\(\int (a+b x^4)^p (c+d x^4)^q \, dx\) [128]

Optimal result

Integrand size = 19, antiderivative size = 79 \[ \int \left (a+b x^4\right )^p \left (c+d x^4\right )^q \, dx=x \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \left (c+d x^4\right )^q \left (1+\frac {d x^4}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{4},-p,-q,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right ) \] Output:

x*(b*x^4+a)^p*(d*x^4+c)^q*AppellF1(1/4,-p,-q,5/4,-b*x^4/a,-d*x^4/c)/((1+b* 
x^4/a)^p)/((1+d*x^4/c)^q)
 

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(172\) vs. \(2(79)=158\).

Time = 0.32 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.18 \[ \int \left (a+b x^4\right )^p \left (c+d x^4\right )^q \, dx=\frac {5 a c x \left (a+b x^4\right )^p \left (c+d x^4\right )^q \operatorname {AppellF1}\left (\frac {1}{4},-p,-q,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{5 a c \operatorname {AppellF1}\left (\frac {1}{4},-p,-q,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+4 x^4 \left (b c p \operatorname {AppellF1}\left (\frac {5}{4},1-p,-q,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )+a d q \operatorname {AppellF1}\left (\frac {5}{4},-p,1-q,\frac {9}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\right )} \] Input:

Integrate[(a + b*x^4)^p*(c + d*x^4)^q,x]
 

Output:

(5*a*c*x*(a + b*x^4)^p*(c + d*x^4)^q*AppellF1[1/4, -p, -q, 5/4, -((b*x^4)/ 
a), -((d*x^4)/c)])/(5*a*c*AppellF1[1/4, -p, -q, 5/4, -((b*x^4)/a), -((d*x^ 
4)/c)] + 4*x^4*(b*c*p*AppellF1[5/4, 1 - p, -q, 9/4, -((b*x^4)/a), -((d*x^4 
)/c)] + a*d*q*AppellF1[5/4, -p, 1 - q, 9/4, -((b*x^4)/a), -((d*x^4)/c)]))
 

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {937, 937, 936}

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.

\(\displaystyle \int \left (a+b x^4\right )^p \left (c+d x^4\right )^q \, dx\)

\(\Big \downarrow \) 937

\(\displaystyle \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \int \left (\frac {b x^4}{a}+1\right )^p \left (d x^4+c\right )^qdx\)

\(\Big \downarrow \) 937

\(\displaystyle \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \left (c+d x^4\right )^q \left (\frac {d x^4}{c}+1\right )^{-q} \int \left (\frac {b x^4}{a}+1\right )^p \left (\frac {d x^4}{c}+1\right )^qdx\)

\(\Big \downarrow \) 936

\(\displaystyle x \left (a+b x^4\right )^p \left (\frac {b x^4}{a}+1\right )^{-p} \left (c+d x^4\right )^q \left (\frac {d x^4}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{4},-p,-q,\frac {5}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )\)

Input:

Int[(a + b*x^4)^p*(c + d*x^4)^q,x]
 

Output:

(x*(a + b*x^4)^p*(c + d*x^4)^q*AppellF1[1/4, -p, -q, 5/4, -((b*x^4)/a), -( 
(d*x^4)/c)])/((1 + (b*x^4)/a)^p*(1 + (d*x^4)/c)^q)
 

Defintions of rubi rules used

rule 936

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

rule 937

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

Maple [F]

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

Input:

int((b*x^4+a)^p*(d*x^4+c)^q,x)
 

Output:

int((b*x^4+a)^p*(d*x^4+c)^q,x)
 

Fricas [F]

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

integrate((b*x^4+a)^p*(d*x^4+c)^q,x, algorithm="fricas")
 

Output:

integral((b*x^4 + a)^p*(d*x^4 + c)^q, x)
 

Sympy [F(-1)]

Timed out. \[ \int \left (a+b x^4\right )^p \left (c+d x^4\right )^q \, dx=\text {Timed out} \] Input:

integrate((b*x**4+a)**p*(d*x**4+c)**q,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((b*x^4+a)^p*(d*x^4+c)^q,x, algorithm="maxima")
 

Output:

integrate((b*x^4 + a)^p*(d*x^4 + c)^q, x)
 

