Integrand size = 28, antiderivative size = 65 \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx=-\frac {\left (a+b x^n\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},-\frac {1-n}{n},-\frac {b x^n}{a}\right )}{a x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \] Output:
-(a+b*x^n)*hypergeom([1, -1/n],[-(1-n)/n],-b*x^n/a)/a/x/(a^2+2*a*b*x^n+b^2 *x^(2*n))^(1/2)
Time = 0.05 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.78 \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx=-\frac {\left (a+b x^n\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},1-\frac {1}{n},-\frac {b x^n}{a}\right )}{a x \sqrt {\left (a+b x^n\right )^2}} \] Input:
Integrate[1/(x^2*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]),x]
Output:
-(((a + b*x^n)*Hypergeometric2F1[1, -n^(-1), 1 - n^(-1), -((b*x^n)/a)])/(a *x*Sqrt[(a + b*x^n)^2]))
Time = 0.20 (sec) , antiderivative size = 72, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {1384, 888}
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.
\(\displaystyle \int \frac {1}{x^2 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx\) |
\(\Big \downarrow \) 1384 |
\(\displaystyle \frac {\left (a b+b^2 x^n\right ) \int \frac {1}{x^2 \left (b^2 x^n+a b\right )}dx}{\sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\) |
\(\Big \downarrow \) 888 |
\(\displaystyle -\frac {\left (a b+b^2 x^n\right ) \operatorname {Hypergeometric2F1}\left (1,-\frac {1}{n},-\frac {1-n}{n},-\frac {b x^n}{a}\right )}{a b x \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}}\) |
Input:
Int[1/(x^2*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]),x]
Output:
-(((a*b + b^2*x^n)*Hypergeometric2F1[1, -n^(-1), -((1 - n)/n), -((b*x^n)/a )])/(a*b*x*Sqrt[a^2 + 2*a*b*x^n + b^2*x^(2*n)]))
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p *((c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/n, (m + 1)/n + 1 , (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0] && (ILt Q[p, 0] || GtQ[a, 0])
Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> S imp[(a + b*x^n + c*x^(2*n))^FracPart[p]/(c^IntPart[p]*(b/2 + c*x^n)^(2*Frac Part[p])) Int[u*(b/2 + c*x^n)^(2*p), x], x] /; FreeQ[{a, b, c, n, p}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p - 1/2] && NeQ[u, x^(n - 1)] && NeQ[u, x^(2*n - 1)] && !(EqQ[p, 1/2] && EqQ[u, x^(-2*n - 1)])
\[\int \frac {1}{x^{2} \sqrt {a^{2}+2 x^{n} a b +b^{2} x^{2 n}}}d x\]
Input:
int(1/x^2/(a^2+2*x^n*a*b+b^2*x^(2*n))^(1/2),x)
Output:
int(1/x^2/(a^2+2*x^n*a*b+b^2*x^(2*n))^(1/2),x)
\[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx=\int { \frac {1}{\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2),x, algorithm="fricas")
Output:
integral(sqrt(b^2*x^(2*n) + 2*a*b*x^n + a^2)/(b^2*x^2*x^(2*n) + 2*a*b*x^2* x^n + a^2*x^2), x)
\[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx=\int \frac {1}{x^{2} \sqrt {\left (a + b x^{n}\right )^{2}}}\, dx \] Input:
integrate(1/x**2/(a**2+2*a*b*x**n+b**2*x**(2*n))**(1/2),x)
Output:
Integral(1/(x**2*sqrt((a + b*x**n)**2)), x)
\[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx=\int { \frac {1}{\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2),x, algorithm="maxima")
Output:
integrate(1/(sqrt(b^2*x^(2*n) + 2*a*b*x^n + a^2)*x^2), x)
\[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx=\int { \frac {1}{\sqrt {b^{2} x^{2 \, n} + 2 \, a b x^{n} + a^{2}} x^{2}} \,d x } \] Input:
integrate(1/x^2/(a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2),x, algorithm="giac")
Output:
integrate(1/(sqrt(b^2*x^(2*n) + 2*a*b*x^n + a^2)*x^2), x)
Timed out. \[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx=\int \frac {1}{x^2\,\sqrt {a^2+b^2\,x^{2\,n}+2\,a\,b\,x^n}} \,d x \] Input:
int(1/(x^2*(a^2 + b^2*x^(2*n) + 2*a*b*x^n)^(1/2)),x)
Output:
int(1/(x^2*(a^2 + b^2*x^(2*n) + 2*a*b*x^n)^(1/2)), x)
\[ \int \frac {1}{x^2 \sqrt {a^2+2 a b x^n+b^2 x^{2 n}}} \, dx=\int \frac {1}{x^{n} b \,x^{2}+a \,x^{2}}d x \] Input:
int(1/x^2/(a^2+2*a*b*x^n+b^2*x^(2*n))^(1/2),x)
Output:
int(1/(x**n*b*x**2 + a*x**2),x)