Integrand size = 33, antiderivative size = 102 \[ \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{\frac {-1-2 n}{2 n}} \, dx=\frac {x \left (a+b x^n\right ) \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{\frac {1}{2} \left (-2-\frac {1}{n}\right )}}{a (1+n)}+\frac {n x \left (a+b x^n\right )^2 \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{\frac {1}{2} \left (-2-\frac {1}{n}\right )}}{a^2 (1+n)} \]
x*(a+b*x^n)*(a^2+2*a*b*x^n+b^2*x^(2*n))^(-1-1/2/n)/a/(1+n)+n*x*(a+b*x^n)^2 *(a^2+2*a*b*x^n+b^2*x^(2*n))^(-1-1/2/n)/a^2/(1+n)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.11 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.58 \[ \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{\frac {-1-2 n}{2 n}} \, dx=\frac {x \left (\left (a+b x^n\right )^2\right )^{\left .-\frac {1}{2}\right /n} \left (1+\frac {b x^n}{a}\right )^{\frac {1}{n}} \operatorname {Hypergeometric2F1}\left (2+\frac {1}{n},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{a^2} \]
(x*(1 + (b*x^n)/a)^n^(-1)*Hypergeometric2F1[2 + n^(-1), n^(-1), 1 + n^(-1) , -((b*x^n)/a)])/(a^2*((a + b*x^n)^2)^(1/(2*n)))
Time = 0.24 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.97, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1385, 777, 746}
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.
\(\displaystyle \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{\frac {-2 n-1}{2 n}} \, dx\) |
\(\Big \downarrow \) 1385 |
\(\displaystyle \left (\frac {b x^n}{a}+1\right )^{\frac {1}{n}+2} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{\frac {1}{2} \left (-\frac {1}{n}-2\right )} \int \left (\frac {b x^n}{a}+1\right )^{-2-\frac {1}{n}}dx\) |
\(\Big \downarrow \) 777 |
\(\displaystyle \left (\frac {b x^n}{a}+1\right )^{\frac {1}{n}+2} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{\frac {1}{2} \left (-\frac {1}{n}-2\right )} \left (\frac {n \int \left (\frac {b x^n}{a}+1\right )^{-1-\frac {1}{n}}dx}{n+1}+\frac {x \left (\frac {b x^n}{a}+1\right )^{-\frac {1}{n}-1}}{n+1}\right )\) |
\(\Big \downarrow \) 746 |
\(\displaystyle \left (\frac {b x^n}{a}+1\right )^{\frac {1}{n}+2} \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{\frac {1}{2} \left (-\frac {1}{n}-2\right )} \left (\frac {x \left (\frac {b x^n}{a}+1\right )^{-\frac {1}{n}-1}}{n+1}+\frac {n x \left (\frac {b x^n}{a}+1\right )^{-1/n}}{n+1}\right )\) |
(1 + (b*x^n)/a)^(2 + n^(-1))*(a^2 + 2*a*b*x^n + b^2*x^(2*n))^((-2 - n^(-1) )/2)*((x*(1 + (b*x^n)/a)^(-1 - n^(-1)))/(1 + n) + (n*x)/((1 + n)*(1 + (b*x ^n)/a)^n^(-1)))
3.6.46.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x*((a + b*x^n)^(p + 1) /a), x] /; FreeQ[{a, b, n, p}, x] && EqQ[1/n + p + 1, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[(n*(p + 1) + 1)/(a*n*(p + 1)) Int[(a + b*x^ n)^(p + 1), x], x] /; FreeQ[{a, b, n, p}, x] && ILtQ[Simplify[1/n + p + 1], 0] && NeQ[p, -1]
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S imp[a^IntPart[p]*((a + b*x^n + c*x^(2*n))^FracPart[p]/(1 + 2*c*(x^n/b))^(2* FracPart[p])) Int[u*(1 + 2*c*(x^n/b))^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && !IntegerQ[2*p] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)]
