3.2557 \(\int x^{-1-2 n} (a+b x^n)^5 \, dx\)

Optimal. Leaf size=85 \[ \frac{10 a^2 b^3 x^n}{n}+10 a^3 b^2 \log (x)-\frac{5 a^4 b x^{-n}}{n}-\frac{a^5 x^{-2 n}}{2 n}+\frac{5 a b^4 x^{2 n}}{2 n}+\frac{b^5 x^{3 n}}{3 n} \]

[Out]

-a^5/(2*n*x^(2*n)) - (5*a^4*b)/(n*x^n) + (10*a^2*b^3*x^n)/n + (5*a*b^4*x^(2*n))/(2*n) + (b^5*x^(3*n))/(3*n) +
10*a^3*b^2*Log[x]

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Rubi [A]  time = 0.0370283, antiderivative size = 85, 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} \[ \frac{10 a^2 b^3 x^n}{n}+10 a^3 b^2 \log (x)-\frac{5 a^4 b x^{-n}}{n}-\frac{a^5 x^{-2 n}}{2 n}+\frac{5 a b^4 x^{2 n}}{2 n}+\frac{b^5 x^{3 n}}{3 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 2*n)*(a + b*x^n)^5,x]

[Out]

-a^5/(2*n*x^(2*n)) - (5*a^4*b)/(n*x^n) + (10*a^2*b^3*x^n)/n + (5*a*b^4*x^(2*n))/(2*n) + (b^5*x^(3*n))/(3*n) +
10*a^3*b^2*Log[x]

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

Mathematica [A]  time = 0.0510785, size = 72, normalized size = 0.85 \[ \frac{x^{-2 n} \left (60 a^2 b^3 x^{3 n}-30 a^4 b x^n-3 a^5+15 a b^4 x^{4 n}+2 b^5 x^{5 n}\right )}{6 n}+10 a^3 b^2 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - 2*n)*(a + b*x^n)^5,x]

[Out]

(-3*a^5 - 30*a^4*b*x^n + 60*a^2*b^3*x^(3*n) + 15*a*b^4*x^(4*n) + 2*b^5*x^(5*n))/(6*n*x^(2*n)) + 10*a^3*b^2*Log
[x]

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Maple [A]  time = 0.016, size = 98, normalized size = 1.2 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}} \left ( 10\,{a}^{3}{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}-{\frac{{a}^{5}}{2\,n}}+{\frac{{b}^{5} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{5}}{3\,n}}+{\frac{5\,a{b}^{4} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{2\,n}}+10\,{\frac{{a}^{2}{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{n}}-5\,{\frac{{a}^{4}b{{\rm e}^{n\ln \left ( x \right ) }}}{n}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(-1-2*n)*(a+b*x^n)^5,x)

[Out]

(10*a^3*b^2*ln(x)*exp(n*ln(x))^2-1/2*a^5/n+1/3*b^5/n*exp(n*ln(x))^5+5/2*a*b^4/n*exp(n*ln(x))^4+10*a^2*b^3/n*ex
p(n*ln(x))^3-5*a^4*b/n*exp(n*ln(x)))/exp(n*ln(x))^2

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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^(-1-2*n)*(a+b*x^n)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.29222, size = 170, normalized size = 2. \begin{align*} \frac{60 \, a^{3} b^{2} n x^{2 \, n} \log \left (x\right ) + 2 \, b^{5} x^{5 \, n} + 15 \, a b^{4} x^{4 \, n} + 60 \, a^{2} b^{3} x^{3 \, n} - 30 \, a^{4} b x^{n} - 3 \, a^{5}}{6 \, n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-2*n)*(a+b*x^n)^5,x, algorithm="fricas")

[Out]

1/6*(60*a^3*b^2*n*x^(2*n)*log(x) + 2*b^5*x^(5*n) + 15*a*b^4*x^(4*n) + 60*a^2*b^3*x^(3*n) - 30*a^4*b*x^n - 3*a^
5)/(n*x^(2*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**(-1-2*n)*(a+b*x**n)**5,x)

[Out]

Timed out

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Giac [A]  time = 1.18681, size = 104, normalized size = 1.22 \begin{align*} \frac{60 \, a^{3} b^{2} n x^{2 \, n} \log \left (x\right ) + 2 \, b^{5} x^{5 \, n} + 15 \, a b^{4} x^{4 \, n} + 60 \, a^{2} b^{3} x^{3 \, n} - 30 \, a^{4} b x^{n} - 3 \, a^{5}}{6 \, n x^{2 \, n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(-1-2*n)*(a+b*x^n)^5,x, algorithm="giac")

[Out]

1/6*(60*a^3*b^2*n*x^(2*n)*log(x) + 2*b^5*x^(5*n) + 15*a*b^4*x^(4*n) + 60*a^2*b^3*x^(3*n) - 30*a^4*b*x^n - 3*a^
5)/(n*x^(2*n))