\(\int (d+e x^n)^q (c d-c e x^n) (c d^2-c e^2 x^{2 n})^p \, dx\) [1]

Optimal result

Integrand size = 40, antiderivative size = 1 \[ \int \left (d+e x^n\right )^q \left (c d-c e x^n\right ) \left (c d^2-c e^2 x^{2 n}\right )^p \, dx=0 \] Output:

0
 

Mathematica [F]

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

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

Output:

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

Rubi [C] (verified)

Result contains higher order function than in optimal. Order 6 vs. order 1 in optimal.

Time = 0.30 (sec) , antiderivative size = 103, normalized size of antiderivative = 103.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1396, 937, 937, 936}

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.

\(\displaystyle \int \left (c d-c e x^n\right ) \left (d+e x^n\right )^q \left (c d^2-c e^2 x^{2 n}\right )^p \, dx\)

\(\Big \downarrow \) 1396

\(\displaystyle \left (d+e x^n\right )^{-p} \left (c d-c e x^n\right )^{-p} \left (c d^2-c e^2 x^{2 n}\right )^p \int \left (e x^n+d\right )^{p+q} \left (c d-c e x^n\right )^{p+1}dx\)

\(\Big \downarrow \) 937

\(\displaystyle \left (d+e x^n\right )^q \left (c d-c e x^n\right )^{-p} \left (c d^2-c e^2 x^{2 n}\right )^p \left (\frac {e x^n}{d}+1\right )^{-p-q} \int \left (c d-c e x^n\right )^{p+1} \left (\frac {e x^n}{d}+1\right )^{p+q}dx\)

\(\Big \downarrow \) 937

\(\displaystyle c d \left (1-\frac {e x^n}{d}\right )^{-p} \left (d+e x^n\right )^q \left (c d^2-c e^2 x^{2 n}\right )^p \left (\frac {e x^n}{d}+1\right )^{-p-q} \int \left (1-\frac {e x^n}{d}\right )^{p+1} \left (\frac {e x^n}{d}+1\right )^{p+q}dx\)

\(\Big \downarrow \) 936

\(\displaystyle c d x \left (1-\frac {e x^n}{d}\right )^{-p} \left (d+e x^n\right )^q \left (c d^2-c e^2 x^{2 n}\right )^p \left (\frac {e x^n}{d}+1\right )^{-p-q} \operatorname {AppellF1}\left (\frac {1}{n},-p-1,-p-q,1+\frac {1}{n},\frac {e x^n}{d},-\frac {e x^n}{d}\right )\)

Input:

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

Output:

(c*d*x*(d + e*x^n)^q*(1 + (e*x^n)/d)^(-p - q)*(c*d^2 - c*e^2*x^(2*n))^p*Ap 
pellF1[n^(-1), -1 - p, -p - q, 1 + n^(-1), (e*x^n)/d, -((e*x^n)/d)])/(1 - 
(e*x^n)/d)^p
 

Defintions of rubi rules used

rule 936

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^p*c^q*x*AppellF1[1/n, -p, -q, 1 + 1/n, (-b)*(x^n/a), (-d)*(x^n/c) 
], x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] 
 && (IntegerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 937

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[a^IntPart[p]*((a + b*x^n)^FracPart[p]/(1 + b*(x^n/a))^FracPart[p]) 
  Int[(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, p, q 
}, x] && NeQ[b*c - a*d, 0] && NeQ[n, -1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 1396

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x 
_Symbol] :> Simp[(a + c*x^(2*n))^FracPart[p]/((d + e*x^n)^FracPart[p]*(a/d 
+ c*(x^n/e))^FracPart[p])   Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, 
x], x] /; FreeQ[{a, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a* 
e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 1] && EqQ[n, 2])
 

Maple [F]

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

Input:

int((d+e*x^n)^q*(c*d-c*e*x^n)*(c*d^2-c*e^2*x^(2*n))^p,x)
 

Output:

int((d+e*x^n)^q*(c*d-c*e*x^n)*(c*d^2-c*e^2*x^(2*n))^p,x)
 

Fricas [F]

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

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

Output:

integral(-(c*e*x^n - c*d)*(-c*e^2*x^(2*n) + c*d^2)^p*(e*x^n + d)^q, x)
 

Sympy [F(-1)]

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

integrate((d+e*x**n)**q*(c*d-c*e*x**n)*(c*d**2-c*e**2*x**(2*n))**p,x)
 

