\(\int (a+b x^2)^p (c+d x^2)^q \, dx\) [1643]

Optimal result

Integrand size = 19, antiderivative size = 79 \[ \int \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (c+d x^2\right )^q \left (1+\frac {d x^2}{c}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right ) \] Output:

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)
 

Mathematica [B] (warning: unable to verify)

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

Time = 0.07 (sec) , antiderivative size = 172, normalized size of antiderivative = 2.18 \[ \int \left (a+b x^2\right )^p \left (c+d x^2\right )^q \, dx=\frac {3 a c x \left (a+b x^2\right )^p \left (c+d x^2\right )^q \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )}{3 a c \operatorname {AppellF1}\left (\frac {1}{2},-p,-q,\frac {3}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+2 x^2 \left (b c p \operatorname {AppellF1}\left (\frac {3}{2},1-p,-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )+a d q \operatorname {AppellF1}\left (\frac {3}{2},-p,1-q,\frac {5}{2},-\frac {b x^2}{a},-\frac {d x^2}{c}\right )\right )} \] Input:

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

Output:

(3*a*c*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)])/(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)]))
 

Rubi [A] (verified)

Time = 0.20 (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 = {334, 334, 333}

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.

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

\(\Big \downarrow \) 334

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

\(\Big \downarrow \) 334

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

\(\Big \downarrow \) 333

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

Input:

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

Output:

(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)
 

Defintions of rubi rules used

rule 333

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[a^p*c^q*x*AppellF1[1/2, -p, -q, 3/2, (-b)*(x^2/a), (-d)*(x^2/c)], x] /; F 
reeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && (IntegerQ[p] || GtQ[a, 
0]) && (IntegerQ[q] || GtQ[c, 0])
 

rule 334

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

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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