3.1605 \(\int \frac {(a+\frac {b}{x})^8}{x^7} \, dx\)

Optimal. Leaf size=106 \[ -\frac {a^8}{6 x^6}-\frac {8 a^7 b}{7 x^7}-\frac {7 a^6 b^2}{2 x^8}-\frac {56 a^5 b^3}{9 x^9}-\frac {7 a^4 b^4}{x^{10}}-\frac {56 a^3 b^5}{11 x^{11}}-\frac {7 a^2 b^6}{3 x^{12}}-\frac {8 a b^7}{13 x^{13}}-\frac {b^8}{14 x^{14}} \]

[Out]

-1/14*b^8/x^14-8/13*a*b^7/x^13-7/3*a^2*b^6/x^12-56/11*a^3*b^5/x^11-7*a^4*b^4/x^10-56/9*a^5*b^3/x^9-7/2*a^6*b^2
/x^8-8/7*a^7*b/x^7-1/6*a^8/x^6

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Rubi [A]  time = 0.04, antiderivative size = 106, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {263, 43} \[ -\frac {7 a^6 b^2}{2 x^8}-\frac {56 a^5 b^3}{9 x^9}-\frac {7 a^4 b^4}{x^{10}}-\frac {56 a^3 b^5}{11 x^{11}}-\frac {7 a^2 b^6}{3 x^{12}}-\frac {8 a^7 b}{7 x^7}-\frac {a^8}{6 x^6}-\frac {8 a b^7}{13 x^{13}}-\frac {b^8}{14 x^{14}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8/x^7,x]

[Out]

-b^8/(14*x^14) - (8*a*b^7)/(13*x^13) - (7*a^2*b^6)/(3*x^12) - (56*a^3*b^5)/(11*x^11) - (7*a^4*b^4)/x^10 - (56*
a^5*b^3)/(9*x^9) - (7*a^6*b^2)/(2*x^8) - (8*a^7*b)/(7*x^7) - a^8/(6*x^6)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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^7} \, dx &=\int \frac {(b+a x)^8}{x^{15}} \, dx\\ &=\int \left (\frac {b^8}{x^{15}}+\frac {8 a b^7}{x^{14}}+\frac {28 a^2 b^6}{x^{13}}+\frac {56 a^3 b^5}{x^{12}}+\frac {70 a^4 b^4}{x^{11}}+\frac {56 a^5 b^3}{x^{10}}+\frac {28 a^6 b^2}{x^9}+\frac {8 a^7 b}{x^8}+\frac {a^8}{x^7}\right ) \, dx\\ &=-\frac {b^8}{14 x^{14}}-\frac {8 a b^7}{13 x^{13}}-\frac {7 a^2 b^6}{3 x^{12}}-\frac {56 a^3 b^5}{11 x^{11}}-\frac {7 a^4 b^4}{x^{10}}-\frac {56 a^5 b^3}{9 x^9}-\frac {7 a^6 b^2}{2 x^8}-\frac {8 a^7 b}{7 x^7}-\frac {a^8}{6 x^6}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 106, normalized size = 1.00 \[ -\frac {a^8}{6 x^6}-\frac {8 a^7 b}{7 x^7}-\frac {7 a^6 b^2}{2 x^8}-\frac {56 a^5 b^3}{9 x^9}-\frac {7 a^4 b^4}{x^{10}}-\frac {56 a^3 b^5}{11 x^{11}}-\frac {7 a^2 b^6}{3 x^{12}}-\frac {8 a b^7}{13 x^{13}}-\frac {b^8}{14 x^{14}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8/x^7,x]

[Out]

-1/14*b^8/x^14 - (8*a*b^7)/(13*x^13) - (7*a^2*b^6)/(3*x^12) - (56*a^3*b^5)/(11*x^11) - (7*a^4*b^4)/x^10 - (56*
a^5*b^3)/(9*x^9) - (7*a^6*b^2)/(2*x^8) - (8*a^7*b)/(7*x^7) - a^8/(6*x^6)

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fricas [A]  time = 0.84, size = 90, normalized size = 0.85 \[ -\frac {3003 \, a^{8} x^{8} + 20592 \, a^{7} b x^{7} + 63063 \, a^{6} b^{2} x^{6} + 112112 \, a^{5} b^{3} x^{5} + 126126 \, a^{4} b^{4} x^{4} + 91728 \, a^{3} b^{5} x^{3} + 42042 \, a^{2} b^{6} x^{2} + 11088 \, a b^{7} x + 1287 \, b^{8}}{18018 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8/x^7,x, algorithm="fricas")

[Out]

-1/18018*(3003*a^8*x^8 + 20592*a^7*b*x^7 + 63063*a^6*b^2*x^6 + 112112*a^5*b^3*x^5 + 126126*a^4*b^4*x^4 + 91728
*a^3*b^5*x^3 + 42042*a^2*b^6*x^2 + 11088*a*b^7*x + 1287*b^8)/x^14

