3.1586 \(\int (a+\frac{b}{x})^8 x^{12} \, dx\)

Optimal. Leaf size=81 \[ \frac{6 b^2 (a x+b)^{11}}{11 a^5}-\frac{2 b^3 (a x+b)^{10}}{5 a^5}+\frac{b^4 (a x+b)^9}{9 a^5}+\frac{(a x+b)^{13}}{13 a^5}-\frac{b (a x+b)^{12}}{3 a^5} \]

[Out]

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

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Rubi [A]  time = 0.0376675, antiderivative size = 81, 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} \[ \frac{6 b^2 (a x+b)^{11}}{11 a^5}-\frac{2 b^3 (a x+b)^{10}}{5 a^5}+\frac{b^4 (a x+b)^9}{9 a^5}+\frac{(a x+b)^{13}}{13 a^5}-\frac{b (a x+b)^{12}}{3 a^5} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^8*x^12,x]

[Out]

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

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

Mathematica [A]  time = 0.0027473, size = 104, normalized size = 1.28 \[ \frac{28}{11} a^6 b^2 x^{11}+\frac{28}{5} a^5 b^3 x^{10}+\frac{70}{9} a^4 b^4 x^9+7 a^3 b^5 x^8+4 a^2 b^6 x^7+\frac{2}{3} a^7 b x^{12}+\frac{a^8 x^{13}}{13}+\frac{4}{3} a b^7 x^6+\frac{b^8 x^5}{5} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^8*x^12,x]

[Out]

(b^8*x^5)/5 + (4*a*b^7*x^6)/3 + 4*a^2*b^6*x^7 + 7*a^3*b^5*x^8 + (70*a^4*b^4*x^9)/9 + (28*a^5*b^3*x^10)/5 + (28
*a^6*b^2*x^11)/11 + (2*a^7*b*x^12)/3 + (a^8*x^13)/13

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Maple [A]  time = 0.002, size = 91, normalized size = 1.1 \begin{align*}{\frac{{a}^{8}{x}^{13}}{13}}+{\frac{2\,{a}^{7}b{x}^{12}}{3}}+{\frac{28\,{a}^{6}{b}^{2}{x}^{11}}{11}}+{\frac{28\,{a}^{5}{b}^{3}{x}^{10}}{5}}+{\frac{70\,{a}^{4}{b}^{4}{x}^{9}}{9}}+7\,{a}^{3}{b}^{5}{x}^{8}+4\,{b}^{6}{a}^{2}{x}^{7}+{\frac{4\,a{b}^{7}{x}^{6}}{3}}+{\frac{{b}^{8}{x}^{5}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^8*x^12,x)

[Out]

1/13*a^8*x^13+2/3*a^7*b*x^12+28/11*a^6*b^2*x^11+28/5*a^5*b^3*x^10+70/9*a^4*b^4*x^9+7*a^3*b^5*x^8+4*b^6*a^2*x^7
+4/3*a*b^7*x^6+1/5*b^8*x^5

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Maxima [A]  time = 0.970348, size = 122, normalized size = 1.51 \begin{align*} \frac{1}{13} \, a^{8} x^{13} + \frac{2}{3} \, a^{7} b x^{12} + \frac{28}{11} \, a^{6} b^{2} x^{11} + \frac{28}{5} \, a^{5} b^{3} x^{10} + \frac{70}{9} \, a^{4} b^{4} x^{9} + 7 \, a^{3} b^{5} x^{8} + 4 \, a^{2} b^{6} x^{7} + \frac{4}{3} \, a b^{7} x^{6} + \frac{1}{5} \, b^{8} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^12,x, algorithm="maxima")

[Out]

1/13*a^8*x^13 + 2/3*a^7*b*x^12 + 28/11*a^6*b^2*x^11 + 28/5*a^5*b^3*x^10 + 70/9*a^4*b^4*x^9 + 7*a^3*b^5*x^8 + 4
*a^2*b^6*x^7 + 4/3*a*b^7*x^6 + 1/5*b^8*x^5

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Fricas [A]  time = 1.37681, size = 208, normalized size = 2.57 \begin{align*} \frac{1}{13} \, a^{8} x^{13} + \frac{2}{3} \, a^{7} b x^{12} + \frac{28}{11} \, a^{6} b^{2} x^{11} + \frac{28}{5} \, a^{5} b^{3} x^{10} + \frac{70}{9} \, a^{4} b^{4} x^{9} + 7 \, a^{3} b^{5} x^{8} + 4 \, a^{2} b^{6} x^{7} + \frac{4}{3} \, a b^{7} x^{6} + \frac{1}{5} \, b^{8} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^12,x, algorithm="fricas")

[Out]

1/13*a^8*x^13 + 2/3*a^7*b*x^12 + 28/11*a^6*b^2*x^11 + 28/5*a^5*b^3*x^10 + 70/9*a^4*b^4*x^9 + 7*a^3*b^5*x^8 + 4
*a^2*b^6*x^7 + 4/3*a*b^7*x^6 + 1/5*b^8*x^5

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Sympy [A]  time = 0.080558, size = 104, normalized size = 1.28 \begin{align*} \frac{a^{8} x^{13}}{13} + \frac{2 a^{7} b x^{12}}{3} + \frac{28 a^{6} b^{2} x^{11}}{11} + \frac{28 a^{5} b^{3} x^{10}}{5} + \frac{70 a^{4} b^{4} x^{9}}{9} + 7 a^{3} b^{5} x^{8} + 4 a^{2} b^{6} x^{7} + \frac{4 a b^{7} x^{6}}{3} + \frac{b^{8} x^{5}}{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**8*x**12,x)

[Out]

a**8*x**13/13 + 2*a**7*b*x**12/3 + 28*a**6*b**2*x**11/11 + 28*a**5*b**3*x**10/5 + 70*a**4*b**4*x**9/9 + 7*a**3
*b**5*x**8 + 4*a**2*b**6*x**7 + 4*a*b**7*x**6/3 + b**8*x**5/5

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Giac [A]  time = 1.17509, size = 122, normalized size = 1.51 \begin{align*} \frac{1}{13} \, a^{8} x^{13} + \frac{2}{3} \, a^{7} b x^{12} + \frac{28}{11} \, a^{6} b^{2} x^{11} + \frac{28}{5} \, a^{5} b^{3} x^{10} + \frac{70}{9} \, a^{4} b^{4} x^{9} + 7 \, a^{3} b^{5} x^{8} + 4 \, a^{2} b^{6} x^{7} + \frac{4}{3} \, a b^{7} x^{6} + \frac{1}{5} \, b^{8} x^{5} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^8*x^12,x, algorithm="giac")

[Out]

1/13*a^8*x^13 + 2/3*a^7*b*x^12 + 28/11*a^6*b^2*x^11 + 28/5*a^5*b^3*x^10 + 70/9*a^4*b^4*x^9 + 7*a^3*b^5*x^8 + 4
*a^2*b^6*x^7 + 4/3*a*b^7*x^6 + 1/5*b^8*x^5