3.2626 \(\int \frac{x^{-1+2 n}}{(a+b x^n)^2} \, dx\)

Optimal. Leaf size=33 \[ \frac{a}{b^2 n \left (a+b x^n\right )}+\frac{\log \left (a+b x^n\right )}{b^2 n} \]

[Out]

a/(b^2*n*(a + b*x^n)) + Log[a + b*x^n]/(b^2*n)

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Rubi [A]  time = 0.0230451, antiderivative size = 33, 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} \[ \frac{a}{b^2 n \left (a+b x^n\right )}+\frac{\log \left (a+b x^n\right )}{b^2 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 + 2*n)/(a + b*x^n)^2,x]

[Out]

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

Mathematica [A]  time = 0.0242485, size = 27, normalized size = 0.82 \[ \frac{\frac{a}{a+b x^n}+\log \left (a+b x^n\right )}{b^2 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a/(a + b*x^n) + Log[a + b*x^n])/(b^2*n)

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Maple [A]  time = 0.014, size = 38, normalized size = 1.2 \begin{align*}{\frac{a}{{b}^{2}n \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }}+{\frac{\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{b}^{2}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.971322, size = 53, normalized size = 1.61 \begin{align*} \frac{a}{b^{3} n x^{n} + a b^{2} n} + \frac{\log \left (\frac{b x^{n} + a}{b}\right )}{b^{2} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.0092, size = 76, normalized size = 2.3 \begin{align*} \frac{{\left (b x^{n} + a\right )} \log \left (b x^{n} + a\right ) + a}{b^{3} n x^{n} + a b^{2} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 81.0982, size = 99, normalized size = 3. \begin{align*} \begin{cases} \frac{\log{\left (x \right )}}{a^{2}} & \text{for}\: b = 0 \wedge n = 0 \\\frac{x^{2 n}}{2 a^{2} n} & \text{for}\: b = 0 \\\frac{\log{\left (x \right )}}{\left (a + b\right )^{2}} & \text{for}\: n = 0 \\\frac{a \log{\left (\frac{a}{b} + x^{n} \right )}}{a b^{2} n + b^{3} n x^{n}} + \frac{b x^{n} \log{\left (\frac{a}{b} + x^{n} \right )}}{a b^{2} n + b^{3} n x^{n}} - \frac{b x^{n}}{a b^{2} n + b^{3} n x^{n}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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