3.2629 \(\int \frac{x^{-1-n}}{(a+b x^n)^2} \, dx\)

Optimal. Leaf size=57 \[ -\frac{b}{a^2 n \left (a+b x^n\right )}+\frac{2 b \log \left (a+b x^n\right )}{a^3 n}-\frac{2 b \log (x)}{a^3}-\frac{x^{-n}}{a^2 n} \]

[Out]

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

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Rubi [A]  time = 0.0330685, antiderivative size = 57, 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, 44} \[ -\frac{b}{a^2 n \left (a+b x^n\right )}+\frac{2 b \log \left (a+b x^n\right )}{a^3 n}-\frac{2 b \log (x)}{a^3}-\frac{x^{-n}}{a^2 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n)/(a + b*x^n)^2,x]

[Out]

-(1/(a^2*n*x^n)) - b/(a^2*n*(a + b*x^n)) - (2*b*Log[x])/a^3 + (2*b*Log[a + b*x^n])/(a^3*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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

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

Mathematica [A]  time = 0.104938, size = 45, normalized size = 0.79 \[ -\frac{a \left (\frac{b}{a+b x^n}+x^{-n}\right )-2 b \log \left (a+b x^n\right )+2 b n \log (x)}{a^3 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - n)/(a + b*x^n)^2,x]

[Out]

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

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Maple [A]  time = 0.018, size = 97, normalized size = 1.7 \begin{align*}{\frac{1}{{{\rm e}^{n\ln \left ( x \right ) }} \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) } \left ( 2\,{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}n}}-{\frac{1}{an}}-2\,{\frac{b\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}}{{a}^{2}}}-2\,{\frac{{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}}} \right ) }+2\,{\frac{b\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{3}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.99635, size = 84, normalized size = 1.47 \begin{align*} -\frac{2 \, b x^{n} + a}{a^{2} b n x^{2 \, n} + a^{3} n x^{n}} - \frac{2 \, b \log \left (x\right )}{a^{3}} + \frac{2 \, b \log \left (\frac{b x^{n} + a}{b}\right )}{a^{3} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.02976, size = 182, normalized size = 3.19 \begin{align*} -\frac{2 \, b^{2} n x^{2 \, n} \log \left (x\right ) + a^{2} + 2 \,{\left (a b n \log \left (x\right ) + a b\right )} x^{n} - 2 \,{\left (b^{2} x^{2 \, n} + a b x^{n}\right )} \log \left (b x^{n} + a\right )}{a^{3} b n x^{2 \, n} + a^{4} n x^{n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-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-n)/(a+b*x^n)^2,x, algorithm="giac")

[Out]

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