\(\int \frac {(a+\frac {b}{x})^8}{x^6} \, dx\) [1604]

Optimal result

Integrand size = 13, antiderivative size = 96 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^6} \, dx=-\frac {(b+a x)^9}{13 b x^{13}}+\frac {a (b+a x)^9}{39 b^2 x^{12}}-\frac {a^2 (b+a x)^9}{143 b^3 x^{11}}+\frac {a^3 (b+a x)^9}{715 b^4 x^{10}}-\frac {a^4 (b+a x)^9}{6435 b^5 x^9} \]

[Out]

-1/13*(a*x+b)^9/b/x^13+1/39*a*(a*x+b)^9/b^2/x^12-1/143*a^2*(a*x+b)^9/b^3/x^11+1/715*a^3*(a*x+b)^9/b^4/x^10-1/6
435*a^4*(a*x+b)^9/b^5/x^9

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {269, 47, 37} \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^6} \, dx=-\frac {a^4 (a x+b)^9}{6435 b^5 x^9}+\frac {a^3 (a x+b)^9}{715 b^4 x^{10}}-\frac {a^2 (a x+b)^9}{143 b^3 x^{11}}+\frac {a (a x+b)^9}{39 b^2 x^{12}}-\frac {(a x+b)^9}{13 b x^{13}} \]

[In]

Int[(a + b/x)^8/x^6,x]

[Out]

-1/13*(b + a*x)^9/(b*x^13) + (a*(b + a*x)^9)/(39*b^2*x^12) - (a^2*(b + a*x)^9)/(143*b^3*x^11) + (a^3*(b + a*x)
^9)/(715*b^4*x^10) - (a^4*(b + a*x)^9)/(6435*b^5*x^9)

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n +
1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*(Simplify[m + n + 2]/((b*c - a*d)*(m + 1))), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 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 \frac {(b+a x)^8}{x^{14}} \, dx \\ & = -\frac {(b+a x)^9}{13 b x^{13}}-\frac {(4 a) \int \frac {(b+a x)^8}{x^{13}} \, dx}{13 b} \\ & = -\frac {(b+a x)^9}{13 b x^{13}}+\frac {a (b+a x)^9}{39 b^2 x^{12}}+\frac {a^2 \int \frac {(b+a x)^8}{x^{12}} \, dx}{13 b^2} \\ & = -\frac {(b+a x)^9}{13 b x^{13}}+\frac {a (b+a x)^9}{39 b^2 x^{12}}-\frac {a^2 (b+a x)^9}{143 b^3 x^{11}}-\frac {\left (2 a^3\right ) \int \frac {(b+a x)^8}{x^{11}} \, dx}{143 b^3} \\ & = -\frac {(b+a x)^9}{13 b x^{13}}+\frac {a (b+a x)^9}{39 b^2 x^{12}}-\frac {a^2 (b+a x)^9}{143 b^3 x^{11}}+\frac {a^3 (b+a x)^9}{715 b^4 x^{10}}+\frac {a^4 \int \frac {(b+a x)^8}{x^{10}} \, dx}{715 b^4} \\ & = -\frac {(b+a x)^9}{13 b x^{13}}+\frac {a (b+a x)^9}{39 b^2 x^{12}}-\frac {a^2 (b+a x)^9}{143 b^3 x^{11}}+\frac {a^3 (b+a x)^9}{715 b^4 x^{10}}-\frac {a^4 (b+a x)^9}{6435 b^5 x^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^6} \, dx=-\frac {b^8}{13 x^{13}}-\frac {2 a b^7}{3 x^{12}}-\frac {28 a^2 b^6}{11 x^{11}}-\frac {28 a^3 b^5}{5 x^{10}}-\frac {70 a^4 b^4}{9 x^9}-\frac {7 a^5 b^3}{x^8}-\frac {4 a^6 b^2}{x^7}-\frac {4 a^7 b}{3 x^6}-\frac {a^8}{5 x^5} \]

[In]

Integrate[(a + b/x)^8/x^6,x]

[Out]

