\(\int (a+b x^3+c x^6)^p (A+B x^3+C x^6+D x^9) \, dx\) [30]

Optimal result

Integrand size = 32, antiderivative size = 497 \[ \int \left (a+b x^3+c x^6\right )^p \left (A+B x^3+C x^6+D x^9\right ) \, dx=\frac {(2 c C (5+3 p)-b D (7+3 p)) x \left (a+b x^3+c x^6\right )^{1+p}}{2 c^2 (5+3 p) (7+6 p)}+\frac {D x^4 \left (a+b x^3+c x^6\right )^{1+p}}{2 c (5+3 p)}+\frac {\left (2 A c^2 \left (35+51 p+18 p^2\right )-a (2 c C (5+3 p)-b D (7+3 p))\right ) x \left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (\frac {1}{3},-p,-p,\frac {4}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{2 c^2 (5+3 p) (7+6 p)}-\frac {\left (4 a c D (7+6 p)+2 b c C \left (20+27 p+9 p^2\right )-b^2 D \left (28+33 p+9 p^2\right )-2 B c^2 \left (35+51 p+18 p^2\right )\right ) x^4 \left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \operatorname {AppellF1}\left (\frac {4}{3},-p,-p,\frac {7}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{8 c^2 (5+3 p) (7+6 p)} \] Output:

1/2*(2*c*C*(5+3*p)-b*D*(7+3*p))*x*(c*x^6+b*x^3+a)^(p+1)/c^2/(5+3*p)/(7+6*p 
)+1/2*D*x^4*(c*x^6+b*x^3+a)^(p+1)/c/(5+3*p)+1/2*(2*A*c^2*(18*p^2+51*p+35)- 
a*(2*c*C*(5+3*p)-b*D*(7+3*p)))*x*(c*x^6+b*x^3+a)^p*AppellF1(1/3,-p,-p,4/3, 
-2*c*x^3/(b-(-4*a*c+b^2)^(1/2)),-2*c*x^3/(b+(-4*a*c+b^2)^(1/2)))/c^2/(5+3* 
p)/(7+6*p)/((1+2*c*x^3/(b-(-4*a*c+b^2)^(1/2)))^p)/((1+2*c*x^3/(b+(-4*a*c+b 
^2)^(1/2)))^p)-1/8*(4*a*c*D*(7+6*p)+2*b*c*C*(9*p^2+27*p+20)-b^2*D*(9*p^2+3 
3*p+28)-2*B*c^2*(18*p^2+51*p+35))*x^4*(c*x^6+b*x^3+a)^p*AppellF1(4/3,-p,-p 
,7/3,-2*c*x^3/(b-(-4*a*c+b^2)^(1/2)),-2*c*x^3/(b+(-4*a*c+b^2)^(1/2)))/c^2/ 
(5+3*p)/(7+6*p)/((1+2*c*x^3/(b-(-4*a*c+b^2)^(1/2)))^p)/((1+2*c*x^3/(b+(-4* 
a*c+b^2)^(1/2)))^p)
 

Mathematica [A] (warning: unable to verify)

Time = 1.50 (sec) , antiderivative size = 363, normalized size of antiderivative = 0.73 \[ \int \left (a+b x^3+c x^6\right )^p \left (A+B x^3+C x^6+D x^9\right ) \, dx=\frac {1}{140} x \left (\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{-p} \left (\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{-p} \left (a+b x^3+c x^6\right )^p \left (140 A \operatorname {AppellF1}\left (\frac {1}{3},-p,-p,\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+35 B x^3 \operatorname {AppellF1}\left (\frac {4}{3},-p,-p,\frac {7}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+20 C x^6 \operatorname {AppellF1}\left (\frac {7}{3},-p,-p,\frac {10}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+14 D x^9 \operatorname {AppellF1}\left (\frac {10}{3},-p,-p,\frac {13}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )\right ) \] Input:

