Integrand size = 15, antiderivative size = 37 \[ \int \frac {1}{\left (a x^n+b x^{1+n}\right )^2} \, dx=\frac {x^{1-2 n} \operatorname {Hypergeometric2F1}\left (2,1-2 n,2-2 n,-\frac {b x}{a}\right )}{a^2 (1-2 n)} \] Output:
x^(1-2*n)*hypergeom([2, 1-2*n],[2-2*n],-b*x/a)/a^2/(1-2*n)
Time = 0.02 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a x^n+b x^{1+n}\right )^2} \, dx=\frac {x^{1-2 n} \operatorname {Hypergeometric2F1}\left (2,1-2 n,2-2 n,-\frac {b x}{a}\right )}{a^2 (1-2 n)} \] Input:
Integrate[(a*x^n + b*x^(1 + n))^(-2),x]
Output:
(x^(1 - 2*n)*Hypergeometric2F1[2, 1 - 2*n, 2 - 2*n, -((b*x)/a)])/(a^2*(1 - 2*n))
Time = 0.29 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {2027, 74}
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.
\(\displaystyle \int \frac {1}{\left (a x^n+b x^{n+1}\right )^2} \, dx\) |
\(\Big \downarrow \) 2027 |
\(\displaystyle \int \frac {x^{-2 n}}{(a+b x)^2}dx\) |
\(\Big \downarrow \) 74 |
\(\displaystyle \frac {x^{1-2 n} \operatorname {Hypergeometric2F1}\left (2,1-2 n,2-2 n,-\frac {b x}{a}\right )}{a^2 (1-2 n)}\) |
Input:
Int[(a*x^n + b*x^(1 + n))^(-2),x]
Output:
(x^(1 - 2*n)*Hypergeometric2F1[2, 1 - 2*n, 2 - 2*n, -((b*x)/a)])/(a^2*(1 - 2*n))
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[c^n*((b*x )^(m + 1)/(b*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[m] && (IntegerQ[n] || (GtQ[c, 0] && !(EqQ[n, -2^(-1)] && EqQ[c^2 - d^2, 0] && GtQ[-d/(b*c), 0])))
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
\[\int \frac {1}{\left (a \,x^{n}+b \,x^{1+n}\right )^{2}}d x\]
Input:
int(1/(a*x^n+b*x^(1+n))^2,x)
Output:
int(1/(a*x^n+b*x^(1+n))^2,x)
\[ \int \frac {1}{\left (a x^n+b x^{1+n}\right )^2} \, dx=\int { \frac {1}{{\left (b x^{n + 1} + a x^{n}\right )}^{2}} \,d x } \] Input:
integrate(1/(a*x^n+b*x^(1+n))^2,x, algorithm="fricas")
Output:
integral(1/(2*a*b*x^(n + 1)*x^n + a^2*x^(2*n) + b^2*x^(2*n + 2)), x)
\[ \int \frac {1}{\left (a x^n+b x^{1+n}\right )^2} \, dx=\int \frac {1}{\left (a x^{n} + b x^{n + 1}\right )^{2}}\, dx \] Input:
integrate(1/(a*x**n+b*x**(1+n))**2,x)
Output:
Integral((a*x**n + b*x**(n + 1))**(-2), x)
\[ \int \frac {1}{\left (a x^n+b x^{1+n}\right )^2} \, dx=\int { \frac {1}{{\left (b x^{n + 1} + a x^{n}\right )}^{2}} \,d x } \] Input:
integrate(1/(a*x^n+b*x^(1+n))^2,x, algorithm="maxima")
Output:
integrate((b*x^(n + 1) + a*x^n)^(-2), x)
\[ \int \frac {1}{\left (a x^n+b x^{1+n}\right )^2} \, dx=\int { \frac {1}{{\left (b x^{n + 1} + a x^{n}\right )}^{2}} \,d x } \] Input:
integrate(1/(a*x^n+b*x^(1+n))^2,x, algorithm="giac")
Output:
integrate((b*x^(n + 1) + a*x^n)^(-2), x)
Time = 9.16 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (a x^n+b x^{1+n}\right )^2} \, dx=\frac {x\,{{}}_2{\mathrm {F}}_1\left (2,1-2\,n;\ 2-2\,n;\ -\frac {b\,x}{a}\right )}{a^2\,x^{2\,n}-2\,a^2\,n\,x^{2\,n}} \] Input:
int(1/(a*x^n + b*x^(n + 1))^2,x)
Output:
(x*hypergeom([2, 1 - 2*n], 2 - 2*n, -(b*x)/a))/(a^2*x^(2*n) - 2*a^2*n*x^(2 *n))
\[ \int \frac {1}{\left (a x^n+b x^{1+n}\right )^2} \, dx=\frac {-4 x^{2 n} \left (\int \frac {x}{2 x^{2 n} a^{2} n -x^{2 n} a^{2}+4 x^{2 n} a b n x -2 x^{2 n} a b x +2 x^{2 n} b^{2} n \,x^{2}-x^{2 n} b^{2} x^{2}}d x \right ) a b \,n^{2}+2 x^{2 n} \left (\int \frac {x}{2 x^{2 n} a^{2} n -x^{2 n} a^{2}+4 x^{2 n} a b n x -2 x^{2 n} a b x +2 x^{2 n} b^{2} n \,x^{2}-x^{2 n} b^{2} x^{2}}d x \right ) a b n -4 x^{2 n} \left (\int \frac {x}{2 x^{2 n} a^{2} n -x^{2 n} a^{2}+4 x^{2 n} a b n x -2 x^{2 n} a b x +2 x^{2 n} b^{2} n \,x^{2}-x^{2 n} b^{2} x^{2}}d x \right ) b^{2} n^{2} x +2 x^{2 n} \left (\int \frac {x}{2 x^{2 n} a^{2} n -x^{2 n} a^{2}+4 x^{2 n} a b n x -2 x^{2 n} a b x +2 x^{2 n} b^{2} n \,x^{2}-x^{2 n} b^{2} x^{2}}d x \right ) b^{2} n x -x}{x^{2 n} a \left (2 b n x +2 a n -b x -a \right )} \] Input:
int(1/(a*x^n+b*x^(1+n))^2,x)
Output:
( - 4*x**(2*n)*int(x/(2*x**(2*n)*a**2*n - x**(2*n)*a**2 + 4*x**(2*n)*a*b*n *x - 2*x**(2*n)*a*b*x + 2*x**(2*n)*b**2*n*x**2 - x**(2*n)*b**2*x**2),x)*a* b*n**2 + 2*x**(2*n)*int(x/(2*x**(2*n)*a**2*n - x**(2*n)*a**2 + 4*x**(2*n)* a*b*n*x - 2*x**(2*n)*a*b*x + 2*x**(2*n)*b**2*n*x**2 - x**(2*n)*b**2*x**2), x)*a*b*n - 4*x**(2*n)*int(x/(2*x**(2*n)*a**2*n - x**(2*n)*a**2 + 4*x**(2*n )*a*b*n*x - 2*x**(2*n)*a*b*x + 2*x**(2*n)*b**2*n*x**2 - x**(2*n)*b**2*x**2 ),x)*b**2*n**2*x + 2*x**(2*n)*int(x/(2*x**(2*n)*a**2*n - x**(2*n)*a**2 + 4 *x**(2*n)*a*b*n*x - 2*x**(2*n)*a*b*x + 2*x**(2*n)*b**2*n*x**2 - x**(2*n)*b **2*x**2),x)*b**2*n*x - x)/(x**(2*n)*a*(2*a*n - a + 2*b*n*x - b*x))