3.1593 \(\int (a+\frac {b}{x})^8 x^5 \, dx\)

Optimal. Leaf size=95 \[ \frac {a^8 x^6}{6}+\frac {8}{5} a^7 b x^5+7 a^6 b^2 x^4+\frac {56}{3} a^5 b^3 x^3+35 a^4 b^4 x^2+56 a^3 b^5 x+28 a^2 b^6 \log (x)-\frac {8 a b^7}{x}-\frac {b^8}{2 x^2} \]

[Out]

-1/2*b^8/x^2-8*a*b^7/x+56*a^3*b^5*x+35*a^4*b^4*x^2+56/3*a^5*b^3*x^3+7*a^6*b^2*x^4+8/5*a^7*b*x^5+1/6*a^8*x^6+28
*a^2*b^6*ln(x)

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 95, 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 = {263, 43} \[ 7 a^6 b^2 x^4+\frac {56}{3} a^5 b^3 x^3+35 a^4 b^4 x^2+56 a^3 b^5 x+28 a^2 b^6 \log (x)+\frac {8}{5} a^7 b x^5+\frac {a^8 x^6}{6}-\frac {8 a b^7}{x}-\frac {b^8}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8*x^5,x]

[Out]

-b^8/(2*x^2) - (8*a*b^7)/x + 56*a^3*b^5*x + 35*a^4*b^4*x^2 + (56*a^5*b^3*x^3)/3 + 7*a^6*b^2*x^4 + (8*a^7*b*x^5
)/5 + (a^8*x^6)/6 + 28*a^2*b^6*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 263

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*} \int \left (a+\frac {b}{x}\right )^8 x^5 \, dx &=\int \frac {(b+a x)^8}{x^3} \, dx\\ &=\int \left (56 a^3 b^5+\frac {b^8}{x^3}+\frac {8 a b^7}{x^2}+\frac {28 a^2 b^6}{x}+70 a^4 b^4 x+56 a^5 b^3 x^2+28 a^6 b^2 x^3+8 a^7 b x^4+a^8 x^5\right ) \, dx\\ &=-\frac {b^8}{2 x^2}-\frac {8 a b^7}{x}+56 a^3 b^5 x+35 a^4 b^4 x^2+\frac {56}{3} a^5 b^3 x^3+7 a^6 b^2 x^4+\frac {8}{5} a^7 b x^5+\frac {a^8 x^6}{6}+28 a^2 b^6 \log (x)\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.01, size = 95, normalized size = 1.00 \[ \frac {a^8 x^6}{6}+\frac {8}{5} a^7 b x^5+7 a^6 b^2 x^4+\frac {56}{3} a^5 b^3 x^3+35 a^4 b^4 x^2+56 a^3 b^5 x+28 a^2 b^6 \log (x)-\frac {8 a b^7}{x}-\frac {b^8}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8*x^5,x]

[Out]

-1/2*b^8/x^2 - (8*a*b^7)/x + 56*a^3*b^5*x + 35*a^4*b^4*x^2 + (56*a^5*b^3*x^3)/3 + 7*a^6*b^2*x^4 + (8*a^7*b*x^5
)/5 + (a^8*x^6)/6 + 28*a^2*b^6*Log[x]

________________________________________________________________________________________

fricas [A]  time = 0.63, size = 92, normalized size = 0.97 \[ \frac {5 \, a^{8} x^{8} + 48 \, a^{7} b x^{7} + 210 \, a^{6} b^{2} x^{6} + 560 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 1680 \, a^{3} b^{5} x^{3} + 840 \, a^{2} b^{6} x^{2} \log \relax (x) - 240 \, a b^{7} x - 15 \, b^{8}}{30 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^5,x, algorithm="fricas")

[Out]

1/30*(5*a^8*x^8 + 48*a^7*b*x^7 + 210*a^6*b^2*x^6 + 560*a^5*b^3*x^5 + 1050*a^4*b^4*x^4 + 1680*a^3*b^5*x^3 + 840
*a^2*b^6*x^2*log(x) - 240*a*b^7*x - 15*b^8)/x^2

