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]

-1/10*(a*x+b)^9/b/x^10+1/90*a*(a*x+b)^9/b^2/x^9

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Rubi [A]  time = 0.01, antiderivative size = 36, normalized size of antiderivative = 1.00, 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 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^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*}

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Mathematica [B]  time = 0.00, size = 104, normalized size = 2.89 \[ -\frac {a^8}{2 x^2}-\frac {8 a^7 b}{3 x^3}-\frac {7 a^6 b^2}{x^4}-\frac {56 a^5 b^3}{5 x^5}-\frac {35 a^4 b^4}{3 x^6}-\frac {8 a^3 b^5}{x^7}-\frac {7 a^2 b^6}{2 x^8}-\frac {8 a b^7}{9 x^9}-\frac {b^8}{10 x^{10}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8/x^3,x]

[Out]

-1/10*b^8/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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fricas [B]  time = 1.00, size = 90, normalized size = 2.50 \[ -\frac {45 \, a^{8} x^{8} + 240 \, a^{7} b x^{7} + 630 \, a^{6} b^{2} x^{6} + 1008 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 720 \, a^{3} b^{5} x^{3} + 315 \, a^{2} b^{6} x^{2} + 80 \, a b^{7} x + 9 \, b^{8}}{90 \, x^{10}} \]

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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giac [B]  time = 0.21, size = 90, normalized size = 2.50 \[ -\frac {45 \, a^{8} x^{8} + 240 \, a^{7} b x^{7} + 630 \, a^{6} b^{2} x^{6} + 1008 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 720 \, a^{3} b^{5} x^{3} + 315 \, a^{2} b^{6} x^{2} + 80 \, a b^{7} x + 9 \, b^{8}}{90 \, x^{10}} \]

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

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maple [B]  time = 0.00, size = 91, normalized size = 2.53 \[ -\frac {a^{8}}{2 x^{2}}-\frac {8 a^{7} b}{3 x^{3}}-\frac {7 a^{6} b^{2}}{x^{4}}-\frac {56 a^{5} b^{3}}{5 x^{5}}-\frac {35 a^{4} b^{4}}{3 x^{6}}-\frac {8 a^{3} b^{5}}{x^{7}}-\frac {7 a^{2} b^{6}}{2 x^{8}}-\frac {8 a \,b^{7}}{9 x^{9}}-\frac {b^{8}}{10 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 0.98, size = 90, normalized size = 2.50 \[ -\frac {45 \, a^{8} x^{8} + 240 \, a^{7} b x^{7} + 630 \, a^{6} b^{2} x^{6} + 1008 \, a^{5} b^{3} x^{5} + 1050 \, a^{4} b^{4} x^{4} + 720 \, a^{3} b^{5} x^{3} + 315 \, a^{2} b^{6} x^{2} + 80 \, a b^{7} x + 9 \, b^{8}}{90 \, x^{10}} \]

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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mupad [B]  time = 0.07, size = 90, normalized size = 2.50 \[ -\frac {\frac {a^8\,x^8}{2}+\frac {8\,a^7\,b\,x^7}{3}+7\,a^6\,b^2\,x^6+\frac {56\,a^5\,b^3\,x^5}{5}+\frac {35\,a^4\,b^4\,x^4}{3}+8\,a^3\,b^5\,x^3+\frac {7\,a^2\,b^6\,x^2}{2}+\frac {8\,a\,b^7\,x}{9}+\frac {b^8}{10}}{x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^8/x^3,x)

[Out]

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

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sympy [B]  time = 0.71, size = 97, normalized size = 2.69 \[ \frac {- 45 a^{8} x^{8} - 240 a^{7} b x^{7} - 630 a^{6} b^{2} x^{6} - 1008 a^{5} b^{3} x^{5} - 1050 a^{4} b^{4} x^{4} - 720 a^{3} b^{5} x^{3} - 315 a^{2} b^{6} x^{2} - 80 a b^{7} x - 9 b^{8}}{90 x^{10}} \]

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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