3.2568 \(\int \frac{\left (a+b x^n\right )^8}{x} \, dx\)

Optimal. Leaf size=138 \[ a^8 \log (x)+\frac{8 a^7 b x^n}{n}+\frac{14 a^6 b^2 x^{2 n}}{n}+\frac{56 a^5 b^3 x^{3 n}}{3 n}+\frac{35 a^4 b^4 x^{4 n}}{2 n}+\frac{56 a^3 b^5 x^{5 n}}{5 n}+\frac{14 a^2 b^6 x^{6 n}}{3 n}+\frac{8 a b^7 x^{7 n}}{7 n}+\frac{b^8 x^{8 n}}{8 n} \]

[Out]

(8*a^7*b*x^n)/n + (14*a^6*b^2*x^(2*n))/n + (56*a^5*b^3*x^(3*n))/(3*n) + (35*a^4*
b^4*x^(4*n))/(2*n) + (56*a^3*b^5*x^(5*n))/(5*n) + (14*a^2*b^6*x^(6*n))/(3*n) + (
8*a*b^7*x^(7*n))/(7*n) + (b^8*x^(8*n))/(8*n) + a^8*Log[x]

_______________________________________________________________________________________

Rubi [A]  time = 0.140255, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ a^8 \log (x)+\frac{8 a^7 b x^n}{n}+\frac{14 a^6 b^2 x^{2 n}}{n}+\frac{56 a^5 b^3 x^{3 n}}{3 n}+\frac{35 a^4 b^4 x^{4 n}}{2 n}+\frac{56 a^3 b^5 x^{5 n}}{5 n}+\frac{14 a^2 b^6 x^{6 n}}{3 n}+\frac{8 a b^7 x^{7 n}}{7 n}+\frac{b^8 x^{8 n}}{8 n} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^n)^8/x,x]

[Out]

(8*a^7*b*x^n)/n + (14*a^6*b^2*x^(2*n))/n + (56*a^5*b^3*x^(3*n))/(3*n) + (35*a^4*
b^4*x^(4*n))/(2*n) + (56*a^3*b^5*x^(5*n))/(5*n) + (14*a^2*b^6*x^(6*n))/(3*n) + (
8*a*b^7*x^(7*n))/(7*n) + (b^8*x^(8*n))/(8*n) + a^8*Log[x]

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{8} \log{\left (x^{n} \right )}}{n} + \frac{8 a^{7} b x^{n}}{n} + \frac{28 a^{6} b^{2} \int ^{x^{n}} x\, dx}{n} + \frac{56 a^{5} b^{3} x^{3 n}}{3 n} + \frac{35 a^{4} b^{4} x^{4 n}}{2 n} + \frac{56 a^{3} b^{5} x^{5 n}}{5 n} + \frac{14 a^{2} b^{6} x^{6 n}}{3 n} + \frac{8 a b^{7} x^{7 n}}{7 n} + \frac{b^{8} x^{8 n}}{8 n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**8*log(x**n)/n + 8*a**7*b*x**n/n + 28*a**6*b**2*Integral(x, (x, x**n))/n + 56*
a**5*b**3*x**(3*n)/(3*n) + 35*a**4*b**4*x**(4*n)/(2*n) + 56*a**3*b**5*x**(5*n)/(
5*n) + 14*a**2*b**6*x**(6*n)/(3*n) + 8*a*b**7*x**(7*n)/(7*n) + b**8*x**(8*n)/(8*
n)

_______________________________________________________________________________________

Mathematica [A]  time = 0.0838256, size = 106, normalized size = 0.77 \[ a^8 \log (x)+\frac{b x^n \left (6720 a^7+11760 a^6 b x^n+15680 a^5 b^2 x^{2 n}+14700 a^4 b^3 x^{3 n}+9408 a^3 b^4 x^{4 n}+3920 a^2 b^5 x^{5 n}+960 a b^6 x^{6 n}+105 b^7 x^{7 n}\right )}{840 n} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^n)^8/x,x]

[Out]

(b*x^n*(6720*a^7 + 11760*a^6*b*x^n + 15680*a^5*b^2*x^(2*n) + 14700*a^4*b^3*x^(3*
n) + 9408*a^3*b^4*x^(4*n) + 3920*a^2*b^5*x^(5*n) + 960*a*b^6*x^(6*n) + 105*b^7*x
^(7*n)))/(840*n) + a^8*Log[x]

