\(\int (a+\frac {b}{x})^8 x \, dx\) [1597]

Optimal result

Integrand size = 11, antiderivative size = 95 \[ \int \left (a+\frac {b}{x}\right )^8 x \, dx=-\frac {b^8}{6 x^6}-\frac {8 a b^7}{5 x^5}-\frac {7 a^2 b^6}{x^4}-\frac {56 a^3 b^5}{3 x^3}-\frac {35 a^4 b^4}{x^2}-\frac {56 a^5 b^3}{x}+8 a^7 b x+\frac {a^8 x^2}{2}+28 a^6 b^2 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {269, 45} \[ \int \left (a+\frac {b}{x}\right )^8 x \, dx=\frac {a^8 x^2}{2}+8 a^7 b x+28 a^6 b^2 \log (x)-\frac {56 a^5 b^3}{x}-\frac {35 a^4 b^4}{x^2}-\frac {56 a^3 b^5}{3 x^3}-\frac {7 a^2 b^6}{x^4}-\frac {8 a b^7}{5 x^5}-\frac {b^8}{6 x^6} \]

[In]

Int[(a + b/x)^8*x,x]

[Out]

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

Rule 45

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x}\right )^8 x \, dx=-\frac {b^8}{6 x^6}-\frac {8 a b^7}{5 x^5}-\frac {7 a^2 b^6}{x^4}-\frac {56 a^3 b^5}{3 x^3}-\frac {35 a^4 b^4}{x^2}-\frac {56 a^5 b^3}{x}+8 a^7 b x+\frac {a^8 x^2}{2}+28 a^6 b^2 \log (x) \]

[In]

Integrate[(a + b/x)^8*x,x]

[Out]

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

Maple [A] (verified)

Time = 0.03 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93

method	result	size
default	\(-\frac {b^{8}}{6 x^{6}}-\frac {8 a \,b^{7}}{5 x^{5}}-\frac {7 a^{2} b^{6}}{x^{4}}-\frac {56 a^{3} b^{5}}{3 x^{3}}-\frac {35 a^{4} b^{4}}{x^{2}}-\frac {56 a^{5} b^{3}}{x}+8 a^{7} x b +\frac {a^{8} x^{2}}{2}+28 a^{6} b^{2} \ln \left (x \right )\)	\(88\)
risch	\(\frac {a^{8} x^{2}}{2}+8 a^{7} x b +\frac {-56 a^{5} b^{3} x^{5}-35 a^{4} x^{4} b^{4}-\frac {56}{3} a^{3} b^{5} x^{3}-7 a^{2} b^{6} x^{2}-\frac {8}{5} a \,b^{7} x -\frac {1}{6} b^{8}}{x^{6}}+28 a^{6} b^{2} \ln \left (x \right )\)	\(88\)
norman	\(\frac {-\frac {1}{6} b^{8} x +\frac {1}{2} x^{9} a^{8}-\frac {8}{5} a \,b^{7} x^{2}-7 a^{2} b^{6} x^{3}-\frac {56}{3} a^{3} b^{5} x^{4}-35 a^{4} b^{4} x^{5}+8 a^{7} b \,x^{8}-56 x^{6} b^{3} a^{5}}{x^{7}}+28 a^{6} b^{2} \ln \left (x \right )\)	\(93\)
parallelrisch	\(\frac {15 a^{8} x^{8}+840 a^{6} b^{2} \ln \left (x \right ) x^{6}+240 x^{7} b \,a^{7}-1680 a^{5} b^{3} x^{5}-1050 a^{4} x^{4} b^{4}-560 a^{3} b^{5} x^{3}-210 a^{2} b^{6} x^{2}-48 a \,b^{7} x -5 b^{8}}{30 x^{6}}\)	\(93\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.97 \[ \int \left (a+\frac {b}{x}\right )^8 x \, dx=\frac {15 \, a^{8} x^{8} + 240 \, a^{7} b x^{7} + 840 \, a^{6} b^{2} x^{6} \log \left (x\right ) - 1680 \, a^{5} b^{3} x^{5} - 1050 \, a^{4} b^{4} x^{4} - 560 \, a^{3} b^{5} x^{3} - 210 \, a^{2} b^{6} x^{2} - 48 \, a b^{7} x - 5 \, b^{8}}{30 \, x^{6}} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x}\right )^8 x \, dx=\frac {a^{8} x^{2}}{2} + 8 a^{7} b x + 28 a^{6} b^{2} \log {\left (x \right )} + \frac {- 1680 a^{5} b^{3} x^{5} - 1050 a^{4} b^{4} x^{4} - 560 a^{3} b^{5} x^{3} - 210 a^{2} b^{6} x^{2} - 48 a b^{7} x - 5 b^{8}}{30 x^{6}} \]

[In]

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

[Out]

a**8*x**2/2 + 8*a**7*b*x + 28*a**6*b**2*log(x) + (-1680*a**5*b**3*x**5 - 1050*a**4*b**4*x**4 - 560*a**3*b**5*x
**3 - 210*a**2*b**6*x**2 - 48*a*b**7*x - 5*b**8)/(30*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int \left (a+\frac {b}{x}\right )^8 x \, dx=\frac {1}{2} \, a^{8} x^{2} + 8 \, a^{7} b x + 28 \, a^{6} b^{2} \log \left (x\right ) - \frac {1680 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 560 \, a^{3} b^{5} x^{3} + 210 \, a^{2} b^{6} x^{2} + 48 \, a b^{7} x + 5 \, b^{8}}{30 \, x^{6}} \]

[In]

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

[Out]

1/2*a^8*x^2 + 8*a^7*b*x + 28*a^6*b^2*log(x) - 1/30*(1680*a^5*b^3*x^5 + 1050*a^4*b^4*x^4 + 560*a^3*b^5*x^3 + 21
0*a^2*b^6*x^2 + 48*a*b^7*x + 5*b^8)/x^6

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.94 \[ \int \left (a+\frac {b}{x}\right )^8 x \, dx=\frac {1}{2} \, a^{8} x^{2} + 8 \, a^{7} b x + 28 \, a^{6} b^{2} \log \left ({\left | x \right |}\right ) - \frac {1680 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 560 \, a^{3} b^{5} x^{3} + 210 \, a^{2} b^{6} x^{2} + 48 \, a b^{7} x + 5 \, b^{8}}{30 \, x^{6}} \]

[In]

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

[Out]

1/2*a^8*x^2 + 8*a^7*b*x + 28*a^6*b^2*log(abs(x)) - 1/30*(1680*a^5*b^3*x^5 + 1050*a^4*b^4*x^4 + 560*a^3*b^5*x^3
+ 210*a^2*b^6*x^2 + 48*a*b^7*x + 5*b^8)/x^6

Mupad [B] (verification not implemented)

Time = 5.41 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.93 \[ \int \left (a+\frac {b}{x}\right )^8 x \, dx=\frac {a^8\,x^2}{2}-\frac {56\,a^5\,b^3\,x^5+35\,a^4\,b^4\,x^4+\frac {56\,a^3\,b^5\,x^3}{3}+7\,a^2\,b^6\,x^2+\frac {8\,a\,b^7\,x}{5}+\frac {b^8}{6}}{x^6}+28\,a^6\,b^2\,\ln \left (x\right )+8\,a^7\,b\,x \]

[In]

int(x*(a + b/x)^8,x)

[Out]

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