\(\int (a+b x^n+c x^{2 n})^p (A+B x^n+C x^{2 n}) \, dx\) [34]

Optimal result

Integrand size = 31, antiderivative size = 387 \[ \int \left (a+b x^n+c x^{2 n}\right )^p \left (A+B x^n+C x^{2 n}\right ) \, dx=\frac {C x \left (a+b x^n+c x^{2 n}\right )^{1+p}}{c (1+2 n+2 n p)}-\frac {(b C (1+n+n p)-B c (1+2 n (1+p))) x^{1+n} \left (1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p \operatorname {AppellF1}\left (1+\frac {1}{n},-p,-p,2+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{c (1+n) (1+2 n (1+p))}-\frac {(a C-A c (1+2 n (1+p))) x \left (1+\frac {2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^n+c x^{2 n}\right )^p \operatorname {AppellF1}\left (\frac {1}{n},-p,-p,1+\frac {1}{n},-\frac {2 c x^n}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}}\right )}{c (1+2 n (1+p))} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 1.01 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.86 \[ \int \left (a+b x^n+c x^{2 n}\right )^p \left (A+B x^n+C x^{2 n}\right ) \, dx=\frac {x \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^n}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^n}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+x^n \left (b+c x^n\right )\right )^p \left (B (1+2 n) x^n \operatorname {AppellF1}\left (1+\frac {1}{n},-p,-p,2+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )+(1+n) \left (C x^{2 n} \operatorname {AppellF1}\left (2+\frac {1}{n},-p,-p,3+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )+A (1+2 n) \operatorname {AppellF1}\left (\frac {1}{n},-p,-p,1+\frac {1}{n},-\frac {2 c x^n}{b+\sqrt {b^2-4 a c}},\frac {2 c x^n}{-b+\sqrt {b^2-4 a c}}\right )\right )\right )}{(1+n) (1+2 n)} \] Input:

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

Output:

(x*(a + x^n*(b + c*x^n))^p*(B*(1 + 2*n)*x^n*AppellF1[1 + n^(-1), -p, -p, 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)*(C*x^(2*n)*AppellF1[2 + n^(-1), -p, -p, 3 + 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*( 
1 + 2*n)*AppellF1[n^(-1), -p, -p, 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])])))/((1 + n)*(1 + 2*n)*((b - Sq 
rt[b^2 - 4*a*c] + 2*c*x^n)/(b - Sqrt[b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a 
*c] + 2*c*x^n)/(b + Sqrt[b^2 - 4*a*c]))^p)
 

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.

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

\(\Big \downarrow \) 2329

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

Input:

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

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

Input:

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

Output:

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

Fricas [F]

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

integrate((a+b*x^n+c*x^(2*n))^p*(A+B*x^n+C*x^(2*n)),x, algorithm="fricas")
 

Output:

integral((C*x^(2*n) + B*x^n + A)*(c*x^(2*n) + b*x^n + a)^p, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

integrate((a+b*x^n+c*x^(2*n))^p*(A+B*x^n+C*x^(2*n)),x, algorithm="maxima")
 

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro 
unding error%%%{-128,[1,0,5,3,6,4,1,6,0,0,1]%%%}+%%%{256,[1,0,5,3,6,4,0,7, 
0,1,0]%%%
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*x**(2*n)*(x**(2*n)*c + x**n*b + a)**p*c*n**2*p**2*x + x**(2*n)*(x**(2*n 
)*c + x**n*b + a)**p*c*n**2*p*x + 3*x**(2*n)*(x**(2*n)*c + x**n*b + a)**p* 
c*n*p*x + x**(2*n)*(x**(2*n)*c + x**n*b + a)**p*c*n*x + x**(2*n)*(x**(2*n) 
*c + x**n*b + a)**p*c*x + 3*x**n*(x**(2*n)*c + x**n*b + a)**p*b*n**2*p**2* 
x + 2*x**n*(x**(2*n)*c + x**n*b + a)**p*b*n**2*p*x + 4*x**n*(x**(2*n)*c + 
x**n*b + a)**p*b*n*p*x + 2*x**n*(x**(2*n)*c + x**n*b + a)**p*b*n*x + x**n* 
(x**(2*n)*c + x**n*b + a)**p*b*x + 8*(x**(2*n)*c + x**n*b + a)**p*a*n**2*p 
**2*x + 9*(x**(2*n)*c + x**n*b + a)**p*a*n**2*p*x + 2*(x**(2*n)*c + x**n*b 
 + a)**p*a*n**2*x + 5*(x**(2*n)*c + x**n*b + a)**p*a*n*p*x + 3*(x**(2*n)*c 
 + x**n*b + a)**p*a*n*x + (x**(2*n)*c + x**n*b + a)**p*a*x + 16*int((x**(2 
*n)*c + x**n*b + a)**p/(4*x**(2*n)*c*n**3*p**3 + 6*x**(2*n)*c*n**3*p**2 + 
2*x**(2*n)*c*n**3*p + 8*x**(2*n)*c*n**2*p**2 + 9*x**(2*n)*c*n**2*p + 2*x** 
(2*n)*c*n**2 + 5*x**(2*n)*c*n*p + 3*x**(2*n)*c*n + x**(2*n)*c + 4*x**n*b*n 
**3*p**3 + 6*x**n*b*n**3*p**2 + 2*x**n*b*n**3*p + 8*x**n*b*n**2*p**2 + 9*x 
**n*b*n**2*p + 2*x**n*b*n**2 + 5*x**n*b*n*p + 3*x**n*b*n + x**n*b + 4*a*n* 
*3*p**3 + 6*a*n**3*p**2 + 2*a*n**3*p + 8*a*n**2*p**2 + 9*a*n**2*p + 2*a*n* 
*2 + 5*a*n*p + 3*a*n + a),x)*a**2*n**6*p**6 + 48*int((x**(2*n)*c + x**n*b 
+ a)**p/(4*x**(2*n)*c*n**3*p**3 + 6*x**(2*n)*c*n**3*p**2 + 2*x**(2*n)*c*n* 
*3*p + 8*x**(2*n)*c*n**2*p**2 + 9*x**(2*n)*c*n**2*p + 2*x**(2*n)*c*n**2 + 
5*x**(2*n)*c*n*p + 3*x**(2*n)*c*n + x**(2*n)*c + 4*x**n*b*n**3*p**3 + 6...