\(\int x^{-1-3 (2+2 p)} (a+b x^3)^p (c+d x^3)^p \, dx\) [893]

Optimal result

Integrand size = 30, antiderivative size = 159 \[ \int x^{-1-3 (2+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx=-\frac {x^{-6 (1+p)} \left (a+b x^3\right )^{1+p} \left (c+d x^3\right )^{1+p}}{6 a c (1+p)}+\frac {(b c+a d) x^{-3 (1+2 p)} \left (a+b x^3\right )^{1+p} \left (c+d x^3\right )^p \left (\frac {a \left (c+d x^3\right )}{c \left (a+b x^3\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{6 a^2 c (1+2 p)} \] Output:

-1/6*(b*x^3+a)^(p+1)*(d*x^3+c)^(p+1)/a/c/(p+1)/(x^(6*p+6))+1/6*(a*d+b*c)*( 
b*x^3+a)^(p+1)*(d*x^3+c)^p*hypergeom([-p, -1-2*p],[-2*p],(-a*d+b*c)*x^3/c/ 
(b*x^3+a))/a^2/c/(1+2*p)/(x^(3+6*p))/((a*(d*x^3+c)/c/(b*x^3+a))^p)
 

Mathematica [F]

\[ \int x^{-1-3 (2+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx=\int x^{-1-3 (2+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx \] Input:

Integrate[x^(-1 - 3*(2 + 2*p))*(a + b*x^3)^p*(c + d*x^3)^p,x]
 

Output:

Integrate[x^(-1 - 3*(2 + 2*p))*(a + b*x^3)^p*(c + d*x^3)^p, x]
 

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.

\(\displaystyle \int x^{-3 (2 p+2)-1} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx\)

\(\Big \downarrow \) 1013

\(\displaystyle \left (a+b x^3\right )^p \left (\frac {b x^3}{a}+1\right )^{-p} \int x^{-6 p-7} \left (\frac {b x^3}{a}+1\right )^p \left (d x^3+c\right )^pdx\)

\(\Big \downarrow \) 1013

\(\displaystyle \left (a+b x^3\right )^p \left (\frac {b x^3}{a}+1\right )^{-p} \left (c+d x^3\right )^p \left (\frac {d x^3}{c}+1\right )^{-p} \int x^{-6 p-7} \left (\frac {b x^3}{a}+1\right )^p \left (\frac {d x^3}{c}+1\right )^pdx\)

\(\Big \downarrow \) 7299

\(\displaystyle \left (a+b x^3\right )^p \left (\frac {b x^3}{a}+1\right )^{-p} \left (c+d x^3\right )^p \left (\frac {d x^3}{c}+1\right )^{-p} \int x^{-6 p-7} \left (\frac {b x^3}{a}+1\right )^p \left (\frac {d x^3}{c}+1\right )^pdx\)

Input:

Int[x^(-1 - 3*(2 + 2*p))*(a + b*x^3)^p*(c + d*x^3)^p,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 1013

Int[((e_.)*(x_))^(m_.)*((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[(e*x)^m*(1 + b*(x^n/a))^p*(c + d*x^n)^q, x], x] /; 
 FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] & 
& NeQ[m, n - 1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

rule 7299

Int[u_, x_] :> CannotIntegrate[u, x]
 

Maple [F]

\[\int x^{-7-6 p} \left (b \,x^{3}+a \right )^{p} \left (d \,x^{3}+c \right )^{p}d x\]

Input:

int(x^(-7-6*p)*(b*x^3+a)^p*(d*x^3+c)^p,x)
 

Output:

int(x^(-7-6*p)*(b*x^3+a)^p*(d*x^3+c)^p,x)
 

Fricas [F]

\[ \int x^{-1-3 (2+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx=\int { {\left (b x^{3} + a\right )}^{p} {\left (d x^{3} + c\right )}^{p} x^{-6 \, p - 7} \,d x } \] Input:

integrate(x^(-7-6*p)*(b*x^3+a)^p*(d*x^3+c)^p,x, algorithm="fricas")
 

Output:

integral((b*x^3 + a)^p*(d*x^3 + c)^p*x^(-6*p - 7), x)
 

Sympy [F(-1)]

Timed out. \[ \int x^{-1-3 (2+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx=\text {Timed out} \] Input:

integrate(x**(-7-6*p)*(b*x**3+a)**p*(d*x**3+c)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^{-1-3 (2+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx=\int { {\left (b x^{3} + a\right )}^{p} {\left (d x^{3} + c\right )}^{p} x^{-6 \, p - 7} \,d x } \] Input:

integrate(x^(-7-6*p)*(b*x^3+a)^p*(d*x^3+c)^p,x, algorithm="maxima")
 

