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]

-1/11*(a*x+b)^9/b/x^11+1/55*a*(a*x+b)^9/b^2/x^10-1/495*a^2*(a*x+b)^9/b^3/x^9

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Rubi [A]  time = 0.01, antiderivative size = 56, normalized size of antiderivative = 1.00, 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 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]

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 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^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*}

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Mathematica [A]  time = 0.01, size = 102, normalized size = 1.82 \[ -\frac {a^8}{3 x^3}-\frac {2 a^7 b}{x^4}-\frac {28 a^6 b^2}{5 x^5}-\frac {28 a^5 b^3}{3 x^6}-\frac {10 a^4 b^4}{x^7}-\frac {7 a^3 b^5}{x^8}-\frac {28 a^2 b^6}{9 x^9}-\frac {4 a b^7}{5 x^{10}}-\frac {b^8}{11 x^{11}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8/x^4,x]

[Out]

-1/11*b^8/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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fricas [A]  time = 1.24, size = 90, normalized size = 1.61 \[ -\frac {165 \, a^{8} x^{8} + 990 \, a^{7} b x^{7} + 2772 \, a^{6} b^{2} x^{6} + 4620 \, a^{5} b^{3} x^{5} + 4950 \, a^{4} b^{4} x^{4} + 3465 \, a^{3} b^{5} x^{3} + 1540 \, a^{2} b^{6} x^{2} + 396 \, a b^{7} x + 45 \, b^{8}}{495 \, x^{11}} \]

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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giac [A]  time = 0.17, size = 90, normalized size = 1.61 \[ -\frac {165 \, a^{8} x^{8} + 990 \, a^{7} b x^{7} + 2772 \, a^{6} b^{2} x^{6} + 4620 \, a^{5} b^{3} x^{5} + 4950 \, a^{4} b^{4} x^{4} + 3465 \, a^{3} b^{5} x^{3} + 1540 \, a^{2} b^{6} x^{2} + 396 \, a b^{7} x + 45 \, b^{8}}{495 \, x^{11}} \]

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

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maple [A]  time = 0.01, size = 91, normalized size = 1.62 \[ -\frac {a^{8}}{3 x^{3}}-\frac {2 a^{7} b}{x^{4}}-\frac {28 a^{6} b^{2}}{5 x^{5}}-\frac {28 a^{5} b^{3}}{3 x^{6}}-\frac {10 a^{4} b^{4}}{x^{7}}-\frac {7 a^{3} b^{5}}{x^{8}}-\frac {28 a^{2} b^{6}}{9 x^{9}}-\frac {4 a \,b^{7}}{5 x^{10}}-\frac {b^{8}}{11 x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.05, size = 90, normalized size = 1.61 \[ -\frac {165 \, a^{8} x^{8} + 990 \, a^{7} b x^{7} + 2772 \, a^{6} b^{2} x^{6} + 4620 \, a^{5} b^{3} x^{5} + 4950 \, a^{4} b^{4} x^{4} + 3465 \, a^{3} b^{5} x^{3} + 1540 \, a^{2} b^{6} x^{2} + 396 \, a b^{7} x + 45 \, b^{8}}{495 \, x^{11}} \]

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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mupad [B]  time = 1.08, size = 90, normalized size = 1.61 \[ -\frac {\frac {a^8\,x^8}{3}+2\,a^7\,b\,x^7+\frac {28\,a^6\,b^2\,x^6}{5}+\frac {28\,a^5\,b^3\,x^5}{3}+10\,a^4\,b^4\,x^4+7\,a^3\,b^5\,x^3+\frac {28\,a^2\,b^6\,x^2}{9}+\frac {4\,a\,b^7\,x}{5}+\frac {b^8}{11}}{x^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^8/x^4,x)

[Out]

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

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sympy [B]  time = 0.76, size = 97, normalized size = 1.73 \[ \frac {- 165 a^{8} x^{8} - 990 a^{7} b x^{7} - 2772 a^{6} b^{2} x^{6} - 4620 a^{5} b^{3} x^{5} - 4950 a^{4} b^{4} x^{4} - 3465 a^{3} b^{5} x^{3} - 1540 a^{2} b^{6} x^{2} - 396 a b^{7} x - 45 b^{8}}{495 x^{11}} \]

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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