\(\int x^{-1-3 (1+2 p)} (a+b x^3)^p (c+d x^3)^p \, dx\) [894]

Optimal result

Integrand size = 30, antiderivative size = 104 \[ \int x^{-1-3 (1+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx=-\frac {x^{-3 (1+2 p)} \left (a+b x^3\right )^{1+p} \left (c+d x^3\right )^p \left (\frac {a \left (c+d x^3\right )}{c \left (a+b x^3\right )}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {(b c-a d) x^3}{c \left (a+b x^3\right )}\right )}{3 a (1+2 p)} \] Output:

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

Mathematica [A] (warning: unable to verify)

Time = 0.10 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.97 \[ \int x^{-1-3 (1+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx=-\frac {x^{-3-6 p} \left (a+b x^3\right )^p \left (1+\frac {b x^3}{a}\right )^{-p} \left (c+d x^3\right )^{1+p} \left (1+\frac {d x^3}{c}\right )^p \operatorname {Hypergeometric2F1}\left (-1-2 p,-p,-2 p,\frac {(-b c+a d) x^3}{a \left (c+d x^3\right )}\right )}{3 (c+2 c p)} \] Input:

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

Output:

-1/3*(x^(-3 - 6*p)*(a + b*x^3)^p*(c + d*x^3)^(1 + p)*(1 + (d*x^3)/c)^p*Hyp 
ergeometric2F1[-1 - 2*p, -p, -2*p, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/ 
((c + 2*c*p)*(1 + (b*x^3)/a)^p)
 

Rubi [A] (warning: unable to verify)

Time = 0.48 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.04, 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

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

\(\displaystyle -\frac {x^{-3 (2 p+1)} \left (a+b x^3\right )^p \left (\frac {b x^3}{a}+1\right )^{-p} \left (c+d x^3\right )^p \left (\frac {d x^3}{c}+1\right )^{p+1} \operatorname {Hypergeometric2F1}\left (-2 p-1,-p,-2 p,-\frac {c \left (\frac {b x^3}{a}-\frac {d x^3}{c}\right )}{d x^3+c}\right )}{3 (2 p+1)}\)

Input:

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

Output:

-1/3*((a + b*x^3)^p*(c + d*x^3)^p*(1 + (d*x^3)/c)^(1 + p)*Hypergeometric2F 
1[-1 - 2*p, -p, -2*p, -((c*((b*x^3)/a - (d*x^3)/c))/(c + d*x^3))])/((1 + 2 
*p)*x^(3*(1 + 2*p))*(1 + (b*x^3)/a)^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]

\[\int x^{-4-6 p} \left (b \,x^{3}+a \right )^{p} \left (d \,x^{3}+c \right )^{p}d x\]

Input:

int(x^(-4-6*p)*(b*x^3+a)^p*(d*x^3+c)^p,x)
 

Output:

int(x^(-4-6*p)*(b*x^3+a)^p*(d*x^3+c)^p,x)
 

Fricas [F]

\[ \int x^{-1-3 (1+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx=\int { {\left (b x^{3} + a\right )}^{p} {\left (d x^{3} + c\right )}^{p} x^{-6 \, p - 4} \,d x } \] Input:

integrate(x^(-4-6*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 - 4), 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**(-4-6*p)*(b*x**3+a)**p*(d*x**3+c)**p,x)
 

Output:

Timed out
 

Maxima [F]

\[ \int x^{-1-3 (1+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx=\int { {\left (b x^{3} + a\right )}^{p} {\left (d x^{3} + c\right )}^{p} x^{-6 \, p - 4} \,d x } \] Input:

integrate(x^(-4-6*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 - 4), x)
 

Giac [F]

\[ \int x^{-1-3 (1+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx=\int { {\left (b x^{3} + a\right )}^{p} {\left (d x^{3} + c\right )}^{p} x^{-6 \, p - 4} \,d x } \] Input:

integrate(x^(-4-6*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 - 4), x)
 

Mupad [F(-1)]

Timed out. \[ \int x^{-1-3 (1+2 p)} \left (a+b x^3\right )^p \left (c+d x^3\right )^p \, dx=\int \frac {{\left (b\,x^3+a\right )}^p\,{\left (d\,x^3+c\right )}^p}{x^{6\,p+4}} \,d x \] Input:

int(((a + b*x^3)^p*(c + d*x^3)^p)/x^(6*p + 4),x)
 

Output:

int(((a + b*x^3)^p*(c + d*x^3)^p)/x^(6*p + 4), 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^(-4-6*p)*(b*x^3+a)^p*(d*x^3+c)^p,x)
 

Output:

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