Integrand size = 21, antiderivative size = 74 \[ \int \frac {(c x)^{-1-\frac {n}{2}}}{a+b x^n} \, dx=-\frac {2 (c x)^{-n/2}}{a c n}+\frac {2 \sqrt {b} x^{n/2} (c x)^{-n/2} \arctan \left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{3/2} c n} \] Output:
-2/a/c/n/((c*x)^(1/2*n))+2*b^(1/2)*x^(1/2*n)*arctan(a^(1/2)/b^(1/2)/(x^(1/ 2*n)))/a^(3/2)/c/n/((c*x)^(1/2*n))
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.01 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.50 \[ \int \frac {(c x)^{-1-\frac {n}{2}}}{a+b x^n} \, dx=-\frac {2 x (c x)^{-1-\frac {n}{2}} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\frac {b x^n}{a}\right )}{a n} \] Input:
Integrate[(c*x)^(-1 - n/2)/(a + b*x^n),x]
Output:
(-2*x*(c*x)^(-1 - n/2)*Hypergeometric2F1[-1/2, 1, 1/2, -((b*x^n)/a)])/(a*n )
Time = 0.32 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {870, 868, 772, 262, 218}
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.
\(\displaystyle \int \frac {(c x)^{-\frac {n}{2}-1}}{a+b x^n} \, dx\) |
\(\Big \downarrow \) 870 |
\(\displaystyle \frac {x^{n/2} (c x)^{-n/2} \int \frac {x^{-\frac {n}{2}-1}}{b x^n+a}dx}{c}\) |
\(\Big \downarrow \) 868 |
\(\displaystyle -\frac {2 x^{n/2} (c x)^{-n/2} \int \frac {1}{b x^n+a}dx^{-n/2}}{c n}\) |
\(\Big \downarrow \) 772 |
\(\displaystyle -\frac {2 x^{n/2} (c x)^{-n/2} \int \frac {x^{-n}}{a x^{-n}+b}dx^{-n/2}}{c n}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle -\frac {2 x^{n/2} (c x)^{-n/2} \left (\frac {x^{-n/2}}{a}-\frac {b \int \frac {1}{a x^{-n}+b}dx^{-n/2}}{a}\right )}{c n}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle -\frac {2 x^{n/2} (c x)^{-n/2} \left (\frac {x^{-n/2}}{a}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{3/2}}\right )}{c n}\) |
Input:
Int[(c*x)^(-1 - n/2)/(a + b*x^n),x]
Output:
(-2*x^(n/2)*(1/(a*x^(n/2)) - (Sqrt[b]*ArcTan[Sqrt[a]/(Sqrt[b]*x^(n/2))])/a ^(3/2)))/(c*n*(c*x)^(n/2))
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[c*(c*x) ^(m - 1)*((a + b*x^2)^(p + 1)/(b*(m + 2*p + 1))), x] - Simp[a*c^2*((m - 1)/ (b*(m + 2*p + 1))) Int[(c*x)^(m - 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b , c, p}, x] && GtQ[m, 2 - 1] && NeQ[m + 2*p + 1, 0] && IntBinomialQ[a, b, c , 2, m, p, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b}, x] && ILtQ[n, 0] && IntegerQ[p]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/(m + 1) Subst[Int[(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[ {a, b, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] && !IntegerQ[n]
Int[((c_)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^Int Part[m]*((c*x)^FracPart[m]/x^FracPart[m]) Int[x^m*(a + b*x^n)^p, x], x] / ; FreeQ[{a, b, c, m, n, p}, x] && IntegerQ[Simplify[n/(m + 1)]] && !Intege rQ[n]
\[\int \frac {\left (c x \right )^{-1-\frac {n}{2}}}{a +b \,x^{n}}d x\]
Input:
int((c*x)^(-1-1/2*n)/(a+b*x^n),x)
Output:
int((c*x)^(-1-1/2*n)/(a+b*x^n),x)
