\(\int (e x)^m (a+b x^2)^p (A+B x^2) (c+d x^2) \, dx\) [46]

Optimal result

Integrand size = 29, antiderivative size = 217 \[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=-\frac {(a B d (3+m)-b (B c+A d) (5+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{b^2 e (3+m+2 p) (5+m+2 p)}+\frac {B d (e x)^{3+m} \left (a+b x^2\right )^{1+p}}{b e^3 (5+m+2 p)}+\frac {\left (\frac {A c}{1+m}+\frac {a (a B d (3+m)-b (B c+A d) (5+m+2 p))}{b^2 (3+m+2 p) (5+m+2 p)}\right ) (e x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {b x^2}{a}\right )}{e} \] Output:

-(a*B*d*(3+m)-b*(A*d+B*c)*(5+m+2*p))*(e*x)^(1+m)*(b*x^2+a)^(p+1)/b^2/e/(3+ 
m+2*p)/(5+m+2*p)+B*d*(e*x)^(3+m)*(b*x^2+a)^(p+1)/b/e^3/(5+m+2*p)+(A*c/(1+m 
)+a*(a*B*d*(3+m)-b*(A*d+B*c)*(5+m+2*p))/b^2/(3+m+2*p)/(5+m+2*p))*(e*x)^(1+ 
m)*(b*x^2+a)^p*hypergeom([-p, 1/2+1/2*m],[3/2+1/2*m],-b*x^2/a)/e/((1+b*x^2 
/a)^p)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.68 \[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=x (e x)^m \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (\frac {A c \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {b x^2}{a}\right )}{1+m}+\frac {(B c+A d) x^2 \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},-p,\frac {5+m}{2},-\frac {b x^2}{a}\right )}{3+m}+\frac {B d x^4 \operatorname {Hypergeometric2F1}\left (\frac {5+m}{2},-p,\frac {7+m}{2},-\frac {b x^2}{a}\right )}{5+m}\right ) \] Input:

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

Output:

(x*(e*x)^m*(a + b*x^2)^p*((A*c*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, 
 -((b*x^2)/a)])/(1 + m) + ((B*c + A*d)*x^2*Hypergeometric2F1[(3 + m)/2, -p 
, (5 + m)/2, -((b*x^2)/a)])/(3 + m) + (B*d*x^4*Hypergeometric2F1[(5 + m)/2 
, -p, (7 + m)/2, -((b*x^2)/a)])/(5 + m)))/(1 + (b*x^2)/a)^p
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.11, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {443, 25, 363, 279, 278}

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.

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

\(\Big \downarrow \) 443

\(\displaystyle \frac {\int -(e x)^m \left (b x^2+a\right )^p \left (A (a d (m+1)-b c (m+2 p+5))-(2 A b d-a B (m+3) d+b B c (m+2 p+5)) x^2\right )dx}{b (m+2 p+5)}+\frac {d \left (A+B x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+5)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {d \left (A+B x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+5)}-\frac {\int (e x)^m \left (b x^2+a\right )^p \left ((a B d (m+3)-b (2 A d+B c (m+2 p+5))) x^2+A (a d (m+1)-b c (m+2 p+5))\right )dx}{b (m+2 p+5)}\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {d \left (A+B x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+5)}-\frac {\left (-\frac {a (m+1) (a B d (m+3)-b (2 A d+B c (m+2 p+5)))}{b (m+2 p+3)}+a A d (m+1)-A b c (m+2 p+5)\right ) \int (e x)^m \left (b x^2+a\right )^pdx+\frac {(e x)^{m+1} \left (a+b x^2\right )^{p+1} (a B d (m+3)-b (2 A d+B c (m+2 p+5)))}{b e (m+2 p+3)}}{b (m+2 p+5)}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {d \left (A+B x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+5)}-\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (-\frac {a (m+1) (a B d (m+3)-b (2 A d+B c (m+2 p+5)))}{b (m+2 p+3)}+a A d (m+1)-A b c (m+2 p+5)\right ) \int (e x)^m \left (\frac {b x^2}{a}+1\right )^pdx+\frac {(e x)^{m+1} \left (a+b x^2\right )^{p+1} (a B d (m+3)-b (2 A d+B c (m+2 p+5)))}{b e (m+2 p+3)}}{b (m+2 p+5)}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {d \left (A+B x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+5)}-\frac {\frac {(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-p,\frac {m+3}{2},-\frac {b x^2}{a}\right ) \left (-\frac {a (m+1) (a B d (m+3)-b (2 A d+B c (m+2 p+5)))}{b (m+2 p+3)}+a A d (m+1)-A b c (m+2 p+5)\right )}{e (m+1)}+\frac {(e x)^{m+1} \left (a+b x^2\right )^{p+1} (a B d (m+3)-b (2 A d+B c (m+2 p+5)))}{b e (m+2 p+3)}}{b (m+2 p+5)}\)

