3.2630 \(\int \frac{x^{-1-2 n}}{(a+b x^n)^2} \, dx\)

Optimal. Leaf size=78 \[ \frac{b^2}{a^3 n \left (a+b x^n\right )}-\frac{3 b^2 \log \left (a+b x^n\right )}{a^4 n}+\frac{3 b^2 \log (x)}{a^4}+\frac{2 b x^{-n}}{a^3 n}-\frac{x^{-2 n}}{2 a^2 n} \]

[Out]

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

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Rubi [A]  time = 0.0443638, antiderivative size = 78, 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} \[ \frac{b^2}{a^3 n \left (a+b x^n\right )}-\frac{3 b^2 \log \left (a+b x^n\right )}{a^4 n}+\frac{3 b^2 \log (x)}{a^4}+\frac{2 b x^{-n}}{a^3 n}-\frac{x^{-2 n}}{2 a^2 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 2*n)/(a + b*x^n)^2,x]

[Out]

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

Mathematica [A]  time = 0.136932, size = 65, normalized size = 0.83 \[ \frac{a \left (\frac{2 b^2}{a+b x^n}-a x^{-2 n}+4 b x^{-n}\right )-6 b^2 \log \left (a+b x^n\right )+6 b^2 n \log (x)}{2 a^4 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(-(a/x^(2*n)) + (4*b)/x^n + (2*b^2)/(a + b*x^n)) + 6*b^2*n*Log[x] - 6*b^2*Log[a + b*x^n])/(2*a^4*n)

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Maple [A]  time = 0.021, size = 117, normalized size = 1.5 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2} \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) } \left ( -3\,{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}n}}-{\frac{1}{2\,an}}+{\frac{3\,b{{\rm e}^{n\ln \left ( x \right ) }}}{2\,{a}^{2}n}}+3\,{\frac{{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}}}+3\,{\frac{{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}}} \right ) }-3\,{\frac{{b}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{4}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]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \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]

Exception raised: RuntimeError

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Fricas [A]  time = 1.02831, size = 230, normalized size = 2.95 \begin{align*} \frac{6 \, b^{3} n x^{3 \, n} \log \left (x\right ) + 3 \, a^{2} b x^{n} - a^{3} + 6 \,{\left (a b^{2} n \log \left (x\right ) + a b^{2}\right )} x^{2 \, n} - 6 \,{\left (b^{3} x^{3 \, n} + a b^{2} x^{2 \, n}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (a^{4} b n x^{3 \, n} + a^{5} n x^{2 \, n}\right )}} \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]

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

[Out]

Timed out

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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)