\(\int (a+b x^2)^p (c+d x^2)^q (e+f x^2)^2 \, dx\) [82]

Optimal result

Integrand size = 28, antiderivative size = 256 \[ \int \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )^2 \, dx=e^2 x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+\frac {2}{3} e f x^3 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+\frac {1}{5} f^2 x^5 \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {5}{2},-p,-q,\frac {7}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) \] Output:

e^2*x*(b*x^2+a)^p*(d*x^2+c)^q*AppellF1(1/2,-p,-q,3/2,-b*x^2/a,-d*x^2/c)/(( 
1+b*x^2/a)^p)/((1+d*x^2/c)^q)+2/3*e*f*x^3*(b*x^2+a)^p*(d*x^2+c)^q*AppellF1 
(3/2,-p,-q,5/2,-b*x^2/a,-d*x^2/c)/((1+b*x^2/a)^p)/((1+d*x^2/c)^q)+1/5*f^2* 
x^5*(b*x^2+a)^p*(d*x^2+c)^q*AppellF1(5/2,-p,-q,7/2,-b*x^2/a,-d*x^2/c)/((1+ 
b*x^2/a)^p)/((1+d*x^2/c)^q)
 

Mathematica [A] (warning: unable to verify)

Time = 0.41 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.11 \[ \int \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (e+f x^2\right )^2 \, dx=\frac {1}{15} x \left (a+b x^2\right )^p \left (c+d x^2\right )^q \left (\frac {45 a c e^2 \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{3 a c \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+2 x^2 \left (b c p \operatorname {AppellF1}\left (\frac {3}{2},1-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+a d q \operatorname {AppellF1}\left (\frac {3}{2},-p,1-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )}+f x^2 \left (1+\frac {b x^2}{a}\right )^{-p} \left (1+\frac {d x^2}{c}\right )^{-q} \left (10 e \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+3 f x^2 \operatorname {AppellF1}\left (\frac {5}{2},-p,-q,\frac {7}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )\right ) \] Input:

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

Output:

(x*(a + b*x^2)^p*(c + d*x^2)^q*((45*a*c*e^2*AppellF1[1/2, -p, -q, 3/2, -(( 
b*x^2)/a), -((d*x^2)/c)])/(3*a*c*AppellF1[1/2, -p, -q, 3/2, -((b*x^2)/a), 
-((d*x^2)/c)] + 2*x^2*(b*c*p*AppellF1[3/2, 1 - p, -q, 5/2, -((b*x^2)/a), - 
((d*x^2)/c)] + a*d*q*AppellF1[3/2, -p, 1 - q, 5/2, -((b*x^2)/a), -((d*x^2) 
/c)])) + (f*x^2*(10*e*AppellF1[3/2, -p, -q, 5/2, -((b*x^2)/a), -((d*x^2)/c 
)] + 3*f*x^2*AppellF1[5/2, -p, -q, 7/2, -((b*x^2)/a), -((d*x^2)/c)]))/((1 
+ (b*x^2)/a)^p*(1 + (d*x^2)/c)^q)))/15
 

Rubi [A] (verified)

Time = 0.68 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {433, 2009}

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.

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

\(\Big \downarrow \) 433

\(\displaystyle \int \left (e^2 \left (a+b x^2\right )^p \left (c+d x^2\right )^q+2 e f x^2 \left (a+b x^2\right )^p \left (c+d x^2\right )^q+f^2 x^4 \left (a+b x^2\right )^p \left (c+d x^2\right )^q\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle e^2 x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+\frac {2}{3} e f x^3 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {3}{2},-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+\frac {1}{5} f^2 x^5 \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (c+d x^2\right )^q \left (\frac {d x^2}{c}+1\right )^{-q} \operatorname {AppellF1}\left (\frac {5}{2},-p,-q,\frac {7}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\)

Input:

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

Output:

