3.2575 \(\int \frac {(a+b x^n)^8}{x} \, dx\)

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

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Rubi [A]  time = 0.05, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \[ \frac {14 a^6 b^2 x^{2 n}}{n}+\frac {56 a^5 b^3 x^{3 n}}{3 n}+\frac {35 a^4 b^4 x^{4 n}}{2 n}+\frac {56 a^3 b^5 x^{5 n}}{5 n}+\frac {14 a^2 b^6 x^{6 n}}{3 n}+\frac {8 a^7 b x^n}{n}+a^8 \log (x)+\frac {8 a b^7 x^{7 n}}{7 n}+\frac {b^8 x^{8 n}}{8 n} \]

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]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int \frac {\left (a+b x^n\right )^8}{x} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {(a+b x)^8}{x} \, dx,x,x^n\right )}{n}\\ &=\frac {\operatorname {Subst}\left (\int \left (8 a^7 b+\frac {a^8}{x}+28 a^6 b^2 x+56 a^5 b^3 x^2+70 a^4 b^4 x^3+56 a^3 b^5 x^4+28 a^2 b^6 x^5+8 a b^7 x^6+b^8 x^7\right ) \, dx,x,x^n\right )}{n}\\ &=\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}+a^8 \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.08, size = 119, normalized size = 0.86 \[ \frac {a^8 n \log (x)+8 a^7 b x^n+14 a^6 b^2 x^{2 n}+\frac {56}{3} a^5 b^3 x^{3 n}+\frac {35}{2} a^4 b^4 x^{4 n}+\frac {56}{5} a^3 b^5 x^{5 n}+\frac {14}{3} a^2 b^6 x^{6 n}+\frac {8}{7} a b^7 x^{7 n}+\frac {1}{8} b^8 x^{8 n}}{n} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^n)^8/x,x]

[Out]

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

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fricas [A]  time = 0.64, size = 109, normalized size = 0.79 \[ \frac {840 \, a^{8} n \log \relax (x) + 105 \, b^{8} x^{8 \, n} + 960 \, a b^{7} x^{7 \, n} + 3920 \, a^{2} b^{6} x^{6 \, n} + 9408 \, a^{3} b^{5} x^{5 \, n} + 14700 \, a^{4} b^{4} x^{4 \, n} + 15680 \, a^{5} b^{3} x^{3 \, n} + 11760 \, a^{6} b^{2} x^{2 \, n} + 6720 \, a^{7} b x^{n}}{840 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^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) + 11760*a^6*b^2*x^(2*n) + 6720*a^7*b*x^n)/n

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (b x^{n} + a\right )}^{8}}{x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*x^n + a)^8/x, x)

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maple [A]  time = 0.00, size = 132, normalized size = 0.96 \[ \frac {a^{8} \ln \left (x^{n}\right )}{n}+\frac {8 a^{7} b \,x^{n}}{n}+\frac {14 a^{6} b^{2} x^{2 n}}{n}+\frac {56 a^{5} b^{3} x^{3 n}}{3 n}+\frac {35 a^{4} b^{4} x^{4 n}}{2 n}+\frac {56 a^{3} b^{5} x^{5 n}}{5 n}+\frac {14 a^{2} b^{6} x^{6 n}}{3 n}+\frac {8 a \,b^{7} x^{7 n}}{7 n}+\frac {b^{8} x^{8 n}}{8 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.54, size = 113, normalized size = 0.82 \[ \frac {a^{8} \log \left (x^{n}\right )}{n} + \frac {105 \, b^{8} x^{8 \, n} + 960 \, a b^{7} x^{7 \, n} + 3920 \, a^{2} b^{6} x^{6 \, n} + 9408 \, a^{3} b^{5} x^{5 \, n} + 14700 \, a^{4} b^{4} x^{4 \, n} + 15680 \, a^{5} b^{3} x^{3 \, n} + 11760 \, a^{6} b^{2} x^{2 \, n} + 6720 \, a^{7} b x^{n}}{840 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*x^n)^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) + 14
700*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)/n

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mupad [B]  time = 1.29, size = 126, normalized size = 0.91 \[ a^8\,\ln \relax (x)+\frac {b^8\,x^{8\,n}}{8\,n}+\frac {14\,a^6\,b^2\,x^{2\,n}}{n}+\frac {56\,a^5\,b^3\,x^{3\,n}}{3\,n}+\frac {35\,a^4\,b^4\,x^{4\,n}}{2\,n}+\frac {56\,a^3\,b^5\,x^{5\,n}}{5\,n}+\frac {14\,a^2\,b^6\,x^{6\,n}}{3\,n}+\frac {8\,a^7\,b\,x^n}{n}+\frac {8\,a\,b^7\,x^{7\,n}}{7\,n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x^n)^8/x,x)

[Out]

a^8*log(x) + (b^8*x^(8*n))/(8*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^7*b*x^n)/n + (8*a*b^7*x^(7*n))/(7*n)

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sympy [A]  time = 4.05, size = 136, normalized size = 0.99 \[ \begin {cases} a^{8} \log {\relax (x )} + \frac {8 a^{7} b x^{n}}{n} + \frac {14 a^{6} b^{2} x^{2 n}}{n} + \frac {56 a^{5} b^{3} x^{3 n}}{3 n} + \frac {35 a^{4} b^{4} x^{4 n}}{2 n} + \frac {56 a^{3} b^{5} x^{5 n}}{5 n} + \frac {14 a^{2} b^{6} x^{6 n}}{3 n} + \frac {8 a b^{7} x^{7 n}}{7 n} + \frac {b^{8} x^{8 n}}{8 n} & \text {for}\: n \neq 0 \\\left (a + b\right )^{8} \log {\relax (x )} & \text {otherwise} \end {cases} \]

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) + 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), Ne(n, 0)), ((a + b)**8*log(x), True))

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