Integrand size = 24, antiderivative size = 55 \[ \int \frac {1}{\left (a^2+2 a c x^n+c^2 x^{2 n}\right )^{3/2}} \, dx=\frac {x \left (a+c x^n\right ) \operatorname {Hypergeometric2F1}\left (3,\frac {1}{n},1+\frac {1}{n},-\frac {c x^n}{a}\right )}{a^3 \sqrt {a^2+2 a c x^n+c^2 x^{2 n}}} \] Output:
x*(a+c*x^n)*hypergeom([3, 1/n],[1+1/n],-c*x^n/a)/a^3/(a^2+2*a*c*x^n+c^2*x^ (2*n))^(1/2)
Time = 0.03 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (a^2+2 a c x^n+c^2 x^{2 n}\right )^{3/2}} \, dx=\frac {x \left (a+c x^n\right )^3 \operatorname {Hypergeometric2F1}\left (3,\frac {1}{n},1+\frac {1}{n},-\frac {c x^n}{a}\right )}{a^3 \left (\left (a+c x^n\right )^2\right )^{3/2}} \] Input:
Integrate[(a^2 + 2*a*c*x^n + c^2*x^(2*n))^(-3/2),x]
Output:
(x*(a + c*x^n)^3*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -((c*x^n)/a)])/( a^3*((a + c*x^n)^2)^(3/2))
Time = 0.19 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.16, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1384, 778}
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.
\(\displaystyle \int \frac {1}{\left (a^2+2 a c x^n+c^2 x^{2 n}\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1384 |
\(\displaystyle \frac {\left (a c^3+c^4 x^n\right ) \int \frac {1}{\left (c^2 x^n+a c\right )^3}dx}{\sqrt {a^2+2 a c x^n+c^2 x^{2 n}}}\) |
\(\Big \downarrow \) 778 |
\(\displaystyle \frac {x \left (a c^3+c^4 x^n\right ) \operatorname {Hypergeometric2F1}\left (3,\frac {1}{n},1+\frac {1}{n},-\frac {c x^n}{a}\right )}{a^3 c^3 \sqrt {a^2+2 a c x^n+c^2 x^{2 n}}}\) |
Input:
Int[(a^2 + 2*a*c*x^n + c^2*x^(2*n))^(-3/2),x]
Output:
(x*(a*c^3 + c^4*x^n)*Hypergeometric2F1[3, n^(-1), 1 + n^(-1), -((c*x^n)/a) ])/(a^3*c^3*Sqrt[a^2 + 2*a*c*x^n + c^2*x^(2*n)])
Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*x*Hypergeometric2F 1[-p, 1/n, 1/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, n, p}, x] && !IGtQ[p , 0] && !IntegerQ[1/n] && !ILtQ[Simplify[1/n + p], 0] && (IntegerQ[p] || 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)])
\[\int \frac {1}{\left (a^{2}+2 a c \,x^{n}+c^{2} x^{2 n}\right )^{\frac {3}{2}}}d x\]
Input:
int(1/(a^2+2*a*c*x^n+c^2*x^(2*n))^(3/2),x)
Output:
int(1/(a^2+2*a*c*x^n+c^2*x^(2*n))^(3/2),x)
\[ \int \frac {1}{\left (a^2+2 a c x^n+c^2 x^{2 n}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c^{2} x^{2 \, n} + 2 \, a c x^{n} + a^{2}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(a^2+2*a*c*x^n+c^2*x^(2*n))^(3/2),x, algorithm="fricas")
Output:
integral(sqrt(c^2*x^(2*n) + 2*a*c*x^n + a^2)/(c^4*x^(4*n) + 4*a^2*c^2*x^(2 *n) + 4*a^3*c*x^n + a^4 + 2*(2*a*c^3*x^n + a^2*c^2)*x^(2*n)), x)
\[ \int \frac {1}{\left (a^2+2 a c x^n+c^2 x^{2 n}\right )^{3/2}} \, dx=\int \frac {1}{\left (a^{2} + 2 a c x^{n} + c^{2} x^{2 n}\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(a**2+2*a*c*x**n+c**2*x**(2*n))**(3/2),x)
Output:
Integral((a**2 + 2*a*c*x**n + c**2*x**(2*n))**(-3/2), x)
\[ \int \frac {1}{\left (a^2+2 a c x^n+c^2 x^{2 n}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c^{2} x^{2 \, n} + 2 \, a c x^{n} + a^{2}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(a^2+2*a*c*x^n+c^2*x^(2*n))^(3/2),x, algorithm="maxima")
Output:
(2*n^2 - 3*n + 1)*integrate(1/2/(a^2*c*n^2*x^n + a^3*n^2), x) + 1/2*(c*(2* n - 1)*x*x^n + a*(3*n - 1)*x)/(a^2*c^2*n^2*x^(2*n) + 2*a^3*c*n^2*x^n + a^4 *n^2)
\[ \int \frac {1}{\left (a^2+2 a c x^n+c^2 x^{2 n}\right )^{3/2}} \, dx=\int { \frac {1}{{\left (c^{2} x^{2 \, n} + 2 \, a c x^{n} + a^{2}\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(a^2+2*a*c*x^n+c^2*x^(2*n))^(3/2),x, algorithm="giac")
Output:
integrate((c^2*x^(2*n) + 2*a*c*x^n + a^2)^(-3/2), x)
Timed out. \[ \int \frac {1}{\left (a^2+2 a c x^n+c^2 x^{2 n}\right )^{3/2}} \, dx=\int \frac {1}{{\left (a^2+c^2\,x^{2\,n}+2\,a\,c\,x^n\right )}^{3/2}} \,d x \] Input:
int(1/(a^2 + c^2*x^(2*n) + 2*a*c*x^n)^(3/2),x)
Output:
int(1/(a^2 + c^2*x^(2*n) + 2*a*c*x^n)^(3/2), x)
\[ \int \frac {1}{\left (a^2+2 a c x^n+c^2 x^{2 n}\right )^{3/2}} \, dx=\int \frac {1}{x^{3 n} c^{3}+3 x^{2 n} a \,c^{2}+3 x^{n} a^{2} c +a^{3}}d x \] Input:
int(1/(a^2+2*a*c*x^n+c^2*x^(2*n))^(3/2),x)
Output:
int(1/(x**(3*n)*c**3 + 3*x**(2*n)*a*c**2 + 3*x**n*a**2*c + a**3),x)