\(\int \frac {A+B x^n+C x^{2 n}}{(a+b x^n+c x^{2 n})^{3/2}} \, dx\) [26]

Optimal result

Integrand size = 33, antiderivative size = 383 \[ \int \frac {A+B x^n+C x^{2 n}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\frac {C x}{c (1-n) \sqrt {a+b x^n+c x^{2 n}}}+\frac {(2 B c (1-n)-b C (2-n)) x^{1+n} \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {3}{2},\frac {3}{2},2+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 a c (1-n) (1+n) \sqrt {a+b x^n+c x^{2 n}}}-\frac {(a C-A c (1-n)) x \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {3}{2},\frac {3}{2},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{a c (1-n) \sqrt {a+b x^n+c x^{2 n}}} \] Output:

C*x/c/(1-n)/(a+b*x^n+c*x^(2*n))^(1/2)+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,3/2,3/2,2+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)))/a/c/(1-n)/(1+n)/(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,3/2,3/2,1+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)))/a/c/(1-n)/(a+b*x^n+c*x^(2*n))^(1/2 
)
 

Mathematica [A] (warning: unable to verify)

Time = 3.24 (sec) , antiderivative size = 436, normalized size of antiderivative = 1.14 \[ \int \frac {A+B x^n+C x^{2 n}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\frac {x \left (2 (A b c-2 a B c+a b C) x^n \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},\frac {1}{2},2+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )-(1+n) \left (2 \left (A \left (b^2-2 a c+b c x^n\right )+a \left (-b B+2 a C-2 B c x^n+b C x^n\right )\right )+\left (2 a (b B-2 a C)+A b^2 (-2+n)-4 a A c (-1+n)\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )\right )\right )}{a \left (-b^2+4 a c\right ) n (1+n) \sqrt {a+x^n \left (b+c x^n\right )}} \] Input:

Integrate[(A + B*x^n + C*x^(2*n))/(a + b*x^n + c*x^(2*n))^(3/2),x]
 

Output:

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

\(\displaystyle \int \frac {A+B x^n+C x^{2 n}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2329

\(\displaystyle \int \frac {A+B x^n+C x^{2 n}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}}dx\)

Input:

Int[(A + B*x^n + C*x^(2*n))/(a + b*x^n + c*x^(2*n))^(3/2),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]

\[\int \frac {A +B \,x^{n}+C \,x^{2 n}}{\left (a +b \,x^{n}+c \,x^{2 n}\right )^{\frac {3}{2}}}d x\]

Input:

int((A+B*x^n+C*x^(2*n))/(a+b*x^n+c*x^(2*n))^(3/2),x)
 

Output:

int((A+B*x^n+C*x^(2*n))/(a+b*x^n+c*x^(2*n))^(3/2),x)
 

Fricas [F(-2)]

Exception generated. \[ \int \frac {A+B x^n+C x^{2 n}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \] Input:

integrate((A+B*x^n+C*x^(2*n))/(a+b*x^n+c*x^(2*n))^(3/2),x, algorithm="fric 
as")
 

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x^n+C x^{2 n}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\text {Timed out} \] Input:

integrate((A+B*x**n+C*x**(2*n))/(a+b*x**n+c*x**(2*n))**(3/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {A+B x^n+C x^{2 n}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int { \frac {C x^{2 \, n} + B x^{n} + A}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((A+B*x^n+C*x^(2*n))/(a+b*x^n+c*x^(2*n))^(3/2),x, algorithm="maxi 
ma")
 

Output:

integrate((C*x^(2*n) + B*x^n + A)/(c*x^(2*n) + b*x^n + a)^(3/2), x)
 

Giac [F]

\[ \int \frac {A+B x^n+C x^{2 n}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int { \frac {C x^{2 \, n} + B x^{n} + A}{{\left (c x^{2 \, n} + b x^{n} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((A+B*x^n+C*x^(2*n))/(a+b*x^n+c*x^(2*n))^(3/2),x, algorithm="giac 
")
 

Output:

integrate((C*x^(2*n) + B*x^n + A)/(c*x^(2*n) + b*x^n + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x^n+C x^{2 n}}{\left (a+b x^n+c x^{2 n}\right )^{3/2}} \, dx=\int \frac {A+C\,x^{2\,n}+B\,x^n}{{\left (a+b\,x^n+c\,x^{2\,n}\right )}^{3/2}} \,d x \] Input:

int((A + C*x^(2*n) + B*x^n)/(a + b*x^n + c*x^(2*n))^(3/2),x)
 

Output:

int((A + C*x^(2*n) + B*x^n)/(a + b*x^n + c*x^(2*n))^(3/2), x)
 

Reduce [F]

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

int((A+B*x^n+C*x^(2*n))/(a+b*x^n+c*x^(2*n))^(3/2),x)
 

Output:

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