\(\int x^{-1-3 (-1+2 p)} (a+b x^3)^p (c+d x^3)^p \, dx\) [896]

Optimal result

Integrand size = 30, antiderivative size = 103 \[ \int x^{-1-3 (-1+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx=\frac {x^{3 (1-2 p)} \left (a+b x^3\right )^p \left (1+\frac {b x^3}{a}\right )^{-p} \left (c+d x^3\right )^p \left (1+\frac {d x^3}{c}\right )^{-p} \operatorname {AppellF1}\left (1-2 p,-p,-p,2 (1-p),-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{3 (1-2 p)} \] Output:

1/3*x^(3-6*p)*(b*x^3+a)^p*(d*x^3+c)^p*AppellF1(1-2*p,-p,-p,2-2*p,-b*x^3/a, 
-d*x^3/c)/(1-2*p)/((1+b*x^3/a)^p)/((1+d*x^3/c)^p)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.95 \[ \int x^{-1-3 (-1+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx=\frac {x^{3-6 p} \left (a+b x^3\right )^p \left (\frac {a+b x^3}{a}\right )^{-p} \left (c+d x^3\right )^p \left (\frac {c+d x^3}{c}\right )^{-p} \operatorname {AppellF1}\left (1-2 p,-p,-p,2-2 p,-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{3-6 p} \] Input:

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

Output:

(x^(3 - 6*p)*(a + b*x^3)^p*(c + d*x^3)^p*AppellF1[1 - 2*p, -p, -p, 2 - 2*p 
, -((b*x^3)/a), -((d*x^3)/c)])/((3 - 6*p)*((a + b*x^3)/a)^p*((c + d*x^3)/c 
)^p)
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1013, 1013, 1012}

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.

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

\(\displaystyle \frac {x^{3 (1-2 p)} \left (a+b x^3\right )^p \left (\frac {b x^3}{a}+1\right )^{-p} \left (c+d x^3\right )^p \left (\frac {d x^3}{c}+1\right )^{-p} \operatorname {AppellF1}\left (1-2 p,-p,-p,2 (1-p),-\frac {b x^3}{a},-\frac {d x^3}{c}\right )}{3 (1-2 p)}\)

Input:

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

Output:

(x^(3*(1 - 2*p))*(a + b*x^3)^p*(c + d*x^3)^p*AppellF1[1 - 2*p, -p, -p, 2*( 
1 - p), -((b*x^3)/a), -((d*x^3)/c)])/(3*(1 - 2*p)*(1 + (b*x^3)/a)^p*(1 + ( 
d*x^3)/c)^p)
 

Defintions of rubi rules used

rule 1012

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m 
+ 1)/n, -p, -q, 1 + (m + 1)/n, (-b)*(x^n/a), (-d)*(x^n/c)], 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]) && (IntegerQ[q] || GtQ[c, 0])
 

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])
 

Maple [F]

\[\int x^{2-6 p} \left (b \,x^{3}+a \right )^{p} \left (d \,x^{3}+c \right )^{p}d x\]

Input:

int(x^(2-6*p)*(b*x^3+a)^p*(d*x^3+c)^p,x)
 

Output:

int(x^(2-6*p)*(b*x^3+a)^p*(d*x^3+c)^p,x)
 

Fricas [F]

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

integrate(x^(2-6*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 + 2), x)
 

Sympy [F(-1)]

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

integrate(x**(2-6*p)*(b*x**3+a)**p*(d*x**3+c)**p,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate(x^(2-6*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 + 2), x)
 

Giac [F]

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

integrate(x^(2-6*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 + 2), x)
 

Mupad [F(-1)]

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

int(x^(2 - 6*p)*(a + b*x^3)^p*(c + d*x^3)^p,x)
 

Output:

int(x^(2 - 6*p)*(a + b*x^3)^p*(c + d*x^3)^p, x)
 

Reduce [F]

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

int(x^(2-6*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 + (c + d*x**3)**p*(a + b*x**3)** 
p*a*d*x**3 + (c + d*x**3)**p*(a + b*x**3)**p*b*c*x**3 + 3*x**(6*p)*int(((c 
 + d*x**3)**p*(a + b*x**3)**p*x**5)/(x**(6*p)*a**2*c*d + x**(6*p)*a**2*d** 
2*x**3 + x**(6*p)*a*b*c**2 + 2*x**(6*p)*a*b*c*d*x**3 + x**(6*p)*a*b*d**2*x 
**6 + x**(6*p)*b**2*c**2*x**3 + x**(6*p)*b**2*c*d*x**6),x)*a**3*d**3*p + 9 
*x**(6*p)*int(((c + d*x**3)**p*(a + b*x**3)**p*x**5)/(x**(6*p)*a**2*c*d + 
x**(6*p)*a**2*d**2*x**3 + x**(6*p)*a*b*c**2 + 2*x**(6*p)*a*b*c*d*x**3 + x* 
*(6*p)*a*b*d**2*x**6 + x**(6*p)*b**2*c**2*x**3 + x**(6*p)*b**2*c*d*x**6),x 
)*a**2*b*c*d**2*p + 9*x**(6*p)*int(((c + d*x**3)**p*(a + b*x**3)**p*x**5)/ 
(x**(6*p)*a**2*c*d + x**(6*p)*a**2*d**2*x**3 + x**(6*p)*a*b*c**2 + 2*x**(6 
*p)*a*b*c*d*x**3 + x**(6*p)*a*b*d**2*x**6 + x**(6*p)*b**2*c**2*x**3 + x**( 
6*p)*b**2*c*d*x**6),x)*a*b**2*c**2*d*p + 3*x**(6*p)*int(((c + d*x**3)**p*( 
a + b*x**3)**p*x**5)/(x**(6*p)*a**2*c*d + x**(6*p)*a**2*d**2*x**3 + x**(6* 
p)*a*b*c**2 + 2*x**(6*p)*a*b*c*d*x**3 + x**(6*p)*a*b*d**2*x**6 + x**(6*p)* 
b**2*c**2*x**3 + x**(6*p)*b**2*c*d*x**6),x)*b**3*c**3*p - 12*x**(6*p)*int( 
((c + d*x**3)**p*(a + b*x**3)**p)/(x**(6*p)*a**2*c*d*x + x**(6*p)*a**2*d** 
2*x**4 + x**(6*p)*a*b*c**2*x + 2*x**(6*p)*a*b*c*d*x**4 + x**(6*p)*a*b*d**2 
*x**7 + x**(6*p)*b**2*c**2*x**4 + x**(6*p)*b**2*c*d*x**7),x)*a**3*c**2*d*p 
 - 12*x**(6*p)*int(((c + d*x**3)**p*(a + b*x**3)**p)/(x**(6*p)*a**2*c*d*x 
+ x**(6*p)*a**2*d**2*x**4 + x**(6*p)*a*b*c**2*x + 2*x**(6*p)*a*b*c*d*x*...