Time = 0.10 (sec) , antiderivative size = 237, normalized size of antiderivative = 3.20 \[ \int \frac {(c x)^{-1-\frac {n}{2}}}{a+b x^n} \, dx=\left [-\frac {2 \, x e^{\left (-\frac {1}{2} \, {\left (n + 2\right )} \log \left (c\right ) - \frac {1}{2} \, {\left (n + 2\right )} \log \left (x\right )\right )} - \sqrt {-\frac {b c^{-n - 2}}{a}} \log \left (\frac {a x^{2} e^{\left (-{\left (n + 2\right )} \log \left (c\right ) - {\left (n + 2\right )} \log \left (x\right )\right )} + 2 \, a \sqrt {-\frac {b c^{-n - 2}}{a}} x e^{\left (-\frac {1}{2} \, {\left (n + 2\right )} \log \left (c\right ) - \frac {1}{2} \, {\left (n + 2\right )} \log \left (x\right )\right )} - b c^{-n - 2}}{a x^{2} e^{\left (-{\left (n + 2\right )} \log \left (c\right ) - {\left (n + 2\right )} \log \left (x\right )\right )} + b c^{-n - 2}}\right )}{a n}, -\frac {2 \, {\left (x e^{\left (-\frac {1}{2} \, {\left (n + 2\right )} \log \left (c\right ) - \frac {1}{2} \, {\left (n + 2\right )} \log \left (x\right )\right )} - \sqrt {\frac {b c^{-n - 2}}{a}} \arctan \left (\frac {a \sqrt {\frac {b c^{-n - 2}}{a}} x e^{\left (-\frac {1}{2} \, {\left (n + 2\right )} \log \left (c\right ) - \frac {1}{2} \, {\left (n + 2\right )} \log \left (x\right )\right )}}{b c^{-n - 2}}\right )\right )}}{a n}\right ] \] Input:
integrate((c*x)^(-1-1/2*n)/(a+b*x^n),x, algorithm="fricas")
Output:
[-(2*x*e^(-1/2*(n + 2)*log(c) - 1/2*(n + 2)*log(x)) - sqrt(-b*c^(-n - 2)/a )*log((a*x^2*e^(-(n + 2)*log(c) - (n + 2)*log(x)) + 2*a*sqrt(-b*c^(-n - 2) /a)*x*e^(-1/2*(n + 2)*log(c) - 1/2*(n + 2)*log(x)) - b*c^(-n - 2))/(a*x^2* e^(-(n + 2)*log(c) - (n + 2)*log(x)) + b*c^(-n - 2))))/(a*n), -2*(x*e^(-1/ 2*(n + 2)*log(c) - 1/2*(n + 2)*log(x)) - sqrt(b*c^(-n - 2)/a)*arctan(a*sqr t(b*c^(-n - 2)/a)*x*e^(-1/2*(n + 2)*log(c) - 1/2*(n + 2)*log(x))/(b*c^(-n - 2))))/(a*n)]
Time = 0.80 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.78 \[ \int \frac {(c x)^{-1-\frac {n}{2}}}{a+b x^n} \, dx=- \frac {2 c^{- \frac {n}{2} - 1} x^{- \frac {n}{2}}}{a n} - \frac {2 \sqrt {b} c^{- \frac {n}{2} - 1} \operatorname {atan}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{a^{\frac {3}{2}} n} \] Input:
integrate((c*x)**(-1-1/2*n)/(a+b*x**n),x)
Output:
-2*c**(-n/2 - 1)/(a*n*x**(n/2)) - 2*sqrt(b)*c**(-n/2 - 1)*atan(sqrt(b)*x** (n/2)/sqrt(a))/(a**(3/2)*n)
\[ \int \frac {(c x)^{-1-\frac {n}{2}}}{a+b x^n} \, dx=\int { \frac {\left (c x\right )^{-\frac {1}{2} \, n - 1}}{b x^{n} + a} \,d x } \] Input:
integrate((c*x)^(-1-1/2*n)/(a+b*x^n),x, algorithm="maxima")
Output:
-b*integrate(x^(1/2*n)/(a*b*c^(1/2*n + 1)*x*x^n + a^2*c^(1/2*n + 1)*x), x) - 2*c^(-1/2*n - 1)/(a*n*x^(1/2*n))
\[ \int \frac {(c x)^{-1-\frac {n}{2}}}{a+b x^n} \, dx=\int { \frac {\left (c x\right )^{-\frac {1}{2} \, n - 1}}{b x^{n} + a} \,d x } \] Input:
integrate((c*x)^(-1-1/2*n)/(a+b*x^n),x, algorithm="giac")
Output:
integrate((c*x)^(-1/2*n - 1)/(b*x^n + a), x)
Timed out. \[ \int \frac {(c x)^{-1-\frac {n}{2}}}{a+b x^n} \, dx=\int \frac {1}{{\left (c\,x\right )}^{\frac {n}{2}+1}\,\left (a+b\,x^n\right )} \,d x \] Input:
int(1/((c*x)^(n/2 + 1)*(a + b*x^n)),x)
Output:
int(1/((c*x)^(n/2 + 1)*(a + b*x^n)), x)
\[ \int \frac {(c x)^{-1-\frac {n}{2}}}{a+b x^n} \, dx=\frac {\int \frac {1}{x^{\frac {3 n}{2}} b x +x^{\frac {n}{2}} a x}d x}{c^{\frac {n}{2}} c} \] Input:
int((c*x)^(-1-1/2*n)/(a+b*x^n),x)
Output:
int(1/(x**((3*n)/2)*b*x + x**(n/2)*a*x),x)/(c**(n/2)*c)