3.2631 \(\int \frac{x^{-1-3 n}}{(a+b x^n)^2} \, dx\)

Optimal. Leaf size=94 \[ -\frac{3 b^2 x^{-n}}{a^4 n}-\frac{b^3}{a^4 n \left (a+b x^n\right )}+\frac{4 b^3 \log \left (a+b x^n\right )}{a^5 n}-\frac{4 b^3 \log (x)}{a^5}+\frac{b x^{-2 n}}{a^3 n}-\frac{x^{-3 n}}{3 a^2 n} \]

[Out]

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

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Rubi [A]  time = 0.0560242, antiderivative size = 94, 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} \[ -\frac{3 b^2 x^{-n}}{a^4 n}-\frac{b^3}{a^4 n \left (a+b x^n\right )}+\frac{4 b^3 \log \left (a+b x^n\right )}{a^5 n}-\frac{4 b^3 \log (x)}{a^5}+\frac{b x^{-2 n}}{a^3 n}-\frac{x^{-3 n}}{3 a^2 n} \]

Antiderivative was successfully verified.

[In]

Int[x^(-1 - 3*n)/(a + b*x^n)^2,x]

[Out]

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

Mathematica [A]  time = 0.146716, size = 78, normalized size = 0.83 \[ \frac{a \left (-a^2 x^{-3 n}-\frac{3 b^3}{a+b x^n}+3 a b x^{-2 n}-9 b^2 x^{-n}\right )+12 b^3 \log \left (a+b x^n\right )-12 b^3 n \log (x)}{3 a^5 n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(-(a^2/x^(3*n)) + (3*a*b)/x^(2*n) - (9*b^2)/x^n - (3*b^3)/(a + b*x^n)) - 12*b^3*n*Log[x] + 12*b^3*Log[a + b
*x^n])/(3*a^5*n)

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Maple [A]  time = 0.026, size = 135, normalized size = 1.4 \begin{align*}{\frac{1}{ \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3} \left ( a+b{{\rm e}^{n\ln \left ( x \right ) }} \right ) } \left ( 4\,{\frac{{b}^{4} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{{a}^{5}n}}-{\frac{1}{3\,an}}+{\frac{2\,b{{\rm e}^{n\ln \left ( x \right ) }}}{3\,{a}^{2}n}}-2\,{\frac{{b}^{2} \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{2}}{{a}^{3}n}}-4\,{\frac{{b}^{3}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{3}}{{a}^{4}}}-4\,{\frac{{b}^{4}\ln \left ( x \right ) \left ({{\rm e}^{n\ln \left ( x \right ) }} \right ) ^{4}}{{a}^{5}}} \right ) }+4\,{\frac{{b}^{3}\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^(-1-3*n)/(a+b*x^n)^2,x)

[Out]

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

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Maxima [A]  time = 0.969758, size = 127, normalized size = 1.35 \begin{align*} -\frac{12 \, b^{3} x^{3 \, n} + 6 \, a b^{2} x^{2 \, n} - 2 \, a^{2} b x^{n} + a^{3}}{3 \,{\left (a^{4} b n x^{4 \, n} + a^{5} n x^{3 \, n}\right )}} - \frac{4 \, b^{3} \log \left (x\right )}{a^{5}} + \frac{4 \, b^{3} \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^(-1-3*n)/(a+b*x^n)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.07984, size = 262, normalized size = 2.79 \begin{align*} -\frac{12 \, b^{4} n x^{4 \, n} \log \left (x\right ) + 6 \, a^{2} b^{2} x^{2 \, n} - 2 \, a^{3} b x^{n} + a^{4} + 12 \,{\left (a b^{3} n \log \left (x\right ) + a b^{3}\right )} x^{3 \, n} - 12 \,{\left (b^{4} x^{4 \, n} + a b^{3} x^{3 \, n}\right )} \log \left (b x^{n} + a\right )}{3 \,{\left (a^{5} b n x^{4 \, n} + a^{6} n x^{3 \, n}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(12*b^4*n*x^(4*n)*log(x) + 6*a^2*b^2*x^(2*n) - 2*a^3*b*x^n + a^4 + 12*(a*b^3*n*log(x) + a*b^3)*x^(3*n) -
12*(b^4*x^(4*n) + a*b^3*x^(3*n))*log(b*x^n + a))/(a^5*b*n*x^(4*n) + a^6*n*x^(3*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**(-1-3*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^{-3 \, 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^(-1-3*n)/(a+b*x^n)^2,x, algorithm="giac")

[Out]

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