________________________________________________________________________________________

giac [A]  time = 0.15, size = 87, normalized size = 0.92 \[ \frac {1}{6} \, a^{8} x^{6} + \frac {8}{5} \, a^{7} b x^{5} + 7 \, a^{6} b^{2} x^{4} + \frac {56}{3} \, a^{5} b^{3} x^{3} + 35 \, a^{4} b^{4} x^{2} + 56 \, a^{3} b^{5} x + 28 \, a^{2} b^{6} \log \left ({\left | x \right |}\right ) - \frac {16 \, a b^{7} x + b^{8}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^5,x, algorithm="giac")

[Out]

1/6*a^8*x^6 + 8/5*a^7*b*x^5 + 7*a^6*b^2*x^4 + 56/3*a^5*b^3*x^3 + 35*a^4*b^4*x^2 + 56*a^3*b^5*x + 28*a^2*b^6*lo
g(abs(x)) - 1/2*(16*a*b^7*x + b^8)/x^2

________________________________________________________________________________________

maple [A]  time = 0.01, size = 88, normalized size = 0.93 \[ \frac {a^{8} x^{6}}{6}+\frac {8 a^{7} b \,x^{5}}{5}+7 a^{6} b^{2} x^{4}+\frac {56 a^{5} b^{3} x^{3}}{3}+35 a^{4} b^{4} x^{2}+56 a^{3} b^{5} x +28 a^{2} b^{6} \ln \relax (x )-\frac {8 a \,b^{7}}{x}-\frac {b^{8}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8*x^5,x)

[Out]

-1/2*b^8/x^2-8*a*b^7/x+56*a^3*b^5*x+35*a^4*b^4*x^2+56/3*a^5*b^3*x^3+7*a^6*b^2*x^4+8/5*a^7*b*x^5+1/6*a^8*x^6+28
*a^2*b^6*ln(x)

________________________________________________________________________________________

maxima [A]  time = 0.99, size = 86, normalized size = 0.91 \[ \frac {1}{6} \, a^{8} x^{6} + \frac {8}{5} \, a^{7} b x^{5} + 7 \, a^{6} b^{2} x^{4} + \frac {56}{3} \, a^{5} b^{3} x^{3} + 35 \, a^{4} b^{4} x^{2} + 56 \, a^{3} b^{5} x + 28 \, a^{2} b^{6} \log \relax (x) - \frac {16 \, a b^{7} x + b^{8}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^5,x, algorithm="maxima")

[Out]

1/6*a^8*x^6 + 8/5*a^7*b*x^5 + 7*a^6*b^2*x^4 + 56/3*a^5*b^3*x^3 + 35*a^4*b^4*x^2 + 56*a^3*b^5*x + 28*a^2*b^6*lo
g(x) - 1/2*(16*a*b^7*x + b^8)/x^2

________________________________________________________________________________________

mupad [B]  time = 0.05, size = 88, normalized size = 0.93 \[ \frac {a^8\,x^6}{6}-\frac {\frac {b^8}{2}+8\,a\,x\,b^7}{x^2}+56\,a^3\,b^5\,x+\frac {8\,a^7\,b\,x^5}{5}+35\,a^4\,b^4\,x^2+\frac {56\,a^5\,b^3\,x^3}{3}+7\,a^6\,b^2\,x^4+28\,a^2\,b^6\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(a + b/x)^8,x)

[Out]

(a^8*x^6)/6 - (b^8/2 + 8*a*b^7*x)/x^2 + 56*a^3*b^5*x + (8*a^7*b*x^5)/5 + 35*a^4*b^4*x^2 + (56*a^5*b^3*x^3)/3 +
7*a^6*b^2*x^4 + 28*a^2*b^6*log(x)

________________________________________________________________________________________

sympy [A]  time = 0.26, size = 97, normalized size = 1.02 \[ \frac {a^{8} x^{6}}{6} + \frac {8 a^{7} b x^{5}}{5} + 7 a^{6} b^{2} x^{4} + \frac {56 a^{5} b^{3} x^{3}}{3} + 35 a^{4} b^{4} x^{2} + 56 a^{3} b^{5} x + 28 a^{2} b^{6} \log {\relax (x )} + \frac {- 16 a b^{7} x - b^{8}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8*x**5,x)

[Out]

a**8*x**6/6 + 8*a**7*b*x**5/5 + 7*a**6*b**2*x**4 + 56*a**5*b**3*x**3/3 + 35*a**4*b**4*x**2 + 56*a**3*b**5*x +
28*a**2*b**6*log(x) + (-16*a*b**7*x - b**8)/(2*x**2)

________________________________________________________________________________________