Integrand size = 19, antiderivative size = 68 \[ \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=-\frac {2 x^{-3 n/2}}{3 a n}+\frac {2 b x^{-n/2}}{a^2 n}-\frac {2 b^{3/2} \arctan \left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{5/2} n} \] Output:
-2/3/a/n/(x^(3/2*n))+2*b/a^2/n/(x^(1/2*n))-2*b^(3/2)*arctan(a^(1/2)/b^(1/2 )/(x^(1/2*n)))/a^(5/2)/n
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.50 \[ \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=-\frac {2 x^{-3 n/2} \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},1,-\frac {1}{2},-\frac {b x^n}{a}\right )}{3 a n} \] Input:
Integrate[x^(-1 - (3*n)/2)/(a + b*x^n),x]
Output:
(-2*Hypergeometric2F1[-3/2, 1, -1/2, -((b*x^n)/a)])/(3*a*n*x^((3*n)/2))
Time = 0.34 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.03, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {886, 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.
\(\displaystyle \int \frac {x^{-\frac {3 n}{2}-1}}{a+b x^n} \, dx\) |
\(\Big \downarrow \) 886 |
\(\displaystyle -\frac {b \int \frac {x^{-\frac {n}{2}-1}}{b x^n+a}dx}{a}-\frac {2 x^{-3 n/2}}{3 a n}\) |
\(\Big \downarrow \) 868 |
\(\displaystyle \frac {2 b \int \frac {1}{b x^n+a}dx^{-n/2}}{a n}-\frac {2 x^{-3 n/2}}{3 a n}\) |
\(\Big \downarrow \) 772 |
\(\displaystyle \frac {2 b \int \frac {x^{-n}}{a x^{-n}+b}dx^{-n/2}}{a n}-\frac {2 x^{-3 n/2}}{3 a n}\) |
\(\Big \downarrow \) 262 |
\(\displaystyle \frac {2 b \left (\frac {x^{-n/2}}{a}-\frac {b \int \frac {1}{a x^{-n}+b}dx^{-n/2}}{a}\right )}{a n}-\frac {2 x^{-3 n/2}}{3 a n}\) |
\(\Big \downarrow \) 218 |
\(\displaystyle \frac {2 b \left (\frac {x^{-n/2}}{a}-\frac {\sqrt {b} \arctan \left (\frac {\sqrt {a} x^{-n/2}}{\sqrt {b}}\right )}{a^{3/2}}\right )}{a n}-\frac {2 x^{-3 n/2}}{3 a n}\) |
Input:
Int[x^(-1 - (3*n)/2)/(a + b*x^n),x]
Output:
-2/(3*a*n*x^((3*n)/2)) + (2*b*(1/(a*x^(n/2)) - (Sqrt[b]*ArcTan[Sqrt[a]/(Sq rt[b]*x^(n/2))])/a^(3/2)))/(a*n)
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[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[x^(m + 1)/(a*(m + 1)), x] - Simp[b/a Int[x^Simplify[m + n]/(a + b*x^n), x], x] /; FreeQ[{a , b, m, n}, x] && FractionQ[Simplify[(m + 1)/n]] && SumSimplerQ[m, n]
Time = 0.67 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.43
method | result | size |
risch | \(\frac {2 b \,x^{-\frac {n}{2}}}{a^{2} n}-\frac {2 x^{-\frac {3 n}{2}}}{3 a n}+\frac {\sqrt {-a b}\, b \ln \left (x^{\frac {n}{2}}+\frac {\sqrt {-a b}}{b}\right )}{a^{3} n}-\frac {\sqrt {-a b}\, b \ln \left (x^{\frac {n}{2}}-\frac {\sqrt {-a b}}{b}\right )}{a^{3} n}\) | \(97\) |
Input:
int(x^(-1-3/2*n)/(a+b*x^n),x,method=_RETURNVERBOSE)
Output:
2*b/a^2/n/(x^(1/2*n))-2/3/a/n/(x^(1/2*n))^3+1/a^3*(-a*b)^(1/2)*b/n*ln(x^(1 /2*n)+1/b*(-a*b)^(1/2))-1/a^3*(-a*b)^(1/2)*b/n*ln(x^(1/2*n)-1/b*(-a*b)^(1/ 2))
