3.2638 \(\int \frac{x^{-1-2 n}}{(a+b x^n)^3} \, dx\)

Optimal. Leaf size=101 \[ \frac{3 b^2}{a^4 n \left (a+b x^n\right )}+\frac{b^2}{2 a^3 n \left (a+b x^n\right )^2}-\frac{6 b^2 \log \left (a+b x^n\right )}{a^5 n}+\frac{6 b^2 \log (x)}{a^5}+\frac{3 b x^{-n}}{a^4 n}-\frac{x^{-2 n}}{2 a^3 n} \]

[Out]

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

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Rubi [A]  time = 0.0586034, antiderivative size = 101, 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} \[ \frac{3 b^2}{a^4 n \left (a+b x^n\right )}+\frac{b^2}{2 a^3 n \left (a+b x^n\right )^2}-\frac{6 b^2 \log \left (a+b x^n\right )}{a^5 n}+\frac{6 b^2 \log (x)}{a^5}+\frac{3 b x^{-n}}{a^4 n}-\frac{x^{-2 n}}{2 a^3 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 2*n)/(a + b*x^n)^3,x]

[Out]

-1/(2*a^3*n*x^(2*n)) + (3*b)/(a^4*n*x^n) + b^2/(2*a^3*n*(a + b*x^n)^2) + (3*b^2)/(a^4*n*(a + b*x^n)) + (6*b^2*
Log[x])/a^5 - (6*b^2*Log[a + b*x^n])/(a^5*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 )^3} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{x^3 (a+b x)^3} \, dx,x,x^n\right )}{n}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^3}-\frac{3 b}{a^4 x^2}+\frac{6 b^2}{a^5 x}-\frac{b^3}{a^3 (a+b x)^3}-\frac{3 b^3}{a^4 (a+b x)^2}-\frac{6 b^3}{a^5 (a+b x)}\right ) \, dx,x,x^n\right )}{n}\\ &=-\frac{x^{-2 n}}{2 a^3 n}+\frac{3 b x^{-n}}{a^4 n}+\frac{b^2}{2 a^3 n \left (a+b x^n\right )^2}+\frac{3 b^2}{a^4 n \left (a+b x^n\right )}+\frac{6 b^2 \log (x)}{a^5}-\frac{6 b^2 \log \left (a+b x^n\right )}{a^5 n}\\ \end{align*}

Mathematica [A]  time = 0.190257, size = 79, normalized size = 0.78 \[ \frac{\frac{a x^{-2 n} \left (a+2 b x^n\right ) \left (-a^2+6 a b x^n+6 b^2 x^{2 n}\right )}{\left (a+b x^n\right )^2}-12 b^2 \log \left (a+b x^n\right )+12 b^2 n \log (x)}{2 a^5 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((a*(a + 2*b*x^n)*(-a^2 + 6*a*b*x^n + 6*b^2*x^(2*n)))/(x^(2*n)*(a + b*x^n)^2) + 12*b^2*n*Log[x] - 12*b^2*Log[a
+ b*x^n])/(2*a^5*n)

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Maple [A]  time = 0.033, size = 152, 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 ) ^{2}} \left ( 9\,{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}n}}-{\frac{1}{2\,an}}+6\,{\frac{{b}^{2}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}}}+2\,{\frac{b{{\rm e}^{n\ln \left ( x \right ) }}}{{a}^{2}n}}+12\,{\frac{{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}}}+6\,{\frac{{b}^{4}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{{a}^{5}}}+6\,{\frac{{b}^{3} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}n}} \right ) }-6\,{\frac{{b}^{2}\ln \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) }{{a}^{5}n}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(9*b^2/a^3/n*exp(n*ln(x))^2-1/2/a/n+6*b^2/a^3*ln(x)*exp(n*ln(x))^2+2*b/a^2/n*exp(n*ln(x))+12*b^3/a^4*ln(x)*exp
(n*ln(x))^3+6*b^4/a^5*ln(x)*exp(n*ln(x))^4+6*b^3/a^4/n*exp(n*ln(x))^3)/exp(n*ln(x))^2/(a+b*exp(n*ln(x)))^2-6*b
^2/a^5/n*ln(a+b*exp(n*ln(x)))

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Maxima [A]  time = 0.971325, size = 149, normalized size = 1.48 \begin{align*} \frac{12 \, b^{3} x^{3 \, n} + 18 \, a b^{2} x^{2 \, n} + 4 \, a^{2} b x^{n} - a^{3}}{2 \,{\left (a^{4} b^{2} n x^{4 \, n} + 2 \, a^{5} b n x^{3 \, n} + a^{6} n x^{2 \, n}\right )}} + \frac{6 \, b^{2} \log \left (x\right )}{a^{5}} - \frac{6 \, b^{2} \log \left (\frac{b x^{n} + a}{b}\right )}{a^{5} n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(12*b^3*x^(3*n) + 18*a*b^2*x^(2*n) + 4*a^2*b*x^n - a^3)/(a^4*b^2*n*x^(4*n) + 2*a^5*b*n*x^(3*n) + a^6*n*x^(
2*n)) + 6*b^2*log(x)/a^5 - 6*b^2*log((b*x^n + a)/b)/(a^5*n)

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Fricas [A]  time = 1.05156, size = 354, normalized size = 3.5 \begin{align*} \frac{12 \, b^{4} n x^{4 \, n} \log \left (x\right ) + 4 \, a^{3} b x^{n} - a^{4} + 12 \,{\left (2 \, a b^{3} n \log \left (x\right ) + a b^{3}\right )} x^{3 \, n} + 6 \,{\left (2 \, a^{2} b^{2} n \log \left (x\right ) + 3 \, a^{2} b^{2}\right )} x^{2 \, n} - 12 \,{\left (b^{4} x^{4 \, n} + 2 \, a b^{3} x^{3 \, n} + a^{2} b^{2} x^{2 \, n}\right )} \log \left (b x^{n} + a\right )}{2 \,{\left (a^{5} b^{2} n x^{4 \, n} + 2 \, a^{6} b n x^{3 \, n} + a^{7} 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)^3,x, algorithm="fricas")

[Out]

1/2*(12*b^4*n*x^(4*n)*log(x) + 4*a^3*b*x^n - a^4 + 12*(2*a*b^3*n*log(x) + a*b^3)*x^(3*n) + 6*(2*a^2*b^2*n*log(
x) + 3*a^2*b^2)*x^(2*n) - 12*(b^4*x^(4*n) + 2*a*b^3*x^(3*n) + a^2*b^2*x^(2*n))*log(b*x^n + a))/(a^5*b^2*n*x^(4
*n) + 2*a^6*b*n*x^(3*n) + a^7*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)**3,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 )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)