3.1606 \(\int \frac {(a+\frac {b}{x})^8}{x^8} \, dx\)

Optimal. Leaf size=106 \[ -\frac {a^8}{7 x^7}-\frac {a^7 b}{x^8}-\frac {28 a^6 b^2}{9 x^9}-\frac {28 a^5 b^3}{5 x^{10}}-\frac {70 a^4 b^4}{11 x^{11}}-\frac {14 a^3 b^5}{3 x^{12}}-\frac {28 a^2 b^6}{13 x^{13}}-\frac {4 a b^7}{7 x^{14}}-\frac {b^8}{15 x^{15}} \]

[Out]

-1/15*b^8/x^15-4/7*a*b^7/x^14-28/13*a^2*b^6/x^13-14/3*a^3*b^5/x^12-70/11*a^4*b^4/x^11-28/5*a^5*b^3/x^10-28/9*a
^6*b^2/x^9-a^7*b/x^8-1/7*a^8/x^7

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Rubi [A]  time = 0.04, antiderivative size = 106, 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} \[ -\frac {28 a^6 b^2}{9 x^9}-\frac {28 a^5 b^3}{5 x^{10}}-\frac {70 a^4 b^4}{11 x^{11}}-\frac {14 a^3 b^5}{3 x^{12}}-\frac {28 a^2 b^6}{13 x^{13}}-\frac {a^7 b}{x^8}-\frac {a^8}{7 x^7}-\frac {4 a b^7}{7 x^{14}}-\frac {b^8}{15 x^{15}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8/x^8,x]

[Out]

-b^8/(15*x^15) - (4*a*b^7)/(7*x^14) - (28*a^2*b^6)/(13*x^13) - (14*a^3*b^5)/(3*x^12) - (70*a^4*b^4)/(11*x^11)
- (28*a^5*b^3)/(5*x^10) - (28*a^6*b^2)/(9*x^9) - (a^7*b)/x^8 - a^8/(7*x^7)

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 \frac {\left (a+\frac {b}{x}\right )^8}{x^8} \, dx &=\int \frac {(b+a x)^8}{x^{16}} \, dx\\ &=\int \left (\frac {b^8}{x^{16}}+\frac {8 a b^7}{x^{15}}+\frac {28 a^2 b^6}{x^{14}}+\frac {56 a^3 b^5}{x^{13}}+\frac {70 a^4 b^4}{x^{12}}+\frac {56 a^5 b^3}{x^{11}}+\frac {28 a^6 b^2}{x^{10}}+\frac {8 a^7 b}{x^9}+\frac {a^8}{x^8}\right ) \, dx\\ &=-\frac {b^8}{15 x^{15}}-\frac {4 a b^7}{7 x^{14}}-\frac {28 a^2 b^6}{13 x^{13}}-\frac {14 a^3 b^5}{3 x^{12}}-\frac {70 a^4 b^4}{11 x^{11}}-\frac {28 a^5 b^3}{5 x^{10}}-\frac {28 a^6 b^2}{9 x^9}-\frac {a^7 b}{x^8}-\frac {a^8}{7 x^7}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 106, normalized size = 1.00 \[ -\frac {a^8}{7 x^7}-\frac {a^7 b}{x^8}-\frac {28 a^6 b^2}{9 x^9}-\frac {28 a^5 b^3}{5 x^{10}}-\frac {70 a^4 b^4}{11 x^{11}}-\frac {14 a^3 b^5}{3 x^{12}}-\frac {28 a^2 b^6}{13 x^{13}}-\frac {4 a b^7}{7 x^{14}}-\frac {b^8}{15 x^{15}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8/x^8,x]

[Out]

-1/15*b^8/x^15 - (4*a*b^7)/(7*x^14) - (28*a^2*b^6)/(13*x^13) - (14*a^3*b^5)/(3*x^12) - (70*a^4*b^4)/(11*x^11)
- (28*a^5*b^3)/(5*x^10) - (28*a^6*b^2)/(9*x^9) - (a^7*b)/x^8 - a^8/(7*x^7)

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fricas [A]  time = 0.98, size = 90, normalized size = 0.85 \[ -\frac {6435 \, a^{8} x^{8} + 45045 \, a^{7} b x^{7} + 140140 \, a^{6} b^{2} x^{6} + 252252 \, a^{5} b^{3} x^{5} + 286650 \, a^{4} b^{4} x^{4} + 210210 \, a^{3} b^{5} x^{3} + 97020 \, a^{2} b^{6} x^{2} + 25740 \, a b^{7} x + 3003 \, b^{8}}{45045 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/45045*(6435*a^8*x^8 + 45045*a^7*b*x^7 + 140140*a^6*b^2*x^6 + 252252*a^5*b^3*x^5 + 286650*a^4*b^4*x^4 + 2102
10*a^3*b^5*x^3 + 97020*a^2*b^6*x^2 + 25740*a*b^7*x + 3003*b^8)/x^15

