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

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

[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]

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Rubi [A]  time = 0.0545366, 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, 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 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]]

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])

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*}

Mathematica [A]  time = 0.0594588, size = 119, normalized size = 0.86 \[ \frac{14 a^6 b^2 x^{2 n}+\frac{56}{3} a^5 b^3 x^{3 n}+\frac{35}{2} a^4 b^4 x^{4 n}+\frac{56}{5} a^3 b^5 x^{5 n}+\frac{14}{3} a^2 b^6 x^{6 n}+8 a^7 b x^n+a^8 n \log (x)+\frac{8}{7} a b^7 x^{7 n}+\frac{1}{8} b^8 x^{8 n}}{n} \]

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

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*b^7*a*(x^n)^7+14/3/n*b^6*a^2*(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 = 0.96894, size = 153, normalized size = 1.11 \begin{align*} \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} \end{align*}

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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Fricas [A]  time = 1.38854, size = 271, normalized size = 1.96 \begin{align*} \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} \end{align*}

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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Sympy [A]  time = 2.14189, size = 136, normalized size = 0.99 \begin{align*} \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} \end{align*}

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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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b x^{n} + a\right )}^{8}}{x}\,{d x} \end{align*}

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)