Integrand size = 19, antiderivative size = 87 \[ \int \frac {x^{-1-2 n}}{\sqrt {a+b x^n}} \, dx=-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 a n}+\frac {3 b x^{-n} \sqrt {a+b x^n}}{4 a^2 n}-\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{4 a^{5/2} n} \] Output:
-1/2*(a+b*x^n)^(1/2)/a/n/(x^(2*n))+3/4*b*(a+b*x^n)^(1/2)/a^2/n/(x^n)-3/4*b ^2*arctanh((a+b*x^n)^(1/2)/a^(1/2))/a^(5/2)/n
Time = 0.12 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.78 \[ \int \frac {x^{-1-2 n}}{\sqrt {a+b x^n}} \, dx=\frac {\sqrt {a} x^{-2 n} \sqrt {a+b x^n} \left (-2 a+3 b x^n\right )-3 b^2 \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{4 a^{5/2} n} \] Input:
Integrate[x^(-1 - 2*n)/Sqrt[a + b*x^n],x]
Output:
((Sqrt[a]*Sqrt[a + b*x^n]*(-2*a + 3*b*x^n))/x^(2*n) - 3*b^2*ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/(4*a^(5/2)*n)
Time = 0.32 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.95, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {798, 52, 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.
\(\displaystyle \int \frac {x^{-2 n-1}}{\sqrt {a+b x^n}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {\int \frac {x^{-3 n}}{\sqrt {b x^n+a}}dx^n}{n}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {-\frac {3 b \int \frac {x^{-2 n}}{\sqrt {b x^n+a}}dx^n}{4 a}-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 a}}{n}\) |
\(\Big \downarrow \) 52 |
\(\displaystyle \frac {-\frac {3 b \left (-\frac {b \int \frac {x^{-n}}{\sqrt {b x^n+a}}dx^n}{2 a}-\frac {x^{-n} \sqrt {a+b x^n}}{a}\right )}{4 a}-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 a}}{n}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {-\frac {3 b \left (-\frac {\int \frac {1}{\frac {x^{2 n}}{b}-\frac {a}{b}}d\sqrt {b x^n+a}}{a}-\frac {x^{-n} \sqrt {a+b x^n}}{a}\right )}{4 a}-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 a}}{n}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {-\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {a+b x^n}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {x^{-n} \sqrt {a+b x^n}}{a}\right )}{4 a}-\frac {x^{-2 n} \sqrt {a+b x^n}}{2 a}}{n}\) |
Input:
Int[x^(-1 - 2*n)/Sqrt[a + b*x^n],x]
Output:
(-1/2*Sqrt[a + b*x^n]/(a*x^(2*n)) - (3*b*(-(Sqrt[a + b*x^n]/(a*x^n)) + (b* ArcTanh[Sqrt[a + b*x^n]/Sqrt[a]])/a^(3/2)))/(4*a))/n
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]]
\[\int \frac {x^{-2 n -1}}{\sqrt {a +b \,x^{n}}}d x\]
