3.1601 \(\int \frac{(a+\frac{b}{x})^8}{x^3} \, dx\)

Optimal. Leaf size=36 \[ \frac{a (a x+b)^9}{90 b^2 x^9}-\frac{(a x+b)^9}{10 b x^{10}} \]

[Out]

-(b + a*x)^9/(10*b*x^10) + (a*(b + a*x)^9)/(90*b^2*x^9)

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Rubi [A]  time = 0.0070418, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 45, 37} \[ \frac{a (a x+b)^9}{90 b^2 x^9}-\frac{(a x+b)^9}{10 b x^{10}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8/x^3,x]

[Out]

-(b + a*x)^9/(10*b*x^10) + (a*(b + a*x)^9)/(90*b^2*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^3} \, dx &=\int \frac{(b+a x)^8}{x^{11}} \, dx\\ &=-\frac{(b+a x)^9}{10 b x^{10}}-\frac{a \int \frac{(b+a x)^8}{x^{10}} \, dx}{10 b}\\ &=-\frac{(b+a x)^9}{10 b x^{10}}+\frac{a (b+a x)^9}{90 b^2 x^9}\\ \end{align*}

Mathematica [B]  time = 0.0039121, size = 104, normalized size = 2.89 \[ -\frac{7 a^6 b^2}{x^4}-\frac{56 a^5 b^3}{5 x^5}-\frac{35 a^4 b^4}{3 x^6}-\frac{8 a^3 b^5}{x^7}-\frac{7 a^2 b^6}{2 x^8}-\frac{8 a^7 b}{3 x^3}-\frac{a^8}{2 x^2}-\frac{8 a b^7}{9 x^9}-\frac{b^8}{10 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8/x^3,x]

[Out]

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

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Maple [B]  time = 0.006, size = 91, normalized size = 2.5 \begin{align*} -{\frac{8\,{a}^{7}b}{3\,{x}^{3}}}-{\frac{56\,{a}^{5}{b}^{3}}{5\,{x}^{5}}}-7\,{\frac{{a}^{6}{b}^{2}}{{x}^{4}}}-{\frac{{b}^{8}}{10\,{x}^{10}}}-{\frac{{a}^{8}}{2\,{x}^{2}}}-{\frac{7\,{a}^{2}{b}^{6}}{2\,{x}^{8}}}-{\frac{35\,{a}^{4}{b}^{4}}{3\,{x}^{6}}}-8\,{\frac{{a}^{3}{b}^{5}}{{x}^{7}}}-{\frac{8\,{b}^{7}a}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8/x^3,x)

[Out]

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

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Maxima [B]  time = 0.963343, size = 122, normalized size = 3.39 \begin{align*} -\frac{45 \, a^{8} x^{8} + 240 \, a^{7} b x^{7} + 630 \, a^{6} b^{2} x^{6} + 1008 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 720 \, a^{3} b^{5} x^{3} + 315 \, a^{2} b^{6} x^{2} + 80 \, a b^{7} x + 9 \, b^{8}}{90 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8/x^3,x, algorithm="maxima")

[Out]

-1/90*(45*a^8*x^8 + 240*a^7*b*x^7 + 630*a^6*b^2*x^6 + 1008*a^5*b^3*x^5 + 1050*a^4*b^4*x^4 + 720*a^3*b^5*x^3 +
315*a^2*b^6*x^2 + 80*a*b^7*x + 9*b^8)/x^10

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Fricas [B]  time = 1.359, size = 208, normalized size = 5.78 \begin{align*} -\frac{45 \, a^{8} x^{8} + 240 \, a^{7} b x^{7} + 630 \, a^{6} b^{2} x^{6} + 1008 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 720 \, a^{3} b^{5} x^{3} + 315 \, a^{2} b^{6} x^{2} + 80 \, a b^{7} x + 9 \, b^{8}}{90 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8/x^3,x, algorithm="fricas")

[Out]

-1/90*(45*a^8*x^8 + 240*a^7*b*x^7 + 630*a^6*b^2*x^6 + 1008*a^5*b^3*x^5 + 1050*a^4*b^4*x^4 + 720*a^3*b^5*x^3 +
315*a^2*b^6*x^2 + 80*a*b^7*x + 9*b^8)/x^10

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Sympy [B]  time = 0.800579, size = 97, normalized size = 2.69 \begin{align*} - \frac{45 a^{8} x^{8} + 240 a^{7} b x^{7} + 630 a^{6} b^{2} x^{6} + 1008 a^{5} b^{3} x^{5} + 1050 a^{4} b^{4} x^{4} + 720 a^{3} b^{5} x^{3} + 315 a^{2} b^{6} x^{2} + 80 a b^{7} x + 9 b^{8}}{90 x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8/x**3,x)

[Out]

-(45*a**8*x**8 + 240*a**7*b*x**7 + 630*a**6*b**2*x**6 + 1008*a**5*b**3*x**5 + 1050*a**4*b**4*x**4 + 720*a**3*b
**5*x**3 + 315*a**2*b**6*x**2 + 80*a*b**7*x + 9*b**8)/(90*x**10)

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Giac [B]  time = 1.12148, size = 122, normalized size = 3.39 \begin{align*} -\frac{45 \, a^{8} x^{8} + 240 \, a^{7} b x^{7} + 630 \, a^{6} b^{2} x^{6} + 1008 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 720 \, a^{3} b^{5} x^{3} + 315 \, a^{2} b^{6} x^{2} + 80 \, a b^{7} x + 9 \, b^{8}}{90 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8/x^3,x, algorithm="giac")

[Out]

-1/90*(45*a^8*x^8 + 240*a^7*b*x^7 + 630*a^6*b^2*x^6 + 1008*a^5*b^3*x^5 + 1050*a^4*b^4*x^4 + 720*a^3*b^5*x^3 +
315*a^2*b^6*x^2 + 80*a*b^7*x + 9*b^8)/x^10