3.1602 \(\int \frac{(a+\frac{b}{x})^8}{x^4} \, dx\)

Optimal. Leaf size=56 \[ -\frac{a^2 (a x+b)^9}{495 b^3 x^9}+\frac{a (a x+b)^9}{55 b^2 x^{10}}-\frac{(a x+b)^9}{11 b x^{11}} \]

[Out]

-(b + a*x)^9/(11*b*x^11) + (a*(b + a*x)^9)/(55*b^2*x^10) - (a^2*(b + a*x)^9)/(495*b^3*x^9)

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Rubi [A]  time = 0.0122673, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {263, 45, 37} \[ -\frac{a^2 (a x+b)^9}{495 b^3 x^9}+\frac{a (a x+b)^9}{55 b^2 x^{10}}-\frac{(a x+b)^9}{11 b x^{11}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8/x^4,x]

[Out]

-(b + a*x)^9/(11*b*x^11) + (a*(b + a*x)^9)/(55*b^2*x^10) - (a^2*(b + a*x)^9)/(495*b^3*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^4} \, dx &=\int \frac{(b+a x)^8}{x^{12}} \, dx\\ &=-\frac{(b+a x)^9}{11 b x^{11}}-\frac{(2 a) \int \frac{(b+a x)^8}{x^{11}} \, dx}{11 b}\\ &=-\frac{(b+a x)^9}{11 b x^{11}}+\frac{a (b+a x)^9}{55 b^2 x^{10}}+\frac{a^2 \int \frac{(b+a x)^8}{x^{10}} \, dx}{55 b^2}\\ &=-\frac{(b+a x)^9}{11 b x^{11}}+\frac{a (b+a x)^9}{55 b^2 x^{10}}-\frac{a^2 (b+a x)^9}{495 b^3 x^9}\\ \end{align*}

Mathematica [A]  time = 0.0075336, size = 102, normalized size = 1.82 \[ -\frac{28 a^6 b^2}{5 x^5}-\frac{28 a^5 b^3}{3 x^6}-\frac{10 a^4 b^4}{x^7}-\frac{7 a^3 b^5}{x^8}-\frac{28 a^2 b^6}{9 x^9}-\frac{2 a^7 b}{x^4}-\frac{a^8}{3 x^3}-\frac{4 a b^7}{5 x^{10}}-\frac{b^8}{11 x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8/x^4,x]

[Out]

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

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Maple [A]  time = 0.007, size = 91, normalized size = 1.6 \begin{align*} -{\frac{{a}^{8}}{3\,{x}^{3}}}-{\frac{28\,{a}^{6}{b}^{2}}{5\,{x}^{5}}}-{\frac{{b}^{8}}{11\,{x}^{11}}}-2\,{\frac{{a}^{7}b}{{x}^{4}}}-{\frac{4\,{b}^{7}a}{5\,{x}^{10}}}-7\,{\frac{{a}^{3}{b}^{5}}{{x}^{8}}}-{\frac{28\,{a}^{5}{b}^{3}}{3\,{x}^{6}}}-10\,{\frac{{a}^{4}{b}^{4}}{{x}^{7}}}-{\frac{28\,{a}^{2}{b}^{6}}{9\,{x}^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8/x^4,x)

[Out]

-1/3*a^8/x^3-28/5*a^6*b^2/x^5-1/11*b^8/x^11-2*a^7*b/x^4-4/5*b^7*a/x^10-7*a^3*b^5/x^8-28/3*a^5*b^3/x^6-10*a^4*b
^4/x^7-28/9*a^2*b^6/x^9

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Maxima [A]  time = 0.996769, size = 122, normalized size = 2.18 \begin{align*} -\frac{165 \, a^{8} x^{8} + 990 \, a^{7} b x^{7} + 2772 \, a^{6} b^{2} x^{6} + 4620 \, a^{5} b^{3} x^{5} + 4950 \, a^{4} b^{4} x^{4} + 3465 \, a^{3} b^{5} x^{3} + 1540 \, a^{2} b^{6} x^{2} + 396 \, a b^{7} x + 45 \, b^{8}}{495 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8/x^4,x, algorithm="maxima")

[Out]

-1/495*(165*a^8*x^8 + 990*a^7*b*x^7 + 2772*a^6*b^2*x^6 + 4620*a^5*b^3*x^5 + 4950*a^4*b^4*x^4 + 3465*a^3*b^5*x^
3 + 1540*a^2*b^6*x^2 + 396*a*b^7*x + 45*b^8)/x^11

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Fricas [A]  time = 1.39164, size = 217, normalized size = 3.88 \begin{align*} -\frac{165 \, a^{8} x^{8} + 990 \, a^{7} b x^{7} + 2772 \, a^{6} b^{2} x^{6} + 4620 \, a^{5} b^{3} x^{5} + 4950 \, a^{4} b^{4} x^{4} + 3465 \, a^{3} b^{5} x^{3} + 1540 \, a^{2} b^{6} x^{2} + 396 \, a b^{7} x + 45 \, b^{8}}{495 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8/x^4,x, algorithm="fricas")

[Out]

-1/495*(165*a^8*x^8 + 990*a^7*b*x^7 + 2772*a^6*b^2*x^6 + 4620*a^5*b^3*x^5 + 4950*a^4*b^4*x^4 + 3465*a^3*b^5*x^
3 + 1540*a^2*b^6*x^2 + 396*a*b^7*x + 45*b^8)/x^11

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Sympy [B]  time = 0.843198, size = 97, normalized size = 1.73 \begin{align*} - \frac{165 a^{8} x^{8} + 990 a^{7} b x^{7} + 2772 a^{6} b^{2} x^{6} + 4620 a^{5} b^{3} x^{5} + 4950 a^{4} b^{4} x^{4} + 3465 a^{3} b^{5} x^{3} + 1540 a^{2} b^{6} x^{2} + 396 a b^{7} x + 45 b^{8}}{495 x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8/x**4,x)

[Out]

-(165*a**8*x**8 + 990*a**7*b*x**7 + 2772*a**6*b**2*x**6 + 4620*a**5*b**3*x**5 + 4950*a**4*b**4*x**4 + 3465*a**
3*b**5*x**3 + 1540*a**2*b**6*x**2 + 396*a*b**7*x + 45*b**8)/(495*x**11)

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Giac [A]  time = 1.1364, size = 122, normalized size = 2.18 \begin{align*} -\frac{165 \, a^{8} x^{8} + 990 \, a^{7} b x^{7} + 2772 \, a^{6} b^{2} x^{6} + 4620 \, a^{5} b^{3} x^{5} + 4950 \, a^{4} b^{4} x^{4} + 3465 \, a^{3} b^{5} x^{3} + 1540 \, a^{2} b^{6} x^{2} + 396 \, a b^{7} x + 45 \, b^{8}}{495 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8/x^4,x, algorithm="giac")

[Out]

-1/495*(165*a^8*x^8 + 990*a^7*b*x^7 + 2772*a^6*b^2*x^6 + 4620*a^5*b^3*x^5 + 4950*a^4*b^4*x^4 + 3465*a^3*b^5*x^
3 + 1540*a^2*b^6*x^2 + 396*a*b^7*x + 45*b^8)/x^11