_______________________________________________________________________________________

Maple [A]  time = 0.003, size = 132, normalized size = 1. \[{\frac{{b}^{8} \left ({x}^{n} \right ) ^{8}}{8\,n}}+{\frac{8\,a{b}^{7} \left ({x}^{n} \right ) ^{7}}{7\,n}}+{\frac{14\,{a}^{2}{b}^{6} \left ({x}^{n} \right ) ^{6}}{3\,n}}+{\frac{56\,{a}^{3}{b}^{5} \left ({x}^{n} \right ) ^{5}}{5\,n}}+{\frac{35\,{a}^{4}{b}^{4} \left ({x}^{n} \right ) ^{4}}{2\,n}}+{\frac{56\,{a}^{5}{b}^{3} \left ({x}^{n} \right ) ^{3}}{3\,n}}+14\,{\frac{{a}^{6}{b}^{2} \left ({x}^{n} \right ) ^{2}}{n}}+8\,{\frac{b{a}^{7}{x}^{n}}{n}}+{\frac{{a}^{8}\ln \left ({x}^{n} \right ) }{n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b*x^n)^8/x,x)

[Out]

1/8/n*b^8*(x^n)^8+8/7/n*a*b^7*(x^n)^7+14/3/n*a^2*b^6*(x^n)^6+56/5/n*a^3*b^5*(x^n
)^5+35/2/n*a^4*b^4*(x^n)^4+56/3/n*a^5*b^3*(x^n)^3+14/n*a^6*b^2*(x^n)^2+8*a^7*b*x
^n/n+1/n*a^8*ln(x^n)

_______________________________________________________________________________________

Maxima [A]  time = 1.44968, size = 153, normalized size = 1.11 \[ \frac{a^{8} \log \left (x^{n}\right )}{n} + \frac{105 \, b^{8} x^{8 \, n} + 960 \, a b^{7} x^{7 \, n} + 3920 \, a^{2} b^{6} x^{6 \, n} + 9408 \, a^{3} b^{5} x^{5 \, n} + 14700 \, a^{4} b^{4} x^{4 \, n} + 15680 \, a^{5} b^{3} x^{3 \, n} + 11760 \, a^{6} b^{2} x^{2 \, n} + 6720 \, a^{7} b x^{n}}{840 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a^8*log(x^n)/n + 1/840*(105*b^8*x^(8*n) + 960*a*b^7*x^(7*n) + 3920*a^2*b^6*x^(6*
n) + 9408*a^3*b^5*x^(5*n) + 14700*a^4*b^4*x^(4*n) + 15680*a^5*b^3*x^(3*n) + 1176
0*a^6*b^2*x^(2*n) + 6720*a^7*b*x^n)/n

_______________________________________________________________________________________

Fricas [A]  time = 0.226953, size = 147, normalized size = 1.07 \[ \frac{840 \, a^{8} n \log \left (x\right ) + 105 \, b^{8} x^{8 \, n} + 960 \, a b^{7} x^{7 \, n} + 3920 \, a^{2} b^{6} x^{6 \, n} + 9408 \, a^{3} b^{5} x^{5 \, n} + 14700 \, a^{4} b^{4} x^{4 \, n} + 15680 \, a^{5} b^{3} x^{3 \, n} + 11760 \, a^{6} b^{2} x^{2 \, n} + 6720 \, a^{7} b x^{n}}{840 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/840*(840*a^8*n*log(x) + 105*b^8*x^(8*n) + 960*a*b^7*x^(7*n) + 3920*a^2*b^6*x^(
6*n) + 9408*a^3*b^5*x^(5*n) + 14700*a^4*b^4*x^(4*n) + 15680*a^5*b^3*x^(3*n) + 11
760*a^6*b^2*x^(2*n) + 6720*a^7*b*x^n)/n

_______________________________________________________________________________________

Sympy [A]  time = 6.77068, size = 136, normalized size = 0.99 \[ \begin{cases} a^{8} \log{\left (x \right )} + \frac{8 a^{7} b x^{n}}{n} + \frac{14 a^{6} b^{2} x^{2 n}}{n} + \frac{56 a^{5} b^{3} x^{3 n}}{3 n} + \frac{35 a^{4} b^{4} x^{4 n}}{2 n} + \frac{56 a^{3} b^{5} x^{5 n}}{5 n} + \frac{14 a^{2} b^{6} x^{6 n}}{3 n} + \frac{8 a b^{7} x^{7 n}}{7 n} + \frac{b^{8} x^{8 n}}{8 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{8} \log{\left (x \right )} & \text{otherwise} \end{cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b*x**n)**8/x,x)

[Out]

Piecewise((a**8*log(x) + 8*a**7*b*x**n/n + 14*a**6*b**2*x**(2*n)/n + 56*a**5*b**
3*x**(3*n)/(3*n) + 35*a**4*b**4*x**(4*n)/(2*n) + 56*a**3*b**5*x**(5*n)/(5*n) + 1
4*a**2*b**6*x**(6*n)/(3*n) + 8*a*b**7*x**(7*n)/(7*n) + b**8*x**(8*n)/(8*n), Ne(n
, 0)), ((a + b)**8*log(x), True))

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^n + a)^8/x, x)