3.2615 \(\int \frac{x^{-1+5 n}}{2+b x^n} \, dx\)

Optimal. Leaf size=71 \[ -\frac{8 x^n}{b^4 n}+\frac{2 x^{2 n}}{b^3 n}-\frac{2 x^{3 n}}{3 b^2 n}+\frac{16 \log \left (b x^n+2\right )}{b^5 n}+\frac{x^{4 n}}{4 b n} \]

[Out]

(-8*x^n)/(b^4*n) + (2*x^(2*n))/(b^3*n) - (2*x^(3*n))/(3*b^2*n) + x^(4*n)/(4*b*n) + (16*Log[2 + b*x^n])/(b^5*n)

________________________________________________________________________________________

Rubi [A]  time = 0.0342742, antiderivative size = 71, 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} \[ -\frac{8 x^n}{b^4 n}+\frac{2 x^{2 n}}{b^3 n}-\frac{2 x^{3 n}}{3 b^2 n}+\frac{16 \log \left (b x^n+2\right )}{b^5 n}+\frac{x^{4 n}}{4 b n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 5*n)/(2 + b*x^n),x]

[Out]

(-8*x^n)/(b^4*n) + (2*x^(2*n))/(b^3*n) - (2*x^(3*n))/(3*b^2*n) + x^(4*n)/(4*b*n) + (16*Log[2 + b*x^n])/(b^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 \frac{x^{-1+5 n}}{2+b x^n} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^4}{2+b x} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{8}{b^4}+\frac{4 x}{b^3}-\frac{2 x^2}{b^2}+\frac{x^3}{b}+\frac{16}{b^4 (2+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{8 x^n}{b^4 n}+\frac{2 x^{2 n}}{b^3 n}-\frac{2 x^{3 n}}{3 b^2 n}+\frac{x^{4 n}}{4 b n}+\frac{16 \log \left (2+b x^n\right )}{b^5 n}\\ \end{align*}

Mathematica [A]  time = 0.0313866, size = 54, normalized size = 0.76 \[ \frac{b x^n \left (-8 b^2 x^{2 n}+3 b^3 x^{3 n}+24 b x^n-96\right )+192 \log \left (b x^n+2\right )}{12 b^5 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 + 5*n)/(2 + b*x^n),x]

[Out]

(b*x^n*(-96 + 24*b*x^n - 8*b^2*x^(2*n) + 3*b^3*x^(3*n)) + 192*Log[2 + b*x^n])/(12*b^5*n)

________________________________________________________________________________________

Maple [A]  time = 0.024, size = 78, normalized size = 1.1 \begin{align*} -8\,{\frac{{{\rm e}^{n\ln \left ( x \right ) }}}{{b}^{4}n}}+2\,{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{b}^{3}n}}-{\frac{2\, \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{3\,{b}^{2}n}}+{\frac{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{4\,bn}}+16\,{\frac{\ln \left ( 2+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{5}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-8/b^4/n*exp(n*ln(x))+2/b^3/n*exp(n*ln(x))^2-2/3/b^2/n*exp(n*ln(x))^3+1/4/b/n*exp(n*ln(x))^4+16/b^5/n*ln(2+b*e
xp(n*ln(x)))

________________________________________________________________________________________

Maxima [A]  time = 0.981099, size = 85, normalized size = 1.2 \begin{align*} \frac{3 \, b^{3} x^{4 \, n} - 8 \, b^{2} x^{3 \, n} + 24 \, b x^{2 \, n} - 96 \, x^{n}}{12 \, b^{4} n} + \frac{16 \, \log \left (\frac{b x^{n} + 2}{b}\right )}{b^{5} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^3*x^(4*n) - 8*b^2*x^(3*n) + 24*b*x^(2*n) - 96*x^n)/(b^4*n) + 16*log((b*x^n + 2)/b)/(b^5*n)

________________________________________________________________________________________

Fricas [A]  time = 1.04028, size = 128, normalized size = 1.8 \begin{align*} \frac{3 \, b^{4} x^{4 \, n} - 8 \, b^{3} x^{3 \, n} + 24 \, b^{2} x^{2 \, n} - 96 \, b x^{n} + 192 \, \log \left (b x^{n} + 2\right )}{12 \, b^{5} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*b^4*x^(4*n) - 8*b^3*x^(3*n) + 24*b^2*x^(2*n) - 96*b*x^n + 192*log(b*x^n + 2))/(b^5*n)

________________________________________________________________________________________

Sympy [A]  time = 105.216, size = 78, normalized size = 1.1 \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{2} & \text{for}\: b = 0 \wedge n = 0 \\\frac{x^{5 n}}{10 n} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{b + 2} & \text{for}\: n = 0 \\\frac{x^{4 n}}{4 b n} - \frac{2 x^{3 n}}{3 b^{2} n} + \frac{2 x^{2 n}}{b^{3} n} - \frac{8 x^{n}}{b^{4} n} + \frac{16 \log{\left (x^{n} + \frac{2}{b} \right )}}{b^{5} n} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(-1+5*n)/(2+b*x**n),x)

[Out]

Piecewise((log(x)/2, Eq(b, 0) & Eq(n, 0)), (x**(5*n)/(10*n), Eq(b, 0)), (log(x)/(b + 2), Eq(n, 0)), (x**(4*n)/
(4*b*n) - 2*x**(3*n)/(3*b**2*n) + 2*x**(2*n)/(b**3*n) - 8*x**n/(b**4*n) + 16*log(x**n + 2/b)/(b**5*n), True))

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{5 \, n - 1}}{b x^{n} + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(x^(5*n - 1)/(b*x^n + 2), x)