3.1587 \(\int (a+\frac{b}{x})^8 x^{11} \, dx\)

Optimal. Leaf size=64 \[ \frac{3 b^2 (a x+b)^{10}}{10 a^4}-\frac{b^3 (a x+b)^9}{9 a^4}+\frac{(a x+b)^{12}}{12 a^4}-\frac{3 b (a x+b)^{11}}{11 a^4} \]

[Out]

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

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Rubi [A]  time = 0.0334422, antiderivative size = 64, normalized size of antiderivative = 1., 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} \[ \frac{3 b^2 (a x+b)^{10}}{10 a^4}-\frac{b^3 (a x+b)^9}{9 a^4}+\frac{(a x+b)^{12}}{12 a^4}-\frac{3 b (a x+b)^{11}}{11 a^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8*x^11,x]

[Out]

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

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]

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])

Rubi steps

\begin{align*} \int \left (a+\frac{b}{x}\right )^8 x^{11} \, dx &=\int x^3 (b+a x)^8 \, dx\\ &=\int \left (-\frac{b^3 (b+a x)^8}{a^3}+\frac{3 b^2 (b+a x)^9}{a^3}-\frac{3 b (b+a x)^{10}}{a^3}+\frac{(b+a x)^{11}}{a^3}\right ) \, dx\\ &=-\frac{b^3 (b+a x)^9}{9 a^4}+\frac{3 b^2 (b+a x)^{10}}{10 a^4}-\frac{3 b (b+a x)^{11}}{11 a^4}+\frac{(b+a x)^{12}}{12 a^4}\\ \end{align*}

Mathematica [A]  time = 0.0024917, size = 106, normalized size = 1.66 \[ \frac{14}{5} a^6 b^2 x^{10}+\frac{56}{9} a^5 b^3 x^9+\frac{35}{4} a^4 b^4 x^8+8 a^3 b^5 x^7+\frac{14}{3} a^2 b^6 x^6+\frac{8}{11} a^7 b x^{11}+\frac{a^8 x^{12}}{12}+\frac{8}{5} a b^7 x^5+\frac{b^8 x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8*x^11,x]

[Out]

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

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Maple [A]  time = 0.002, size = 91, normalized size = 1.4 \begin{align*}{\frac{{a}^{8}{x}^{12}}{12}}+{\frac{8\,{a}^{7}b{x}^{11}}{11}}+{\frac{14\,{a}^{6}{b}^{2}{x}^{10}}{5}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{9}}{9}}+{\frac{35\,{a}^{4}{b}^{4}{x}^{8}}{4}}+8\,{a}^{3}{b}^{5}{x}^{7}+{\frac{14\,{a}^{2}{b}^{6}{x}^{6}}{3}}+{\frac{8\,{b}^{7}a{x}^{5}}{5}}+{\frac{{b}^{8}{x}^{4}}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8*x^11,x)

[Out]

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

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Maxima [A]  time = 0.953033, size = 122, normalized size = 1.91 \begin{align*} \frac{1}{12} \, a^{8} x^{12} + \frac{8}{11} \, a^{7} b x^{11} + \frac{14}{5} \, a^{6} b^{2} x^{10} + \frac{56}{9} \, a^{5} b^{3} x^{9} + \frac{35}{4} \, a^{4} b^{4} x^{8} + 8 \, a^{3} b^{5} x^{7} + \frac{14}{3} \, a^{2} b^{6} x^{6} + \frac{8}{5} \, a b^{7} x^{5} + \frac{1}{4} \, b^{8} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^11,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.40023, size = 211, normalized size = 3.3 \begin{align*} \frac{1}{12} \, a^{8} x^{12} + \frac{8}{11} \, a^{7} b x^{11} + \frac{14}{5} \, a^{6} b^{2} x^{10} + \frac{56}{9} \, a^{5} b^{3} x^{9} + \frac{35}{4} \, a^{4} b^{4} x^{8} + 8 \, a^{3} b^{5} x^{7} + \frac{14}{3} \, a^{2} b^{6} x^{6} + \frac{8}{5} \, a b^{7} x^{5} + \frac{1}{4} \, b^{8} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^11,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.07894, size = 105, normalized size = 1.64 \begin{align*} \frac{a^{8} x^{12}}{12} + \frac{8 a^{7} b x^{11}}{11} + \frac{14 a^{6} b^{2} x^{10}}{5} + \frac{56 a^{5} b^{3} x^{9}}{9} + \frac{35 a^{4} b^{4} x^{8}}{4} + 8 a^{3} b^{5} x^{7} + \frac{14 a^{2} b^{6} x^{6}}{3} + \frac{8 a b^{7} x^{5}}{5} + \frac{b^{8} x^{4}}{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8*x**11,x)

[Out]

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

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Giac [A]  time = 1.12946, size = 122, normalized size = 1.91 \begin{align*} \frac{1}{12} \, a^{8} x^{12} + \frac{8}{11} \, a^{7} b x^{11} + \frac{14}{5} \, a^{6} b^{2} x^{10} + \frac{56}{9} \, a^{5} b^{3} x^{9} + \frac{35}{4} \, a^{4} b^{4} x^{8} + 8 \, a^{3} b^{5} x^{7} + \frac{14}{3} \, a^{2} b^{6} x^{6} + \frac{8}{5} \, a b^{7} x^{5} + \frac{1}{4} \, b^{8} x^{4} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^11,x, algorithm="giac")

[Out]

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