Input:

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

Output:

(d*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(A + B*x^2))/(b*e*(5 + m + 2*p)) - (( 
(a*B*d*(3 + m) - b*(2*A*d + B*c*(5 + m + 2*p)))*(e*x)^(1 + m)*(a + b*x^2)^ 
(1 + p))/(b*e*(3 + m + 2*p)) + ((a*A*d*(1 + m) - A*b*c*(5 + m + 2*p) - (a* 
(1 + m)*(a*B*d*(3 + m) - b*(2*A*d + B*c*(5 + m + 2*p))))/(b*(3 + m + 2*p)) 
)*(e*x)^(1 + m)*(a + b*x^2)^p*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, 
-((b*x^2)/a)])/(e*(1 + m)*(1 + (b*x^2)/a)^p))/(b*(5 + m + 2*p))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 278

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^p*(( 
c*x)^(m + 1)/(c*(m + 1)))*Hypergeometric2F1[-p, (m + 1)/2, (m + 1)/2 + 1, ( 
-b)*(x^2/a)], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && (ILtQ[p, 0 
] || GtQ[a, 0])
 

rule 279

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[a^IntP 
art[p]*((a + b*x^2)^FracPart[p]/(1 + b*(x^2/a))^FracPart[p])   Int[(c*x)^m* 
(1 + b*(x^2/a))^p, x], x] /; FreeQ[{a, b, c, m, p}, x] &&  !IGtQ[p, 0] && 
!(ILtQ[p, 0] || GtQ[a, 0])
 

rule 363

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2), x 
_Symbol] :> Simp[d*(e*x)^(m + 1)*((a + b*x^2)^(p + 1)/(b*e*(m + 2*p + 3))), 
 x] - Simp[(a*d*(m + 1) - b*c*(m + 2*p + 3))/(b*(m + 2*p + 3))   Int[(e*x)^ 
m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b*c - a*d 
, 0] && NeQ[m + 2*p + 3, 0]
 

rule 443

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q 
_.)*((e_) + (f_.)*(x_)^2), x_Symbol] :> Simp[f*(g*x)^(m + 1)*(a + b*x^2)^(p 
 + 1)*((c + d*x^2)^q/(b*g*(m + 2*(p + q + 1) + 1))), x] + Simp[1/(b*(m + 2* 
(p + q + 1) + 1))   Int[(g*x)^m*(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(( 
b*e - a*f)*(m + 1) + b*e*2*(p + q + 1)) + (d*(b*e - a*f)*(m + 1) + f*2*q*(b 
*c - a*d) + b*e*d*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f 
, g, m, p}, x] && GtQ[q, 0] &&  !(EqQ[q, 1] && SimplerQ[e + f*x^2, c + d*x^ 
2])
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((B*d*x^4 + (B*c + A*d)*x^2 + A*c)*(b*x^2 + a)^p*(e*x)^m, x)
 

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 177.90 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.09 \[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right ) \, dx=\frac {A a^{p} c e^{m} x^{m + 1} \Gamma \left (\frac {m}{2} + \frac {1}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {1}{2} \\ \frac {m}{2} + \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {3}{2}\right )} + \frac {A a^{p} d e^{m} x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {B a^{p} c e^{m} x^{m + 3} \Gamma \left (\frac {m}{2} + \frac {3}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {3}{2} \\ \frac {m}{2} + \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {5}{2}\right )} + \frac {B a^{p} d e^{m} x^{m + 5} \Gamma \left (\frac {m}{2} + \frac {5}{2}\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, \frac {m}{2} + \frac {5}{2} \\ \frac {m}{2} + \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \Gamma \left (\frac {m}{2} + \frac {7}{2}\right )} \] Input:

