\(\int (a+b x^4+c x^8)^p (A+B x^4+C x^8+D x^{12}) \, dx\) [31]

Optimal result

Integrand size = 32, antiderivative size = 484 \[ \int \left (a+b x^4+c x^8\right )^p \left (A+B x^4+C x^8+D x^{12}\right ) \, dx=-\frac {(b D (9+4 p)-c C (13+8 p)) x \left (a+b x^4+c x^8\right )^{1+p}}{c^2 (9+8 p) (13+8 p)}+\frac {D x^5 \left (a+b x^4+c x^8\right )^{1+p}}{c (13+8 p)}+\frac {\left (A c^2 \left (117+176 p+64 p^2\right )+a (b D (9+4 p)-c C (13+8 p))\right ) x \left (1+\frac {2 c x^4}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^4}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^4+c x^8\right )^p \operatorname {AppellF1}\left (\frac {1}{4},-p,-p,\frac {5}{4},-\frac {2 c x^4}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^4}{b+\sqrt {b^2-4 a c}}\right )}{c^2 (9+8 p) (13+8 p)}-\frac {\left (5 a c D (9+8 p)-b^2 D \left (45+56 p+16 p^2\right )+b c C \left (65+92 p+32 p^2\right )-B c^2 \left (117+176 p+64 p^2\right )\right ) x^5 \left (1+\frac {2 c x^4}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^4}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^4+c x^8\right )^p \operatorname {AppellF1}\left (\frac {5}{4},-p,-p,\frac {9}{4},-\frac {2 c x^4}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^4}{b+\sqrt {b^2-4 a c}}\right )}{5 c^2 (9+8 p) (13+8 p)} \] Output:

-(b*D*(9+4*p)-c*C*(13+8*p))*x*(c*x^8+b*x^4+a)^(p+1)/c^2/(9+8*p)/(13+8*p)+D 
*x^5*(c*x^8+b*x^4+a)^(p+1)/c/(13+8*p)+(A*c^2*(64*p^2+176*p+117)+a*(b*D*(9+ 
4*p)-c*C*(13+8*p)))*x*(c*x^8+b*x^4+a)^p*AppellF1(1/4,-p,-p,5/4,-2*c*x^4/(b 
-(-4*a*c+b^2)^(1/2)),-2*c*x^4/(b+(-4*a*c+b^2)^(1/2)))/c^2/(9+8*p)/(13+8*p) 
/((1+2*c*x^4/(b-(-4*a*c+b^2)^(1/2)))^p)/((1+2*c*x^4/(b+(-4*a*c+b^2)^(1/2)) 
)^p)-1/5*(5*a*c*D*(9+8*p)-b^2*D*(16*p^2+56*p+45)+b*c*C*(32*p^2+92*p+65)-B* 
c^2*(64*p^2+176*p+117))*x^5*(c*x^8+b*x^4+a)^p*AppellF1(5/4,-p,-p,9/4,-2*c* 
x^4/(b-(-4*a*c+b^2)^(1/2)),-2*c*x^4/(b+(-4*a*c+b^2)^(1/2)))/c^2/(9+8*p)/(1 
3+8*p)/((1+2*c*x^4/(b-(-4*a*c+b^2)^(1/2)))^p)/((1+2*c*x^4/(b+(-4*a*c+b^2)^ 
(1/2)))^p)
 

Mathematica [A] (warning: unable to verify)

Time = 2.45 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.75 \[ \int \left (a+b x^4+c x^8\right )^p \left (A+B x^4+C x^8+D x^{12}\right ) \, dx=\frac {1}{585} x \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^4}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^4}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^4+c x^8\right )^p \left (585 A \operatorname {AppellF1}\left (\frac {1}{4},-p,-p,\frac {5}{4},-\frac {2 c x^4}{b+\sqrt {b^2-4 a c}},\frac {2 c x^4}{-b+\sqrt {b^2-4 a c}}\right )+117 B x^4 \operatorname {AppellF1}\left (\frac {5}{4},-p,-p,\frac {9}{4},-\frac {2 c x^4}{b+\sqrt {b^2-4 a c}},\frac {2 c x^4}{-b+\sqrt {b^2-4 a c}}\right )+65 C x^8 \operatorname {AppellF1}\left (\frac {9}{4},-p,-p,\frac {13}{4},-\frac {2 c x^4}{b+\sqrt {b^2-4 a c}},\frac {2 c x^4}{-b+\sqrt {b^2-4 a c}}\right )+45 D x^{12} \operatorname {AppellF1}\left (\frac {13}{4},-p,-p,\frac {17}{4},-\frac {2 c x^4}{b+\sqrt {b^2-4 a c}},\frac {2 c x^4}{-b+\sqrt {b^2-4 a c}}\right )\right ) \] Input:

