\(\int (d+e x^n)^q (c d^2-c e^2 x^{2 n})^p \, dx\) [54]

Optimal result

Integrand size = 29, antiderivative size = 99 \[ \int \left (d+e x^n\right )^q \left (c d^2-c e^2 x^{2 n}\right )^p \, dx=x \left (d+e x^n\right )^q \left (1-\frac {e x^n}{d}\right )^{-p} \left (1+\frac {e x^n}{d}\right )^{-p-q} \left (c d^2-c e^2 x^{2 n}\right )^p \operatorname {AppellF1}\left (\frac {1}{n},-p,-p-q,1+\frac {1}{n},\frac {e x^n}{d},-\frac {e x^n}{d}\right ) \] Output:

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

Mathematica [F]

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

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

Output:

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

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, 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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 937

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

\(\Big \downarrow \) 936

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

Input:

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

Output:

(x*(d + e*x^n)^q*(1 + (e*x^n)/d)^(-p - q)*(c*d^2 - c*e^2*x^(2*n))^p*Appell 
F1[n^(-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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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