Integrate[(a + b*x^3 + c*x^6)^p*(A + B*x^3 + C*x^6 + D*x^9),x]
 

Output:

(x*(a + b*x^3 + c*x^6)^p*(140*A*AppellF1[1/3, -p, -p, 4/3, (-2*c*x^3)/(b + 
 Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 35*B*x^3*Appell 
F1[4/3, -p, -p, 7/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + S 
qrt[b^2 - 4*a*c])] + 20*C*x^6*AppellF1[7/3, -p, -p, 10/3, (-2*c*x^3)/(b + 
Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + Sqrt[b^2 - 4*a*c])] + 14*D*x^9*AppellF 
1[10/3, -p, -p, 13/3, (-2*c*x^3)/(b + Sqrt[b^2 - 4*a*c]), (2*c*x^3)/(-b + 
Sqrt[b^2 - 4*a*c])]))/(140*((b - Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(b - Sqrt[b^ 
2 - 4*a*c]))^p*((b + Sqrt[b^2 - 4*a*c] + 2*c*x^3)/(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.

\(\displaystyle \int \left (a+b x^3+c x^6\right )^p \left (A+B x^3+C x^6+D x^9\right ) \, dx\)

\(\Big \downarrow \) 2329

\(\displaystyle \int \left (a+b x^3+c x^6\right )^p \left (A+B x^3+C x^6+D x^9\right )dx\)

Input:

Int[(a + b*x^3 + c*x^6)^p*(A + B*x^3 + C*x^6 + D*x^9),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]

\[\int \left (c \,x^{6}+b \,x^{3}+a \right )^{p} \left (D x^{9}+C \,x^{6}+B \,x^{3}+A \right )d x\]

Input:

int((c*x^6+b*x^3+a)^p*(D*x^9+C*x^6+B*x^3+A),x)
 

Output:

int((c*x^6+b*x^3+a)^p*(D*x^9+C*x^6+B*x^3+A),x)
 

Fricas [F]

\[ \int \left (a+b x^3+c x^6\right )^p \left (A+B x^3+C x^6+D x^9\right ) \, dx=\int { {\left (D x^{9} + C x^{6} + B x^{3} + A\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p} \,d x } \] Input:

integrate((c*x^6+b*x^3+a)^p*(D*x^9+C*x^6+B*x^3+A),x, algorithm="fricas")
 

Output:

integral((D*x^9 + C*x^6 + B*x^3 + A)*(c*x^6 + b*x^3 + a)^p, x)
 

Sympy [F(-1)]

Timed out. \[ \int \left (a+b x^3+c x^6\right )^p \left (A+B x^3+C x^6+D x^9\right ) \, dx=\text {Timed out} \] Input:

integrate((c*x**6+b*x**3+a)**p*(D*x**9+C*x**6+B*x**3+A),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \left (a+b x^3+c x^6\right )^p \left (A+B x^3+C x^6+D x^9\right ) \, dx=\int { {\left (D x^{9} + C x^{6} + B x^{3} + A\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p} \,d x } \] Input:

integrate((c*x^6+b*x^3+a)^p*(D*x^9+C*x^6+B*x^3+A),x, algorithm="maxima")
 

Output:

integrate((D*x^9 + C*x^6 + B*x^3 + A)*(c*x^6 + b*x^3 + a)^p, x)
 

Giac [F]

\[ \int \left (a+b x^3+c x^6\right )^p \left (A+B x^3+C x^6+D x^9\right ) \, dx=\int { {\left (D x^{9} + C x^{6} + B x^{3} + A\right )} {\left (c x^{6} + b x^{3} + a\right )}^{p} \,d x } \] Input:

integrate((c*x^6+b*x^3+a)^p*(D*x^9+C*x^6+B*x^3+A),x, algorithm="giac")
 

