Integrand size = 17, antiderivative size = 68 \[ \int \frac {x^{-1-3 n}}{2+b x^n} \, dx=-\frac {x^{-3 n}}{6 n}+\frac {b x^{-2 n}}{8 n}-\frac {b^2 x^{-n}}{8 n}-\frac {1}{16} b^3 \log (x)+\frac {b^3 \log \left (2+b x^n\right )}{16 n} \]
Time = 0.10 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.82 \[ \int \frac {x^{-1-3 n}}{2+b x^n} \, dx=-\frac {x^{-3 n} \left (8-6 b x^n+6 b^2 x^{2 n}\right )+3 b^3 \log \left (x^n\right )-3 b^3 \log \left (n \left (2+b x^n\right )\right )}{48 n} \]
Time = 0.20 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.91, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {798, 54, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x^{-3 n-1}}{b x^n+2} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int \frac {x^{-4 n}}{b x^n+2}dx^n}{n}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle \frac {\int \left (\frac {x^{-4 n}}{2}-\frac {1}{4} b x^{-3 n}+\frac {1}{8} b^2 x^{-2 n}-\frac {1}{16} b^3 x^{-n}+\frac {b^4}{16 \left (b x^n+2\right )}\right )dx^n}{n}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {1}{16} b^3 \log \left (x^n\right )+\frac {1}{16} b^3 \log \left (b x^n+2\right )-\frac {1}{8} b^2 x^{-n}+\frac {1}{8} b x^{-2 n}-\frac {1}{6} x^{-3 n}}{n}\) |
(-1/6*1/x^(3*n) + b/(8*x^(2*n)) - b^2/(8*x^n) - (b^3*Log[x^n])/16 + (b^3*L og[2 + b*x^n])/16)/n
3.27.23.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Time = 3.83 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90
method | result | size |
risch | \(-\frac {b^{2} x^{-n}}{8 n}+\frac {b \,x^{-2 n}}{8 n}-\frac {x^{-3 n}}{6 n}-\frac {b^{3} \ln \left (x \right )}{16}+\frac {b^{3} \ln \left (x^{n}+\frac {2}{b}\right )}{16 n}\) | \(61\) |
norman | \(\left (-\frac {b^{3} \ln \left (x \right ) {\mathrm e}^{3 n \ln \left (x \right )}}{16}-\frac {1}{6 n}+\frac {b \,{\mathrm e}^{n \ln \left (x \right )}}{8 n}-\frac {b^{2} {\mathrm e}^{2 n \ln \left (x \right )}}{8 n}\right ) {\mathrm e}^{-3 n \ln \left (x \right )}+\frac {b^{3} \ln \left (2+b \,{\mathrm e}^{n \ln \left (x \right )}\right )}{16 n}\) | \(74\) |
meijerg | \(\frac {i b^{3} \left (-1\right )^{\frac {3 \,\operatorname {csgn}\left (i b \right )}{2}+\frac {3 \,\operatorname {csgn}\left (i x^{n}\right )}{2}-\frac {3 \,\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i b \right )}{2}} \left (-\frac {\left (-1\right )^{\frac {\left (\frac {-1-3 n}{n}+\frac {1}{n}\right ) \operatorname {csgn}\left (i b \right )}{2}+\frac {\left (\frac {-1-3 n}{n}+\frac {1}{n}\right ) \operatorname {csgn}\left (i x^{n}\right )}{2}-\frac {\left (\frac {-1-3 n}{n}+\frac {1}{n}\right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i b \right )}{2}-\frac {-1-3 n}{2 n}-\frac {1}{2 n}} x^{\left (\frac {-1-3 n}{n}+\frac {1}{n}\right ) n} 2^{-\frac {-1-3 n}{n}-\frac {1}{n}} b^{\frac {-1-3 n}{n}+\frac {1}{n}}}{3}-\frac {\left (-1\right )^{\frac {\left (1+\frac {-1-3 n}{n}+\frac {1}{n}\right ) \operatorname {csgn}\left (i b \right )}{2}+\frac {\left (1+\frac {-1-3 n}{n}+\frac {1}{n}\right ) \operatorname {csgn}\left (i x^{n}\right )}{2}-\frac {\left (1+\frac {-1-3 n}{n}+\frac {1}{n}\right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i b \right )}{2}-\frac {-1-3 n}{2 n}-\frac {1}{2 n}+\frac {1}{2}} x^{\left (1+\frac {-1-3 n}{n}+\frac {1}{n}\right ) n} 2^{-1-\frac {-1-3 n}{n}-\frac {1}{n}} b^{1+\frac {-1-3 n}{n}+\frac {1}{n}}}{2}+x^{\left (\frac {-1-3 n}{n}+\frac {1}{n}+2\right ) n} 2^{-\frac {-1-3 n}{n}-\frac {1}{n}-2} b^{\frac {-1-3 n}{n}+\frac {1}{n}+2} \left (-1\right )^{\frac {\left (\frac {-1-3 n}{n}+\frac {1}{n}+2\right ) \operatorname {csgn}\left (i b \right )}{2}+\frac {\left (\frac {-1-3 n}{n}+\frac {1}{n}+2\right ) \operatorname {csgn}\left (i x^{n}\right )}{2}-\frac {\left (\frac {-1-3 n}{n}+\frac {1}{n}+2\right ) \operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i b \right )}{2}-\frac {-1-3 n}{2 n}-\frac {1}{2 n}}+\left (-1\right )^{-\frac {-1-3 n}{n}-\frac {1}{n}} \left (n \ln \left (x \right )-\ln \left (2\right )+\ln \left (b \right )+i \left (\frac {\operatorname {csgn}\left (i b \right )}{2}+\frac {\operatorname {csgn}\left (i x^{n}\right )}{2}-\frac {\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i b \right )}{2}-\frac {1}{2}\right ) \pi \right )-\left (-1\right )^{-\frac {-1-3 n}{n}-\frac {1}{n}} \ln \left (1-\frac {i x^{n} b \left (-1\right )^{\frac {\operatorname {csgn}\left (i b \right )}{2}+\frac {\operatorname {csgn}\left (i x^{n}\right )}{2}-\frac {\operatorname {csgn}\left (i x^{n}\right ) \operatorname {csgn}\left (i b \right )}{2}}}{2}\right )\right )}{16 n}\) | \(602\) |
Time = 0.30 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.90 \[ \int \frac {x^{-1-3 n}}{2+b x^n} \, dx=-\frac {3 \, b^{3} n x^{3 \, n} \log \left (x\right ) - 3 \, b^{3} x^{3 \, n} \log \left (b x^{n} + 2\right ) + 6 \, b^{2} x^{2 \, n} - 6 \, b x^{n} + 8}{48 \, n x^{3 \, n}} \]
-1/48*(3*b^3*n*x^(3*n)*log(x) - 3*b^3*x^(3*n)*log(b*x^n + 2) + 6*b^2*x^(2* n) - 6*b*x^n + 8)/(n*x^(3*n))
Time = 4.25 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.15 \[ \int \frac {x^{-1-3 n}}{2+b x^n} \, dx=\begin {cases} \frac {\log {\left (x \right )}}{2} & \text {for}\: b = 0 \wedge n = 0 \\- \frac {x x^{- 3 n - 1}}{6 n} & \text {for}\: b = 0 \\\frac {\log {\left (x \right )}}{b + 2} & \text {for}\: n = 0 \\- \frac {b^{3} \log {\left (x \right )}}{16} + \frac {b^{3} \log {\left (x^{n} + \frac {2}{b} \right )}}{16 n} - \frac {b^{2} x^{- n}}{8 n} + \frac {b x^{- 2 n}}{8 n} - \frac {x^{- 3 n}}{6 n} & \text {otherwise} \end {cases} \]
Piecewise((log(x)/2, Eq(b, 0) & Eq(n, 0)), (-x*x**(-3*n - 1)/(6*n), Eq(b, 0)), (log(x)/(b + 2), Eq(n, 0)), (-b**3*log(x)/16 + b**3*log(x**n + 2/b)/( 16*n) - b**2/(8*n*x**n) + b/(8*n*x**(2*n)) - 1/(6*n*x**(3*n)), True))
Time = 0.21 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.85 \[ \int \frac {x^{-1-3 n}}{2+b x^n} \, dx=-\frac {1}{16} \, b^{3} \log \left (x\right ) + \frac {b^{3} \log \left (\frac {b x^{n} + 2}{b}\right )}{16 \, n} - \frac {3 \, b^{2} x^{2 \, n} - 3 \, b x^{n} + 4}{24 \, n x^{3 \, n}} \]
-1/16*b^3*log(x) + 1/16*b^3*log((b*x^n + 2)/b)/n - 1/24*(3*b^2*x^(2*n) - 3 *b*x^n + 4)/(n*x^(3*n))
\[ \int \frac {x^{-1-3 n}}{2+b x^n} \, dx=\int { \frac {x^{-3 \, n - 1}}{b x^{n} + 2} \,d x } \]
Timed out. \[ \int \frac {x^{-1-3 n}}{2+b x^n} \, dx=\int \frac {1}{x^{3\,n+1}\,\left (b\,x^n+2\right )} \,d x \]