3.1585 \(\int (a+\frac{b}{x})^8 x^{13} \, dx\)

Optimal. Leaf size=98 \[ \frac{5 b^2 (a x+b)^{12}}{6 a^6}-\frac{10 b^3 (a x+b)^{11}}{11 a^6}+\frac{b^4 (a x+b)^{10}}{2 a^6}-\frac{b^5 (a x+b)^9}{9 a^6}+\frac{(a x+b)^{14}}{14 a^6}-\frac{5 b (a x+b)^{13}}{13 a^6} \]

[Out]

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

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Rubi [A]  time = 0.044195, antiderivative size = 98, 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} \[ \frac{5 b^2 (a x+b)^{12}}{6 a^6}-\frac{10 b^3 (a x+b)^{11}}{11 a^6}+\frac{b^4 (a x+b)^{10}}{2 a^6}-\frac{b^5 (a x+b)^9}{9 a^6}+\frac{(a x+b)^{14}}{14 a^6}-\frac{5 b (a x+b)^{13}}{13 a^6} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8*x^13,x]

[Out]

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

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

Mathematica [A]  time = 0.0024772, size = 106, normalized size = 1.08 \[ \frac{7}{3} a^6 b^2 x^{12}+\frac{56}{11} a^5 b^3 x^{11}+7 a^4 b^4 x^{10}+\frac{56}{9} a^3 b^5 x^9+\frac{7}{2} a^2 b^6 x^8+\frac{8}{13} a^7 b x^{13}+\frac{a^8 x^{14}}{14}+\frac{8}{7} a b^7 x^7+\frac{b^8 x^6}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8*x^13,x]

[Out]

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

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Maple [A]  time = 0.001, size = 91, normalized size = 0.9 \begin{align*}{\frac{{a}^{8}{x}^{14}}{14}}+{\frac{8\,{a}^{7}b{x}^{13}}{13}}+{\frac{7\,{b}^{2}{a}^{6}{x}^{12}}{3}}+{\frac{56\,{a}^{5}{b}^{3}{x}^{11}}{11}}+7\,{a}^{4}{b}^{4}{x}^{10}+{\frac{56\,{a}^{3}{b}^{5}{x}^{9}}{9}}+{\frac{7\,{b}^{6}{a}^{2}{x}^{8}}{2}}+{\frac{8\,{b}^{7}a{x}^{7}}{7}}+{\frac{{b}^{8}{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8*x^13,x)

[Out]

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

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Maxima [A]  time = 0.957656, size = 122, normalized size = 1.24 \begin{align*} \frac{1}{14} \, a^{8} x^{14} + \frac{8}{13} \, a^{7} b x^{13} + \frac{7}{3} \, a^{6} b^{2} x^{12} + \frac{56}{11} \, a^{5} b^{3} x^{11} + 7 \, a^{4} b^{4} x^{10} + \frac{56}{9} \, a^{3} b^{5} x^{9} + \frac{7}{2} \, a^{2} b^{6} x^{8} + \frac{8}{7} \, a b^{7} x^{7} + \frac{1}{6} \, b^{8} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^13,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.38676, size = 212, normalized size = 2.16 \begin{align*} \frac{1}{14} \, a^{8} x^{14} + \frac{8}{13} \, a^{7} b x^{13} + \frac{7}{3} \, a^{6} b^{2} x^{12} + \frac{56}{11} \, a^{5} b^{3} x^{11} + 7 \, a^{4} b^{4} x^{10} + \frac{56}{9} \, a^{3} b^{5} x^{9} + \frac{7}{2} \, a^{2} b^{6} x^{8} + \frac{8}{7} \, a b^{7} x^{7} + \frac{1}{6} \, b^{8} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^13,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.080723, size = 105, normalized size = 1.07 \begin{align*} \frac{a^{8} x^{14}}{14} + \frac{8 a^{7} b x^{13}}{13} + \frac{7 a^{6} b^{2} x^{12}}{3} + \frac{56 a^{5} b^{3} x^{11}}{11} + 7 a^{4} b^{4} x^{10} + \frac{56 a^{3} b^{5} x^{9}}{9} + \frac{7 a^{2} b^{6} x^{8}}{2} + \frac{8 a b^{7} x^{7}}{7} + \frac{b^{8} x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8*x**13,x)

[Out]

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

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Giac [A]  time = 1.18628, size = 122, normalized size = 1.24 \begin{align*} \frac{1}{14} \, a^{8} x^{14} + \frac{8}{13} \, a^{7} b x^{13} + \frac{7}{3} \, a^{6} b^{2} x^{12} + \frac{56}{11} \, a^{5} b^{3} x^{11} + 7 \, a^{4} b^{4} x^{10} + \frac{56}{9} \, a^{3} b^{5} x^{9} + \frac{7}{2} \, a^{2} b^{6} x^{8} + \frac{8}{7} \, a b^{7} x^{7} + \frac{1}{6} \, b^{8} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^13,x, algorithm="giac")

[Out]

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