\[\int \left (a^{2}+2 a b \,x^{n}+b^{2} x^{2 n}\right )^{-\frac {1+2 n}{2 n}}d x\]
Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.80 \[ \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{\frac {-1-2 n}{2 n}} \, dx=\frac {b^{2} n x x^{2 \, n} + {\left (2 \, a b n + a b\right )} x x^{n} + {\left (a^{2} n + a^{2}\right )} x}{{\left (a^{2} n + a^{2}\right )} {\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {2 \, n + 1}{2 \, n}}} \]
(b^2*n*x*x^(2*n) + (2*a*b*n + a*b)*x*x^n + (a^2*n + a^2)*x)/((a^2*n + a^2) *(b^2*x^(2*n) + 2*a*b*x^n + a^2)^(1/2*(2*n + 1)/n))
\[ \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{\frac {-1-2 n}{2 n}} \, dx=\begin {cases} \frac {x}{2 \sqrt {\frac {b^{2}}{x^{2}}}} & \text {for}\: a = 0 \wedge n = -1 \\- x \left (b^{2} x^{2 n}\right )^{-2 - \frac {1}{n}} \left (b^{2} x^{2 n}\right )^{1 + \frac {1}{2 n}} + x \left (b^{2} x^{2 n}\right )^{-1 - \frac {1}{2 n}} - \frac {x \left (b^{2} x^{2 n}\right )^{-2 - \frac {1}{n}} \left (b^{2} x^{2 n}\right )^{1 + \frac {1}{2 n}}}{2 n} & \text {for}\: a = 0 \\\int \frac {1}{\sqrt {a^{2} + \frac {2 a b}{x} + \frac {b^{2}}{x^{2}}}}\, dx & \text {for}\: n = -1 \\\frac {a^{2} n x \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{-1 - \frac {1}{2 n}}}{a^{2} n + a^{2}} + \frac {a^{2} x \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{-1 - \frac {1}{2 n}}}{a^{2} n + a^{2}} + \frac {2 a b n x x^{n} \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{-1 - \frac {1}{2 n}}}{a^{2} n + a^{2}} + \frac {a b x x^{n} \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{-1 - \frac {1}{2 n}}}{a^{2} n + a^{2}} + \frac {b^{2} n x x^{2 n} \left (a^{2} + 2 a b x^{n} + b^{2} x^{2 n}\right )^{-1 - \frac {1}{2 n}}}{a^{2} n + a^{2}} & \text {otherwise} \end {cases} \]
Piecewise((x/(2*sqrt(b**2/x**2)), Eq(a, 0) & Eq(n, -1)), (-x*(b**2*x**(2*n ))**(-2 - 1/n)*(b**2*x**(2*n))**(1 + 1/(2*n)) + x*(b**2*x**(2*n))**(-1 - 1 /(2*n)) - x*(b**2*x**(2*n))**(-2 - 1/n)*(b**2*x**(2*n))**(1 + 1/(2*n))/(2* n), Eq(a, 0)), (Integral(1/sqrt(a**2 + 2*a*b/x + b**2/x**2), x), Eq(n, -1) ), (a**2*n*x*(a**2 + 2*a*b*x**n + b**2*x**(2*n))**(-1 - 1/(2*n))/(a**2*n + a**2) + a**2*x*(a**2 + 2*a*b*x**n + b**2*x**(2*n))**(-1 - 1/(2*n))/(a**2* n + a**2) + 2*a*b*n*x*x**n*(a**2 + 2*a*b*x**n + b**2*x**(2*n))**(-1 - 1/(2 *n))/(a**2*n + a**2) + a*b*x*x**n*(a**2 + 2*a*b*x**n + b**2*x**(2*n))**(-1 - 1/(2*n))/(a**2*n + a**2) + b**2*n*x*x**(2*n)*(a**2 + 2*a*b*x**n + b**2* x**(2*n))**(-1 - 1/(2*n))/(a**2*n + a**2), True))
\[ \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{\frac {-1-2 n}{2 n}} \, dx=\int { \frac {1}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {2 \, n + 1}{2 \, n}}} \,d x } \]
\[ \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{\frac {-1-2 n}{2 n}} \, dx=\int { \frac {1}{{\left (b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}\right )}^{\frac {2 \, n + 1}{2 \, n}}} \,d x } \]
Timed out. \[ \int \left (a^2+2 a b x^n+b^2 x^{2 n}\right )^{\frac {-1-2 n}{2 n}} \, dx=\int \frac {1}{{\left (a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n\right )}^{\frac {n+\frac {1}{2}}{n}}} \,d x \]