Output:

integrate((b*x^3 + a)^p*(d*x^3 + c)^p*x^(-6*p - 7), x)
 

Giac [F]

\[ \int x^{-1-3 (2+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx=\int { {\left (b x^{3} + a\right )}^{p} {\left (d x^{3} + c\right )}^{p} x^{-6 \, p - 7} \,d x } \] Input:

integrate(x^(-7-6*p)*(b*x^3+a)^p*(d*x^3+c)^p,x, algorithm="giac")
 

Output:

integrate((b*x^3 + a)^p*(d*x^3 + c)^p*x^(-6*p - 7), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^{-1-3 (2+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx=\int \frac {{\left (b\,x^3+a\right )}^p\,{\left (d\,x^3+c\right )}^p}{x^{6\,p+7}} \,d x \] Input:

int(((a + b*x^3)^p*(c + d*x^3)^p)/x^(6*p + 7),x)
 

Output:

int(((a + b*x^3)^p*(c + d*x^3)^p)/x^(6*p + 7), x)
 

Reduce [F]

\[ \int x^{-1-3 (2+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx=\text {too large to display} \] Input:

int(x^(-7-6*p)*(b*x^3+a)^p*(d*x^3+c)^p,x)
 

Output:

( - 2*(c + d*x**3)**p*(a + b*x**3)**p*a*c*p - (c + d*x**3)**p*(a + b*x**3) 
**p*a*c - (c + d*x**3)**p*(a + b*x**3)**p*a*d*p*x**3 - (c + d*x**3)**p*(a 
+ b*x**3)**p*b*c*p*x**3 + (c + d*x**3)**p*(a + b*x**3)**p*b*d*x**6 - 6*x** 
(6*p)*int(((c + d*x**3)**p*(a + b*x**3)**p)/(2*x**(6*p)*a**2*c*d*p*x + x** 
(6*p)*a**2*c*d*x + 2*x**(6*p)*a**2*d**2*p*x**4 + x**(6*p)*a**2*d**2*x**4 + 
 2*x**(6*p)*a*b*c**2*p*x + x**(6*p)*a*b*c**2*x + 4*x**(6*p)*a*b*c*d*p*x**4 
 + 2*x**(6*p)*a*b*c*d*x**4 + 2*x**(6*p)*a*b*d**2*p*x**7 + x**(6*p)*a*b*d** 
2*x**7 + 2*x**(6*p)*b**2*c**2*p*x**4 + x**(6*p)*b**2*c**2*x**4 + 2*x**(6*p 
)*b**2*c*d*p*x**7 + x**(6*p)*b**2*c*d*x**7),x)*a**3*d**3*p**3*x**6 - 9*x** 
(6*p)*int(((c + d*x**3)**p*(a + b*x**3)**p)/(2*x**(6*p)*a**2*c*d*p*x + x** 
(6*p)*a**2*c*d*x + 2*x**(6*p)*a**2*d**2*p*x**4 + x**(6*p)*a**2*d**2*x**4 + 
 2*x**(6*p)*a*b*c**2*p*x + x**(6*p)*a*b*c**2*x + 4*x**(6*p)*a*b*c*d*p*x**4 
 + 2*x**(6*p)*a*b*c*d*x**4 + 2*x**(6*p)*a*b*d**2*p*x**7 + x**(6*p)*a*b*d** 
2*x**7 + 2*x**(6*p)*b**2*c**2*p*x**4 + x**(6*p)*b**2*c**2*x**4 + 2*x**(6*p 
)*b**2*c*d*p*x**7 + x**(6*p)*b**2*c*d*x**7),x)*a**3*d**3*p**2*x**6 - 3*x** 
(6*p)*int(((c + d*x**3)**p*(a + b*x**3)**p)/(2*x**(6*p)*a**2*c*d*p*x + x** 
(6*p)*a**2*c*d*x + 2*x**(6*p)*a**2*d**2*p*x**4 + x**(6*p)*a**2*d**2*x**4 + 
 2*x**(6*p)*a*b*c**2*p*x + x**(6*p)*a*b*c**2*x + 4*x**(6*p)*a*b*c*d*p*x**4 
 + 2*x**(6*p)*a*b*c*d*x**4 + 2*x**(6*p)*a*b*d**2*p*x**7 + x**(6*p)*a*b*d** 
2*x**7 + 2*x**(6*p)*b**2*c**2*p*x**4 + x**(6*p)*b**2*c**2*x**4 + 2*x**(...