-1/13*b^8/x^13 - (2*a*b^7)/(3*x^12) - (28*a^2*b^6)/(11*x^11) - (28*a^3*b^5)/(5*x^10) - (70*a^4*b^4)/(9*x^9) -
(7*a^5*b^3)/x^8 - (4*a^6*b^2)/x^7 - (4*a^7*b)/(3*x^6) - a^8/(5*x^5)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94

method	result	size
norman	\(\frac {-\frac {1}{5} a^{8} x^{8}-\frac {4}{3} x^{7} b \,a^{7}-4 a^{6} b^{2} x^{6}-7 a^{5} b^{3} x^{5}-\frac {70}{9} a^{4} x^{4} b^{4}-\frac {28}{5} a^{3} b^{5} x^{3}-\frac {28}{11} a^{2} b^{6} x^{2}-\frac {2}{3} a \,b^{7} x -\frac {1}{13} b^{8}}{x^{13}}\)	\(90\)
risch	\(\frac {-\frac {1}{5} a^{8} x^{8}-\frac {4}{3} x^{7} b \,a^{7}-4 a^{6} b^{2} x^{6}-7 a^{5} b^{3} x^{5}-\frac {70}{9} a^{4} x^{4} b^{4}-\frac {28}{5} a^{3} b^{5} x^{3}-\frac {28}{11} a^{2} b^{6} x^{2}-\frac {2}{3} a \,b^{7} x -\frac {1}{13} b^{8}}{x^{13}}\)	\(90\)
gosper	\(-\frac {1287 a^{8} x^{8}+8580 x^{7} b \,a^{7}+25740 a^{6} b^{2} x^{6}+45045 a^{5} b^{3} x^{5}+50050 a^{4} x^{4} b^{4}+36036 a^{3} b^{5} x^{3}+16380 a^{2} b^{6} x^{2}+4290 a \,b^{7} x +495 b^{8}}{6435 x^{13}}\)	\(91\)
default	\(-\frac {4 a^{7} b}{3 x^{6}}-\frac {2 a \,b^{7}}{3 x^{12}}-\frac {4 a^{6} b^{2}}{x^{7}}-\frac {7 a^{5} b^{3}}{x^{8}}-\frac {28 a^{3} b^{5}}{5 x^{10}}-\frac {28 a^{2} b^{6}}{11 x^{11}}-\frac {b^{8}}{13 x^{13}}-\frac {a^{8}}{5 x^{5}}-\frac {70 a^{4} b^{4}}{9 x^{9}}\)	\(91\)
parallelrisch	\(\frac {-1287 a^{8} x^{8}-8580 x^{7} b \,a^{7}-25740 a^{6} b^{2} x^{6}-45045 a^{5} b^{3} x^{5}-50050 a^{4} x^{4} b^{4}-36036 a^{3} b^{5} x^{3}-16380 a^{2} b^{6} x^{2}-4290 a \,b^{7} x -495 b^{8}}{6435 x^{13}}\)	\(91\)

[In]

int((a+b/x)^8/x^6,x,method=_RETURNVERBOSE)

[Out]

(-1/5*a^8*x^8-4/3*x^7*b*a^7-4*a^6*b^2*x^6-7*a^5*b^3*x^5-70/9*a^4*x^4*b^4-28/5*a^3*b^5*x^3-28/11*a^2*b^6*x^2-2/
3*a*b^7*x-1/13*b^8)/x^13

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^6} \, dx=-\frac {1287 \, a^{8} x^{8} + 8580 \, a^{7} b x^{7} + 25740 \, a^{6} b^{2} x^{6} + 45045 \, a^{5} b^{3} x^{5} + 50050 \, a^{4} b^{4} x^{4} + 36036 \, a^{3} b^{5} x^{3} + 16380 \, a^{2} b^{6} x^{2} + 4290 \, a b^{7} x + 495 \, b^{8}}{6435 \, x^{13}} \]

[In]

integrate((a+b/x)^8/x^6,x, algorithm="fricas")

[Out]

