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]

-1/12*(a*x+b)^9/b/x^12+1/44*a*(a*x+b)^9/b^2/x^11-1/220*a^2*(a*x+b)^9/b^3/x^10+1/1980*a^3*(a*x+b)^9/b^4/x^9

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Rubi [A]  time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, 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 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^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*}

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Mathematica [A]  time = 0.00, size = 106, normalized size = 1.39 \[ -\frac {a^8}{4 x^4}-\frac {8 a^7 b}{5 x^5}-\frac {14 a^6 b^2}{3 x^6}-\frac {8 a^5 b^3}{x^7}-\frac {35 a^4 b^4}{4 x^8}-\frac {56 a^3 b^5}{9 x^9}-\frac {14 a^2 b^6}{5 x^{10}}-\frac {8 a b^7}{11 x^{11}}-\frac {b^8}{12 x^{12}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8/x^5,x]

[Out]

-1/12*b^8/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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fricas [A]  time = 0.86, size = 90, normalized size = 1.18 \[ -\frac {495 \, a^{8} x^{8} + 3168 \, a^{7} b x^{7} + 9240 \, a^{6} b^{2} x^{6} + 15840 \, a^{5} b^{3} x^{5} + 17325 \, a^{4} b^{4} x^{4} + 12320 \, a^{3} b^{5} x^{3} + 5544 \, a^{2} b^{6} x^{2} + 1440 \, a b^{7} x + 165 \, b^{8}}{1980 \, x^{12}} \]

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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giac [A]  time = 0.17, size = 90, normalized size = 1.18 \[ -\frac {495 \, a^{8} x^{8} + 3168 \, a^{7} b x^{7} + 9240 \, a^{6} b^{2} x^{6} + 15840 \, a^{5} b^{3} x^{5} + 17325 \, a^{4} b^{4} x^{4} + 12320 \, a^{3} b^{5} x^{3} + 5544 \, a^{2} b^{6} x^{2} + 1440 \, a b^{7} x + 165 \, b^{8}}{1980 \, x^{12}} \]

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

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maple [A]  time = 0.01, size = 91, normalized size = 1.20 \[ -\frac {a^{8}}{4 x^{4}}-\frac {8 a^{7} b}{5 x^{5}}-\frac {14 a^{6} b^{2}}{3 x^{6}}-\frac {8 a^{5} b^{3}}{x^{7}}-\frac {35 a^{4} b^{4}}{4 x^{8}}-\frac {56 a^{3} b^{5}}{9 x^{9}}-\frac {14 a^{2} b^{6}}{5 x^{10}}-\frac {8 a \,b^{7}}{11 x^{11}}-\frac {b^{8}}{12 x^{12}} \]

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

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maxima [A]  time = 1.16, size = 90, normalized size = 1.18 \[ -\frac {495 \, a^{8} x^{8} + 3168 \, a^{7} b x^{7} + 9240 \, a^{6} b^{2} x^{6} + 15840 \, a^{5} b^{3} x^{5} + 17325 \, a^{4} b^{4} x^{4} + 12320 \, a^{3} b^{5} x^{3} + 5544 \, a^{2} b^{6} x^{2} + 1440 \, a b^{7} x + 165 \, b^{8}}{1980 \, x^{12}} \]

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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mupad [B]  time = 0.07, size = 90, normalized size = 1.18 \[ -\frac {\frac {a^8\,x^8}{4}+\frac {8\,a^7\,b\,x^7}{5}+\frac {14\,a^6\,b^2\,x^6}{3}+8\,a^5\,b^3\,x^5+\frac {35\,a^4\,b^4\,x^4}{4}+\frac {56\,a^3\,b^5\,x^3}{9}+\frac {14\,a^2\,b^6\,x^2}{5}+\frac {8\,a\,b^7\,x}{11}+\frac {b^8}{12}}{x^{12}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^8/x^5,x)

[Out]

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

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sympy [A]  time = 0.85, size = 97, normalized size = 1.28 \[ \frac {- 495 a^{8} x^{8} - 3168 a^{7} b x^{7} - 9240 a^{6} b^{2} x^{6} - 15840 a^{5} b^{3} x^{5} - 17325 a^{4} b^{4} x^{4} - 12320 a^{3} b^{5} x^{3} - 5544 a^{2} b^{6} x^{2} - 1440 a b^{7} x - 165 b^{8}}{1980 x^{12}} \]

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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