3.1603 \(\int \frac{(a+\frac{b}{x})^8}{x^5} \, dx\)

Optimal. Leaf size=76 \[ \frac{a^3 (a x+b)^9}{1980 b^4 x^9}-\frac{a^2 (a x+b)^9}{220 b^3 x^{10}}+\frac{a (a x+b)^9}{44 b^2 x^{11}}-\frac{(a x+b)^9}{12 b x^{12}} \]

[Out]

-(b + a*x)^9/(12*b*x^12) + (a*(b + a*x)^9)/(44*b^2*x^11) - (a^2*(b + a*x)^9)/(220*b^3*x^10) + (a^3*(b + a*x)^9
)/(1980*b^4*x^9)

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Rubi [A]  time = 0.0187091, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 45, 37} \[ \frac{a^3 (a x+b)^9}{1980 b^4 x^9}-\frac{a^2 (a x+b)^9}{220 b^3 x^{10}}+\frac{a (a x+b)^9}{44 b^2 x^{11}}-\frac{(a x+b)^9}{12 b x^{12}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8/x^5,x]

[Out]

-(b + a*x)^9/(12*b*x^12) + (a*(b + a*x)^9)/(44*b^2*x^11) - (a^2*(b + a*x)^9)/(220*b^3*x^10) + (a^3*(b + a*x)^9
)/(1980*b^4*x^9)

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 45

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 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]

Rubi steps

\begin{align*} \int \frac{\left (a+\frac{b}{x}\right )^8}{x^5} \, dx &=\int \frac{(b+a x)^8}{x^{13}} \, dx\\ &=-\frac{(b+a x)^9}{12 b x^{12}}-\frac{a \int \frac{(b+a x)^8}{x^{12}} \, dx}{4 b}\\ &=-\frac{(b+a x)^9}{12 b x^{12}}+\frac{a (b+a x)^9}{44 b^2 x^{11}}+\frac{a^2 \int \frac{(b+a x)^8}{x^{11}} \, dx}{22 b^2}\\ &=-\frac{(b+a x)^9}{12 b x^{12}}+\frac{a (b+a x)^9}{44 b^2 x^{11}}-\frac{a^2 (b+a x)^9}{220 b^3 x^{10}}-\frac{a^3 \int \frac{(b+a x)^8}{x^{10}} \, dx}{220 b^3}\\ &=-\frac{(b+a x)^9}{12 b x^{12}}+\frac{a (b+a x)^9}{44 b^2 x^{11}}-\frac{a^2 (b+a x)^9}{220 b^3 x^{10}}+\frac{a^3 (b+a x)^9}{1980 b^4 x^9}\\ \end{align*}

Mathematica [A]  time = 0.0041369, size = 106, normalized size = 1.39 \[ -\frac{14 a^6 b^2}{3 x^6}-\frac{8 a^5 b^3}{x^7}-\frac{35 a^4 b^4}{4 x^8}-\frac{56 a^3 b^5}{9 x^9}-\frac{14 a^2 b^6}{5 x^{10}}-\frac{8 a^7 b}{5 x^5}-\frac{a^8}{4 x^4}-\frac{8 a b^7}{11 x^{11}}-\frac{b^8}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8/x^5,x]

[Out]

-b^8/(12*x^12) - (8*a*b^7)/(11*x^11) - (14*a^2*b^6)/(5*x^10) - (56*a^3*b^5)/(9*x^9) - (35*a^4*b^4)/(4*x^8) - (
8*a^5*b^3)/x^7 - (14*a^6*b^2)/(3*x^6) - (8*a^7*b)/(5*x^5) - a^8/(4*x^4)