Time = 0.12 (sec) , antiderivative size = 253, normalized size of antiderivative = 3.72 \[ \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=\left [-\frac {2 \, a x x^{-\frac {3}{2} \, n - 1} - 3 \, b \sqrt {-\frac {b}{a}} \log \left (-\frac {2 \, a^{3} x^{\frac {5}{3}} x^{-\frac {5}{2} \, n - \frac {5}{3}} \sqrt {-\frac {b}{a}} - a^{3} x^{2} x^{-3 \, n - 2} - 2 \, a^{2} b x x^{-\frac {3}{2} \, n - 1} \sqrt {-\frac {b}{a}} + 2 \, a^{2} b x^{\frac {4}{3}} x^{-2 \, n - \frac {4}{3}} + 2 \, a b^{2} x^{\frac {1}{3}} x^{-\frac {1}{2} \, n - \frac {1}{3}} \sqrt {-\frac {b}{a}} - 2 \, a b^{2} x^{\frac {2}{3}} x^{-n - \frac {2}{3}} + b^{3}}{a^{3} x^{2} x^{-3 \, n - 2} + b^{3}}\right ) - 6 \, b x^{\frac {1}{3}} x^{-\frac {1}{2} \, n - \frac {1}{3}}}{3 \, a^{2} n}, -\frac {2 \, {\left (a x x^{-\frac {3}{2} \, n - 1} + 3 \, b \sqrt {\frac {b}{a}} \arctan \left (\frac {a x^{\frac {1}{3}} x^{-\frac {1}{2} \, n - \frac {1}{3}} \sqrt {\frac {b}{a}}}{b}\right ) - 3 \, b x^{\frac {1}{3}} x^{-\frac {1}{2} \, n - \frac {1}{3}}\right )}}{3 \, a^{2} n}\right ] \] Input:
integrate(x^(-1-3/2*n)/(a+b*x^n),x, algorithm="fricas")
Output:
[-1/3*(2*a*x*x^(-3/2*n - 1) - 3*b*sqrt(-b/a)*log(-(2*a^3*x^(5/3)*x^(-5/2*n - 5/3)*sqrt(-b/a) - a^3*x^2*x^(-3*n - 2) - 2*a^2*b*x*x^(-3/2*n - 1)*sqrt( -b/a) + 2*a^2*b*x^(4/3)*x^(-2*n - 4/3) + 2*a*b^2*x^(1/3)*x^(-1/2*n - 1/3)* sqrt(-b/a) - 2*a*b^2*x^(2/3)*x^(-n - 2/3) + b^3)/(a^3*x^2*x^(-3*n - 2) + b ^3)) - 6*b*x^(1/3)*x^(-1/2*n - 1/3))/(a^2*n), -2/3*(a*x*x^(-3/2*n - 1) + 3 *b*sqrt(b/a)*arctan(a*x^(1/3)*x^(-1/2*n - 1/3)*sqrt(b/a)/b) - 3*b*x^(1/3)* x^(-1/2*n - 1/3))/(a^2*n)]
Time = 1.09 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.82 \[ \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=- \frac {2 x^{- \frac {3 n}{2}}}{3 a n} + \frac {2 b x^{- \frac {n}{2}}}{a^{2} n} + \frac {2 b^{\frac {3}{2}} \operatorname {atan}{\left (\frac {\sqrt {b} x^{\frac {n}{2}}}{\sqrt {a}} \right )}}{a^{\frac {5}{2}} n} \] Input:
integrate(x**(-1-3/2*n)/(a+b*x**n),x)
Output:
-2/(3*a*n*x**(3*n/2)) + 2*b/(a**2*n*x**(n/2)) + 2*b**(3/2)*atan(sqrt(b)*x* *(n/2)/sqrt(a))/(a**(5/2)*n)
\[ \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=\int { \frac {x^{-\frac {3}{2} \, n - 1}}{b x^{n} + a} \,d x } \] Input:
integrate(x^(-1-3/2*n)/(a+b*x^n),x, algorithm="maxima")
Output:
b^2*integrate(x^(1/2*n)/(a^2*b*x*x^n + a^3*x), x) + 2/3*(3*b*x^n - a)/(a^2 *n*x^(3/2*n))
\[ \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=\int { \frac {x^{-\frac {3}{2} \, n - 1}}{b x^{n} + a} \,d x } \] Input:
integrate(x^(-1-3/2*n)/(a+b*x^n),x, algorithm="giac")
Output:
integrate(x^(-3/2*n - 1)/(b*x^n + a), x)
Timed out. \[ \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=\int \frac {1}{x^{\frac {3\,n}{2}+1}\,\left (a+b\,x^n\right )} \,d x \] Input:
int(1/(x^((3*n)/2 + 1)*(a + b*x^n)),x)
Output:
int(1/(x^((3*n)/2 + 1)*(a + b*x^n)), x)
\[ \int \frac {x^{-1-\frac {3 n}{2}}}{a+b x^n} \, dx=\int \frac {1}{x^{\frac {5 n}{2}} b x +x^{\frac {3 n}{2}} a x}d x \] Input:
int(x^(-1-3/2*n)/(a+b*x^n),x)
Output:
int(1/(x**((5*n)/2)*b*x + x**((3*n)/2)*a*x),x)