\(\int (a+b x^n)^p (c+d x^n)^q (e+f x^n)^2 \, dx\) [7]

Optimal result

Integrand size = 28, antiderivative size = 279 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^q \left (e+f x^n\right )^2 \, dx=\frac {2 e f x^{1+n} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \operatorname {AppellF1}\left (1+\frac {1}{n},-p,-q,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{1+n}+\frac {f^2 x^{1+2 n} \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \operatorname {AppellF1}\left (2+\frac {1}{n},-p,-q,3+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{1+2 n}+e^2 x \left (a+b x^n\right )^p \left (1+\frac {b x^n}{a}\right )^{-p} \left (c+d x^n\right )^q \left (1+\frac {d x^n}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{n},-p,-q,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right ) \] Output:

2*e*f*x^(1+n)*(a+b*x^n)^p*(c+d*x^n)^q*AppellF1(1+1/n,-p,-q,2+1/n,-b*x^n/a, 
-d*x^n/c)/(1+n)/((1+b*x^n/a)^p)/((1+d*x^n/c)^q)+f^2*x^(1+2*n)*(a+b*x^n)^p* 
(c+d*x^n)^q*AppellF1(2+1/n,-p,-q,3+1/n,-b*x^n/a,-d*x^n/c)/(1+2*n)/((1+b*x^ 
n/a)^p)/((1+d*x^n/c)^q)+e^2*x*(a+b*x^n)^p*(c+d*x^n)^q*AppellF1(1/n,-p,-q,1 
+1/n,-b*x^n/a,-d*x^n/c)/((1+b*x^n/a)^p)/((1+d*x^n/c)^q)
 

Mathematica [A] (warning: unable to verify)

Time = 0.81 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.25 \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^q \left (e+f x^n\right )^2 \, dx=x \left (a+b x^n\right )^p \left (c+d x^n\right )^q \left (\frac {2 e f x^n \left (1+\frac {b x^n}{a}\right )^{-p} \left (1+\frac {d x^n}{c}\right )^{-q} \operatorname {AppellF1}\left (1+\frac {1}{n},-p,-q,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{1+n}+\frac {f^2 x^{2 n} \left (1+\frac {b x^n}{a}\right )^{-p} \left (1+\frac {d x^n}{c}\right )^{-q} \operatorname {AppellF1}\left (2+\frac {1}{n},-p,-q,3+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{1+2 n}+\frac {a c e^2 (1+n) \operatorname {AppellF1}\left (\frac {1}{n},-p,-q,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}{b c n p x^n \operatorname {AppellF1}\left (1+\frac {1}{n},1-p,-q,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+a d n q x^n \operatorname {AppellF1}\left (1+\frac {1}{n},-p,1-q,2+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )+a c (1+n) \operatorname {AppellF1}\left (\frac {1}{n},-p,-q,1+\frac {1}{n},-\frac {b x^n}{a},-\frac {d x^n}{c}\right )}\right ) \] Input:

Integrate[(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^2,x]
 

Output:

x*(a + b*x^n)^p*(c + d*x^n)^q*((2*e*f*x^n*AppellF1[1 + n^(-1), -p, -q, 2 + 
 n^(-1), -((b*x^n)/a), -((d*x^n)/c)])/((1 + n)*(1 + (b*x^n)/a)^p*(1 + (d*x 
^n)/c)^q) + (f^2*x^(2*n)*AppellF1[2 + n^(-1), -p, -q, 3 + n^(-1), -((b*x^n 
)/a), -((d*x^n)/c)])/((1 + 2*n)*(1 + (b*x^n)/a)^p*(1 + (d*x^n)/c)^q) + (a* 
c*e^2*(1 + n)*AppellF1[n^(-1), -p, -q, 1 + n^(-1), -((b*x^n)/a), -((d*x^n) 
/c)])/(b*c*n*p*x^n*AppellF1[1 + n^(-1), 1 - p, -q, 2 + n^(-1), -((b*x^n)/a 
), -((d*x^n)/c)] + a*d*n*q*x^n*AppellF1[1 + n^(-1), -p, 1 - q, 2 + n^(-1), 
 -((b*x^n)/a), -((d*x^n)/c)] + a*c*(1 + n)*AppellF1[n^(-1), -p, -q, 1 + n^ 
(-1), -((b*x^n)/a), -((d*x^n)/c)]))
 

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.

\(\displaystyle \int \left (e+f x^n\right )^2 \left (a+b x^n\right )^p \left (c+d x^n\right )^q \, dx\)

\(\Big \downarrow \) 1032

\(\displaystyle \int \left (e+f x^n\right )^2 \left (a+b x^n\right )^p \left (c+d x^n\right )^qdx\)

Input:

Int[(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^2,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 1032

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + ( 
f_.)*(x_)^(n_))^(r_.), x_Symbol] :> Unintegrable[(a + b*x^n)^p*(c + d*x^n)^ 
q*(e + f*x^n)^r, x] /; FreeQ[{a, b, c, d, e, f, n, p, q, r}, x]
 

Maple [F]

\[\int \left (a +b \,x^{n}\right )^{p} \left (c +d \,x^{n}\right )^{q} \left (e +f \,x^{n}\right )^{2}d x\]

Input:

int((a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^2,x)
 

Output:

int((a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^2,x)
 

Fricas [F]

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

integrate((a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^2,x, algorithm="fricas")
 

Output:

integral((f^2*x^(2*n) + 2*e*f*x^n + e^2)*(b*x^n + a)^p*(d*x^n + c)^q, x)
 

Sympy [F(-2)]

Exception generated. \[ \int \left (a+b x^n\right )^p \left (c+d x^n\right )^q \left (e+f x^n\right )^2 \, dx=\text {Exception raised: HeuristicGCDFailed} \] Input:

integrate((a+b*x**n)**p*(c+d*x**n)**q*(e+f*x**n)**2,x)
 

Output:

Exception raised: HeuristicGCDFailed >> no luck
 

Maxima [F]

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

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

Output:

integrate((f*x^n + e)^2*(b*x^n + a)^p*(d*x^n + c)^q, x)
 

Giac [F(-2)]

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

integrate((a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^2,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%%%{-1,[1,0,5,3,10,3,4,4,10,3,0,2]%%%}+%%%{-4,[1,0,5,3,10,3,4, 
3,10,3,0,
 

Mupad [F(-1)]

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

int((a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^2,x)
 

Output:

int((a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^2, x)
 

Reduce [F]

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

int((a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^2,x)
 

Output:

int((a+b*x^n)^p*(c+d*x^n)^q*(e+f*x^n)^2,x)