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

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

[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)

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Rubi [A]  time = 0.0407175, antiderivative size = 106, normalized size of antiderivative = 1., 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 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]

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])

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*}

Mathematica [A]  time = 0.0064143, size = 106, normalized size = 1. \[ -\frac{28 a^6 b^2}{9 x^9}-\frac{28 a^5 b^3}{5 x^{10}}-\frac{70 a^4 b^4}{11 x^{11}}-\frac{14 a^3 b^5}{3 x^{12}}-\frac{28 a^2 b^6}{13 x^{13}}-\frac{a^7 b}{x^8}-\frac{a^8}{7 x^7}-\frac{4 a b^7}{7 x^{14}}-\frac{b^8}{15 x^{15}} \]

Antiderivative was successfully verified.

[In]

Integrate[(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)

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Maple [A]  time = 0.006, size = 91, normalized size = 0.9 \begin{align*} -{\frac{{b}^{8}}{15\,{x}^{15}}}-{\frac{4\,{b}^{7}a}{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*}

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 = 1.0127, size = 122, normalized size = 1.15 \begin{align*} -\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}} \end{align*}

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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Fricas [A]  time = 1.41895, size = 242, normalized size = 2.28 \begin{align*} -\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}} \end{align*}

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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Sympy [A]  time = 1.03046, size = 97, normalized size = 0.92 \begin{align*} - \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}} \end{align*}

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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Giac [A]  time = 1.17508, size = 122, normalized size = 1.15 \begin{align*} -\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}} \end{align*}

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