\(\int \frac {x^{-1-n}}{(a+b x^n)^3} \, dx\) [2637]

Optimal result

Integrand size = 17, antiderivative size = 77 \[ \int \frac {x^{-1-n}}{\left (a+b x^n\right )^3} \, dx=-\frac {x^{-n}}{a^3 n}-\frac {b}{2 a^2 n \left (a+b x^n\right )^2}-\frac {2 b}{a^3 n \left (a+b x^n\right )}-\frac {3 b \log (x)}{a^4}+\frac {3 b \log \left (a+b x^n\right )}{a^4 n} \]

[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] (verified)

Time = 0.03 (sec) , antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {272, 46} \[ \int \frac {x^{-1-n}}{\left (a+b x^n\right )^3} \, dx=\frac {3 b \log \left (a+b x^n\right )}{a^4 n}-\frac {3 b \log (x)}{a^4}-\frac {2 b}{a^3 n \left (a+b x^n\right )}-\frac {x^{-n}}{a^3 n}-\frac {b}{2 a^2 n \left (a+b x^n\right )^2} \]

[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 46

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] && Lt
Q[m + n + 2, 0])

Rule 272

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*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^2 (a+b x)^3} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {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] (verified)

Time = 0.21 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.88 \[ \int \frac {x^{-1-n}}{\left (a+b x^n\right )^3} \, dx=-\frac {\frac {a x^{-n} \left (2 a^2+9 a b x^n+6 b^2 x^{2 n}\right )}{\left (a+b x^n\right )^2}+6 b \log \left (x^n\right )-6 b \log \left (a+b x^n\right )}{2 a^4 n} \]

[In]

Integrate[x^(-1 - n)/(a + b*x^n)^3,x]

[Out]

-1/2*((a*(2*a^2 + 9*a*b*x^n + 6*b^2*x^(2*n)))/(x^n*(a + b*x^n)^2) + 6*b*Log[x^n] - 6*b*Log[a + b*x^n])/(a^4*n)

Maple [A] (verified)

Time = 3.94 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.91

method	result	size
risch	\(-\frac {x^{-n}}{a^{3} n}-\frac {3 b \ln \left (x \right )}{a^{4}}-\frac {b \left (4 b \,x^{n}+5 a \right )}{2 a^{3} n \left (a +b \,x^{n}\right )^{2}}+\frac {3 b \ln \left (x^{n}+\frac {a}{b}\right )}{a^{4} n}\)	\(70\)
norman	\(\frac {\left (-\frac {1}{a n}-\frac {3 b \ln \left (x \right ) {\mathrm e}^{n \ln \left (x \right )}}{a^{2}}-\frac {6 b^{2} \ln \left (x \right ) {\mathrm e}^{2 n \ln \left (x \right )}}{a^{3}}+\frac {6 b^{2} {\mathrm e}^{2 n \ln \left (x \right )}}{a^{3} n}-\frac {3 b^{3} \ln \left (x \right ) {\mathrm e}^{3 n \ln \left (x \right )}}{a^{4}}+\frac {9 b^{3} {\mathrm e}^{3 n \ln \left (x \right )}}{2 a^{4} n}\right ) {\mathrm e}^{-n \ln \left (x \right )}}{\left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )^{2}}+\frac {3 b \ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}{a^{4} n}\)	\(132\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.81 \[ \int \frac {x^{-1-n}}{\left (a+b x^n\right )^3} \, dx=-\frac {6 \, b^{3} n x^{3 \, n} \log \left (x\right ) + 2 \, a^{3} + 6 \, {\left (2 \, a b^{2} n \log \left (x\right ) + a b^{2}\right )} x^{2 \, n} + 3 \, {\left (2 \, a^{2} b n \log \left (x\right ) + 3 \, a^{2} b\right )} x^{n} - 6 \, {\left (b^{3} x^{3 \, n} + 2 \, a b^{2} x^{2 \, n} + a^{2} b x^{n}\right )} \log \left (b x^{n} + a\right )}{2 \, {\left (a^{4} b^{2} n x^{3 \, n} + 2 \, a^{5} b n x^{2 \, n} + a^{6} n x^{n}\right )}} \]

[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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (68) = 136\).

