\(\int (d+e x^4)^q (d^2-e^2 x^8)^p \, dx\) [24]

Optimal result

Integrand size = 24, antiderivative size = 92 \[ \int \left (d+e x^4\right )^q \left (d^2-e^2 x^8\right )^p \, dx=x \left (d+e x^4\right )^q \left (1-\frac {e x^4}{d}\right )^{-p} \left (1+\frac {e x^4}{d}\right )^{-p-q} \left (d^2-e^2 x^8\right )^p \operatorname {AppellF1}\left (\frac {1}{4},-p,-p-q,\frac {5}{4},\frac {e x^4}{d},-\frac {e x^4}{d}\right ) \] Output:

x*(e*x^4+d)^q*(1+e*x^4/d)^(-p-q)*(-e^2*x^8+d^2)^p*AppellF1(1/4,-p,-p-q,5/4 
,e*x^4/d,-e*x^4/d)/((1-e*x^4/d)^p)
 

Mathematica [F]

\[ \int \left (d+e x^4\right )^q \left (d^2-e^2 x^8\right )^p \, dx=\int \left (d+e x^4\right )^q \left (d^2-e^2 x^8\right )^p \, dx \] Input:

Integrate[(d + e*x^4)^q*(d^2 - e^2*x^8)^p,x]
 

Output:

Integrate[(d + e*x^4)^q*(d^2 - e^2*x^8)^p, x]
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, 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.

\(\displaystyle \int \left (d^2-e^2 x^8\right )^p \left (d+e x^4\right )^q \, dx\)

\(\Big \downarrow \) 1396

\(\displaystyle \left (d-e x^4\right )^{-p} \left (d+e x^4\right )^{-p} \left (d^2-e^2 x^8\right )^p \int \left (d-e x^4\right )^p \left (e x^4+d\right )^{p+q}dx\)

\(\Big \downarrow \) 937

\(\displaystyle \left (d+e x^4\right )^{-p} \left (1-\frac {e x^4}{d}\right )^{-p} \left (d^2-e^2 x^8\right )^p \int \left (e x^4+d\right )^{p+q} \left (1-\frac {e x^4}{d}\right )^pdx\)

\(\Big \downarrow \) 937

\(\displaystyle \left (1-\frac {e x^4}{d}\right )^{-p} \left (d^2-e^2 x^8\right )^p \left (d+e x^4\right )^q \left (\frac {e x^4}{d}+1\right )^{-p-q} \int \left (1-\frac {e x^4}{d}\right )^p \left (\frac {e x^4}{d}+1\right )^{p+q}dx\)

\(\Big \downarrow \) 936

\(\displaystyle x \left (1-\frac {e x^4}{d}\right )^{-p} \left (d^2-e^2 x^8\right )^p \left (d+e x^4\right )^q \left (\frac {e x^4}{d}+1\right )^{-p-q} \operatorname {AppellF1}\left (\frac {1}{4},-p,-p-q,\frac {5}{4},\frac {e x^4}{d},-\frac {e x^4}{d}\right )\)

Input:

Int[(d + e*x^4)^q*(d^2 - e^2*x^8)^p,x]
 

Output:

(x*(d + e*x^4)^q*(1 + (e*x^4)/d)^(-p - q)*(d^2 - e^2*x^8)^p*AppellF1[1/4, 
-p, -p - q, 5/4, (e*x^4)/d, -((e*x^4)/d)])/(1 - (e*x^4)/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]

\[\int \left (x^{4} e +d \right )^{q} \left (-e^{2} x^{8}+d^{2}\right )^{p}d x\]

Input:

int((e*x^4+d)^q*(-e^2*x^8+d^2)^p,x)
 

Output:

int((e*x^4+d)^q*(-e^2*x^8+d^2)^p,x)
 

Fricas [F]

\[ \int \left (d+e x^4\right )^q \left (d^2-e^2 x^8\right )^p \, dx=\int { {\left (-e^{2} x^{8} + d^{2}\right )}^{p} {\left (e x^{4} + d\right )}^{q} \,d x } \] Input:

integrate((e*x^4+d)^q*(-e^2*x^8+d^2)^p,x, algorithm="fricas")
 

Output:

integral((-e^2*x^8 + d^2)^p*(e*x^4 + d)^q, x)
 

Sympy [F(-1)]

Timed out. \[ \int \left (d+e x^4\right )^q \left (d^2-e^2 x^8\right )^p \, dx=\text {Timed out} \] Input:

integrate((e*x**4+d)**q*(-e**2*x**8+d**2)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \left (d+e x^4\right )^q \left (d^2-e^2 x^8\right )^p \, dx=\int { {\left (-e^{2} x^{8} + d^{2}\right )}^{p} {\left (e x^{4} + d\right )}^{q} \,d x } \] Input:

integrate((e*x^4+d)^q*(-e^2*x^8+d^2)^p,x, algorithm="maxima")
 

Output:

integrate((-e^2*x^8 + d^2)^p*(e*x^4 + d)^q, x)
 

Giac [F]

\[ \int \left (d+e x^4\right )^q \left (d^2-e^2 x^8\right )^p \, dx=\int { {\left (-e^{2} x^{8} + d^{2}\right )}^{p} {\left (e x^{4} + d\right )}^{q} \,d x } \] Input:

integrate((e*x^4+d)^q*(-e^2*x^8+d^2)^p,x, algorithm="giac")
 

Output:

integrate((-e^2*x^8 + d^2)^p*(e*x^4 + d)^q, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (d+e x^4\right )^q \left (d^2-e^2 x^8\right )^p \, dx=\int {\left (d^2-e^2\,x^8\right )}^p\,{\left (e\,x^4+d\right )}^q \,d x \] Input:

int((d^2 - e^2*x^8)^p*(d + e*x^4)^q,x)
 

