\(\int (a+\frac {b}{x})^8 x^8 \, dx\) [1590]

Optimal result

Integrand size = 13, antiderivative size = 14 \[ \int \left (a+\frac {b}{x}\right )^8 x^8 \, dx=\frac {(b+a x)^9}{9 a} \]

[Out]

1/9*(a*x+b)^9/a

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 32} \[ \int \left (a+\frac {b}{x}\right )^8 x^8 \, dx=\frac {(a x+b)^9}{9 a} \]

[In]

Int[(a + b/x)^8*x^8,x]

[Out]

(b + a*x)^9/(9*a)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 269

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

Rubi steps \begin{align*} \text {integral}& = \int (b+a x)^8 \, dx \\ & = \frac {(b+a x)^9}{9 a} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x}\right )^8 x^8 \, dx=\frac {(b+a x)^9}{9 a} \]

[In]

Integrate[(a + b/x)^8*x^8,x]

[Out]

(b + a*x)^9/(9*a)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
default	\(\frac {\left (a x +b \right )^{9}}{9 a}\)	\(13\)
parallelrisch	\(\frac {1}{9} x^{9} a^{8}+a^{7} b \,x^{8}+4 x^{7} b^{2} a^{6}+\frac {28}{3} x^{6} b^{3} a^{5}+14 a^{4} b^{4} x^{5}+14 a^{3} b^{5} x^{4}+\frac {28}{3} a^{2} b^{6} x^{3}+4 a \,b^{7} x^{2}+b^{8} x\)	\(87\)
gosper	\(\frac {x \left (a^{8} x^{8}+9 x^{7} b \,a^{7}+36 a^{6} b^{2} x^{6}+84 a^{5} b^{3} x^{5}+126 a^{4} x^{4} b^{4}+126 a^{3} b^{5} x^{3}+84 a^{2} b^{6} x^{2}+36 a \,b^{7} x +9 b^{8}\right )}{9}\)	\(88\)
norman	\(\frac {b^{8} x^{8}+a^{7} b \,x^{15}+\frac {1}{9} a^{8} x^{16}+4 a \,b^{7} x^{9}+\frac {28}{3} a^{2} b^{6} x^{10}+14 a^{4} b^{4} x^{12}+4 a^{6} b^{2} x^{14}+14 x^{11} b^{5} a^{3}+\frac {28}{3} x^{13} b^{3} a^{5}}{x^{7}}\)	\(93\)
risch	\(\frac {x^{9} a^{8}}{9}+a^{7} b \,x^{8}+4 x^{7} b^{2} a^{6}+\frac {28 x^{6} b^{3} a^{5}}{3}+14 a^{4} b^{4} x^{5}+14 a^{3} b^{5} x^{4}+\frac {28 a^{2} b^{6} x^{3}}{3}+4 a \,b^{7} x^{2}+b^{8} x +\frac {b^{9}}{9 a}\)	\(95\)

[In]

int((a+b/x)^8*x^8,x,method=_RETURNVERBOSE)

[Out]

1/9*(a*x+b)^9/a

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (12) = 24\).

Time = 0.26 (sec) , antiderivative size = 86, normalized size of antiderivative = 6.14 \[ \int \left (a+\frac {b}{x}\right )^8 x^8 \, dx=\frac {1}{9} \, a^{8} x^{9} + a^{7} b x^{8} + 4 \, a^{6} b^{2} x^{7} + \frac {28}{3} \, a^{5} b^{3} x^{6} + 14 \, a^{4} b^{4} x^{5} + 14 \, a^{3} b^{5} x^{4} + \frac {28}{3} \, a^{2} b^{6} x^{3} + 4 \, a b^{7} x^{2} + b^{8} x \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 94 vs. \(2 (8) = 16\).

Time = 0.03 (sec) , antiderivative size = 94, normalized size of antiderivative = 6.71 \[ \int \left (a+\frac {b}{x}\right )^8 x^8 \, dx=\frac {a^{8} x^{9}}{9} + a^{7} b x^{8} + 4 a^{6} b^{2} x^{7} + \frac {28 a^{5} b^{3} x^{6}}{3} + 14 a^{4} b^{4} x^{5} + 14 a^{3} b^{5} x^{4} + \frac {28 a^{2} b^{6} x^{3}}{3} + 4 a b^{7} x^{2} + b^{8} x \]

[In]

integrate((a+b/x)**8*x**8,x)

[Out]

a**8*x**9/9 + a**7*b*x**8 + 4*a**6*b**2*x**7 + 28*a**5*b**3*x**6/3 + 14*a**4*b**4*x**5 + 14*a**3*b**5*x**4 + 2
8*a**2*b**6*x**3/3 + 4*a*b**7*x**2 + b**8*x

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (12) = 24\).

Time = 0.21 (sec) , antiderivative size = 86, normalized size of antiderivative = 6.14 \[ \int \left (a+\frac {b}{x}\right )^8 x^8 \, dx=\frac {1}{9} \, a^{8} x^{9} + a^{7} b x^{8} + 4 \, a^{6} b^{2} x^{7} + \frac {28}{3} \, a^{5} b^{3} x^{6} + 14 \, a^{4} b^{4} x^{5} + 14 \, a^{3} b^{5} x^{4} + \frac {28}{3} \, a^{2} b^{6} x^{3} + 4 \, a b^{7} x^{2} + b^{8} x \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 86 vs. \(2 (12) = 24\).

Time = 0.28 (sec) , antiderivative size = 86, normalized size of antiderivative = 6.14 \[ \int \left (a+\frac {b}{x}\right )^8 x^8 \, dx=\frac {1}{9} \, a^{8} x^{9} + a^{7} b x^{8} + 4 \, a^{6} b^{2} x^{7} + \frac {28}{3} \, a^{5} b^{3} x^{6} + 14 \, a^{4} b^{4} x^{5} + 14 \, a^{3} b^{5} x^{4} + \frac {28}{3} \, a^{2} b^{6} x^{3} + 4 \, a b^{7} x^{2} + b^{8} x \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 86, normalized size of antiderivative = 6.14 \[ \int \left (a+\frac {b}{x}\right )^8 x^8 \, dx=\frac {a^8\,x^9}{9}+a^7\,b\,x^8+4\,a^6\,b^2\,x^7+\frac {28\,a^5\,b^3\,x^6}{3}+14\,a^4\,b^4\,x^5+14\,a^3\,b^5\,x^4+\frac {28\,a^2\,b^6\,x^3}{3}+4\,a\,b^7\,x^2+b^8\,x \]

[In]

int(x^8*(a + b/x)^8,x)

[Out]

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