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 = 154, normalized size of antiderivative = 1.77 \[ \int \frac {x^{-1-2 n}}{\sqrt {a+b x^n}} \, dx=\left [\frac {3 \, \sqrt {a} b^{2} x^{2 \, n} \log \left (\frac {b x^{n} - 2 \, \sqrt {b x^{n} + a} \sqrt {a} + 2 \, a}{x^{n}}\right ) + 2 \, {\left (3 \, a b x^{n} - 2 \, a^{2}\right )} \sqrt {b x^{n} + a}}{8 \, a^{3} n x^{2 \, n}}, \frac {3 \, \sqrt {-a} b^{2} x^{2 \, n} \arctan \left (\frac {\sqrt {-a}}{\sqrt {b x^{n} + a}}\right ) + {\left (3 \, a b x^{n} - 2 \, a^{2}\right )} \sqrt {b x^{n} + a}}{4 \, a^{3} n x^{2 \, n}}\right ] \] Input:
integrate(x^(-1-2*n)/(a+b*x^n)^(1/2),x, algorithm="fricas")
Output:
[1/8*(3*sqrt(a)*b^2*x^(2*n)*log((b*x^n - 2*sqrt(b*x^n + a)*sqrt(a) + 2*a)/ x^n) + 2*(3*a*b*x^n - 2*a^2)*sqrt(b*x^n + a))/(a^3*n*x^(2*n)), 1/4*(3*sqrt (-a)*b^2*x^(2*n)*arctan(sqrt(-a)/sqrt(b*x^n + a)) + (3*a*b*x^n - 2*a^2)*sq rt(b*x^n + a))/(a^3*n*x^(2*n))]
Time = 9.81 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.34 \[ \int \frac {x^{-1-2 n}}{\sqrt {a+b x^n}} \, dx=- \frac {x^{- \frac {5 n}{2}}}{2 \sqrt {b} n \sqrt {\frac {a x^{- n}}{b} + 1}} + \frac {\sqrt {b} x^{- \frac {3 n}{2}}}{4 a n \sqrt {\frac {a x^{- n}}{b} + 1}} + \frac {3 b^{\frac {3}{2}} x^{- \frac {n}{2}}}{4 a^{2} n \sqrt {\frac {a x^{- n}}{b} + 1}} - \frac {3 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a} x^{- \frac {n}{2}}}{\sqrt {b}} \right )}}{4 a^{\frac {5}{2}} n} \] Input:
integrate(x**(-1-2*n)/(a+b*x**n)**(1/2),x)
Output:
-1/(2*sqrt(b)*n*x**(5*n/2)*sqrt(a/(b*x**n) + 1)) + sqrt(b)/(4*a*n*x**(3*n/ 2)*sqrt(a/(b*x**n) + 1)) + 3*b**(3/2)/(4*a**2*n*x**(n/2)*sqrt(a/(b*x**n) + 1)) - 3*b**2*asinh(sqrt(a)/(sqrt(b)*x**(n/2)))/(4*a**(5/2)*n)
\[ \int \frac {x^{-1-2 n}}{\sqrt {a+b x^n}} \, dx=\int { \frac {x^{-2 \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \] Input:
integrate(x^(-1-2*n)/(a+b*x^n)^(1/2),x, algorithm="maxima")
Output:
integrate(x^(-2*n - 1)/sqrt(b*x^n + a), x)
\[ \int \frac {x^{-1-2 n}}{\sqrt {a+b x^n}} \, dx=\int { \frac {x^{-2 \, n - 1}}{\sqrt {b x^{n} + a}} \,d x } \] Input:
integrate(x^(-1-2*n)/(a+b*x^n)^(1/2),x, algorithm="giac")
Output:
integrate(x^(-2*n - 1)/sqrt(b*x^n + a), x)
Timed out. \[ \int \frac {x^{-1-2 n}}{\sqrt {a+b x^n}} \, dx=\int \frac {1}{x^{2\,n+1}\,\sqrt {a+b\,x^n}} \,d x \] Input:
int(1/(x^(2*n + 1)*(a + b*x^n)^(1/2)),x)
Output:
int(1/(x^(2*n + 1)*(a + b*x^n)^(1/2)), x)
\[ \int \frac {x^{-1-2 n}}{\sqrt {a+b x^n}} \, dx=\frac {6 x^{n} \sqrt {x^{n} b +a}\, b -4 \sqrt {x^{n} b +a}\, a +3 x^{2 n} \left (\int \frac {\sqrt {x^{n} b +a}}{x^{n} b x +a x}d x \right ) b^{2} n}{8 x^{2 n} a^{2} n} \] Input:
int(x^(-1-2*n)/(a+b*x^n)^(1/2),x)
Output:
(6*x**n*sqrt(x**n*b + a)*b - 4*sqrt(x**n*b + a)*a + 3*x**(2*n)*int(sqrt(x* *n*b + a)/(x**n*b*x + a*x),x)*b**2*n)/(8*x**(2*n)*a**2*n)