Integrate[(a + b*x^4 + c*x^8)^p*(A + B*x^4 + C*x^8 + D*x^12),x]
 

Output:

(x*(a + b*x^4 + c*x^8)^p*(585*A*AppellF1[1/4, -p, -p, 5/4, (-2*c*x^4)/(b + 
 Sqrt[b^2 - 4*a*c]), (2*c*x^4)/(-b + Sqrt[b^2 - 4*a*c])] + 117*B*x^4*Appel 
lF1[5/4, -p, -p, 9/4, (-2*c*x^4)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^4)/(-b + 
Sqrt[b^2 - 4*a*c])] + 65*C*x^8*AppellF1[9/4, -p, -p, 13/4, (-2*c*x^4)/(b + 
 Sqrt[b^2 - 4*a*c]), (2*c*x^4)/(-b + Sqrt[b^2 - 4*a*c])] + 45*D*x^12*Appel 
lF1[13/4, -p, -p, 17/4, (-2*c*x^4)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^4)/(-b 
+ Sqrt[b^2 - 4*a*c])]))/(585*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^4)/(b - Sqrt[ 
b^2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^4)/(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.

\(\displaystyle \int \left (a+b x^4+c x^8\right )^p \left (A+B x^4+C x^8+D x^{12}\right ) \, dx\)

\(\Big \downarrow \) 2329

\(\displaystyle \int \left (a+b x^4+c x^8\right )^p \left (A+B x^4+C x^8+D x^{12}\right )dx\)

Input:

Int[(a + b*x^4 + c*x^8)^p*(A + B*x^4 + C*x^8 + D*x^12),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]

\[\int \left (c \,x^{8}+b \,x^{4}+a \right )^{p} \left (D x^{12}+C \,x^{8}+B \,x^{4}+A \right )d x\]

Input:

int((c*x^8+b*x^4+a)^p*(D*x^12+C*x^8+B*x^4+A),x)
 

Output:

int((c*x^8+b*x^4+a)^p*(D*x^12+C*x^8+B*x^4+A),x)
 

Fricas [F]

\[ \int \left (a+b x^4+c x^8\right )^p \left (A+B x^4+C x^8+D x^{12}\right ) \, dx=\int { {\left (D x^{12} + C x^{8} + B x^{4} + A\right )} {\left (c x^{8} + b x^{4} + a\right )}^{p} \,d x } \] Input:

integrate((c*x^8+b*x^4+a)^p*(D*x^12+C*x^8+B*x^4+A),x, algorithm="fricas")
 

Output:

integral((D*x^12 + C*x^8 + B*x^4 + A)*(c*x^8 + b*x^4 + a)^p, x)
 

Sympy [F(-1)]

Timed out. \[ \int \left (a+b x^4+c x^8\right )^p \left (A+B x^4+C x^8+D x^{12}\right ) \, dx=\text {Timed out} \] Input:

integrate((c*x**8+b*x**4+a)**p*(D*x**12+C*x**8+B*x**4+A),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \left (a+b x^4+c x^8\right )^p \left (A+B x^4+C x^8+D x^{12}\right ) \, dx=\int { {\left (D x^{12} + C x^{8} + B x^{4} + A\right )} {\left (c x^{8} + b x^{4} + a\right )}^{p} \,d x } \] Input:

integrate((c*x^8+b*x^4+a)^p*(D*x^12+C*x^8+B*x^4+A),x, algorithm="maxima")
 

Output:

integrate((D*x^12 + C*x^8 + B*x^4 + A)*(c*x^8 + b*x^4 + a)^p, x)
 