Giac [F]

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

integrate((b*x^4+a)^p*(d*x^4+c)^q,x, algorithm="giac")
 

Output:

integrate((b*x^4 + a)^p*(d*x^4 + c)^q, x)
 

Mupad [F(-1)]

Timed out. \[ \int \left (a+b x^4\right )^p \left (c+d x^4\right )^q \, dx=\int {\left (b\,x^4+a\right )}^p\,{\left (d\,x^4+c\right )}^q \,d x \] Input:

int((a + b*x^4)^p*(c + d*x^4)^q,x)
 

Output:

int((a + b*x^4)^p*(c + d*x^4)^q, x)
 

Reduce [F]

\[ \int \left (a+b x^4\right )^p \left (c+d x^4\right )^q \, dx =\text {Too large to display} \] Input:

int((b*x^4+a)^p*(d*x^4+c)^q,x)
 

Output:

((c + d*x**4)**q*(a + b*x**4)**p*x + 16*int(((c + d*x**4)**q*(a + b*x**4)* 
*p*x**4)/(4*a*c*p + 4*a*c*q + a*c + 4*a*d*p*x**4 + 4*a*d*q*x**4 + a*d*x**4 
 + 4*b*c*p*x**4 + 4*b*c*q*x**4 + b*c*x**4 + 4*b*d*p*x**8 + 4*b*d*q*x**8 + 
b*d*x**8),x)*a*d*p**2 + 16*int(((c + d*x**4)**q*(a + b*x**4)**p*x**4)/(4*a 
*c*p + 4*a*c*q + a*c + 4*a*d*p*x**4 + 4*a*d*q*x**4 + a*d*x**4 + 4*b*c*p*x* 
*4 + 4*b*c*q*x**4 + b*c*x**4 + 4*b*d*p*x**8 + 4*b*d*q*x**8 + b*d*x**8),x)* 
a*d*p*q + 4*int(((c + d*x**4)**q*(a + b*x**4)**p*x**4)/(4*a*c*p + 4*a*c*q 
+ a*c + 4*a*d*p*x**4 + 4*a*d*q*x**4 + a*d*x**4 + 4*b*c*p*x**4 + 4*b*c*q*x* 
*4 + b*c*x**4 + 4*b*d*p*x**8 + 4*b*d*q*x**8 + b*d*x**8),x)*a*d*p + 16*int( 
((c + d*x**4)**q*(a + b*x**4)**p*x**4)/(4*a*c*p + 4*a*c*q + a*c + 4*a*d*p* 
x**4 + 4*a*d*q*x**4 + a*d*x**4 + 4*b*c*p*x**4 + 4*b*c*q*x**4 + b*c*x**4 + 
4*b*d*p*x**8 + 4*b*d*q*x**8 + b*d*x**8),x)*b*c*p*q + 16*int(((c + d*x**4)* 
*q*(a + b*x**4)**p*x**4)/(4*a*c*p + 4*a*c*q + a*c + 4*a*d*p*x**4 + 4*a*d*q 
*x**4 + a*d*x**4 + 4*b*c*p*x**4 + 4*b*c*q*x**4 + b*c*x**4 + 4*b*d*p*x**8 + 
 4*b*d*q*x**8 + b*d*x**8),x)*b*c*q**2 + 4*int(((c + d*x**4)**q*(a + b*x**4 
)**p*x**4)/(4*a*c*p + 4*a*c*q + a*c + 4*a*d*p*x**4 + 4*a*d*q*x**4 + a*d*x* 
*4 + 4*b*c*p*x**4 + 4*b*c*q*x**4 + b*c*x**4 + 4*b*d*p*x**8 + 4*b*d*q*x**8 
+ b*d*x**8),x)*b*c*q + 16*int(((c + d*x**4)**q*(a + b*x**4)**p)/(4*a*c*p + 
 4*a*c*q + a*c + 4*a*d*p*x**4 + 4*a*d*q*x**4 + a*d*x**4 + 4*b*c*p*x**4 + 4 
*b*c*q*x**4 + b*c*x**4 + 4*b*d*p*x**8 + 4*b*d*q*x**8 + b*d*x**8),x)*a*c...