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

Output:

A*a**p*c*e**m*x**(m + 1)*gamma(m/2 + 1/2)*hyper((-p, m/2 + 1/2), (m/2 + 3/ 
2,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(m/2 + 3/2)) + A*a**p*d*e**m*x**(m 
+ 3)*gamma(m/2 + 3/2)*hyper((-p, m/2 + 3/2), (m/2 + 5/2,), b*x**2*exp_pola 
r(I*pi)/a)/(2*gamma(m/2 + 5/2)) + B*a**p*c*e**m*x**(m + 3)*gamma(m/2 + 3/2 
)*hyper((-p, m/2 + 3/2), (m/2 + 5/2,), b*x**2*exp_polar(I*pi)/a)/(2*gamma( 
m/2 + 5/2)) + B*a**p*d*e**m*x**(m + 5)*gamma(m/2 + 5/2)*hyper((-p, m/2 + 5 
/2), (m/2 + 7/2,), b*x**2*exp_polar(I*pi)/a)/(2*gamma(m/2 + 7/2))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(e**m*(4*x**m*(a + b*x**2)**p*a**2*d*p**2*x + 4*x**m*(a + b*x**2)**p*a**2* 
d*p*x + x**m*(a + b*x**2)**p*a*b*c*m**2*x + 6*x**m*(a + b*x**2)**p*a*b*c*m 
*p*x + 8*x**m*(a + b*x**2)**p*a*b*c*m*x + 8*x**m*(a + b*x**2)**p*a*b*c*p** 
2*x + 26*x**m*(a + b*x**2)**p*a*b*c*p*x + 15*x**m*(a + b*x**2)**p*a*b*c*x 
+ x**m*(a + b*x**2)**p*a*b*d*m**2*x**3 + 6*x**m*(a + b*x**2)**p*a*b*d*m*p* 
x**3 + 6*x**m*(a + b*x**2)**p*a*b*d*m*x**3 + 8*x**m*(a + b*x**2)**p*a*b*d* 
p**2*x**3 + 14*x**m*(a + b*x**2)**p*a*b*d*p*x**3 + 5*x**m*(a + b*x**2)**p* 
a*b*d*x**3 + x**m*(a + b*x**2)**p*b**2*c*m**2*x**3 + 4*x**m*(a + b*x**2)** 
p*b**2*c*m*p*x**3 + 6*x**m*(a + b*x**2)**p*b**2*c*m*x**3 + 4*x**m*(a + b*x 
**2)**p*b**2*c*p**2*x**3 + 12*x**m*(a + b*x**2)**p*b**2*c*p*x**3 + 5*x**m* 
(a + b*x**2)**p*b**2*c*x**3 + x**m*(a + b*x**2)**p*b**2*d*m**2*x**5 + 4*x* 
*m*(a + b*x**2)**p*b**2*d*m*p*x**5 + 4*x**m*(a + b*x**2)**p*b**2*d*m*x**5 
+ 4*x**m*(a + b*x**2)**p*b**2*d*p**2*x**5 + 8*x**m*(a + b*x**2)**p*b**2*d* 
p*x**5 + 3*x**m*(a + b*x**2)**p*b**2*d*x**5 - 4*int((x**m*(a + b*x**2)**p) 
/(a*m**3 + 6*a*m**2*p + 9*a*m**2 + 12*a*m*p**2 + 36*a*m*p + 23*a*m + 8*a*p 
**3 + 36*a*p**2 + 46*a*p + 15*a + b*m**3*x**2 + 6*b*m**2*p*x**2 + 9*b*m**2 
*x**2 + 12*b*m*p**2*x**2 + 36*b*m*p*x**2 + 23*b*m*x**2 + 8*b*p**3*x**2 + 3 
6*b*p**2*x**2 + 46*b*p*x**2 + 15*b*x**2),x)*a**3*d*m**4*p**2 - 4*int((x**m 
*(a + b*x**2)**p)/(a*m**3 + 6*a*m**2*p + 9*a*m**2 + 12*a*m*p**2 + 36*a*m*p 
 + 23*a*m + 8*a*p**3 + 36*a*p**2 + 46*a*p + 15*a + b*m**3*x**2 + 6*b*m*...