3.2564 \(\int x^{-1-9 n} (a+b x^n)^5 \, dx\)

Optimal. Leaf size=97 \[ -\frac{10 a^3 b^2 x^{-7 n}}{7 n}-\frac{5 a^2 b^3 x^{-6 n}}{3 n}-\frac{5 a^4 b x^{-8 n}}{8 n}-\frac{a^5 x^{-9 n}}{9 n}-\frac{a b^4 x^{-5 n}}{n}-\frac{b^5 x^{-4 n}}{4 n} \]

[Out]

-a^5/(9*n*x^(9*n)) - (5*a^4*b)/(8*n*x^(8*n)) - (10*a^3*b^2)/(7*n*x^(7*n)) - (5*a^2*b^3)/(3*n*x^(6*n)) - (a*b^4
)/(n*x^(5*n)) - b^5/(4*n*x^(4*n))

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Rubi [A]  time = 0.0385956, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {266, 43} \[ -\frac{10 a^3 b^2 x^{-7 n}}{7 n}-\frac{5 a^2 b^3 x^{-6 n}}{3 n}-\frac{5 a^4 b x^{-8 n}}{8 n}-\frac{a^5 x^{-9 n}}{9 n}-\frac{a b^4 x^{-5 n}}{n}-\frac{b^5 x^{-4 n}}{4 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 9*n)*(a + b*x^n)^5,x]

[Out]

-a^5/(9*n*x^(9*n)) - (5*a^4*b)/(8*n*x^(8*n)) - (10*a^3*b^2)/(7*n*x^(7*n)) - (5*a^2*b^3)/(3*n*x^(6*n)) - (a*b^4
)/(n*x^(5*n)) - b^5/(4*n*x^(4*n))

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/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 x^{-1-9 n} \left (a+b x^n\right )^5 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^{10}} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^5}{x^{10}}+\frac{5 a^4 b}{x^9}+\frac{10 a^3 b^2}{x^8}+\frac{10 a^2 b^3}{x^7}+\frac{5 a b^4}{x^6}+\frac{b^5}{x^5}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^5 x^{-9 n}}{9 n}-\frac{5 a^4 b x^{-8 n}}{8 n}-\frac{10 a^3 b^2 x^{-7 n}}{7 n}-\frac{5 a^2 b^3 x^{-6 n}}{3 n}-\frac{a b^4 x^{-5 n}}{n}-\frac{b^5 x^{-4 n}}{4 n}\\ \end{align*}

Mathematica [A]  time = 0.0310675, size = 74, normalized size = 0.76 \[ -\frac{x^{-9 n} \left (720 a^3 b^2 x^{2 n}+840 a^2 b^3 x^{3 n}+315 a^4 b x^n+56 a^5+504 a b^4 x^{4 n}+126 b^5 x^{5 n}\right )}{504 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 9*n)*(a + b*x^n)^5,x]

[Out]

-(56*a^5 + 315*a^4*b*x^n + 720*a^3*b^2*x^(2*n) + 840*a^2*b^3*x^(3*n) + 504*a*b^4*x^(4*n) + 126*b^5*x^(5*n))/(5
04*n*x^(9*n))

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Maple [A]  time = 0.02, size = 88, normalized size = 0.9 \begin{align*} -{\frac{{b}^{5}}{4\,n \left ({x}^{n} \right ) ^{4}}}-{\frac{a{b}^{4}}{n \left ({x}^{n} \right ) ^{5}}}-{\frac{5\,{a}^{2}{b}^{3}}{3\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{10\,{a}^{3}{b}^{2}}{7\,n \left ({x}^{n} \right ) ^{7}}}-{\frac{5\,{a}^{4}b}{8\,n \left ({x}^{n} \right ) ^{8}}}-{\frac{{a}^{5}}{9\,n \left ({x}^{n} \right ) ^{9}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-9*n)*(a+b*x^n)^5,x)

[Out]

-1/4*b^5/n/(x^n)^4-a*b^4/n/(x^n)^5-5/3*a^2*b^3/n/(x^n)^6-10/7*a^3*b^2/n/(x^n)^7-5/8*a^4*b/n/(x^n)^8-1/9*a^5/n/
(x^n)^9

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-9*n)*(a+b*x^n)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.35517, size = 171, normalized size = 1.76 \begin{align*} -\frac{126 \, b^{5} x^{5 \, n} + 504 \, a b^{4} x^{4 \, n} + 840 \, a^{2} b^{3} x^{3 \, n} + 720 \, a^{3} b^{2} x^{2 \, n} + 315 \, a^{4} b x^{n} + 56 \, a^{5}}{504 \, n x^{9 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-9*n)*(a+b*x^n)^5,x, algorithm="fricas")

[Out]

-1/504*(126*b^5*x^(5*n) + 504*a*b^4*x^(4*n) + 840*a^2*b^3*x^(3*n) + 720*a^3*b^2*x^(2*n) + 315*a^4*b*x^n + 56*a
^5)/(n*x^(9*n))

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Sympy [A]  time = 174.117, size = 94, normalized size = 0.97 \begin{align*} \begin{cases} - \frac{a^{5} x^{- 9 n}}{9 n} - \frac{5 a^{4} b x^{- 8 n}}{8 n} - \frac{10 a^{3} b^{2} x^{- 7 n}}{7 n} - \frac{5 a^{2} b^{3} x^{- 6 n}}{3 n} - \frac{a b^{4} x^{- 5 n}}{n} - \frac{b^{5} x^{- 4 n}}{4 n} & \text{for}\: n \neq 0 \\\left (a + b\right )^{5} \log{\left (x \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-9*n)*(a+b*x**n)**5,x)

[Out]

Piecewise((-a**5*x**(-9*n)/(9*n) - 5*a**4*b*x**(-8*n)/(8*n) - 10*a**3*b**2*x**(-7*n)/(7*n) - 5*a**2*b**3*x**(-
6*n)/(3*n) - a*b**4*x**(-5*n)/n - b**5*x**(-4*n)/(4*n), Ne(n, 0)), ((a + b)**5*log(x), True))

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Giac [A]  time = 1.22793, size = 100, normalized size = 1.03 \begin{align*} -\frac{126 \, b^{5} x^{5 \, n} + 504 \, a b^{4} x^{4 \, n} + 840 \, a^{2} b^{3} x^{3 \, n} + 720 \, a^{3} b^{2} x^{2 \, n} + 315 \, a^{4} b x^{n} + 56 \, a^{5}}{504 \, n x^{9 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-9*n)*(a+b*x^n)^5,x, algorithm="giac")

[Out]

-1/504*(126*b^5*x^(5*n) + 504*a*b^4*x^(4*n) + 840*a^2*b^3*x^(3*n) + 720*a^3*b^2*x^(2*n) + 315*a^4*b*x^n + 56*a
^5)/(n*x^(9*n))