\(\int x^{\frac {1}{n}} (a+b x^n+c x^{2 n})^p \, dx\) [259]

Optimal result

Integrand size = 22, antiderivative size = 156 \[ \int x^{\frac {1}{n}} \left (a+b x^n+c x^{2 n}\right )^p \, dx=\frac {n x^{1+\frac {1}{n}} \left (1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p \operatorname {AppellF1}\left (\frac {1+n}{n^2},-p,-p,1+\frac {1}{n^2}+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{1+n} \] Output:

n*x^(1+1/n)*(a+b*x^n+c*x^(2*n))^p*AppellF1((1+n)/n^2,-p,-p,1+1/n^2+1/n,-2* 
c*x^n/(b-(-4*a*c+b^2)^(1/2)),-2*c*x^n/(b+(-4*a*c+b^2)^(1/2)))/(1+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 = 0.56 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.17 \[ \int x^{\frac {1}{n}} \left (a+b x^n+c x^{2 n}\right )^p \, dx=\frac {n x^{1+\frac {1}{n}} \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+x^n \left (b+c x^n\right )\right )^p \operatorname {AppellF1}\left (\frac {1+n}{n^2},-p,-p,1+\frac {1}{n^2}+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )}{1+n} \] Input:

Integrate[x^n^(-1)*(a + b*x^n + c*x^(2*n))^p,x]
 

Output:

(n*x^(1 + n^(-1))*(a + x^n*(b + c*x^n))^p*AppellF1[(1 + n)/n^2, -p, -p, 1 
+ n^(-2) + n^(-1), (-2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^n)/(-b + Sqr 
t[b^2 - 4*a*c])])/((1 + 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 = 0.33 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.

\(\displaystyle \int x^{\frac {1}{n}} \left (a+b x^n+c x^{2 n}\right )^p \, dx\)

\(\Big \downarrow \) 1721

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

\(\Big \downarrow \) 1012

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

Input:

Int[x^n^(-1)*(a + b*x^n + c*x^(2*n))^p,x]
 

Output:

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

\[\int x^{\frac {1}{n}} \left (a +b \,x^{n}+c \,x^{2 n}\right )^{p}d x\]

Input:

int(x^(1/n)*(a+b*x^n+c*x^(2*n))^p,x)
 

Output:

int(x^(1/n)*(a+b*x^n+c*x^(2*n))^p,x)
 

Fricas [F]

\[ \int x^{\frac {1}{n}} \left (a+b x^n+c x^{2 n}\right )^p \, dx=\int { {\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} x^{\left (\frac {1}{n}\right )} \,d x } \] Input:

integrate(x^(1/n)*(a+b*x^n+c*x^(2*n))^p,x, algorithm="fricas")
 

Output:

integral((c*x^(2*n) + b*x^n + a)^p*x^(1/n), x)
 

Sympy [F(-1)]

Timed out. \[ \int x^{\frac {1}{n}} \left (a+b x^n+c x^{2 n}\right )^p \, dx=\text {Timed out} \] Input:

integrate(x**(1/n)*(a+b*x**n+c*x**(2*n))**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^{\frac {1}{n}} \left (a+b x^n+c x^{2 n}\right )^p \, dx=\int { {\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} x^{\left (\frac {1}{n}\right )} \,d x } \] Input:

integrate(x^(1/n)*(a+b*x^n+c*x^(2*n))^p,x, algorithm="maxima")
 

Output:

integrate((c*x^(2*n) + b*x^n + a)^p*x^(1/n), x)
 

Giac [F]

\[ \int x^{\frac {1}{n}} \left (a+b x^n+c x^{2 n}\right )^p \, dx=\int { {\left (c x^{2 \, n} + b x^{n} + a\right )}^{p} x^{\left (\frac {1}{n}\right )} \,d x } \] Input:

integrate(x^(1/n)*(a+b*x^n+c*x^(2*n))^p,x, algorithm="giac")
 

Output:

integrate((c*x^(2*n) + b*x^n + a)^p*x^(1/n), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^{\frac {1}{n}} \left (a+b x^n+c x^{2 n}\right )^p \, dx=\int x^{1/n}\,{\left (a+b\,x^n+c\,x^{2\,n}\right )}^p \,d x \] Input:

int(x^(1/n)*(a + b*x^n + c*x^(2*n))^p,x)
 

Output:

int(x^(1/n)*(a + b*x^n + c*x^(2*n))^p, x)
 

