Integrand size = 17, antiderivative size = 66 \[ \int (c x)^{-2 n} \left (a+b x^n\right )^p \, dx=\frac {(c x)^{1-2 n} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2+\frac {1}{n},-p,-1+\frac {1}{n},-\frac {b x^n}{a}\right )}{c (1-2 n)} \] Output:
(c*x)^(1-2*n)*(a+b*x^n)^p*hypergeom([-p, -2+1/n],[-1+1/n],-b*x^n/a)/c/(1-2 *n)/((1+b*x^n/a)^p)
Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.95 \[ \int (c x)^{-2 n} \left (a+b x^n\right )^p \, dx=-\frac {x (c x)^{-2 n} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-2+\frac {1}{n},-p,-1+\frac {1}{n},-\frac {b x^n}{a}\right )}{-1+2 n} \] Input:
Integrate[(a + b*x^n)^p/(c*x)^(2*n),x]
Output:
-((x*(a + b*x^n)^p*Hypergeometric2F1[-2 + n^(-1), -p, -1 + n^(-1), -((b*x^ n)/a)])/((-1 + 2*n)*(c*x)^(2*n)*(1 + (b*x^n)/a)^p))
Time = 0.33 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {889, 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.
\(\displaystyle \int (c x)^{-2 n} \left (a+b x^n\right )^p \, dx\) |
\(\Big \downarrow \) 889 |
\(\displaystyle \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \int (c x)^{-2 n} \left (\frac {b x^n}{a}+1\right )^pdx\) |
\(\Big \downarrow \) 888 |
\(\displaystyle \frac {(c x)^{1-2 n} \left (a+b x^n\right )^p \left (\frac {b x^n}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{n}-2,-p,\frac {1}{n}-1,-\frac {b x^n}{a}\right )}{c (1-2 n)}\) |
Input:
Int[(a + b*x^n)^p/(c*x)^(2*n),x]
Output:
((c*x)^(1 - 2*n)*(a + b*x^n)^p*Hypergeometric2F1[-2 + n^(-1), -p, -1 + n^( -1), -((b*x^n)/a)])/(c*(1 - 2*n)*(1 + (b*x^n)/a)^p)
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[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^I ntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) Int[(c*x) ^m*(1 + b*(x^n/a))^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] && !IGtQ[p, 0 ] && !(ILtQ[p, 0] || GtQ[a, 0])
\[\int \left (a +b \,x^{n}\right )^{p} \left (c x \right )^{-2 n}d x\]
Input:
int((a+b*x^n)^p/((c*x)^(2*n)),x)
Output:
int((a+b*x^n)^p/((c*x)^(2*n)),x)
\[ \int (c x)^{-2 n} \left (a+b x^n\right )^p \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p}}{\left (c x\right )^{2 \, n}} \,d x } \] Input:
integrate((a+b*x^n)^p/((c*x)^(2*n)),x, algorithm="fricas")
Output:
integral((b*x^n + a)^p/(c*x)^(2*n), x)
Result contains complex when optimal does not.
Time = 5.72 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.15 \[ \int (c x)^{-2 n} \left (a+b x^n\right )^p \, dx=\frac {a^{-2 + \frac {1}{n}} a^{p + 2 - \frac {1}{n}} b^{-2 + \frac {1}{n}} b^{2 - \frac {1}{n}} c^{- 2 n} x^{1 - 2 n} \Gamma \left (-2 + \frac {1}{n}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, -2 + \frac {1}{n} \\ -1 + \frac {1}{n} \end {matrix}\middle | {\frac {b x^{n} e^{i \pi }}{a}} \right )}}{n \Gamma \left (-1 + \frac {1}{n}\right )} \] Input:
integrate((a+b*x**n)**p/((c*x)**(2*n)),x)
Output:
a**(-2 + 1/n)*a**(p + 2 - 1/n)*b**(-2 + 1/n)*b**(2 - 1/n)*x**(1 - 2*n)*gam ma(-2 + 1/n)*hyper((-p, -2 + 1/n), (-1 + 1/n,), b*x**n*exp_polar(I*pi)/a)/ (c**(2*n)*n*gamma(-1 + 1/n))
\[ \int (c x)^{-2 n} \left (a+b x^n\right )^p \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p}}{\left (c x\right )^{2 \, n}} \,d x } \] Input:
integrate((a+b*x^n)^p/((c*x)^(2*n)),x, algorithm="maxima")
Output:
integrate((b*x^n + a)^p/(c*x)^(2*n), x)
\[ \int (c x)^{-2 n} \left (a+b x^n\right )^p \, dx=\int { \frac {{\left (b x^{n} + a\right )}^{p}}{\left (c x\right )^{2 \, n}} \,d x } \] Input:
integrate((a+b*x^n)^p/((c*x)^(2*n)),x, algorithm="giac")
Output:
integrate((b*x^n + a)^p/(c*x)^(2*n), x)
Timed out. \[ \int (c x)^{-2 n} \left (a+b x^n\right )^p \, dx=\int \frac {{\left (a+b\,x^n\right )}^p}{{\left (c\,x\right )}^{2\,n}} \,d x \] Input:
int((a + b*x^n)^p/(c*x)^(2*n),x)
Output:
int((a + b*x^n)^p/(c*x)^(2*n), x)
\[ \int (c x)^{-2 n} \left (a+b x^n\right )^p \, dx=\frac {\left (x^{n} b +a \right )^{p} x +x^{2 n} \left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{3 n} b n p -2 x^{3 n} b n +x^{3 n} b +x^{2 n} a n p -2 x^{2 n} a n +x^{2 n} a}d x \right ) a \,n^{2} p^{2}-2 x^{2 n} \left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{3 n} b n p -2 x^{3 n} b n +x^{3 n} b +x^{2 n} a n p -2 x^{2 n} a n +x^{2 n} a}d x \right ) a \,n^{2} p +x^{2 n} \left (\int \frac {\left (x^{n} b +a \right )^{p}}{x^{3 n} b n p -2 x^{3 n} b n +x^{3 n} b +x^{2 n} a n p -2 x^{2 n} a n +x^{2 n} a}d x \right ) a n p}{x^{2 n} c^{2 n} \left (n p -2 n +1\right )} \] Input:
int((a+b*x^n)^p/((c*x)^(2*n)),x)
Output:
((x**n*b + a)**p*x + x**(2*n)*int((x**n*b + a)**p/(x**(3*n)*b*n*p - 2*x**( 3*n)*b*n + x**(3*n)*b + x**(2*n)*a*n*p - 2*x**(2*n)*a*n + x**(2*n)*a),x)*a *n**2*p**2 - 2*x**(2*n)*int((x**n*b + a)**p/(x**(3*n)*b*n*p - 2*x**(3*n)*b *n + x**(3*n)*b + x**(2*n)*a*n*p - 2*x**(2*n)*a*n + x**(2*n)*a),x)*a*n**2* p + x**(2*n)*int((x**n*b + a)**p/(x**(3*n)*b*n*p - 2*x**(3*n)*b*n + x**(3* n)*b + x**(2*n)*a*n*p - 2*x**(2*n)*a*n + x**(2*n)*a),x)*a*n*p)/(x**(2*n)*c **(2*n)*(n*p - 2*n + 1))