(e^2*x*(a + b*x^2)^p*(c + d*x^2)^q*AppellF1[1/2, -p, -q, 3/2, -((b*x^2)/a) 
, -((d*x^2)/c)])/((1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q) + (2*e*f*x^3*(a + b 
*x^2)^p*(c + d*x^2)^q*AppellF1[3/2, -p, -q, 5/2, -((b*x^2)/a), -((d*x^2)/c 
)])/(3*(1 + (b*x^2)/a)^p*(1 + (d*x^2)/c)^q) + (f^2*x^5*(a + b*x^2)^p*(c + 
d*x^2)^q*AppellF1[5/2, -p, -q, 7/2, -((b*x^2)/a), -((d*x^2)/c)])/(5*(1 + ( 
b*x^2)/a)^p*(1 + (d*x^2)/c)^q)
 

Defintions of rubi rules used

rule 433

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_)*((e_) + (f_.)*(x_ 
)^2)^(r_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*x^2)^p*(c + d*x^2) 
^q*(e + f*x^2)^r, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, f, p, 
 q, r}, x]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((f^2*x^4 + 2*e*f*x^2 + e^2)*(b*x^2 + a)^p*(d*x^2 + c)^q, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 4*(c + d*x**2)**q*(a + b*x**2)**p*a**2*d**2*f**2*p*q*x - 6*(c + d*x**2 
)**q*(a + b*x**2)**p*a**2*d**2*f**2*p*x + 8*(c + d*x**2)**q*(a + b*x**2)** 
p*a*b*c*d*f**2*p*q*x + 8*(c + d*x**2)**q*(a + b*x**2)**p*a*b*d**2*e*f*p**2 
*x + 8*(c + d*x**2)**q*(a + b*x**2)**p*a*b*d**2*e*f*p*q*x + 20*(c + d*x**2 
)**q*(a + b*x**2)**p*a*b*d**2*e*f*p*x + 4*(c + d*x**2)**q*(a + b*x**2)**p* 
a*b*d**2*f**2*p**2*x**3 + 4*(c + d*x**2)**q*(a + b*x**2)**p*a*b*d**2*f**2* 
p*q*x**3 + 2*(c + d*x**2)**q*(a + b*x**2)**p*a*b*d**2*f**2*p*x**3 - 4*(c + 
 d*x**2)**q*(a + b*x**2)**p*b**2*c**2*f**2*p*q*x - 6*(c + d*x**2)**q*(a + 
b*x**2)**p*b**2*c**2*f**2*q*x + 8*(c + d*x**2)**q*(a + b*x**2)**p*b**2*c*d 
*e*f*p*q*x + 8*(c + d*x**2)**q*(a + b*x**2)**p*b**2*c*d*e*f*q**2*x + 20*(c 
 + d*x**2)**q*(a + b*x**2)**p*b**2*c*d*e*f*q*x + 4*(c + d*x**2)**q*(a + b* 
x**2)**p*b**2*c*d*f**2*p*q*x**3 + 4*(c + d*x**2)**q*(a + b*x**2)**p*b**2*c 
*d*f**2*q**2*x**3 + 2*(c + d*x**2)**q*(a + b*x**2)**p*b**2*c*d*f**2*q*x**3 
 + 4*(c + d*x**2)**q*(a + b*x**2)**p*b**2*d**2*e**2*p**2*x + 8*(c + d*x**2 
)**q*(a + b*x**2)**p*b**2*d**2*e**2*p*q*x + 16*(c + d*x**2)**q*(a + b*x**2 
)**p*b**2*d**2*e**2*p*x + 4*(c + d*x**2)**q*(a + b*x**2)**p*b**2*d**2*e**2 
*q**2*x + 16*(c + d*x**2)**q*(a + b*x**2)**p*b**2*d**2*e**2*q*x + 15*(c + 
d*x**2)**q*(a + b*x**2)**p*b**2*d**2*e**2*x + 8*(c + d*x**2)**q*(a + b*x** 
2)**p*b**2*d**2*e*f*p**2*x**3 + 16*(c + d*x**2)**q*(a + b*x**2)**p*b**2*d* 
*2*e*f*p*q*x**3 + 24*(c + d*x**2)**q*(a + b*x**2)**p*b**2*d**2*e*f*p*x*...