\(\int x^{-2/n} (a+b x^n+c x^{2 n})^p \, dx\) [262]

Optimal result

Integrand size = 24, antiderivative size = 174 \[ \int x^{-2/n} \left (a+b x^n+c x^{2 n}\right )^p \, dx=-\frac {n x^{-\frac {2-n}{n}} \left (1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p \operatorname {AppellF1}\left (-\frac {2-n}{n^2},-p,-p,-\frac {2-n-n^2}{n^2},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2-n} \] Output:

-n*(a+b*x^n+c*x^(2*n))^p*AppellF1(-(2-n)/n^2,-p,-p,-(-n^2-n+2)/n^2,-2*c*x^ 
n/(b-(-4*a*c+b^2)^(1/2)),-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))/(2-n)/(x^((2-n)/ 
n))/((1+2*c*x^n/(b-(-4*a*c+b^2)^(1/2)))^p)/((1+2*c*x^n/(b+(-4*a*c+b^2)^(1/ 
2)))^p)
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.07 \[ \int x^{-2/n} \left (a+b x^n+c x^{2 n}\right )^p \, dx=\frac {n x^{\frac {-2+n}{n}} \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+x^n \left (b+c x^n\right )\right )^p \operatorname {AppellF1}\left (\frac {-2+n}{n^2},-p,-p,1+\frac {-2+n}{n^2},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )}{-2+n} \] Input:

Integrate[(a + b*x^n + c*x^(2*n))^p/x^(2/n),x]
 

Output:

(n*x^((-2 + n)/n)*(a + x^n*(b + c*x^n))^p*AppellF1[(-2 + n)/n^2, -p, -p, 1 
 + (-2 + n)/n^2, (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqrt[ 
b^2 - 4*a*c])])/((-2 + n)*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 
 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^ 
p)
 

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1721, 1012}

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.

\(\displaystyle \int x^{-2/n} \left (a+b x^n+c x^{2 n}\right )^p \, dx\)

\(\Big \downarrow \) 1721

\(\displaystyle \left (\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p \int x^{-2/n} \left (\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1\right )^p \left (\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}+1\right )^pdx\)

\(\Big \downarrow \) 1012

\(\displaystyle -\frac {n x^{-\frac {2-n}{n}} \left (\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}+1\right )^{-p} \left (\frac {2 c x^n}{\sqrt {b^2-4 a c}+b}+1\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p \operatorname {AppellF1}\left (-\frac {2-n}{n^2},-p,-p,-\frac {-n^2-n+2}{n^2},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2-n}\)

Input:

Int[(a + b*x^n + c*x^(2*n))^p/x^(2/n),x]
 

Output:

-((n*(a + b*x^n + c*x^(2*n))^p*AppellF1[-((2 - n)/n^2), -p, -p, -((2 - n - 
 n^2)/n^2), (-2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]), (-2*c*x^n)/(b + Sqrt[b^2 - 
 4*a*c])])/((2 - n)*x^((2 - n)/n)*(1 + (2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^ 
p*(1 + (2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p))
 

Defintions of rubi rules used

rule 1012

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m 
+ 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], x] /; FreeQ[{a, 
 b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n 
 - 1] && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 1721

Int[((d_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_), x 
_Symbol] :> Simp[a^IntPart[p]*((a + b*x^n + c*x^(2*n))^FracPart[p]/((1 + 2* 
c*(x^n/(b + Rt[b^2 - 4*a*c, 2])))^FracPart[p]*(1 + 2*c*(x^n/(b - Rt[b^2 - 4 
*a*c, 2])))^FracPart[p]))   Int[(d*x)^m*(1 + 2*c*(x^n/(b + Sqrt[b^2 - 4*a*c 
])))^p*(1 + 2*c*(x^n/(b - Sqrt[b^2 - 4*a*c])))^p, x], x] /; FreeQ[{a, b, c, 
 d, m, n, p}, x] && EqQ[n2, 2*n]
 

Maple [F]

\[\int \left (a +b \,x^{n}+c \,x^{2 n}\right )^{p} x^{-\frac {2}{n}}d x\]

Input:

int((a+b*x^n+c*x^(2*n))^p/(x^(2/n)),x)
 

Output:

int((a+b*x^n+c*x^(2*n))^p/(x^(2/n)),x)
 

Fricas [F]

\[ \int x^{-2/n} \left (a+b x^n+c x^{2 n}\right )^p \, dx=\int { \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p}}{x^{\frac {2}{n}}} \,d x } \] Input:

integrate((a+b*x^n+c*x^(2*n))^p/(x^(2/n)),x, algorithm="fricas")
 

Output:

integral((c*x^(2*n) + b*x^n + a)^p/x^(2/n), x)
 

Sympy [F(-1)]