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giac [A]  time = 0.17, size = 90, normalized size = 0.85 \[ -\frac {3003 \, a^{8} x^{8} + 20592 \, a^{7} b x^{7} + 63063 \, a^{6} b^{2} x^{6} + 112112 \, a^{5} b^{3} x^{5} + 126126 \, a^{4} b^{4} x^{4} + 91728 \, a^{3} b^{5} x^{3} + 42042 \, a^{2} b^{6} x^{2} + 11088 \, a b^{7} x + 1287 \, b^{8}}{18018 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8/x^7,x, algorithm="giac")

[Out]

-1/18018*(3003*a^8*x^8 + 20592*a^7*b*x^7 + 63063*a^6*b^2*x^6 + 112112*a^5*b^3*x^5 + 126126*a^4*b^4*x^4 + 91728
*a^3*b^5*x^3 + 42042*a^2*b^6*x^2 + 11088*a*b^7*x + 1287*b^8)/x^14

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maple [A]  time = 0.01, size = 91, normalized size = 0.86 \[ -\frac {a^{8}}{6 x^{6}}-\frac {8 a^{7} b}{7 x^{7}}-\frac {7 a^{6} b^{2}}{2 x^{8}}-\frac {56 a^{5} b^{3}}{9 x^{9}}-\frac {7 a^{4} b^{4}}{x^{10}}-\frac {56 a^{3} b^{5}}{11 x^{11}}-\frac {7 a^{2} b^{6}}{3 x^{12}}-\frac {8 a \,b^{7}}{13 x^{13}}-\frac {b^{8}}{14 x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8/x^7,x)

[Out]

-1/14*b^8/x^14-8/13*a*b^7/x^13-7/3*a^2*b^6/x^12-56/11*a^3*b^5/x^11-7*a^4*b^4/x^10-56/9*a^5*b^3/x^9-7/2*a^6*b^2
/x^8-8/7*a^7*b/x^7-1/6*a^8/x^6

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maxima [A]  time = 1.08, size = 90, normalized size = 0.85 \[ -\frac {3003 \, a^{8} x^{8} + 20592 \, a^{7} b x^{7} + 63063 \, a^{6} b^{2} x^{6} + 112112 \, a^{5} b^{3} x^{5} + 126126 \, a^{4} b^{4} x^{4} + 91728 \, a^{3} b^{5} x^{3} + 42042 \, a^{2} b^{6} x^{2} + 11088 \, a b^{7} x + 1287 \, b^{8}}{18018 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8/x^7,x, algorithm="maxima")

[Out]

-1/18018*(3003*a^8*x^8 + 20592*a^7*b*x^7 + 63063*a^6*b^2*x^6 + 112112*a^5*b^3*x^5 + 126126*a^4*b^4*x^4 + 91728
*a^3*b^5*x^3 + 42042*a^2*b^6*x^2 + 11088*a*b^7*x + 1287*b^8)/x^14

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mupad [B]  time = 0.07, size = 90, normalized size = 0.85 \[ -\frac {\frac {a^8\,x^8}{6}+\frac {8\,a^7\,b\,x^7}{7}+\frac {7\,a^6\,b^2\,x^6}{2}+\frac {56\,a^5\,b^3\,x^5}{9}+7\,a^4\,b^4\,x^4+\frac {56\,a^3\,b^5\,x^3}{11}+\frac {7\,a^2\,b^6\,x^2}{3}+\frac {8\,a\,b^7\,x}{13}+\frac {b^8}{14}}{x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/x)^8/x^7,x)

[Out]

-(b^8/14 + (a^8*x^8)/6 + (8*a^7*b*x^7)/7 + (7*a^2*b^6*x^2)/3 + (56*a^3*b^5*x^3)/11 + 7*a^4*b^4*x^4 + (56*a^5*b
^3*x^5)/9 + (7*a^6*b^2*x^6)/2 + (8*a*b^7*x)/13)/x^14

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sympy [A]  time = 0.93, size = 97, normalized size = 0.92 \[ \frac {- 3003 a^{8} x^{8} - 20592 a^{7} b x^{7} - 63063 a^{6} b^{2} x^{6} - 112112 a^{5} b^{3} x^{5} - 126126 a^{4} b^{4} x^{4} - 91728 a^{3} b^{5} x^{3} - 42042 a^{2} b^{6} x^{2} - 11088 a b^{7} x - 1287 b^{8}}{18018 x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8/x**7,x)

[Out]

(-3003*a**8*x**8 - 20592*a**7*b*x**7 - 63063*a**6*b**2*x**6 - 112112*a**5*b**3*x**5 - 126126*a**4*b**4*x**4 -
91728*a**3*b**5*x**3 - 42042*a**2*b**6*x**2 - 11088*a*b**7*x - 1287*b**8)/(18018*x**14)

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