\(\int \frac {x^{4 n}}{\sqrt {a+b x^n} (c+d x^n) (e+f x^n)} \, dx\) [66]

Optimal result

Integrand size = 35, antiderivative size = 381 \[ \int \frac {x^{4 n}}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (e+f x^n\right )} \, dx=\frac {c^3 \left (4+8 n+3 n^2\right ) x \sqrt {1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{d^3 (d e-c f) (2+n) (2+3 n) \sqrt {a+b x^n}}-\frac {e^3 \left (4+8 n+3 n^2\right ) x \sqrt {1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {f x^n}{e}\right )}{f^3 (d e-c f) (2+n) (2+3 n) \sqrt {a+b x^n}}+\frac {x \left (-2 d f \left (a+b x^n\right ) \left (2 a d f (1+n)+b \left (d e (2+3 n)+c f (2+3 n)-d f (2+n) x^n\right )\right )+\left (4 a^2 d^2 f^2 (1+n)+2 a b d f (d e+c f) (2+3 n)+b^2 \left (d^2 e^2+c d e f+c^2 f^2\right ) \left (4+8 n+3 n^2\right )\right ) \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )\right )}{b^2 d^3 f^3 (2+n) (2+3 n) \sqrt {a+b x^n}} \] Output:

c^3*(3*n^2+8*n+4)*x*(1+b*x^n/a)^(1/2)*AppellF1(1/n,1/2,1,1+1/n,-b*x^n/a,-d 
*x^n/c)/d^3/(-c*f+d*e)/(2+n)/(2+3*n)/(a+b*x^n)^(1/2)-e^3*(3*n^2+8*n+4)*x*( 
1+b*x^n/a)^(1/2)*AppellF1(1/n,1/2,1,1+1/n,-b*x^n/a,-f*x^n/e)/f^3/(-c*f+d*e 
)/(2+n)/(2+3*n)/(a+b*x^n)^(1/2)+x*(-2*d*f*(a+b*x^n)*(2*a*d*f*(1+n)+b*(d*e* 
(2+3*n)+c*f*(2+3*n)-d*f*(2+n)*x^n))+(4*a^2*d^2*f^2*(1+n)+2*a*b*d*f*(c*f+d* 
e)*(2+3*n)+b^2*(c^2*f^2+c*d*e*f+d^2*e^2)*(3*n^2+8*n+4))*(1+b*x^n/a)^(1/2)* 
hypergeom([1/2, 1/n],[1+1/n],-b*x^n/a))/b^2/d^3/f^3/(2+n)/(2+3*n)/(a+b*x^n 
)^(1/2)
 

Mathematica [F]

\[ \int \frac {x^{4 n}}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (e+f x^n\right )} \, dx=\int \frac {x^{4 n}}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (e+f x^n\right )} \, dx \] Input:

Integrate[x^(4*n)/(Sqrt[a + b*x^n]*(c + d*x^n)*(e + f*x^n)),x]
 

Output:

Integrate[x^(4*n)/(Sqrt[a + b*x^n]*(c + d*x^n)*(e + f*x^n)), x]
 

Rubi [F]

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.

\(\displaystyle \int \frac {x^{4 n}}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (e+f x^n\right )} \, dx\)

\(\Big \downarrow \) 1073

\(\displaystyle \int \frac {x^{4 n}}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (e+f x^n\right )}dx\)

Input:

Int[x^(4*n)/(Sqrt[a + b*x^n]*(c + d*x^n)*(e + f*x^n)),x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 1073

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :> Unintegrable[(g*x)^m 
*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, 
g, m, n, p, q, r}, x]
 

Maple [F]

\[\int \frac {x^{4 n}}{\sqrt {a +b \,x^{n}}\, \left (c +d \,x^{n}\right ) \left (e +f \,x^{n}\right )}d x\]

Input:

int(x^(4*n)/(a+b*x^n)^(1/2)/(c+d*x^n)/(e+f*x^n),x)
 

Output:

int(x^(4*n)/(a+b*x^n)^(1/2)/(c+d*x^n)/(e+f*x^n),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {x^{4 n}}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (e+f x^n\right )} \, dx=\text {Exception raised: TypeError} \] Input:

integrate(x^(4*n)/(a+b*x^n)^(1/2)/(c+d*x^n)/(e+f*x^n),x, algorithm="fricas 
")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (has polynomial part)
 

Sympy [F(-2)]

Exception generated. \[ \int \frac {x^{4 n}}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (e+f x^n\right )} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate(x**(4*n)/(a+b*x**n)**(1/2)/(c+d*x**n)/(e+f*x**n),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

\[ \int \frac {x^{4 n}}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (e+f x^n\right )} \, dx=\int { \frac {x^{4 \, n}}{\sqrt {b x^{n} + a} {\left (d x^{n} + c\right )} {\left (f x^{n} + e\right )}} \,d x } \] Input:

integrate(x^(4*n)/(a+b*x^n)^(1/2)/(c+d*x^n)/(e+f*x^n),x, algorithm="maxima 
")
 

Output:

integrate(x^(4*n)/(sqrt(b*x^n + a)*(d*x^n + c)*(f*x^n + e)), x)
 

Giac [F]

\[ \int \frac {x^{4 n}}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (e+f x^n\right )} \, dx=\int { \frac {x^{4 \, n}}{\sqrt {b x^{n} + a} {\left (d x^{n} + c\right )} {\left (f x^{n} + e\right )}} \,d x } \] Input:

integrate(x^(4*n)/(a+b*x^n)^(1/2)/(c+d*x^n)/(e+f*x^n),x, algorithm="giac")
 

Output:

integrate(x^(4*n)/(sqrt(b*x^n + a)*(d*x^n + c)*(f*x^n + e)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^{4 n}}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (e+f x^n\right )} \, dx=\int \frac {x^{4\,n}}{\sqrt {a+b\,x^n}\,\left (c+d\,x^n\right )\,\left (e+f\,x^n\right )} \,d x \] Input:

int(x^(4*n)/((a + b*x^n)^(1/2)*(c + d*x^n)*(e + f*x^n)),x)
 

Output:

int(x^(4*n)/((a + b*x^n)^(1/2)*(c + d*x^n)*(e + f*x^n)), x)
 

Reduce [F]

\[ \int \frac {x^{4 n}}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (e+f x^n\right )} \, dx=\int \frac {x^{4 n}}{\sqrt {x^{n} b +a}\, \left (x^{n} d +c \right ) \left (e +f \,x^{n}\right )}d x \] Input:

int(x^(4*n)/(a+b*x^n)^(1/2)/(c+d*x^n)/(e+f*x^n),x)
 

Output:

int(x^(4*n)/(a+b*x^n)^(1/2)/(c+d*x^n)/(e+f*x^n),x)