\(\int \frac {x^{-1-3 n}}{(a+b x^n)^2} \, dx\) [2631]

Optimal result

Integrand size = 17, antiderivative size = 94 \[ \int \frac {x^{-1-3 n}}{\left (a+b x^n\right )^2} \, dx=-\frac {x^{-3 n}}{3 a^2 n}+\frac {b x^{-2 n}}{a^3 n}-\frac {3 b^2 x^{-n}}{a^4 n}-\frac {b^3}{a^4 n \left (a+b x^n\right )}-\frac {4 b^3 \log (x)}{a^5}+\frac {4 b^3 \log \left (a+b x^n\right )}{a^5 n} \]

[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*ln(x)/a^5+4*b^3*ln(a+b*x^n)
/a^5/n

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 94, 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} \[ \int \frac {x^{-1-3 n}}{\left (a+b x^n\right )^2} \, dx=\frac {4 b^3 \log \left (a+b x^n\right )}{a^5 n}-\frac {4 b^3 \log (x)}{a^5}-\frac {b^3}{a^4 n \left (a+b x^n\right )}-\frac {3 b^2 x^{-n}}{a^4 n}+\frac {b x^{-2 n}}{a^3 n}-\frac {x^{-3 n}}{3 a^2 n} \]

[In]

Int[x^(-1 - 3*n)/(a + b*x^n)^2,x]

[Out]

-1/3*1/(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 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^4 (a+b x)^2} \, dx,x,x^n\right )}{n} \\ & = \frac {\text {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] (verified)

Time = 0.24 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.88 \[ \int \frac {x^{-1-3 n}}{\left (a+b x^n\right )^2} \, dx=-\frac {\frac {a x^{-3 n} \left (a^3-2 a^2 b x^n+6 a b^2 x^{2 n}+12 b^3 x^{3 n}\right )}{a+b x^n}+12 b^3 \log \left (x^n\right )-12 b^3 \log \left (a+b x^n\right )}{3 a^5 n} \]

[In]

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

[Out]

-1/3*((a*(a^3 - 2*a^2*b*x^n + 6*a*b^2*x^(2*n) + 12*b^3*x^(3*n)))/(x^(3*n)*(a + b*x^n)) + 12*b^3*Log[x^n] - 12*
b^3*Log[a + b*x^n])/(a^5*n)

Maple [A] (verified)

Time = 4.04 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.01

method	result	size
risch	\(-\frac {3 b^{2} x^{-n}}{a^{4} n}+\frac {b \,x^{-2 n}}{a^{3} n}-\frac {x^{-3 n}}{3 a^{2} n}-\frac {4 b^{3} \ln \left (x \right )}{a^{5}}-\frac {b^{3}}{a^{4} n \left (a +b \,x^{n}\right )}+\frac {4 b^{3} \ln \left (x^{n}+\frac {a}{b}\right )}{a^{5} n}\)	\(95\)
norman	\(\frac {\left (\frac {4 b^{4} {\mathrm e}^{4 n \ln \left (x \right )}}{a^{5} n}-\frac {1}{3 a n}+\frac {2 b \,{\mathrm e}^{n \ln \left (x \right )}}{3 a^{2} n}-\frac {2 b^{2} {\mathrm e}^{2 n \ln \left (x \right )}}{a^{3} n}-\frac {4 b^{3} \ln \left (x \right ) {\mathrm e}^{3 n \ln \left (x \right )}}{a^{4}}-\frac {4 b^{4} \ln \left (x \right ) {\mathrm e}^{4 n \ln \left (x \right )}}{a^{5}}\right ) {\mathrm e}^{-3 n \ln \left (x \right )}}{a +b \,{\mathrm e}^{n \ln \left (x \right )}}+\frac {4 b^{3} \ln \left (a +b \,{\mathrm e}^{n \ln \left (x \right )}\right )}{a^{5} n}\)	\(135\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.23 \[ \int \frac {x^{-1-3 n}}{\left (a+b x^n\right )^2} \, dx=-\frac {12 \, b^{4} n x^{4 \, n} \log \left (x\right ) + 6 \, a^{2} b^{2} x^{2 \, n} - 2 \, a^{3} b x^{n} + a^{4} + 12 \, {\left (a b^{3} n \log \left (x\right ) + a b^{3}\right )} x^{3 \, n} - 12 \, {\left (b^{4} x^{4 \, n} + a b^{3} x^{3 \, n}\right )} \log \left (b x^{n} + a\right )}{3 \, {\left (a^{5} b n x^{4 \, n} + a^{6} n x^{3 \, n}\right )}} \]