Time = 21.30 (sec) , antiderivative size = 547, normalized size of antiderivative = 7.10 \[ \int \frac {x^{-1-n}}{\left (a+b x^n\right )^3} \, dx=\begin {cases} \tilde {\infty } \log {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac {x x^{- n - 1}}{a^{3} n} & \text {for}\: b = 0 \\- \frac {x x^{- 3 n} x^{- n - 1}}{4 b^{3} n} & \text {for}\: a = 0 \\\frac {\tilde {\infty } x x^{- n - 1}}{n} & \text {for}\: b = - a x^{- n} \\\frac {\log {\left (x \right )}}{\left (a + b\right )^{3}} & \text {for}\: n = 0 \\- \frac {2 a^{3}}{2 a^{6} n x^{n} + 4 a^{5} b n x^{2 n} + 2 a^{4} b^{2} n x^{3 n}} - \frac {6 a^{2} b n x^{n} \log {\left (x \right )}}{2 a^{6} n x^{n} + 4 a^{5} b n x^{2 n} + 2 a^{4} b^{2} n x^{3 n}} + \frac {6 a^{2} b x^{n} \log {\left (\frac {a}{b} + x^{n} \right )}}{2 a^{6} n x^{n} + 4 a^{5} b n x^{2 n} + 2 a^{4} b^{2} n x^{3 n}} - \frac {9 a^{2} b x^{n}}{2 a^{6} n x^{n} + 4 a^{5} b n x^{2 n} + 2 a^{4} b^{2} n x^{3 n}} - \frac {12 a b^{2} n x^{2 n} \log {\left (x \right )}}{2 a^{6} n x^{n} + 4 a^{5} b n x^{2 n} + 2 a^{4} b^{2} n x^{3 n}} + \frac {12 a b^{2} x^{2 n} \log {\left (\frac {a}{b} + x^{n} \right )}}{2 a^{6} n x^{n} + 4 a^{5} b n x^{2 n} + 2 a^{4} b^{2} n x^{3 n}} - \frac {6 a b^{2} x^{2 n}}{2 a^{6} n x^{n} + 4 a^{5} b n x^{2 n} + 2 a^{4} b^{2} n x^{3 n}} - \frac {6 b^{3} n x^{3 n} \log {\left (x \right )}}{2 a^{6} n x^{n} + 4 a^{5} b n x^{2 n} + 2 a^{4} b^{2} n x^{3 n}} + \frac {6 b^{3} x^{3 n} \log {\left (\frac {a}{b} + x^{n} \right )}}{2 a^{6} n x^{n} + 4 a^{5} b n x^{2 n} + 2 a^{4} b^{2} n x^{3 n}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo*log(x), Eq(a, 0) & Eq(b, 0) & Eq(n, 0)), (-x*x**(-n - 1)/(a**3*n), Eq(b, 0)), (-x*x**(-n - 1)/(
4*b**3*n*x**(3*n)), Eq(a, 0)), (zoo*x*x**(-n - 1)/n, Eq(b, -a/x**n)), (log(x)/(a + b)**3, Eq(n, 0)), (-2*a**3/
(2*a**6*n*x**n + 4*a**5*b*n*x**(2*n) + 2*a**4*b**2*n*x**(3*n)) - 6*a**2*b*n*x**n*log(x)/(2*a**6*n*x**n + 4*a**
5*b*n*x**(2*n) + 2*a**4*b**2*n*x**(3*n)) + 6*a**2*b*x**n*log(a/b + x**n)/(2*a**6*n*x**n + 4*a**5*b*n*x**(2*n)
+ 2*a**4*b**2*n*x**(3*n)) - 9*a**2*b*x**n/(2*a**6*n*x**n + 4*a**5*b*n*x**(2*n) + 2*a**4*b**2*n*x**(3*n)) - 12*
a*b**2*n*x**(2*n)*log(x)/(2*a**6*n*x**n + 4*a**5*b*n*x**(2*n) + 2*a**4*b**2*n*x**(3*n)) + 12*a*b**2*x**(2*n)*l
og(a/b + x**n)/(2*a**6*n*x**n + 4*a**5*b*n*x**(2*n) + 2*a**4*b**2*n*x**(3*n)) - 6*a*b**2*x**(2*n)/(2*a**6*n*x*
*n + 4*a**5*b*n*x**(2*n) + 2*a**4*b**2*n*x**(3*n)) - 6*b**3*n*x**(3*n)*log(x)/(2*a**6*n*x**n + 4*a**5*b*n*x**(
2*n) + 2*a**4*b**2*n*x**(3*n)) + 6*b**3*x**(3*n)*log(a/b + x**n)/(2*a**6*n*x**n + 4*a**5*b*n*x**(2*n) + 2*a**4
*b**2*n*x**(3*n)), True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.18 \[ \int \frac {x^{-1-n}}{\left (a+b x^n\right )^3} \, dx=-\frac {6 \, b^{2} x^{2 \, n} + 9 \, a b x^{n} + 2 \, a^{2}}{2 \, {\left (a^{3} b^{2} n x^{3 \, n} + 2 \, a^{4} b n x^{2 \, n} + a^{5} n x^{n}\right )}} - \frac {3 \, b \log \left (x\right )}{a^{4}} + \frac {3 \, b \log \left (\frac {b x^{n} + a}{b}\right )}{a^{4} n} \]

[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)

Giac [F]

\[ \int \frac {x^{-1-n}}{\left (a+b x^n\right )^3} \, dx=\int { \frac {x^{-n - 1}}{{\left (b x^{n} + a\right )}^{3}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1-n}}{\left (a+b x^n\right )^3} \, dx=\int \frac {1}{x^{n+1}\,{\left (a+b\,x^n\right )}^3} \,d x \]

[In]

int(1/(x^(n + 1)*(a + b*x^n)^3),x)

[Out]

int(1/(x^(n + 1)*(a + b*x^n)^3), x)