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giac [A]  time = 0.15, size = 90, normalized size = 0.85 \[ -\frac {6435 \, a^{8} x^{8} + 45045 \, a^{7} b x^{7} + 140140 \, a^{6} b^{2} x^{6} + 252252 \, a^{5} b^{3} x^{5} + 286650 \, a^{4} b^{4} x^{4} + 210210 \, a^{3} b^{5} x^{3} + 97020 \, a^{2} b^{6} x^{2} + 25740 \, a b^{7} x + 3003 \, b^{8}}{45045 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/45045*(6435*a^8*x^8 + 45045*a^7*b*x^7 + 140140*a^6*b^2*x^6 + 252252*a^5*b^3*x^5 + 286650*a^4*b^4*x^4 + 2102
10*a^3*b^5*x^3 + 97020*a^2*b^6*x^2 + 25740*a*b^7*x + 3003*b^8)/x^15

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maple [A]  time = 0.01, size = 91, normalized size = 0.86 \[ -\frac {a^{8}}{7 x^{7}}-\frac {a^{7} b}{x^{8}}-\frac {28 a^{6} b^{2}}{9 x^{9}}-\frac {28 a^{5} b^{3}}{5 x^{10}}-\frac {70 a^{4} b^{4}}{11 x^{11}}-\frac {14 a^{3} b^{5}}{3 x^{12}}-\frac {28 a^{2} b^{6}}{13 x^{13}}-\frac {4 a \,b^{7}}{7 x^{14}}-\frac {b^{8}}{15 x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/15*b^8/x^15-4/7*a*b^7/x^14-28/13*a^2*b^6/x^13-14/3*a^3*b^5/x^12-70/11*a^4*b^4/x^11-28/5*a^5*b^3/x^10-28/9*a
^6*b^2/x^9-a^7*b/x^8-1/7*a^8/x^7

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maxima [A]  time = 1.01, size = 90, normalized size = 0.85 \[ -\frac {6435 \, a^{8} x^{8} + 45045 \, a^{7} b x^{7} + 140140 \, a^{6} b^{2} x^{6} + 252252 \, a^{5} b^{3} x^{5} + 286650 \, a^{4} b^{4} x^{4} + 210210 \, a^{3} b^{5} x^{3} + 97020 \, a^{2} b^{6} x^{2} + 25740 \, a b^{7} x + 3003 \, b^{8}}{45045 \, x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/45045*(6435*a^8*x^8 + 45045*a^7*b*x^7 + 140140*a^6*b^2*x^6 + 252252*a^5*b^3*x^5 + 286650*a^4*b^4*x^4 + 2102
10*a^3*b^5*x^3 + 97020*a^2*b^6*x^2 + 25740*a*b^7*x + 3003*b^8)/x^15

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mupad [B]  time = 0.07, size = 89, normalized size = 0.84 \[ -\frac {\frac {a^8\,x^8}{7}+a^7\,b\,x^7+\frac {28\,a^6\,b^2\,x^6}{9}+\frac {28\,a^5\,b^3\,x^5}{5}+\frac {70\,a^4\,b^4\,x^4}{11}+\frac {14\,a^3\,b^5\,x^3}{3}+\frac {28\,a^2\,b^6\,x^2}{13}+\frac {4\,a\,b^7\,x}{7}+\frac {b^8}{15}}{x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^8/x^8,x)

[Out]

-(b^8/15 + (a^8*x^8)/7 + a^7*b*x^7 + (28*a^2*b^6*x^2)/13 + (14*a^3*b^5*x^3)/3 + (70*a^4*b^4*x^4)/11 + (28*a^5*
b^3*x^5)/5 + (28*a^6*b^2*x^6)/9 + (4*a*b^7*x)/7)/x^15

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sympy [A]  time = 0.92, size = 97, normalized size = 0.92 \[ \frac {- 6435 a^{8} x^{8} - 45045 a^{7} b x^{7} - 140140 a^{6} b^{2} x^{6} - 252252 a^{5} b^{3} x^{5} - 286650 a^{4} b^{4} x^{4} - 210210 a^{3} b^{5} x^{3} - 97020 a^{2} b^{6} x^{2} - 25740 a b^{7} x - 3003 b^{8}}{45045 x^{15}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-6435*a**8*x**8 - 45045*a**7*b*x**7 - 140140*a**6*b**2*x**6 - 252252*a**5*b**3*x**5 - 286650*a**4*b**4*x**4 -
210210*a**3*b**5*x**3 - 97020*a**2*b**6*x**2 - 25740*a*b**7*x - 3003*b**8)/(45045*x**15)

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