\(\int (e x)^m (a+b x^4)^p (c+d x^4)^p \, dx\) [297]

Optimal result

Integrand size = 24, antiderivative size = 101 \[ \int (e x)^m \left (a+b x^4\right )^p \left (c+d x^4\right )^p \, dx=\frac {(e x)^{1+m} \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \left (c+d x^4\right )^p \left (1+\frac {d x^4}{c}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m}{4},-p,-p,\frac {5+m}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{e (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.96 \[ \int (e x)^m \left (a+b x^4\right )^p \left (c+d x^4\right )^p \, dx=\frac {x (e x)^m \left (a+b x^4\right )^p \left (1+\frac {b x^4}{a}\right )^{-p} \left (c+d x^4\right )^p \left (1+\frac {d x^4}{c}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m}{4},-p,-p,\frac {5+m}{4},-\frac {b x^4}{a},-\frac {d x^4}{c}\right )}{1+m} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, 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.

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1013

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

\(\Big \downarrow \) 1012

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

Input:

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

Output:

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

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(e**m*(x**m*(c + d*x**4)**p*(a + b*x**4)**p*x + 4*int((x**m*(c + d*x**4)** 
p*(a + b*x**4)**p*x**4)/(a*c*m + 8*a*c*p + a*c + a*d*m*x**4 + 8*a*d*p*x**4 
 + a*d*x**4 + b*c*m*x**4 + 8*b*c*p*x**4 + b*c*x**4 + b*d*m*x**8 + 8*b*d*p* 
x**8 + b*d*x**8),x)*a*d*m*p + 32*int((x**m*(c + d*x**4)**p*(a + b*x**4)**p 
*x**4)/(a*c*m + 8*a*c*p + a*c + a*d*m*x**4 + 8*a*d*p*x**4 + a*d*x**4 + b*c 
*m*x**4 + 8*b*c*p*x**4 + b*c*x**4 + b*d*m*x**8 + 8*b*d*p*x**8 + b*d*x**8), 
x)*a*d*p**2 + 4*int((x**m*(c + d*x**4)**p*(a + b*x**4)**p*x**4)/(a*c*m + 8 
*a*c*p + a*c + a*d*m*x**4 + 8*a*d*p*x**4 + a*d*x**4 + b*c*m*x**4 + 8*b*c*p 
*x**4 + b*c*x**4 + b*d*m*x**8 + 8*b*d*p*x**8 + b*d*x**8),x)*a*d*p + 4*int( 
(x**m*(c + d*x**4)**p*(a + b*x**4)**p*x**4)/(a*c*m + 8*a*c*p + a*c + a*d*m 
*x**4 + 8*a*d*p*x**4 + a*d*x**4 + b*c*m*x**4 + 8*b*c*p*x**4 + b*c*x**4 + b 
*d*m*x**8 + 8*b*d*p*x**8 + b*d*x**8),x)*b*c*m*p + 32*int((x**m*(c + d*x**4 
)**p*(a + b*x**4)**p*x**4)/(a*c*m + 8*a*c*p + a*c + a*d*m*x**4 + 8*a*d*p*x 
**4 + a*d*x**4 + b*c*m*x**4 + 8*b*c*p*x**4 + b*c*x**4 + b*d*m*x**8 + 8*b*d 
*p*x**8 + b*d*x**8),x)*b*c*p**2 + 4*int((x**m*(c + d*x**4)**p*(a + b*x**4) 
**p*x**4)/(a*c*m + 8*a*c*p + a*c + a*d*m*x**4 + 8*a*d*p*x**4 + a*d*x**4 + 
b*c*m*x**4 + 8*b*c*p*x**4 + b*c*x**4 + b*d*m*x**8 + 8*b*d*p*x**8 + b*d*x** 
8),x)*b*c*p + 8*int((x**m*(c + d*x**4)**p*(a + b*x**4)**p)/(a*c*m + 8*a*c* 
p + a*c + a*d*m*x**4 + 8*a*d*p*x**4 + a*d*x**4 + b*c*m*x**4 + 8*b*c*p*x**4 
 + b*c*x**4 + b*d*m*x**8 + 8*b*d*p*x**8 + b*d*x**8),x)*a*c*m*p + 64*int...