-1/6435*(1287*a^8*x^8 + 8580*a^7*b*x^7 + 25740*a^6*b^2*x^6 + 45045*a^5*b^3*x^5 + 50050*a^4*b^4*x^4 + 36036*a^3
*b^5*x^3 + 16380*a^2*b^6*x^2 + 4290*a*b^7*x + 495*b^8)/x^13

Sympy [A] (verification not implemented)

Time = 0.41 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.01 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^6} \, dx=\frac {- 1287 a^{8} x^{8} - 8580 a^{7} b x^{7} - 25740 a^{6} b^{2} x^{6} - 45045 a^{5} b^{3} x^{5} - 50050 a^{4} b^{4} x^{4} - 36036 a^{3} b^{5} x^{3} - 16380 a^{2} b^{6} x^{2} - 4290 a b^{7} x - 495 b^{8}}{6435 x^{13}} \]

[In]

integrate((a+b/x)**8/x**6,x)

[Out]

(-1287*a**8*x**8 - 8580*a**7*b*x**7 - 25740*a**6*b**2*x**6 - 45045*a**5*b**3*x**5 - 50050*a**4*b**4*x**4 - 360
36*a**3*b**5*x**3 - 16380*a**2*b**6*x**2 - 4290*a*b**7*x - 495*b**8)/(6435*x**13)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^6} \, dx=-\frac {1287 \, a^{8} x^{8} + 8580 \, a^{7} b x^{7} + 25740 \, a^{6} b^{2} x^{6} + 45045 \, a^{5} b^{3} x^{5} + 50050 \, a^{4} b^{4} x^{4} + 36036 \, a^{3} b^{5} x^{3} + 16380 \, a^{2} b^{6} x^{2} + 4290 \, a b^{7} x + 495 \, b^{8}}{6435 \, x^{13}} \]

[In]

integrate((a+b/x)^8/x^6,x, algorithm="maxima")

[Out]

-1/6435*(1287*a^8*x^8 + 8580*a^7*b*x^7 + 25740*a^6*b^2*x^6 + 45045*a^5*b^3*x^5 + 50050*a^4*b^4*x^4 + 36036*a^3
*b^5*x^3 + 16380*a^2*b^6*x^2 + 4290*a*b^7*x + 495*b^8)/x^13

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^6} \, dx=-\frac {1287 \, a^{8} x^{8} + 8580 \, a^{7} b x^{7} + 25740 \, a^{6} b^{2} x^{6} + 45045 \, a^{5} b^{3} x^{5} + 50050 \, a^{4} b^{4} x^{4} + 36036 \, a^{3} b^{5} x^{3} + 16380 \, a^{2} b^{6} x^{2} + 4290 \, a b^{7} x + 495 \, b^{8}}{6435 \, x^{13}} \]

[In]

integrate((a+b/x)^8/x^6,x, algorithm="giac")

[Out]

-1/6435*(1287*a^8*x^8 + 8580*a^7*b*x^7 + 25740*a^6*b^2*x^6 + 45045*a^5*b^3*x^5 + 50050*a^4*b^4*x^4 + 36036*a^3
*b^5*x^3 + 16380*a^2*b^6*x^2 + 4290*a*b^7*x + 495*b^8)/x^13

Mupad [B] (verification not implemented)

Time = 5.42 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.94 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^6} \, dx=-\frac {\frac {a^8\,x^8}{5}+\frac {4\,a^7\,b\,x^7}{3}+4\,a^6\,b^2\,x^6+7\,a^5\,b^3\,x^5+\frac {70\,a^4\,b^4\,x^4}{9}+\frac {28\,a^3\,b^5\,x^3}{5}+\frac {28\,a^2\,b^6\,x^2}{11}+\frac {2\,a\,b^7\,x}{3}+\frac {b^8}{13}}{x^{13}} \]

[In]

int((a + b/x)^8/x^6,x)

[Out]

-(b^8/13 + (a^8*x^8)/5 + (4*a^7*b*x^7)/3 + (28*a^2*b^6*x^2)/11 + (28*a^3*b^5*x^3)/5 + (70*a^4*b^4*x^4)/9 + 7*a
^5*b^3*x^5 + 4*a^6*b^2*x^6 + (2*a*b^7*x)/3)/x^13