3.2565 \(\int x^{-1-10 n} (a+b x^n)^5 \, dx\)

Optimal. Leaf size=99 \[ -\frac{5 a^3 b^2 x^{-8 n}}{4 n}-\frac{10 a^2 b^3 x^{-7 n}}{7 n}-\frac{5 a^4 b x^{-9 n}}{9 n}-\frac{a^5 x^{-10 n}}{10 n}-\frac{5 a b^4 x^{-6 n}}{6 n}-\frac{b^5 x^{-5 n}}{5 n} \]

[Out]

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

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Rubi [A]  time = 0.040428, antiderivative size = 99, 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} \[ -\frac{5 a^3 b^2 x^{-8 n}}{4 n}-\frac{10 a^2 b^3 x^{-7 n}}{7 n}-\frac{5 a^4 b x^{-9 n}}{9 n}-\frac{a^5 x^{-10 n}}{10 n}-\frac{5 a b^4 x^{-6 n}}{6 n}-\frac{b^5 x^{-5 n}}{5 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 10*n)*(a + b*x^n)^5,x]

[Out]

-a^5/(10*n*x^(10*n)) - (5*a^4*b)/(9*n*x^(9*n)) - (5*a^3*b^2)/(4*n*x^(8*n)) - (10*a^2*b^3)/(7*n*x^(7*n)) - (5*a
*b^4)/(6*n*x^(6*n)) - b^5/(5*n*x^(5*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^{-1-10 n} \left (a+b x^n\right )^5 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^5}{x^{11}} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a^5}{x^{11}}+\frac{5 a^4 b}{x^{10}}+\frac{10 a^3 b^2}{x^9}+\frac{10 a^2 b^3}{x^8}+\frac{5 a b^4}{x^7}+\frac{b^5}{x^6}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{a^5 x^{-10 n}}{10 n}-\frac{5 a^4 b x^{-9 n}}{9 n}-\frac{5 a^3 b^2 x^{-8 n}}{4 n}-\frac{10 a^2 b^3 x^{-7 n}}{7 n}-\frac{5 a b^4 x^{-6 n}}{6 n}-\frac{b^5 x^{-5 n}}{5 n}\\ \end{align*}

Mathematica [A]  time = 0.032089, size = 74, normalized size = 0.75 \[ -\frac{x^{-10 n} \left (1575 a^3 b^2 x^{2 n}+1800 a^2 b^3 x^{3 n}+700 a^4 b x^n+126 a^5+1050 a b^4 x^{4 n}+252 b^5 x^{5 n}\right )}{1260 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 10*n)*(a + b*x^n)^5,x]

[Out]

-(126*a^5 + 700*a^4*b*x^n + 1575*a^3*b^2*x^(2*n) + 1800*a^2*b^3*x^(3*n) + 1050*a*b^4*x^(4*n) + 252*b^5*x^(5*n)
)/(1260*n*x^(10*n))

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Maple [A]  time = 0.019, size = 88, normalized size = 0.9 \begin{align*} -{\frac{{b}^{5}}{5\,n \left ({x}^{n} \right ) ^{5}}}-{\frac{5\,a{b}^{4}}{6\,n \left ({x}^{n} \right ) ^{6}}}-{\frac{10\,{a}^{2}{b}^{3}}{7\,n \left ({x}^{n} \right ) ^{7}}}-{\frac{5\,{a}^{3}{b}^{2}}{4\,n \left ({x}^{n} \right ) ^{8}}}-{\frac{5\,{a}^{4}b}{9\,n \left ({x}^{n} \right ) ^{9}}}-{\frac{{a}^{5}}{10\,n \left ({x}^{n} \right ) ^{10}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-10*n)*(a+b*x^n)^5,x)

[Out]

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

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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^(-1-10*n)*(a+b*x^n)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.25522, size = 180, normalized size = 1.82 \begin{align*} -\frac{252 \, b^{5} x^{5 \, n} + 1050 \, a b^{4} x^{4 \, n} + 1800 \, a^{2} b^{3} x^{3 \, n} + 1575 \, a^{3} b^{2} x^{2 \, n} + 700 \, a^{4} b x^{n} + 126 \, a^{5}}{1260 \, n x^{10 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-10*n)*(a+b*x^n)^5,x, algorithm="fricas")

[Out]

-1/1260*(252*b^5*x^(5*n) + 1050*a*b^4*x^(4*n) + 1800*a^2*b^3*x^(3*n) + 1575*a^3*b^2*x^(2*n) + 700*a^4*b*x^n +
126*a^5)/(n*x^(10*n))

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1-10*n)*(a+b*x**n)**5,x)

[Out]

Timed out

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Giac [A]  time = 1.23201, size = 100, normalized size = 1.01 \begin{align*} -\frac{252 \, b^{5} x^{5 \, n} + 1050 \, a b^{4} x^{4 \, n} + 1800 \, a^{2} b^{3} x^{3 \, n} + 1575 \, a^{3} b^{2} x^{2 \, n} + 700 \, a^{4} b x^{n} + 126 \, a^{5}}{1260 \, n x^{10 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-10*n)*(a+b*x^n)^5,x, algorithm="giac")

[Out]

-1/1260*(252*b^5*x^(5*n) + 1050*a*b^4*x^(4*n) + 1800*a^2*b^3*x^(3*n) + 1575*a^3*b^2*x^(2*n) + 700*a^4*b*x^n +
126*a^5)/(n*x^(10*n))