3.2637 \(\int \frac{x^{-1-n}}{(a+b x^n)^3} \, dx\)

Optimal. Leaf size=77 \[ -\frac{2 b}{a^3 n \left (a+b x^n\right )}-\frac{b}{2 a^2 n \left (a+b x^n\right )^2}+\frac{3 b \log \left (a+b x^n\right )}{a^4 n}-\frac{3 b \log (x)}{a^4}-\frac{x^{-n}}{a^3 n} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0473548, antiderivative size = 77, 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} \[ -\frac{2 b}{a^3 n \left (a+b x^n\right )}-\frac{b}{2 a^2 n \left (a+b x^n\right )^2}+\frac{3 b \log \left (a+b x^n\right )}{a^4 n}-\frac{3 b \log (x)}{a^4}-\frac{x^{-n}}{a^3 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - n)/(a + b*x^n)^3,x]

[Out]

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

Mathematica [A]  time = 0.196964, size = 58, normalized size = 0.75 \[ -\frac{\frac{a b \left (5 a+4 b x^n\right )}{\left (a+b x^n\right )^2}-6 b \log \left (a+b x^n\right )+2 a x^{-n}+6 b n \log (x)}{2 a^4 n} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(-1 - n)/(a + b*x^n)^3,x]

[Out]

-((2*a)/x^n + (a*b*(5*a + 4*b*x^n))/(a + b*x^n)^2 + 6*b*n*Log[x] - 6*b*Log[a + b*x^n])/(2*a^4*n)

________________________________________________________________________________________

Maple [A]  time = 0.03, size = 132, 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 ) ^{2}} \left ( -{\frac{1}{an}}-3\,{\frac{b\ln \left ( x \right ){{\rm e}^{n\ln \left ( x \right ) }}}{{a}^{2}}}-6\,{\frac{{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}}}+6\,{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}n}}-3\,{\frac{{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}}}+{\frac{9\,{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{2\,{a}^{4}n}} \right ) }+3\,{\frac{b\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-n)/(a+b*x^n)^3,x)

[Out]

(-1/a/n-3*b/a^2*ln(x)*exp(n*ln(x))-6*b^2/a^3*ln(x)*exp(n*ln(x))^2+6*b^2/a^3/n*exp(n*ln(x))^2-3*b^3/a^4*ln(x)*e
xp(n*ln(x))^3+9/2*b^3/a^4/n*exp(n*ln(x))^3)/exp(n*ln(x))/(a+b*exp(n*ln(x)))^2+3*b/a^4/n*ln(a+b*exp(n*ln(x)))

________________________________________________________________________________________

Maxima [A]  time = 0.97778, size = 123, normalized size = 1.6 \begin{align*} -\frac{6 \, b^{2} x^{2 \, n} + 9 \, a b x^{n} + 2 \, a^{2}}{2 \,{\left (a^{3} b^{2} n x^{3 \, n} + 2 \, a^{4} b n x^{2 \, n} + a^{5} n x^{n}\right )}} - \frac{3 \, b \log \left (x\right )}{a^{4}} + \frac{3 \, b \log \left (\frac{b x^{n} + a}{b}\right )}{a^{4} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(6*b^2*x^(2*n) + 9*a*b*x^n + 2*a^2)/(a^3*b^2*n*x^(3*n) + 2*a^4*b*n*x^(2*n) + a^5*n*x^n) - 3*b*log(x)/a^4
+ 3*b*log((b*x^n + a)/b)/(a^4*n)

________________________________________________________________________________________

Fricas [A]  time = 1.01348, size = 311, normalized size = 4.04 \begin{align*} -\frac{6 \, b^{3} n x^{3 \, n} \log \left (x\right ) + 2 \, a^{3} + 6 \,{\left (2 \, a b^{2} n \log \left (x\right ) + a b^{2}\right )} x^{2 \, n} + 3 \,{\left (2 \, a^{2} b n \log \left (x\right ) + 3 \, a^{2} b\right )} x^{n} - 6 \,{\left (b^{3} x^{3 \, n} + 2 \, a b^{2} x^{2 \, n} + a^{2} b x^{n}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (a^{4} b^{2} n x^{3 \, n} + 2 \, a^{5} b n x^{2 \, n} + a^{6} n x^{n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2*(6*b^3*n*x^(3*n)*log(x) + 2*a^3 + 6*(2*a*b^2*n*log(x) + a*b^2)*x^(2*n) + 3*(2*a^2*b*n*log(x) + 3*a^2*b)*x
^n - 6*(b^3*x^(3*n) + 2*a*b^2*x^(2*n) + a^2*b*x^n)*log(b*x^n + a))/(a^4*b^2*n*x^(3*n) + 2*a^5*b*n*x^(2*n) + a^
6*n*x^n)

________________________________________________________________________________________

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)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{-n - 1}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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