Integrand size = 19, antiderivative size = 84 \[ \int x^{-1-2 n} \sqrt {a+b x^n} \, dx=-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 n}-\frac {b x^{-n} \sqrt {a+b x^n}}{4 a n}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{4 a^{3/2} n} \] Output:
-1/2*(a+b*x^n)^(1/2)/n/(x^(2*n))-1/4*b*(a+b*x^n)^(1/2)/a/n/(x^n)+1/4*b^2*a rctanh((a+b*x^n)^(1/2)/a^(1/2))/a^(3/2)/n
Time = 0.11 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.82 \[ \int x^{-1-2 n} \sqrt {a+b x^n} \, dx=-\frac {x^{-2 n} \sqrt {a+b x^n} \left (2 a+b x^n\right )}{4 a n}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{4 a^{3/2} n} \] Input:
Integrate[x^(-1 - 2*n)*Sqrt[a + b*x^n],x]
Output:
-1/4*(Sqrt[a + b*x^n]*(2*a + b*x^n))/(a*n*x^(2*n)) + (b^2*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(4*a^(3/2)*n)
Time = 0.30 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.92, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {798, 51, 52, 73, 221}
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.
\(\displaystyle \int x^{-2 n-1} \sqrt {a+b x^n} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int x^{-3 n} \sqrt {b x^n+a}dx^n}{n}\) |
\(\Big \downarrow \) 51 |
\(\displaystyle \frac {\frac {1}{4} b \int \frac {x^{-2 n}}{\sqrt {b x^n+a}}dx^n-\frac {1}{2} x^{-2 n} \sqrt {a+b x^n}}{n}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {\frac {1}{4} b \left (-\frac {b \int \frac {x^{-n}}{\sqrt {b x^n+a}}dx^n}{2 a}-\frac {x^{-n} \sqrt {a+b x^n}}{a}\right )-\frac {1}{2} x^{-2 n} \sqrt {a+b x^n}}{n}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {\frac {1}{4} b \left (-\frac {\int \frac {1}{\frac {x^{2 n}}{b}-\frac {a}{b}}d\sqrt {b x^n+a}}{a}-\frac {x^{-n} \sqrt {a+b x^n}}{a}\right )-\frac {1}{2} x^{-2 n} \sqrt {a+b x^n}}{n}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {\frac {1}{4} b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {x^{-n} \sqrt {a+b x^n}}{a}\right )-\frac {1}{2} x^{-2 n} \sqrt {a+b x^n}}{n}\) |
Input:
Int[x^(-1 - 2*n)*Sqrt[a + b*x^n],x]
Output:
(-1/2*Sqrt[a + b*x^n]/x^(2*n) + (b*(-(Sqrt[a + b*x^n]/(a*x^n)) + (b*ArcTan h[Sqrt[a + b*x^n]/Sqrt[a]])/a^(3/2)))/4)/n
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x ] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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]]
\[\int x^{-2 n -1} \sqrt {a +b \,x^{n}}d x\]
Input:
int(x^(-2*n-1)*(a+b*x^n)^(1/2),x)
Output:
int(x^(-2*n-1)*(a+b*x^n)^(1/2),x)
Time = 0.09 (sec) , antiderivative size = 150, normalized size of antiderivative = 1.79 \[ \int x^{-1-2 n} \sqrt {a+b x^n} \, dx=\left [\frac {\sqrt {a} b^{2} x^{2 \, n} \log \left (\frac {b x^{n} + 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) - 2 \, {\left (a b x^{n} + 2 \, a^{2}\right )} \sqrt {b x^{n} + a}}{8 \, a^{2} n x^{2 \, n}}, -\frac {\sqrt {-a} b^{2} x^{2 \, n} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{n} + a}}\right ) + {\left (a b x^{n} + 2 \, a^{2}\right )} \sqrt {b x^{n} + a}}{4 \, a^{2} n x^{2 \, n}}\right ] \] Input:
integrate(x^(-1-2*n)*(a+b*x^n)^(1/2),x, algorithm="fricas")
Output:
[1/8*(sqrt(a)*b^2*x^(2*n)*log((b*x^n + 2*sqrt(b*x^n + a)*sqrt(a) + 2*a)/x^ n) - 2*(a*b*x^n + 2*a^2)*sqrt(b*x^n + a))/(a^2*n*x^(2*n)), -1/4*(sqrt(-a)* b^2*x^(2*n)*arctan(sqrt(-a)/sqrt(b*x^n + a)) + (a*b*x^n + 2*a^2)*sqrt(b*x^ n + a))/(a^2*n*x^(2*n))]
Time = 6.53 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.33 \[ \int x^{-1-2 n} \sqrt {a+b x^n} \, dx=- \frac {a x^{- \frac {5 n}{2}}}{2 \sqrt {b} n \sqrt {\frac {a x^{- n}}{b} + 1}} - \frac {3 \sqrt {b} x^{- \frac {3 n}{2}}}{4 n \sqrt {\frac {a x^{- n}}{b} + 1}} - \frac {b^{\frac {3}{2}} x^{- \frac {n}{2}}}{4 a n \sqrt {\frac {a x^{- n}}{b} + 1}} + \frac {b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} x^{- \frac {n}{2}}}{\sqrt {b}} \right )}}{4 a^{\frac {3}{2}} n} \] Input:
integrate(x**(-1-2*n)*(a+b*x**n)**(1/2),x)
Output:
-a/(2*sqrt(b)*n*x**(5*n/2)*sqrt(a/(b*x**n) + 1)) - 3*sqrt(b)/(4*n*x**(3*n/ 2)*sqrt(a/(b*x**n) + 1)) - b**(3/2)/(4*a*n*x**(n/2)*sqrt(a/(b*x**n) + 1)) + b**2*asinh(sqrt(a)/(sqrt(b)*x**(n/2)))/(4*a**(3/2)*n)
\[ \int x^{-1-2 n} \sqrt {a+b x^n} \, dx=\int { \sqrt {b x^{n} + a} x^{-2 \, n - 1} \,d x } \] Input:
integrate(x^(-1-2*n)*(a+b*x^n)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(b*x^n + a)*x^(-2*n - 1), x)
\[ \int x^{-1-2 n} \sqrt {a+b x^n} \, dx=\int { \sqrt {b x^{n} + a} x^{-2 \, n - 1} \,d x } \] Input:
integrate(x^(-1-2*n)*(a+b*x^n)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(b*x^n + a)*x^(-2*n - 1), x)
Timed out. \[ \int x^{-1-2 n} \sqrt {a+b x^n} \, dx=\int \frac {\sqrt {a+b\,x^n}}{x^{2\,n+1}} \,d x \] Input:
int((a + b*x^n)^(1/2)/x^(2*n + 1),x)
Output:
int((a + b*x^n)^(1/2)/x^(2*n + 1), x)
\[ \int x^{-1-2 n} \sqrt {a+b x^n} \, dx=\frac {-2 x^{n} \sqrt {x^{n} b +a}\, b -4 \sqrt {x^{n} b +a}\, a -x^{2 n} \left (\int \frac {\sqrt {x^{n} b +a}}{x^{n} b x +a x}d x \right ) b^{2} n}{8 x^{2 n} a n} \] Input:
int(x^(-1-2*n)*(a+b*x^n)^(1/2),x)
Output:
( - 2*x**n*sqrt(x**n*b + a)*b - 4*sqrt(x**n*b + a)*a - x**(2*n)*int(sqrt(x **n*b + a)/(x**n*b*x + a*x),x)*b**2*n)/(8*x**(2*n)*a*n)