[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))

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 384 vs. \(2 (83) = 166\).

Time = 35.93 (sec) , antiderivative size = 384, normalized size of antiderivative = 4.09 \[ \int \frac {x^{-1-3 n}}{\left (a+b x^n\right )^2} \, dx=\begin {cases} \tilde {\infty } \log {\left (x \right )} & \text {for}\: a = 0 \wedge b = 0 \wedge n = 0 \\- \frac {x x^{- 3 n - 1}}{3 a^{2} n} & \text {for}\: b = 0 \\- \frac {x x^{- 2 n} x^{- 3 n - 1}}{5 b^{2} n} & \text {for}\: a = 0 \\\frac {\tilde {\infty } x x^{- 3 n - 1}}{n} & \text {for}\: b = - a x^{- n} \\\frac {\log {\left (x \right )}}{\left (a + b\right )^{2}} & \text {for}\: n = 0 \\- \frac {a^{4}}{3 a^{6} n x^{3 n} + 3 a^{5} b n x^{4 n}} + \frac {2 a^{3} b x^{n}}{3 a^{6} n x^{3 n} + 3 a^{5} b n x^{4 n}} - \frac {6 a^{2} b^{2} x^{2 n}}{3 a^{6} n x^{3 n} + 3 a^{5} b n x^{4 n}} - \frac {12 a b^{3} n x^{3 n} \log {\left (x \right )}}{3 a^{6} n x^{3 n} + 3 a^{5} b n x^{4 n}} + \frac {12 a b^{3} x^{3 n} \log {\left (\frac {a}{b} + x^{n} \right )}}{3 a^{6} n x^{3 n} + 3 a^{5} b n x^{4 n}} - \frac {12 a b^{3} x^{3 n}}{3 a^{6} n x^{3 n} + 3 a^{5} b n x^{4 n}} - \frac {12 b^{4} n x^{4 n} \log {\left (x \right )}}{3 a^{6} n x^{3 n} + 3 a^{5} b n x^{4 n}} + \frac {12 b^{4} x^{4 n} \log {\left (\frac {a}{b} + x^{n} \right )}}{3 a^{6} n x^{3 n} + 3 a^{5} b n x^{4 n}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 94, normalized size of antiderivative = 1.00 \[ \int \frac {x^{-1-3 n}}{\left (a+b x^n\right )^2} \, dx=-\frac {12 \, b^{3} x^{3 \, n} + 6 \, a b^{2} x^{2 \, n} - 2 \, a^{2} b x^{n} + a^{3}}{3 \, {\left (a^{4} b n x^{4 \, n} + a^{5} n x^{3 \, n}\right )}} - \frac {4 \, b^{3} \log \left (x\right )}{a^{5}} + \frac {4 \, b^{3} \log \left (\frac {b x^{n} + a}{b}\right )}{a^{5} n} \]

[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)

Giac [F]

\[ \int \frac {x^{-1-3 n}}{\left (a+b x^n\right )^2} \, dx=\int { \frac {x^{-3 \, n - 1}}{{\left (b x^{n} + a\right )}^{2}} \,d x } \]

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{-1-3 n}}{\left (a+b x^n\right )^2} \, dx=\int \frac {1}{x^{3\,n+1}\,{\left (a+b\,x^n\right )}^2} \,d x \]

[In]

int(1/(x^(3*n + 1)*(a + b*x^n)^2),x)

[Out]

int(1/(x^(3*n + 1)*(a + b*x^n)^2), x)