Timed out. \[ \int x^{-2/n} \left (a+b x^n+c x^{2 n}\right )^p \, dx=\text {Timed out} \] Input:

integrate((a+b*x**n+c*x**(2*n))**p/(x**(2/n)),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^{-2/n} \left (a+b x^n+c x^{2 n}\right )^p \, dx=\int { \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p}}{x^{\frac {2}{n}}} \,d x } \] Input:

integrate((a+b*x^n+c*x^(2*n))^p/(x^(2/n)),x, algorithm="maxima")
 

Output:

integrate((c*x^(2*n) + b*x^n + a)^p/x^(2/n), x)
 

Giac [F]

\[ \int x^{-2/n} \left (a+b x^n+c x^{2 n}\right )^p \, dx=\int { \frac {{\left (c x^{2 \, n} + b x^{n} + a\right )}^{p}}{x^{\frac {2}{n}}} \,d x } \] Input:

integrate((a+b*x^n+c*x^(2*n))^p/(x^(2/n)),x, algorithm="giac")
 

Output:

integrate((c*x^(2*n) + b*x^n + a)^p/x^(2/n), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^{-2/n} \left (a+b x^n+c x^{2 n}\right )^p \, dx=\int \frac {{\left (a+b\,x^n+c\,x^{2\,n}\right )}^p}{x^{2/n}} \,d x \] Input:

int((a + b*x^n + c*x^(2*n))^p/x^(2/n),x)
 

Output:

int((a + b*x^n + c*x^(2*n))^p/x^(2/n), x)
 

Reduce [F]

\[ \int x^{-2/n} \left (a+b x^n+c x^{2 n}\right )^p \, dx =\text {Too large to display} \] Input:

int((a+b*x^n+c*x^(2*n))^p/(x^(2/n)),x)
 

Output:

(n*((x**(2*n)*c + x**n*b + a)**p*x + x**(2/n)*int((x**(2*n)*c + x**n*b + a 
)**p/(x**((2*n**2 + 2)/n)*c*n**2*p + x**((2*n**2 + 2)/n)*c*n - 2*x**((2*n* 
*2 + 2)/n)*c + x**((n**2 + 2)/n)*b*n**2*p + x**((n**2 + 2)/n)*b*n - 2*x**( 
(n**2 + 2)/n)*b + x**(2/n)*a*n**2*p + x**(2/n)*a*n - 2*x**(2/n)*a),x)*a*n* 
*3*p**2 + x**(2/n)*int((x**(2*n)*c + x**n*b + a)**p/(x**((2*n**2 + 2)/n)*c 
*n**2*p + x**((2*n**2 + 2)/n)*c*n - 2*x**((2*n**2 + 2)/n)*c + x**((n**2 + 
2)/n)*b*n**2*p + x**((n**2 + 2)/n)*b*n - 2*x**((n**2 + 2)/n)*b + x**(2/n)* 
a*n**2*p + x**(2/n)*a*n - 2*x**(2/n)*a),x)*a*n**2*p - 2*x**(2/n)*int((x**( 
2*n)*c + x**n*b + a)**p/(x**((2*n**2 + 2)/n)*c*n**2*p + x**((2*n**2 + 2)/n 
)*c*n - 2*x**((2*n**2 + 2)/n)*c + x**((n**2 + 2)/n)*b*n**2*p + x**((n**2 + 
 2)/n)*b*n - 2*x**((n**2 + 2)/n)*b + x**(2/n)*a*n**2*p + x**(2/n)*a*n - 2* 
x**(2/n)*a),x)*a*n*p - x**(2/n)*int((x**(2*n)*(x**(2*n)*c + x**n*b + a)**p 
)/(x**((2*n**2 + 2)/n)*c*n**2*p + x**((2*n**2 + 2)/n)*c*n - 2*x**((2*n**2 
+ 2)/n)*c + x**((n**2 + 2)/n)*b*n**2*p + x**((n**2 + 2)/n)*b*n - 2*x**((n* 
*2 + 2)/n)*b + x**(2/n)*a*n**2*p + x**(2/n)*a*n - 2*x**(2/n)*a),x)*c*n**3* 
p**2 - x**(2/n)*int((x**(2*n)*(x**(2*n)*c + x**n*b + a)**p)/(x**((2*n**2 + 
 2)/n)*c*n**2*p + x**((2*n**2 + 2)/n)*c*n - 2*x**((2*n**2 + 2)/n)*c + x**( 
(n**2 + 2)/n)*b*n**2*p + x**((n**2 + 2)/n)*b*n - 2*x**((n**2 + 2)/n)*b + x 
**(2/n)*a*n**2*p + x**(2/n)*a*n - 2*x**(2/n)*a),x)*c*n**2*p + 2*x**(2/n)*i 
nt((x**(2*n)*(x**(2*n)*c + x**n*b + a)**p)/(x**((2*n**2 + 2)/n)*c*n**2*...