\(\int \frac {x^{-1+2 n}}{(a+b x^n)^3} \, dx\) [2634]

Optimal result

Integrand size = 17, antiderivative size = 24 \[ \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3} \, dx=\frac {x^{2 n}}{2 a n \left (a+b x^n\right )^2} \]

[Out]

1/2*x^(2*n)/a/n/(a+b*x^n)^2

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {270} \[ \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3} \, dx=\frac {x^{2 n}}{2 a n \left (a+b x^n\right )^2} \]

[In]

Int[x^(-1 + 2*n)/(a + b*x^n)^3,x]

[Out]

x^(2*n)/(2*a*n*(a + b*x^n)^2)

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*
c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {x^{2 n}}{2 a n \left (a+b x^n\right )^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3} \, dx=\frac {-a-2 b x^n}{2 b^2 n \left (a+b x^n\right )^2} \]

[In]

Integrate[x^(-1 + 2*n)/(a + b*x^n)^3,x]

[Out]

(-a - 2*b*x^n)/(2*b^2*n*(a + b*x^n)^2)

Maple [A] (verified)

Time = 3.89 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08

method	result	size
risch	\(-\frac {2 b \,x^{n}+a}{2 b^{2} n \left (a +b \,x^{n}\right )^{2}}\)	\(26\)
parallelrisch	\(\frac {x \,x^{-1+2 n}}{2 a n \left (a +b \,x^{n}\right )^{2}}\)	\(26\)
norman	\(\frac {-\frac {{\mathrm e}^{n \ln \left (x \right )}}{b n}-\frac {a}{2 b^{2} n}}{\left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}}\)	\(36\)

[In]

int(x^(-1+2*n)/(a+b*x^n)^3,x,method=_RETURNVERBOSE)

[Out]

-1/2*(2*b*x^n+a)/b^2/n/(a+b*x^n)^2

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3} \, dx=-\frac {2 \, b x^{n} + a}{2 \, {\left (b^{4} n x^{2 \, n} + 2 \, a b^{3} n x^{n} + a^{2} b^{2} n\right )}} \]

[In]

integrate(x^(-1+2*n)/(a+b*x^n)^3,x, algorithm="fricas")

[Out]

-1/2*(2*b*x^n + a)/(b^4*n*x^(2*n) + 2*a*b^3*n*x^n + a^2*b^2*n)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (17) = 34\).

Time = 1.43 (sec) , antiderivative size = 88, normalized size of antiderivative = 3.67 \[ \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3} \, dx=\begin {cases} \tilde {\infty } \log {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac {x x^{- 3 n} x^{2 n - 1}}{b^{3} n} & \text {for}\: a = 0 \\\frac {\tilde {\infty } x x^{2 n - 1}}{n} & \text {for}\: b = - a x^{- n} \\\frac {\log {\left (x \right )}}{\left (a + b\right )^{3}} & \text {for}\: n = 0 \\\frac {x x^{2 n - 1}}{2 a^{3} n + 4 a^{2} b n x^{n} + 2 a b^{2} n x^{2 n}} & \text {otherwise} \end {cases} \]

[In]

integrate(x**(-1+2*n)/(a+b*x**n)**3,x)

[Out]

Piecewise((zoo*log(x), Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (-x*x**(2*n - 1)/(b**3*n*x**(3*n)), Eq(a, 0)), (zoo*x*
x**(2*n - 1)/n, Eq(b, -a/x**n)), (log(x)/(a + b)**3, Eq(n, 0)), (x*x**(2*n - 1)/(2*a**3*n + 4*a**2*b*n*x**n +
2*a*b**2*n*x**(2*n)), True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3} \, dx=-\frac {2 \, b x^{n} + a}{2 \, {\left (b^{4} n x^{2 \, n} + 2 \, a b^{3} n x^{n} + a^{2} b^{2} n\right )}} \]

[In]

integrate(x^(-1+2*n)/(a+b*x^n)^3,x, algorithm="maxima")

[Out]

-1/2*(2*b*x^n + a)/(b^4*n*x^(2*n) + 2*a*b^3*n*x^n + a^2*b^2*n)

Giac [F]

\[ \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3} \, dx=\int { \frac {x^{2 \, n - 1}}{{\left (b x^{n} + a\right )}^{3}} \,d x } \]

[In]

integrate(x^(-1+2*n)/(a+b*x^n)^3,x, algorithm="giac")

[Out]

integrate(x^(2*n - 1)/(b*x^n + a)^3, x)

Mupad [B] (verification not implemented)

Time = 5.71 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.54 \[ \int \frac {x^{-1+2 n}}{\left (a+b x^n\right )^3} \, dx=\frac {x^{2\,n}}{2\,\left (a^3\,n+2\,a^2\,b\,n\,x^n+a\,b^2\,n\,x^{2\,n}\right )} \]

[In]

int(x^(2*n - 1)/(a + b*x^n)^3,x)

[Out]

x^(2*n)/(2*(a^3*n + 2*a^2*b*n*x^n + a*b^2*n*x^(2*n)))