \frac{70 a^{4} b^{4} x^{3 n}}{3 n} + \frac{14 a^{3} b^{5} x^{4 n}}{n} + \frac{28 a^{2} b^{6} x^{5 n}}{5 n} + \frac{4 a b^{7} x^{6 n}}{3 n} + \frac{b^{8} x^{7 n}}{7 n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**8*x**(-n)/n + 8*a**7*b*log(x**n)/n + 28*a**6*b**2*x**n/n + 56*a**5*b**3*Inte
gral(x, (x, x**n))/n + 70*a**4*b**4*x**(3*n)/(3*n) + 14*a**3*b**5*x**(4*n)/n + 2
8*a**2*b**6*x**(5*n)/(5*n) + 4*a*b**7*x**(6*n)/(3*n) + b**8*x**(7*n)/(7*n)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0532273, size = 116, normalized size = 0.86 \[ \frac{x^{-n} \left (-105 a^8+840 a^7 b n x^n \log (x)+2940 a^6 b^2 x^{2 n}+2940 a^5 b^3 x^{3 n}+2450 a^4 b^4 x^{4 n}+1470 a^3 b^5 x^{5 n}+588 a^2 b^6 x^{6 n}+140 a b^7 x^{7 n}+15 b^8 x^{8 n}\right )}{105 n} \]

Antiderivative was successfully verified.

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

[Out]

(-105*a^8 + 2940*a^6*b^2*x^(2*n) + 2940*a^5*b^3*x^(3*n) + 2450*a^4*b^4*x^(4*n) +
1470*a^3*b^5*x^(5*n) + 588*a^2*b^6*x^(6*n) + 140*a*b^7*x^(7*n) + 15*b^8*x^(8*n)
+ 840*a^7*b*n*x^n*Log[x])/(105*n*x^n)

_______________________________________________________________________________________

Maple [A]  time = 0.044, size = 128, normalized size = 1. \[ 8\,{a}^{7}b\ln \left ( x \right ) +{\frac{{b}^{8} \left ({x}^{n} \right ) ^{7}}{7\,n}}+{\frac{4\,a{b}^{7} \left ({x}^{n} \right ) ^{6}}{3\,n}}+{\frac{28\,{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{5}}{5\,n}}+14\,{\frac{{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{4}}{n}}+{\frac{70\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{3}}{3\,n}}+28\,{\frac{{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{2}}{n}}+28\,{\frac{{a}^{6}{b}^{2}{x}^{n}}{n}}-{\frac{{a}^{8}}{n{x}^{n}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1-n)*(a+b*x^n)^8,x)

[Out]

8*a^7*b*ln(x)+1/7*b^8/n*(x^n)^7+4/3*a*b^7/n*(x^n)^6+28/5*a^2*b^6/n*(x^n)^5+14*a^
3*b^5/n*(x^n)^4+70/3*a^4*b^4/n*(x^n)^3+28*a^5*b^3/n*(x^n)^2+28*a^6*b^2*x^n/n-a^8
/n/(x^n)

_______________________________________________________________________________________

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^(-n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.227965, size = 154, normalized size = 1.14 \[ \frac{840 \, a^{7} b n x^{n} \log \left (x\right ) + 15 \, b^{8} x^{8 \, n} + 140 \, a b^{7} x^{7 \, n} + 588 \, a^{2} b^{6} x^{6 \, n} + 1470 \, a^{3} b^{5} x^{5 \, n} + 2450 \, a^{4} b^{4} x^{4 \, n} + 2940 \, a^{5} b^{3} x^{3 \, n} + 2940 \, a^{6} b^{2} x^{2 \, n} - 105 \, a^{8}}{105 \, n x^{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(840*a^7*b*n*x^n*log(x) + 15*b^8*x^(8*n) + 140*a*b^7*x^(7*n) + 588*a^2*b^6
*x^(6*n) + 1470*a^3*b^5*x^(5*n) + 2450*a^4*b^4*x^(4*n) + 2940*a^5*b^3*x^(3*n) +
2940*a^6*b^2*x^(2*n) - 105*a^8)/(n*x^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 = 0.233057, size = 167, normalized size = 1.24 \[ \frac{{\left (840 \, a^{7} b n e^{\left (n{\rm ln}\left (x\right )\right )}{\rm ln}\left (x\right ) + 15 \, b^{8} e^{\left (8 \, n{\rm ln}\left (x\right )\right )} + 140 \, a b^{7} e^{\left (7 \, n{\rm ln}\left (x\right )\right )} + 588 \, a^{2} b^{6} e^{\left (6 \, n{\rm ln}\left (x\right )\right )} + 1470 \, a^{3} b^{5} e^{\left (5 \, n{\rm ln}\left (x\right )\right )} + 2450 \, a^{4} b^{4} e^{\left (4 \, n{\rm ln}\left (x\right )\right )} + 2940 \, a^{5} b^{3} e^{\left (3 \, n{\rm ln}\left (x\right )\right )} + 2940 \, a^{6} b^{2} e^{\left (2 \, n{\rm ln}\left (x\right )\right )} - 105 \, a^{8}\right )} e^{\left (-n{\rm ln}\left (x\right )\right )}}{105 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(840*a^7*b*n*e^(n*ln(x))*ln(x) + 15*b^8*e^(8*n*ln(x)) + 140*a*b^7*e^(7*n*l
n(x)) + 588*a^2*b^6*e^(6*n*ln(x)) + 1470*a^3*b^5*e^(5*n*ln(x)) + 2450*a^4*b^4*e^
(4*n*ln(x)) + 2940*a^5*b^3*e^(3*n*ln(x)) + 2940*a^6*b^2*e^(2*n*ln(x)) - 105*a^8)
*e^(-n*ln(x))/n