Giac [F]

\[ \int \left (a+b x^4+c x^8\right )^p \left (A+B x^4+C x^8+D x^{12}\right ) \, dx=\int { {\left (D x^{12} + C x^{8} + B x^{4} + A\right )} {\left (c x^{8} + b x^{4} + a\right )}^{p} \,d x } \] Input:

integrate((c*x^8+b*x^4+a)^p*(D*x^12+C*x^8+B*x^4+A),x, algorithm="giac")
 

Output:

integrate((D*x^12 + C*x^8 + B*x^4 + A)*(c*x^8 + b*x^4 + a)^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x^4+c x^8\right )^p \left (A+B x^4+C x^8+D x^{12}\right ) \, dx=\int {\left (c\,x^8+b\,x^4+a\right )}^p\,\left (A+B\,x^4+C\,x^8+x^{12}\,D\right ) \,d x \] Input:

int((a + b*x^4 + c*x^8)^p*(A + B*x^4 + C*x^8 + x^12*D),x)
 

Output:

int((a + b*x^4 + c*x^8)^p*(A + B*x^4 + C*x^8 + x^12*D), x)
 

Reduce [F]

\[ \int \left (a+b x^4+c x^8\right )^p \left (A+B x^4+C x^8+D x^{12}\right ) \, dx=\text {too large to display} \] Input:

int((c*x^8+b*x^4+a)^p*(D*x^12+C*x^8+B*x^4+A),x)
 

Output:

( - 256*(a + b*x**4 + c*x**8)**p*a*b*c*d*p**3*x - 896*(a + b*x**4 + c*x**8 
)**p*a*b*c*d*p**2*x - 540*(a + b*x**4 + c*x**8)**p*a*b*c*d*p*x + 1024*(a + 
 b*x**4 + c*x**8)**p*a*c**3*p**3*x + 2880*(a + b*x**4 + c*x**8)**p*a*c**3* 
p**2*x + 2336*(a + b*x**4 + c*x**8)**p*a*c**3*p*x + 585*(a + b*x**4 + c*x* 
*8)**p*a*c**3*x + 512*(a + b*x**4 + c*x**8)**p*a*c**2*d*p**3*x**5 + 640*(a 
 + b*x**4 + c*x**8)**p*a*c**2*d*p**2*x**5 + 72*(a + b*x**4 + c*x**8)**p*a* 
c**2*d*p*x**5 + 64*(a + b*x**4 + c*x**8)**p*b**3*d*p**3*x + 224*(a + b*x** 
4 + c*x**8)**p*b**3*d*p**2*x + 180*(a + b*x**4 + c*x**8)**p*b**3*d*p*x + 1 
28*(a + b*x**4 + c*x**8)**p*b**2*c**2*p**3*x + 336*(a + b*x**4 + c*x**8)** 
p*b**2*c**2*p**2*x + 208*(a + b*x**4 + c*x**8)**p*b**2*c**2*p*x - 128*(a + 
 b*x**4 + c*x**8)**p*b**2*c*d*p**3*x**5 - 304*(a + b*x**4 + c*x**8)**p*b** 
2*c*d*p**2*x**5 - 36*(a + b*x**4 + c*x**8)**p*b**2*c*d*p*x**5 + 768*(a + b 
*x**4 + c*x**8)**p*b*c**3*p**3*x**5 + 1920*(a + b*x**4 + c*x**8)**p*b*c**3 
*p**2*x**5 + 1164*(a + b*x**4 + c*x**8)**p*b*c**3*p*x**5 + 117*(a + b*x**4 
 + c*x**8)**p*b*c**3*x**5 + 256*(a + b*x**4 + c*x**8)**p*b*c**2*d*p**3*x** 
9 + 192*(a + b*x**4 + c*x**8)**p*b*c**2*d*p**2*x**9 + 20*(a + b*x**4 + c*x 
**8)**p*b*c**2*d*p*x**9 + 512*(a + b*x**4 + c*x**8)**p*c**4*p**3*x**9 + 12 
16*(a + b*x**4 + c*x**8)**p*c**4*p**2*x**9 + 664*(a + b*x**4 + c*x**8)**p* 
c**4*p*x**9 + 65*(a + b*x**4 + c*x**8)**p*c**4*x**9 + 512*(a + b*x**4 + c* 
x**8)**p*c**3*d*p**3*x**13 + 960*(a + b*x**4 + c*x**8)**p*c**3*d*p**2*x...