Output:

integrate((D*x^9 + C*x^6 + B*x^3 + A)*(c*x^6 + b*x^3 + a)^p, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x^3+c x^6\right )^p \left (A+B x^3+C x^6+D x^9\right ) \, dx=\int {\left (c\,x^6+b\,x^3+a\right )}^p\,\left (A+B\,x^3+C\,x^6+x^9\,D\right ) \,d x \] Input:

int((a + b*x^3 + c*x^6)^p*(A + B*x^3 + C*x^6 + x^9*D),x)
 

Output:

int((a + b*x^3 + c*x^6)^p*(A + B*x^3 + C*x^6 + x^9*D), x)
 

Reduce [F]

\[ \int \left (a+b x^3+c x^6\right )^p \left (A+B x^3+C x^6+D x^9\right ) \, dx=\text {too large to display} \] Input:

int((c*x^6+b*x^3+a)^p*(D*x^9+C*x^6+B*x^3+A),x)
 

Output:

( - 108*(a + b*x**3 + c*x**6)**p*a*b*c*d*p**3*x - 396*(a + b*x**3 + c*x**6 
)**p*a*b*c*d*p**2*x - 252*(a + b*x**3 + c*x**6)**p*a*b*c*d*p*x + 432*(a + 
b*x**3 + c*x**6)**p*a*c**3*p**3*x + 1260*(a + b*x**3 + c*x**6)**p*a*c**3*p 
**2*x + 1068*(a + b*x**3 + c*x**6)**p*a*c**3*p*x + 280*(a + b*x**3 + c*x** 
6)**p*a*c**3*x + 216*(a + b*x**3 + c*x**6)**p*a*c**2*d*p**3*x**4 + 288*(a 
+ b*x**3 + c*x**6)**p*a*c**2*d*p**2*x**4 + 42*(a + b*x**3 + c*x**6)**p*a*c 
**2*d*p*x**4 + 27*(a + b*x**3 + c*x**6)**p*b**3*d*p**3*x + 99*(a + b*x**3 
+ c*x**6)**p*b**3*d*p**2*x + 84*(a + b*x**3 + c*x**6)**p*b**3*d*p*x + 54*( 
a + b*x**3 + c*x**6)**p*b**2*c**2*p**3*x + 144*(a + b*x**3 + c*x**6)**p*b* 
*2*c**2*p**2*x + 90*(a + b*x**3 + c*x**6)**p*b**2*c**2*p*x - 54*(a + b*x** 
3 + c*x**6)**p*b**2*c*d*p**3*x**4 - 135*(a + b*x**3 + c*x**6)**p*b**2*c*d* 
p**2*x**4 - 21*(a + b*x**3 + c*x**6)**p*b**2*c*d*p*x**4 + 324*(a + b*x**3 
+ c*x**6)**p*b*c**3*p**3*x**4 + 846*(a + b*x**3 + c*x**6)**p*b*c**3*p**2*x 
**4 + 552*(a + b*x**3 + c*x**6)**p*b*c**3*p*x**4 + 70*(a + b*x**3 + c*x**6 
)**p*b*c**3*x**4 + 108*(a + b*x**3 + c*x**6)**p*b*c**2*d*p**3*x**7 + 90*(a 
 + b*x**3 + c*x**6)**p*b*c**2*d*p**2*x**7 + 12*(a + b*x**3 + c*x**6)**p*b* 
c**2*d*p*x**7 + 216*(a + b*x**3 + c*x**6)**p*c**4*p**3*x**7 + 540*(a + b*x 
**3 + c*x**6)**p*c**4*p**2*x**7 + 324*(a + b*x**3 + c*x**6)**p*c**4*p*x**7 
 + 40*(a + b*x**3 + c*x**6)**p*c**4*x**7 + 216*(a + b*x**3 + c*x**6)**p*c* 
*3*d*p**3*x**10 + 432*(a + b*x**3 + c*x**6)**p*c**3*d*p**2*x**10 + 234*...