3.16.89 \(\int (a+\frac {b}{x})^8 x^9 \, dx\) [1589]

Optimal. Leaf size=30 \[ -\frac {b (b+a x)^9}{9 a^2}+\frac {(b+a x)^{10}}{10 a^2} \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 30, 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 = {269, 45} \begin {gather*} \frac {(a x+b)^{10}}{10 a^2}-\frac {b (a x+b)^9}{9 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8*x^9,x]

[Out]

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

Rule 45

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 269

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 \left (a+\frac {b}{x}\right )^8 x^9 \, dx &=\int x (b+a x)^8 \, dx\\ &=\int \left (-\frac {b (b+a x)^8}{a}+\frac {(b+a x)^9}{a}\right ) \, dx\\ &=-\frac {b (b+a x)^9}{9 a^2}+\frac {(b+a x)^{10}}{10 a^2}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(104\) vs. \(2(30)=60\).
time = 0.00, size = 104, normalized size = 3.47 \begin {gather*} \frac {b^8 x^2}{2}+\frac {8}{3} a b^7 x^3+7 a^2 b^6 x^4+\frac {56}{5} a^3 b^5 x^5+\frac {35}{3} a^4 b^4 x^6+8 a^5 b^3 x^7+\frac {7}{2} a^6 b^2 x^8+\frac {8}{9} a^7 b x^9+\frac {a^8 x^{10}}{10} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8*x^9,x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs. \(2(26)=52\).
time = 0.02, size = 91, normalized size = 3.03

method	result	size
gosper	\(\frac {x^{2} \left (9 a^{8} x^{8}+80 a^{7} b \,x^{7}+315 a^{6} b^{2} x^{6}+720 a^{5} b^{3} x^{5}+1050 a^{4} b^{4} x^{4}+1008 a^{3} b^{5} x^{3}+630 a^{2} b^{6} x^{2}+240 a \,b^{7} x +45 b^{8}\right )}{90}\)	\(91\)
default	\(\frac {1}{10} a^{8} x^{10}+\frac {8}{9} a^{7} b \,x^{9}+\frac {7}{2} a^{6} b^{2} x^{8}+8 a^{5} b^{3} x^{7}+\frac {35}{3} a^{4} b^{4} x^{6}+\frac {56}{5} a^{3} b^{5} x^{5}+7 a^{2} b^{6} x^{4}+\frac {8}{3} a \,b^{7} x^{3}+\frac {1}{2} b^{8} x^{2}\)	\(91\)
risch	\(\frac {1}{10} a^{8} x^{10}+\frac {8}{9} a^{7} b \,x^{9}+\frac {7}{2} a^{6} b^{2} x^{8}+8 a^{5} b^{3} x^{7}+\frac {35}{3} a^{4} b^{4} x^{6}+\frac {56}{5} a^{3} b^{5} x^{5}+7 a^{2} b^{6} x^{4}+\frac {8}{3} a \,b^{7} x^{3}+\frac {1}{2} b^{8} x^{2}\)	\(91\)
norman	\(\frac {\frac {1}{10} a^{8} x^{17}+\frac {1}{2} b^{8} x^{9}+\frac {8}{3} a \,b^{7} x^{10}+7 a^{2} b^{6} x^{11}+\frac {56}{5} a^{3} b^{5} x^{12}+\frac {35}{3} a^{4} b^{4} x^{13}+8 a^{5} b^{3} x^{14}+\frac {7}{2} a^{6} b^{2} x^{15}+\frac {8}{9} a^{7} b \,x^{16}}{x^{7}}\)	\(95\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+1/x*b)^8*x^9,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (26) = 52\).
time = 0.30, size = 90, normalized size = 3.00 \begin {gather*} \frac {1}{10} \, a^{8} x^{10} + \frac {8}{9} \, a^{7} b x^{9} + \frac {7}{2} \, a^{6} b^{2} x^{8} + 8 \, a^{5} b^{3} x^{7} + \frac {35}{3} \, a^{4} b^{4} x^{6} + \frac {56}{5} \, a^{3} b^{5} x^{5} + 7 \, a^{2} b^{6} x^{4} + \frac {8}{3} \, a b^{7} x^{3} + \frac {1}{2} \, b^{8} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^9,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (26) = 52\).
time = 0.35, size = 90, normalized size = 3.00 \begin {gather*} \frac {1}{10} \, a^{8} x^{10} + \frac {8}{9} \, a^{7} b x^{9} + \frac {7}{2} \, a^{6} b^{2} x^{8} + 8 \, a^{5} b^{3} x^{7} + \frac {35}{3} \, a^{4} b^{4} x^{6} + \frac {56}{5} \, a^{3} b^{5} x^{5} + 7 \, a^{2} b^{6} x^{4} + \frac {8}{3} \, a b^{7} x^{3} + \frac {1}{2} \, b^{8} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^9,x, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 104 vs. \(2 (24) = 48\).
time = 0.02, size = 104, normalized size = 3.47 \begin {gather*} \frac {a^{8} x^{10}}{10} + \frac {8 a^{7} b x^{9}}{9} + \frac {7 a^{6} b^{2} x^{8}}{2} + 8 a^{5} b^{3} x^{7} + \frac {35 a^{4} b^{4} x^{6}}{3} + \frac {56 a^{3} b^{5} x^{5}}{5} + 7 a^{2} b^{6} x^{4} + \frac {8 a b^{7} x^{3}}{3} + \frac {b^{8} x^{2}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8*x**9,x)

[Out]

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

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 90 vs. \(2 (26) = 52\).
time = 1.23, size = 90, normalized size = 3.00 \begin {gather*} \frac {1}{10} \, a^{8} x^{10} + \frac {8}{9} \, a^{7} b x^{9} + \frac {7}{2} \, a^{6} b^{2} x^{8} + 8 \, a^{5} b^{3} x^{7} + \frac {35}{3} \, a^{4} b^{4} x^{6} + \frac {56}{5} \, a^{3} b^{5} x^{5} + 7 \, a^{2} b^{6} x^{4} + \frac {8}{3} \, a b^{7} x^{3} + \frac {1}{2} \, b^{8} x^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^9,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.06, size = 90, normalized size = 3.00 \begin {gather*} \frac {a^8\,x^{10}}{10}+\frac {8\,a^7\,b\,x^9}{9}+\frac {7\,a^6\,b^2\,x^8}{2}+8\,a^5\,b^3\,x^7+\frac {35\,a^4\,b^4\,x^6}{3}+\frac {56\,a^3\,b^5\,x^5}{5}+7\,a^2\,b^6\,x^4+\frac {8\,a\,b^7\,x^3}{3}+\frac {b^8\,x^2}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9*(a + b/x)^8,x)

[Out]

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

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