\(\int (e x)^m (a+b x^2)^p (c+d x^2)^p \, dx\) [1651]

Optimal result

Integrand size = 24, antiderivative size = 101 \[ \int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=\frac {(e x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^p \left (1+\frac {d x^2}{c}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m}{2},-p,-p,\frac {3+m}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{e (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.96 \[ \int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx=\frac {x (e x)^m \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^p \left (1+\frac {d x^2}{c}\right )^{-p} \operatorname {AppellF1}\left (\frac {1+m}{2},-p,-p,\frac {3+m}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{1+m} \] Input:

Integrate[(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^p,x]
 

Output:

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

Rubi [A] (verified)

Time = 0.24 (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 = {395, 395, 394}

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.

\(\displaystyle \int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right )^p \, dx\)

\(\Big \downarrow \) 395

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

\(\Big \downarrow \) 395

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

\(\Big \downarrow \) 394

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

Input:

Int[(e*x)^m*(a + b*x^2)^p*(c + d*x^2)^p,x]
 

Output:

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

Defintions of rubi rules used

rule 394

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^p*c^q*((e*x)^(m + 1)/(e*(m + 1)))*AppellF1[(m + 1)/2 
, -p, -q, 1 + (m + 1)/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; FreeQ[{a, b, c, 
 d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 1] && (Int 
egerQ[p] || GtQ[a, 0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 395

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ 
), x_Symbol] :> Simp[a^IntPart[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^ 
FracPart[p])   Int[(e*x)^m*(1 + b*(x^2/a))^p*(c + d*x^2)^q, x], x] /; FreeQ 
[{a, b, c, d, e, m, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, 
1] &&  !(IntegerQ[p] || GtQ[a, 0])
 

Maple [F]

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

Input:

int((e*x)^m*(b*x^2+a)^p*(d*x^2+c)^p,x)
 

Output:

int((e*x)^m*(b*x^2+a)^p*(d*x^2+c)^p,x)
 

Fricas [F]

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

integrate((e*x)^m*(b*x^2+a)^p*(d*x^2+c)^p,x, algorithm="fricas")
 

Output:

integral((b*x^2 + a)^p*(d*x^2 + c)^p*(e*x)^m, x)
 

Sympy [F(-1)]

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

integrate((e*x)**m*(b*x**2+a)**p*(d*x**2+c)**p,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((e*x)^m*(b*x^2+a)^p*(d*x^2+c)^p,x, algorithm="maxima")
 

Output:

integrate((b*x^2 + a)^p*(d*x^2 + c)^p*(e*x)^m, x)
 

Giac [F]

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

integrate((e*x)^m*(b*x^2+a)^p*(d*x^2+c)^p,x, algorithm="giac")
 

Output:

integrate((b*x^2 + a)^p*(d*x^2 + c)^p*(e*x)^m, x)
 

Mupad [F(-1)]

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

int((e*x)^m*(a + b*x^2)^p*(c + d*x^2)^p,x)
 

Output:

int((e*x)^m*(a + b*x^2)^p*(c + d*x^2)^p, x)
 

Reduce [F]

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

int((e*x)^m*(b*x^2+a)^p*(d*x^2+c)^p,x)
 

Output:

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