\(\int \sqrt {a+b x^n+c x^{2 n}} (A+B x^n+C x^{2 n}) \, dx\) [24]

Optimal result

Integrand size = 33, antiderivative size = 376 \[ \int \sqrt {a+b x^n+c x^{2 n}} \left (A+B x^n+C x^{2 n}\right ) \, dx=\frac {C x \left (a+b x^n+c x^{2 n}\right )^{3/2}}{c (1+3 n)}-\frac {(b C (2+3 n)-2 B (c+3 c n)) x^{1+n} \sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (1+\frac {1}{n},-\frac {1}{2},-\frac {1}{2},2+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{2 c (1+n) (1+3 n) \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}}-\frac {(a C-A (c+3 c n)) x \sqrt {a+b x^n+c x^{2 n}} \operatorname {AppellF1}\left (\frac {1}{n},-\frac {1}{2},-\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{c (1+3 n) \sqrt {1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}}} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 4.41 (sec) , antiderivative size = 576, normalized size of antiderivative = 1.53 \[ \int \sqrt {a+b x^n+c x^{2 n}} \left (A+B x^n+C x^{2 n}\right ) \, dx=\frac {x \left (n \left (8 a B c^2 \left (1+4 n+3 n^2\right )-2 b^2 B c \left (2+7 n+3 n^2\right )+b^3 C \left (4+8 n+3 n^2\right )+4 b c \left (-a C \left (3+7 n+3 n^2\right )+A c \left (1+5 n+6 n^2\right )\right )\right ) x^n \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (1+\frac {1}{n},\frac {1}{2},\frac {1}{2},2+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )+2 (1+n) \left (\left (a+x^n \left (b+c x^n\right )\right ) \left (-b^2 C n (2+3 n)+4 A c^2 \left (1+5 n+6 n^2\right )+2 b c n \left (B+3 B n+C (1+n) x^n\right )+4 c \left (a C n (1+2 n)+c (1+n) x^n \left (B+3 B n+C (1+2 n) x^n\right )\right )\right )+a n \left (-2 b B c (1+3 n)+b^2 C (2+3 n)+4 c (1+2 n) (-a C+A (c+3 c n))\right ) \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{n},\frac {1}{2},\frac {1}{2},1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )\right )\right )}{8 c^2 (1+n)^2 \left (1+5 n+6 n^2\right ) \sqrt {a+x^n \left (b+c x^n\right )}} \] Input:

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

Output:

(x*(n*(8*a*B*c^2*(1 + 4*n + 3*n^2) - 2*b^2*B*c*(2 + 7*n + 3*n^2) + b^3*C*( 
4 + 8*n + 3*n^2) + 4*b*c*(-(a*C*(3 + 7*n + 3*n^2)) + A*c*(1 + 5*n + 6*n^2) 
))*x^n*Sqrt[(b - Sqrt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c])]*Sqr 
t[(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)*((a + x^n*(b + c*x^n))*(-(b^2*C*n* 
(2 + 3*n)) + 4*A*c^2*(1 + 5*n + 6*n^2) + 2*b*c*n*(B + 3*B*n + C*(1 + n)*x^ 
n) + 4*c*(a*C*n*(1 + 2*n) + c*(1 + n)*x^n*(B + 3*B*n + C*(1 + 2*n)*x^n))) 
+ a*n*(-2*b*B*c*(1 + 3*n) + b^2*C*(2 + 3*n) + 4*c*(1 + 2*n)*(-(a*C) + A*(c 
 + 3*c*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])])))/(8*c^2*(1 + n)^2*(1 + 5*n + 6*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.

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

\(\Big \downarrow \) 2329

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

Input:

Int[Sqrt[a + b*x^n + c*x^(2*n)]*(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]

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

Input:

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

Output:

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

Fricas [F(-2)]

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

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

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

integrate((a+b*x^n+c*x^(2*n))^(1/2)*(A+B*x^n+C*x^(2*n)),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. \[ \int \sqrt {a+b x^n+c x^{2 n}} \left (A+B x^n+C x^{2 n}\right ) \, dx=\int \sqrt {a+b\,x^n+c\,x^{2\,n}}\,\left (A+C\,x^{2\,n}+B\,x^n\right ) \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \sqrt {a+b x^n+c x^{2 n}} \left (A+B x^n+C x^{2 n}\right ) \, dx=\text {too large to display} \] Input:

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

Output:

(8*x**(2*n)*sqrt(x**(2*n)*c + x**n*b + a)*c*n**2*x + 20*x**(2*n)*sqrt(x**( 
2*n)*c + x**n*b + a)*c*n*x + 8*x**(2*n)*sqrt(x**(2*n)*c + x**n*b + a)*c*x 
+ 14*x**n*sqrt(x**(2*n)*c + x**n*b + a)*b*n**2*x + 32*x**n*sqrt(x**(2*n)*c 
 + x**n*b + a)*b*n*x + 8*x**n*sqrt(x**(2*n)*c + x**n*b + a)*b*x + 68*sqrt( 
x**(2*n)*c + x**n*b + a)*a*n**2*x + 44*sqrt(x**(2*n)*c + x**n*b + a)*a*n*x 
 + 8*sqrt(x**(2*n)*c + x**n*b + a)*a*x + 144*int(sqrt(x**(2*n)*c + x**n*b 
+ a)/(6*x**(2*n)*c*n**3 + 17*x**(2*n)*c*n**2 + 11*x**(2*n)*c*n + 2*x**(2*n 
)*c + 6*x**n*b*n**3 + 17*x**n*b*n**2 + 11*x**n*b*n + 2*x**n*b + 6*a*n**3 + 
 17*a*n**2 + 11*a*n + 2*a),x)*a**2*n**6 + 408*int(sqrt(x**(2*n)*c + x**n*b 
 + a)/(6*x**(2*n)*c*n**3 + 17*x**(2*n)*c*n**2 + 11*x**(2*n)*c*n + 2*x**(2* 
n)*c + 6*x**n*b*n**3 + 17*x**n*b*n**2 + 11*x**n*b*n + 2*x**n*b + 6*a*n**3 
+ 17*a*n**2 + 11*a*n + 2*a),x)*a**2*n**5 + 264*int(sqrt(x**(2*n)*c + x**n* 
b + a)/(6*x**(2*n)*c*n**3 + 17*x**(2*n)*c*n**2 + 11*x**(2*n)*c*n + 2*x**(2 
*n)*c + 6*x**n*b*n**3 + 17*x**n*b*n**2 + 11*x**n*b*n + 2*x**n*b + 6*a*n**3 
 + 17*a*n**2 + 11*a*n + 2*a),x)*a**2*n**4 + 48*int(sqrt(x**(2*n)*c + x**n* 
b + a)/(6*x**(2*n)*c*n**3 + 17*x**(2*n)*c*n**2 + 11*x**(2*n)*c*n + 2*x**(2 
*n)*c + 6*x**n*b*n**3 + 17*x**n*b*n**2 + 11*x**n*b*n + 2*x**n*b + 6*a*n**3 
 + 17*a*n**2 + 11*a*n + 2*a),x)*a**2*n**3 - 216*int((x**(2*n)*sqrt(x**(2*n 
)*c + x**n*b + a))/(6*x**(2*n)*c*n**3 + 17*x**(2*n)*c*n**2 + 11*x**(2*n)*c 
*n + 2*x**(2*n)*c + 6*x**n*b*n**3 + 17*x**n*b*n**2 + 11*x**n*b*n + 2*x*...