Output:

int((d^2 - e^2*x^8)^p*(d + e*x^4)^q, x)
 

Reduce [F]

\[ \int \left (d+e x^4\right )^q \left (d^2-e^2 x^8\right )^p \, dx=\frac {\left (e \,x^{4}+d \right )^{q} \left (-e^{2} x^{8}+d^{2}\right )^{p} x -32 \left (\int \frac {\left (e \,x^{4}+d \right )^{q} \left (-e^{2} x^{8}+d^{2}\right )^{p} x^{4}}{-8 e^{2} p \,x^{8}-4 e^{2} q \,x^{8}-e^{2} x^{8}+8 d^{2} p +4 d^{2} q +d^{2}}d x \right ) d e p q -16 \left (\int \frac {\left (e \,x^{4}+d \right )^{q} \left (-e^{2} x^{8}+d^{2}\right )^{p} x^{4}}{-8 e^{2} p \,x^{8}-4 e^{2} q \,x^{8}-e^{2} x^{8}+8 d^{2} p +4 d^{2} q +d^{2}}d x \right ) d e \,q^{2}-4 \left (\int \frac {\left (e \,x^{4}+d \right )^{q} \left (-e^{2} x^{8}+d^{2}\right )^{p} x^{4}}{-8 e^{2} p \,x^{8}-4 e^{2} q \,x^{8}-e^{2} x^{8}+8 d^{2} p +4 d^{2} q +d^{2}}d x \right ) d e q +64 \left (\int \frac {\left (e \,x^{4}+d \right )^{q} \left (-e^{2} x^{8}+d^{2}\right )^{p}}{-8 e^{2} p \,x^{8}-4 e^{2} q \,x^{8}-e^{2} x^{8}+8 d^{2} p +4 d^{2} q +d^{2}}d x \right ) d^{2} p^{2}+64 \left (\int \frac {\left (e \,x^{4}+d \right )^{q} \left (-e^{2} x^{8}+d^{2}\right )^{p}}{-8 e^{2} p \,x^{8}-4 e^{2} q \,x^{8}-e^{2} x^{8}+8 d^{2} p +4 d^{2} q +d^{2}}d x \right ) d^{2} p q +8 \left (\int \frac {\left (e \,x^{4}+d \right )^{q} \left (-e^{2} x^{8}+d^{2}\right )^{p}}{-8 e^{2} p \,x^{8}-4 e^{2} q \,x^{8}-e^{2} x^{8}+8 d^{2} p +4 d^{2} q +d^{2}}d x \right ) d^{2} p +16 \left (\int \frac {\left (e \,x^{4}+d \right )^{q} \left (-e^{2} x^{8}+d^{2}\right )^{p}}{-8 e^{2} p \,x^{8}-4 e^{2} q \,x^{8}-e^{2} x^{8}+8 d^{2} p +4 d^{2} q +d^{2}}d x \right ) d^{2} q^{2}+4 \left (\int \frac {\left (e \,x^{4}+d \right )^{q} \left (-e^{2} x^{8}+d^{2}\right )^{p}}{-8 e^{2} p \,x^{8}-4 e^{2} q \,x^{8}-e^{2} x^{8}+8 d^{2} p +4 d^{2} q +d^{2}}d x \right ) d^{2} q}{8 p +4 q +1} \] Input:

int((e*x^4+d)^q*(-e^2*x^8+d^2)^p,x)
 

Output:

((d + e*x**4)**q*(d**2 - e**2*x**8)**p*x - 32*int(((d + e*x**4)**q*(d**2 - 
 e**2*x**8)**p*x**4)/(8*d**2*p + 4*d**2*q + d**2 - 8*e**2*p*x**8 - 4*e**2* 
q*x**8 - e**2*x**8),x)*d*e*p*q - 16*int(((d + e*x**4)**q*(d**2 - e**2*x**8 
)**p*x**4)/(8*d**2*p + 4*d**2*q + d**2 - 8*e**2*p*x**8 - 4*e**2*q*x**8 - e 
**2*x**8),x)*d*e*q**2 - 4*int(((d + e*x**4)**q*(d**2 - e**2*x**8)**p*x**4) 
/(8*d**2*p + 4*d**2*q + d**2 - 8*e**2*p*x**8 - 4*e**2*q*x**8 - e**2*x**8), 
x)*d*e*q + 64*int(((d + e*x**4)**q*(d**2 - e**2*x**8)**p)/(8*d**2*p + 4*d* 
*2*q + d**2 - 8*e**2*p*x**8 - 4*e**2*q*x**8 - e**2*x**8),x)*d**2*p**2 + 64 
*int(((d + e*x**4)**q*(d**2 - e**2*x**8)**p)/(8*d**2*p + 4*d**2*q + d**2 - 
 8*e**2*p*x**8 - 4*e**2*q*x**8 - e**2*x**8),x)*d**2*p*q + 8*int(((d + e*x* 
*4)**q*(d**2 - e**2*x**8)**p)/(8*d**2*p + 4*d**2*q + d**2 - 8*e**2*p*x**8 
- 4*e**2*q*x**8 - e**2*x**8),x)*d**2*p + 16*int(((d + e*x**4)**q*(d**2 - e 
**2*x**8)**p)/(8*d**2*p + 4*d**2*q + d**2 - 8*e**2*p*x**8 - 4*e**2*q*x**8 
- e**2*x**8),x)*d**2*q**2 + 4*int(((d + e*x**4)**q*(d**2 - e**2*x**8)**p)/ 
(8*d**2*p + 4*d**2*q + d**2 - 8*e**2*p*x**8 - 4*e**2*q*x**8 - e**2*x**8),x 
)*d**2*q)/(8*p + 4*q + 1)