\(\int \frac {A+B x^n+C x^{2 n}}{\sqrt {a+b x^n+c x^{2 n}}} \, dx\) [25]

Optimal result

Integrand size = 33, antiderivative size = 360 \[ \int \frac {A+B x^n+C x^{2 n}}{\sqrt {a+b x^n+c x^{2 n}}} \, dx=\frac {C x \sqrt {a+b x^n+c x^{2 n}}}{c (1+n)}+\frac {(2 B c (1+n)-b C (2+n)) x^{1+n} \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},\frac {1}{2},2+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 c (1+n)^2 \sqrt {a+b x^n+c x^{2 n}}}-\frac {(a C-A c (1+n)) x \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{c (1+n) \sqrt {a+b x^n+c x^{2 n}}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 1.68 (sec) , antiderivative size = 379, normalized size of antiderivative = 1.05 \[ \int \frac {A+B x^n+C x^{2 n}}{\sqrt {a+b x^n+c x^{2 n}}} \, dx=\frac {x \left (-\left ((-2 B c (1+n)+b C (2+n)) x^n \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},\frac {1}{2},2+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )\right )+2 (1+n) \left (C \left (a+x^n \left (b+c x^n\right )\right )+(-a C+A c (1+n)) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )\right )\right )}{2 c (1+n)^2 \sqrt {a+x^n \left (b+c x^n\right )}} \] Input:

Integrate[(A + B*x^n + C*x^(2*n))/Sqrt[a + b*x^n + c*x^(2*n)],x]
 

Output:

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

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.

\(\displaystyle \int \frac {A+B x^n+C x^{2 n}}{\sqrt {a+b x^n+c x^{2 n}}} \, dx\)

\(\Big \downarrow \) 2329

\(\displaystyle \int \frac {A+B x^n+C x^{2 n}}{\sqrt {a+b x^n+c x^{2 n}}}dx\)

Input:

Int[(A + B*x^n + C*x^(2*n))/Sqrt[a + b*x^n + c*x^(2*n)],x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2329

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> 
 Unintegrable[Pq*(a + b*x^n + c*x^(2*n))^p, x] /; FreeQ[{a, b, c, n, p}, x] 
 && EqQ[n2, 2*n] && (PolyQ[Pq, x] || PolyQ[Pq, x^n])
 

Maple [F]

\[\int \frac {A +B \,x^{n}+C \,x^{2 n}}{\sqrt {a +b \,x^{n}+c \,x^{2 n}}}d x\]

Input:

int((A+B*x^n+C*x^(2*n))/(a+b*x^n+c*x^(2*n))^(1/2),x)
 

Output:

int((A+B*x^n+C*x^(2*n))/(a+b*x^n+c*x^(2*n))^(1/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {A+B x^n+C x^{2 n}}{\sqrt {a+b x^n+c x^{2 n}}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((A+B*x^n+C*x^(2*n))/(a+b*x^n+c*x^(2*n))^(1/2),x, algorithm="fric 
as")
 

Output:

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

Sympy [F]

\[ \int \frac {A+B x^n+C x^{2 n}}{\sqrt {a+b x^n+c x^{2 n}}} \, dx=\int \frac {A + B x^{n} + C x^{2 n}}{\sqrt {a + b x^{n} + c x^{2 n}}}\, dx \] Input:

integrate((A+B*x**n+C*x**(2*n))/(a+b*x**n+c*x**(2*n))**(1/2),x)
 

Output:

Integral((A + B*x**n + C*x**(2*n))/sqrt(a + b*x**n + c*x**(2*n)), x)
 

Maxima [F]

\[ \int \frac {A+B x^n+C x^{2 n}}{\sqrt {a+b x^n+c x^{2 n}}} \, dx=\int { \frac {C x^{2 \, n} + B x^{n} + A}{\sqrt {c x^{2 \, n} + b x^{n} + a}} \,d x } \] Input:

integrate((A+B*x^n+C*x^(2*n))/(a+b*x^n+c*x^(2*n))^(1/2),x, algorithm="maxi 
ma")
 

Output:

integrate((C*x^(2*n) + B*x^n + A)/sqrt(c*x^(2*n) + b*x^n + a), x)
 

Giac [F]

\[ \int \frac {A+B x^n+C x^{2 n}}{\sqrt {a+b x^n+c x^{2 n}}} \, dx=\int { \frac {C x^{2 \, n} + B x^{n} + A}{\sqrt {c x^{2 \, n} + b x^{n} + a}} \,d x } \] Input:

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

Output:

integrate((C*x^(2*n) + B*x^n + A)/sqrt(c*x^(2*n) + b*x^n + a), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^n+C x^{2 n}}{\sqrt {a+b x^n+c x^{2 n}}} \, dx=\int \frac {A+C\,x^{2\,n}+B\,x^n}{\sqrt {a+b\,x^n+c\,x^{2\,n}}} \,d x \] Input:

int((A + C*x^(2*n) + B*x^n)/(a + b*x^n + c*x^(2*n))^(1/2),x)
 

Output:

int((A + C*x^(2*n) + B*x^n)/(a + b*x^n + c*x^(2*n))^(1/2), x)
 

Reduce [F]

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

int((A+B*x^n+C*x^(2*n))/(a+b*x^n+c*x^(2*n))^(1/2),x)
 

Output:

(2*sqrt(x**(2*n)*c + x**n*b + a)*x + int(sqrt(x**(2*n)*c + x**n*b + a)/(x* 
*(2*n)*c*n + 2*x**(2*n)*c + x**n*b*n + 2*x**n*b + a*n + 2*a),x)*a*n**2 + 2 
*int(sqrt(x**(2*n)*c + x**n*b + a)/(x**(2*n)*c*n + 2*x**(2*n)*c + x**n*b*n 
 + 2*x**n*b + a*n + 2*a),x)*a*n - int((x**(2*n)*sqrt(x**(2*n)*c + x**n*b + 
 a))/(x**(2*n)*c*n + 2*x**(2*n)*c + x**n*b*n + 2*x**n*b + a*n + 2*a),x)*c* 
n**2 - 2*int((x**(2*n)*sqrt(x**(2*n)*c + x**n*b + a))/(x**(2*n)*c*n + 2*x* 
*(2*n)*c + x**n*b*n + 2*x**n*b + a*n + 2*a),x)*c*n)/(n + 2)