Reduce [F]

\[ \int x^{\frac {1}{n}} \left (a+b x^n+c x^{2 n}\right )^p \, dx=\frac {n \left (x^{\frac {1}{n}} \left (x^{2 n} c +x^{n} b +a \right )^{p} x -\left (\int \frac {x^{\frac {2 n^{2}+1}{n}} \left (x^{2 n} c +x^{n} b +a \right )^{p}}{x^{2 n} c \,n^{2} p +x^{2 n} c n +x^{2 n} c +x^{n} b \,n^{2} p +x^{n} b n +x^{n} b +a \,n^{2} p +a n +a}d x \right ) c \,n^{3} p^{2}-\left (\int \frac {x^{\frac {2 n^{2}+1}{n}} \left (x^{2 n} c +x^{n} b +a \right )^{p}}{x^{2 n} c \,n^{2} p +x^{2 n} c n +x^{2 n} c +x^{n} b \,n^{2} p +x^{n} b n +x^{n} b +a \,n^{2} p +a n +a}d x \right ) c \,n^{2} p -\left (\int \frac {x^{\frac {2 n^{2}+1}{n}} \left (x^{2 n} c +x^{n} b +a \right )^{p}}{x^{2 n} c \,n^{2} p +x^{2 n} c n +x^{2 n} c +x^{n} b \,n^{2} p +x^{n} b n +x^{n} b +a \,n^{2} p +a n +a}d x \right ) c n p +\left (\int \frac {x^{\frac {1}{n}} \left (x^{2 n} c +x^{n} b +a \right )^{p}}{x^{2 n} c \,n^{2} p +x^{2 n} c n +x^{2 n} c +x^{n} b \,n^{2} p +x^{n} b n +x^{n} b +a \,n^{2} p +a n +a}d x \right ) a \,n^{3} p^{2}+\left (\int \frac {x^{\frac {1}{n}} \left (x^{2 n} c +x^{n} b +a \right )^{p}}{x^{2 n} c \,n^{2} p +x^{2 n} c n +x^{2 n} c +x^{n} b \,n^{2} p +x^{n} b n +x^{n} b +a \,n^{2} p +a n +a}d x \right ) a \,n^{2} p +\left (\int \frac {x^{\frac {1}{n}} \left (x^{2 n} c +x^{n} b +a \right )^{p}}{x^{2 n} c \,n^{2} p +x^{2 n} c n +x^{2 n} c +x^{n} b \,n^{2} p +x^{n} b n +x^{n} b +a \,n^{2} p +a n +a}d x \right ) a n p \right )}{n^{2} p +n +1} \] Input:

int(x^(1/n)*(a+b*x^n+c*x^(2*n))^p,x)
 

Output:

(n*(x**(1/n)*(x**(2*n)*c + x**n*b + a)**p*x - int((x**((2*n**2 + 1)/n)*(x* 
*(2*n)*c + x**n*b + a)**p)/(x**(2*n)*c*n**2*p + x**(2*n)*c*n + x**(2*n)*c 
+ x**n*b*n**2*p + x**n*b*n + x**n*b + a*n**2*p + a*n + a),x)*c*n**3*p**2 - 
 int((x**((2*n**2 + 1)/n)*(x**(2*n)*c + x**n*b + a)**p)/(x**(2*n)*c*n**2*p 
 + x**(2*n)*c*n + x**(2*n)*c + x**n*b*n**2*p + x**n*b*n + x**n*b + a*n**2* 
p + a*n + a),x)*c*n**2*p - int((x**((2*n**2 + 1)/n)*(x**(2*n)*c + x**n*b + 
 a)**p)/(x**(2*n)*c*n**2*p + x**(2*n)*c*n + x**(2*n)*c + x**n*b*n**2*p + x 
**n*b*n + x**n*b + a*n**2*p + a*n + a),x)*c*n*p + int((x**(1/n)*(x**(2*n)* 
c + x**n*b + a)**p)/(x**(2*n)*c*n**2*p + x**(2*n)*c*n + x**(2*n)*c + x**n* 
b*n**2*p + x**n*b*n + x**n*b + a*n**2*p + a*n + a),x)*a*n**3*p**2 + int((x 
**(1/n)*(x**(2*n)*c + x**n*b + a)**p)/(x**(2*n)*c*n**2*p + x**(2*n)*c*n + 
x**(2*n)*c + x**n*b*n**2*p + x**n*b*n + x**n*b + a*n**2*p + a*n + a),x)*a* 
n**2*p + int((x**(1/n)*(x**(2*n)*c + x**n*b + a)**p)/(x**(2*n)*c*n**2*p + 
x**(2*n)*c*n + x**(2*n)*c + x**n*b*n**2*p + x**n*b*n + x**n*b + a*n**2*p + 
 a*n + a),x)*a*n*p))/(n**2*p + n + 1)