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Maple [A]  time = 0.005, size = 91, normalized size = 1.2 \begin{align*} -{\frac{8\,{a}^{7}b}{5\,{x}^{5}}}-{\frac{{b}^{8}}{12\,{x}^{12}}}-{\frac{8\,{b}^{7}a}{11\,{x}^{11}}}-{\frac{{a}^{8}}{4\,{x}^{4}}}-{\frac{14\,{a}^{2}{b}^{6}}{5\,{x}^{10}}}-{\frac{35\,{a}^{4}{b}^{4}}{4\,{x}^{8}}}-{\frac{14\,{a}^{6}{b}^{2}}{3\,{x}^{6}}}-8\,{\frac{{a}^{5}{b}^{3}}{{x}^{7}}}-{\frac{56\,{a}^{3}{b}^{5}}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/5*a^7*b/x^5-1/12*b^8/x^12-8/11*b^7*a/x^11-1/4*a^8/x^4-14/5*a^2*b^6/x^10-35/4*a^4*b^4/x^8-14/3*a^6*b^2/x^6-8
*a^5*b^3/x^7-56/9*a^3*b^5/x^9

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Maxima [A]  time = 1.0639, size = 122, normalized size = 1.61 \begin{align*} -\frac{495 \, a^{8} x^{8} + 3168 \, a^{7} b x^{7} + 9240 \, a^{6} b^{2} x^{6} + 15840 \, a^{5} b^{3} x^{5} + 17325 \, a^{4} b^{4} x^{4} + 12320 \, a^{3} b^{5} x^{3} + 5544 \, a^{2} b^{6} x^{2} + 1440 \, a b^{7} x + 165 \, b^{8}}{1980 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1980*(495*a^8*x^8 + 3168*a^7*b*x^7 + 9240*a^6*b^2*x^6 + 15840*a^5*b^3*x^5 + 17325*a^4*b^4*x^4 + 12320*a^3*b
^5*x^3 + 5544*a^2*b^6*x^2 + 1440*a*b^7*x + 165*b^8)/x^12

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Fricas [A]  time = 1.42626, size = 227, normalized size = 2.99 \begin{align*} -\frac{495 \, a^{8} x^{8} + 3168 \, a^{7} b x^{7} + 9240 \, a^{6} b^{2} x^{6} + 15840 \, a^{5} b^{3} x^{5} + 17325 \, a^{4} b^{4} x^{4} + 12320 \, a^{3} b^{5} x^{3} + 5544 \, a^{2} b^{6} x^{2} + 1440 \, a b^{7} x + 165 \, b^{8}}{1980 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1980*(495*a^8*x^8 + 3168*a^7*b*x^7 + 9240*a^6*b^2*x^6 + 15840*a^5*b^3*x^5 + 17325*a^4*b^4*x^4 + 12320*a^3*b
^5*x^3 + 5544*a^2*b^6*x^2 + 1440*a*b^7*x + 165*b^8)/x^12

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Sympy [A]  time = 0.876523, size = 97, normalized size = 1.28 \begin{align*} - \frac{495 a^{8} x^{8} + 3168 a^{7} b x^{7} + 9240 a^{6} b^{2} x^{6} + 15840 a^{5} b^{3} x^{5} + 17325 a^{4} b^{4} x^{4} + 12320 a^{3} b^{5} x^{3} + 5544 a^{2} b^{6} x^{2} + 1440 a b^{7} x + 165 b^{8}}{1980 x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(495*a**8*x**8 + 3168*a**7*b*x**7 + 9240*a**6*b**2*x**6 + 15840*a**5*b**3*x**5 + 17325*a**4*b**4*x**4 + 12320
*a**3*b**5*x**3 + 5544*a**2*b**6*x**2 + 1440*a*b**7*x + 165*b**8)/(1980*x**12)

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Giac [A]  time = 1.10661, size = 122, normalized size = 1.61 \begin{align*} -\frac{495 \, a^{8} x^{8} + 3168 \, a^{7} b x^{7} + 9240 \, a^{6} b^{2} x^{6} + 15840 \, a^{5} b^{3} x^{5} + 17325 \, a^{4} b^{4} x^{4} + 12320 \, a^{3} b^{5} x^{3} + 5544 \, a^{2} b^{6} x^{2} + 1440 \, a b^{7} x + 165 \, b^{8}}{1980 \, x^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1980*(495*a^8*x^8 + 3168*a^7*b*x^7 + 9240*a^6*b^2*x^6 + 15840*a^5*b^3*x^5 + 17325*a^4*b^4*x^4 + 12320*a^3*b
^5*x^3 + 5544*a^2*b^6*x^2 + 1440*a*b^7*x + 165*b^8)/x^12