Output:

Timed out
 

Maxima [F]

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

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

Output:

-integrate((c*e*x^n - c*d)*(-c*e^2*x^(2*n) + c*d^2)^p*(e*x^n + d)^q, x)
 

Giac [F]

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

integrate((d+e*x^n)^q*(c*d-c*e*x^n)*(c*d^2-c*e^2*x^(2*n))^p,x, algorithm=" 
giac")
 

Output:

integrate(-(c*e*x^n - c*d)*(-c*e^2*x^(2*n) + c*d^2)^p*(e*x^n + d)^q, x)
 

Mupad [F(-1)]

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

int((c*d^2 - c*e^2*x^(2*n))^p*(d + e*x^n)^q*(c*d - c*e*x^n),x)
 

Output:

int((c*d^2 - c*e^2*x^(2*n))^p*(d + e*x^n)^q*(c*d - c*e*x^n), x)
 

Reduce [F]

\[ \int \left (d+e x^n\right )^q \left (c d-c e x^n\right ) \left (c d^2-c e^2 x^{2 n}\right )^p \, dx=\text {too large to display} \] Input:

int((d+e*x^n)^q*(c*d-c*e*x^n)*(c*d^2-c*e^2*x^(2*n))^p,x)
 

Output:

(c*( - x**n*(x**n*e + d)**q*( - x**(2*n)*c*e**2 + c*d**2)**p*e*q*x - 2*(x* 
*n*e + d)**q*( - x**(2*n)*c*e**2 + c*d**2)**p*d*p*x - (x**n*e + d)**q*( - 
x**(2*n)*c*e**2 + c*d**2)**p*d*q*x - 4*int(((x**n*e + d)**q*( - x**(2*n)*c 
*e**2 + c*d**2)**p)/(2*x**(2*n)*e**2*n*p + x**(2*n)*e**2*n*q + x**(2*n)*e* 
*2*n + x**(2*n)*e**2 - 2*d**2*n*p - d**2*n*q - d**2*n - d**2),x)*d**3*n**2 
*p**2*q - 4*int(((x**n*e + d)**q*( - x**(2*n)*c*e**2 + c*d**2)**p)/(2*x**( 
2*n)*e**2*n*p + x**(2*n)*e**2*n*q + x**(2*n)*e**2*n + x**(2*n)*e**2 - 2*d* 
*2*n*p - d**2*n*q - d**2*n - d**2),x)*d**3*n**2*p*q**2 - 4*int(((x**n*e + 
d)**q*( - x**(2*n)*c*e**2 + c*d**2)**p)/(2*x**(2*n)*e**2*n*p + x**(2*n)*e* 
*2*n*q + x**(2*n)*e**2*n + x**(2*n)*e**2 - 2*d**2*n*p - d**2*n*q - d**2*n 
- d**2),x)*d**3*n**2*p*q - int(((x**n*e + d)**q*( - x**(2*n)*c*e**2 + c*d* 
*2)**p)/(2*x**(2*n)*e**2*n*p + x**(2*n)*e**2*n*q + x**(2*n)*e**2*n + x**(2 
*n)*e**2 - 2*d**2*n*p - d**2*n*q - d**2*n - d**2),x)*d**3*n**2*q**3 - 2*in 
t(((x**n*e + d)**q*( - x**(2*n)*c*e**2 + c*d**2)**p)/(2*x**(2*n)*e**2*n*p 
+ x**(2*n)*e**2*n*q + x**(2*n)*e**2*n + x**(2*n)*e**2 - 2*d**2*n*p - d**2* 
n*q - d**2*n - d**2),x)*d**3*n**2*q**2 - int(((x**n*e + d)**q*( - x**(2*n) 
*c*e**2 + c*d**2)**p)/(2*x**(2*n)*e**2*n*p + x**(2*n)*e**2*n*q + x**(2*n)* 
e**2*n + x**(2*n)*e**2 - 2*d**2*n*p - d**2*n*q - d**2*n - d**2),x)*d**3*n* 
*2*q - 4*int(((x**n*e + d)**q*( - x**(2*n)*c*e**2 + c*d**2)**p)/(2*x**(2*n 
)*e**2*n*p + x**(2*n)*e**2*n*q + x**(2*n)*e**2*n + x**(2*n)*e**2 - 2*d*...