\(\int \frac {x^{3 n}}{\sqrt {a+b x^n} (c+d x^n) (e+f x^n)} \, dx\) [65]

Optimal result

Integrand size = 35, antiderivative size = 259 \[ \int \frac {x^{3 n}}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (e+f x^n\right )} \, dx=\frac {2 x \sqrt {a+b x^n}}{b d f (2+n)}-\frac {c^2 x \sqrt {1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{d^2 (d e-c f) \sqrt {a+b x^n}}+\frac {e^2 x \sqrt {1+\frac {b x^n}{a}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},1,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {f x^n}{e}\right )}{f^2 (d e-c f) \sqrt {a+b x^n}}-\frac {(2 a d f+b (d e+c f) (2+n)) x \sqrt {1+\frac {b x^n}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {1}{n},1+\frac {1}{n},-\frac {b x^n}{a}\right )}{b d^2 f^2 (2+n) \sqrt {a+b x^n}} \] Output:

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

Mathematica [F]

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

Integrate[x^(3*n)/(Sqrt[a + b*x^n]*(c + d*x^n)*(e + f*x^n)),x]
 

Output:

Integrate[x^(3*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.

\(\displaystyle \int \frac {x^{3 n}}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (e+f x^n\right )} \, dx\)

\(\Big \downarrow \) 1073

\(\displaystyle \int \frac {x^{3 n}}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (e+f x^n\right )}dx\)

Input:

Int[x^(3*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]

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

Input:

int(x^(3*n)/(a+b*x^n)^(1/2)/(c+d*x^n)/(e+f*x^n),x)
 

Output:

int(x^(3*n)/(a+b*x^n)^(1/2)/(c+d*x^n)/(e+f*x^n),x)
 

Fricas [F(-2)]

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

integrate(x^(3*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. \[ \int \frac {x^{3 n}}{\sqrt {a+b x^n} \left (c+d x^n\right ) \left (e+f x^n\right )} \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate(x**(3*n)/(a+b*x**n)**(1/2)/(c+d*x**n)/(e+f*x**n),x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

integrate(x^(3*n)/(a+b*x^n)^(1/2)/(c+d*x^n)/(e+f*x^n),x, algorithm="maxima 
")
 

Output:

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

Giac [F]

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

integrate(x^(3*n)/(a+b*x^n)^(1/2)/(c+d*x^n)/(e+f*x^n),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int(x^(3*n)/((a + b*x^n)^(1/2)*(c + d*x^n)*(e + f*x^n)),x)
 

Output:

int(x^(3*n)/((a + b*x^n)^(1/2)*(c + d*x^n)*(e + f*x^n)), x)
 

Reduce [F]

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

int(x^(3*n)/(a+b*x^n)^(1/2)/(c+d*x^n)/(e+f*x^n),x)
 

Output:

int((x**(3*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 
)