\(\int \frac {x^n}{\sqrt {a+b x^n} (c+d x^n) (e+f x^n)} \, dx\) [63]

Optimal result

Integrand size = 33, antiderivative size = 138 \[ \int \frac {x^n}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (e+f x^n\right )} \, dx=-\frac {x \sqrt {1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{(d e-c f) \sqrt {a+b x^n}}+\frac {x \sqrt {1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {f x^n}{e}\right )}{(d e-c f) \sqrt {a+b x^n}} \] Output:

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

Mathematica [F]

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

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

Output:

Integrate[x^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.

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

\(\Big \downarrow \) 1073

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

Input:

Int[x^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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral(sqrt(b*x^n + a)*x^n/(b*d*f*x^(3*n) + a*c*e + (b*d*e + (b*c + a*d) 
*f)*x^(2*n) + (a*c*f + (b*c + a*d)*e)*x^n), x)
 

Sympy [F(-2)]

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

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

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((x**n*sqrt(x**n*b + a))/(x**(3*n)*b*d*f + x**(2*n)*a*d*f + x**(2*n)*b* 
c*f + x**(2*n)*b*d*e + x**n*a*c*f + x**n*a*d*e + x**n*b*c*e + a*c*e),x)