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

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.05, antiderivative size = 77, normalized size of antiderivative = 1.00, 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 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])

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]]

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 = 0.22, 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]

-1/2*((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])/(a^4*n)

________________________________________________________________________________________

fricas [A]  time = 1.17, size = 139, normalized size = 1.81 \[ -\frac {6 \, b^{3} n x^{3 \, n} \log \relax (x) + 2 \, a^{3} + 6 \, {\left (2 \, a b^{2} n \log \relax (x) + a b^{2}\right )} x^{2 \, n} + 3 \, {\left (2 \, a^{2} b n \log \relax (x) + 3 \, a^{2} b\right )} x^{n} - 6 \, {\left (b^{3} x^{3 \, n} + 2 \, a b^{2} x^{2 \, n} + a^{2} b x^{n}\right )} \log \left (b x^{n} + a\right )}{2 \, {\left (a^{4} b^{2} n x^{3 \, n} + 2 \, a^{5} b n x^{2 \, n} + a^{6} n x^{n}\right )}} \]

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)

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{-n - 1}}{{\left (b x^{n} + a\right )}^{3}}\,{d x} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.04, size = 132, normalized size = 1.71 \[ \frac {\left (-\frac {3 b \,{\mathrm e}^{n \ln \relax (x )} \ln \relax (x )}{a^{2}}-\frac {6 b^{2} {\mathrm e}^{2 n \ln \relax (x )} \ln \relax (x )}{a^{3}}-\frac {3 b^{3} {\mathrm e}^{3 n \ln \relax (x )} \ln \relax (x )}{a^{4}}+\frac {6 b^{2} {\mathrm e}^{2 n \ln \relax (x )}}{a^{3} n}+\frac {9 b^{3} {\mathrm e}^{3 n \ln \relax (x )}}{2 a^{4} n}-\frac {1}{a n}\right ) {\mathrm e}^{-n \ln \relax (x )}}{\left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )^{2}}+\frac {3 b \ln \left (b \,{\mathrm e}^{n \ln \relax (x )}+a \right )}{a^{4} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.56, size = 91, normalized size = 1.18 \[ -\frac {6 \, b^{2} x^{2 \, n} + 9 \, a b x^{n} + 2 \, a^{2}}{2 \, {\left (a^{3} b^{2} n x^{3 \, n} + 2 \, a^{4} b n x^{2 \, n} + a^{5} n x^{n}\right )}} - \frac {3 \, b \log \relax (x)}{a^{4}} + \frac {3 \, b \log \left (\frac {b x^{n} + a}{b}\right )}{a^{4} n} \]

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{x^{n+1}\,{\left (a+b\,x^n\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(n + 1)*(a + b*x^n)^3),x)

[Out]

int(1/(x^(n + 1)*(a + b*x^n)^3), x)

________________________________________________________________________________________

sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: HeuristicGCDFailed

________________________________________________________________________________________