\(\int (e x)^m (a+b x^2)^p (A+B x^2) (c+d x^2)^2 \, dx\) [45]

Optimal result

Integrand size = 31, antiderivative size = 399 \[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=\frac {\left (a^2 B d^2 \left (15+8 m+m^2\right )-a b d (2 B c+A d) (3+m) (7+m+2 p)+b^2 c (B c+2 A d) \left (35+m^2+24 p+4 p^2+4 m (3+p)\right )\right ) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{b^3 e (3+m+2 p) (5+m+2 p) (7+m+2 p)}-\frac {d (a B d (5+m)-b (2 B c+A d) (7+m+2 p)) (e x)^{3+m} \left (a+b x^2\right )^{1+p}}{b^2 e^3 (5+m+2 p) (7+m+2 p)}+\frac {B d^2 (e x)^{5+m} \left (a+b x^2\right )^{1+p}}{b e^5 (7+m+2 p)}+\frac {\left (\frac {A c^2}{1+m}-\frac {a \left (a^2 B d^2 \left (15+8 m+m^2\right )-a b d (2 B c+A d) (3+m) (7+m+2 p)+b^2 c (B c+2 A d) \left (35+m^2+24 p+4 p^2+4 m (3+p)\right )\right )}{b^3 (3+m+2 p) (5+m+2 p) (7+m+2 p)}\right ) (e x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {b x^2}{a}\right )}{e} \] Output:

(a^2*B*d^2*(m^2+8*m+15)-a*b*d*(A*d+2*B*c)*(3+m)*(7+m+2*p)+b^2*c*(2*A*d+B*c 
)*(35+m^2+24*p+4*p^2+4*m*(3+p)))*(e*x)^(1+m)*(b*x^2+a)^(p+1)/b^3/e/(3+m+2* 
p)/(5+m+2*p)/(7+m+2*p)-d*(a*B*d*(5+m)-b*(A*d+2*B*c)*(7+m+2*p))*(e*x)^(3+m) 
*(b*x^2+a)^(p+1)/b^2/e^3/(5+m+2*p)/(7+m+2*p)+B*d^2*(e*x)^(5+m)*(b*x^2+a)^( 
p+1)/b/e^5/(7+m+2*p)+(A*c^2/(1+m)-a*(a^2*B*d^2*(m^2+8*m+15)-a*b*d*(A*d+2*B 
*c)*(3+m)*(7+m+2*p)+b^2*c*(2*A*d+B*c)*(35+m^2+24*p+4*p^2+4*m*(3+p)))/b^3/( 
3+m+2*p)/(5+m+2*p)/(7+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 = 0.16 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.50 \[ \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^2 \, dx=x (e x)^m \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (\frac {A c^2 \operatorname {Hypergeometric2F1}\left (\frac {1+m}{2},-p,\frac {3+m}{2},-\frac {b x^2}{a}\right )}{1+m}+\frac {c (B c+2 A d) x^2 \operatorname {Hypergeometric2F1}\left (\frac {3+m}{2},-p,\frac {5+m}{2},-\frac {b x^2}{a}\right )}{3+m}+d x^4 \left (\frac {(2 B c+A d) \operatorname {Hypergeometric2F1}\left (\frac {5+m}{2},-p,\frac {7+m}{2},-\frac {b x^2}{a}\right )}{5+m}+\frac {B d x^2 \operatorname {Hypergeometric2F1}\left (\frac {7+m}{2},-p,\frac {9+m}{2},-\frac {b x^2}{a}\right )}{7+m}\right )\right ) \] Input:

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

Output:

(x*(e*x)^m*(a + b*x^2)^p*((A*c^2*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/ 
2, -((b*x^2)/a)])/(1 + m) + (c*(B*c + 2*A*d)*x^2*Hypergeometric2F1[(3 + m) 
/2, -p, (5 + m)/2, -((b*x^2)/a)])/(3 + m) + d*x^4*(((2*B*c + A*d)*Hypergeo 
metric2F1[(5 + m)/2, -p, (7 + m)/2, -((b*x^2)/a)])/(5 + m) + (B*d*x^2*Hype 
rgeometric2F1[(7 + m)/2, -p, (9 + m)/2, -((b*x^2)/a)])/(7 + m))))/(1 + (b* 
x^2)/a)^p
 

Rubi [A] (verified)

Time = 0.81 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {443, 25, 443, 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.

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

\(\Big \downarrow \) 443

\(\displaystyle \frac {\int -(e x)^m \left (b x^2+a\right )^p \left (d x^2+c\right ) \left (c (a B (m+1)-A b (m+2 p+7))-(4 b B c-a B d (m+5)+A b d (m+2 p+7)) x^2\right )dx}{b (m+2 p+7)}+\frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)}-\frac {\int (e x)^m \left (b x^2+a\right )^p \left (d x^2+c\right ) \left ((a B d (m+5)-b (4 B c+A d (m+2 p+7))) x^2+c (a B (m+1)-A b (m+2 p+7))\right )dx}{b (m+2 p+7)}\)

\(\Big \downarrow \) 443

\(\displaystyle \frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)}-\frac {\frac {\int (e x)^m \left (b x^2+a\right )^p \left ((2 b c d (p+2) (a B (m+1)-A b (m+2 p+7))+d (b c-a d) (m+1) (a B (m+5)-A b (m+2 p+7))+2 (b c-a d) (a B d (m+5)-b (4 B c+A d (m+2 p+7)))) x^2+c (2 b c (p+2) (a B (m+1)-A b (m+2 p+7))+(b c-a d) (m+1) (a B (m+5)-A b (m+2 p+7)))\right )dx}{b (m+2 p+5)}-\frac {\left (c+d x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1} (-a B d (m+5)+A b d (m+2 p+7)+4 b B c)}{b e (m+2 p+5)}}{b (m+2 p+7)}\)

\(\Big \downarrow \) 363

\(\displaystyle \frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)}-\frac {\frac {\left (\frac {a (m+1) \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b (m+2 p+3)}+c ((m+1) (b c-a d) (a B (m+5)-A b (m+2 p+7))+2 b c (p+2) (a B (m+1)-A b (m+2 p+7)))\right ) \int (e x)^m \left (b x^2+a\right )^pdx-\frac {(e x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b e (m+2 p+3)}}{b (m+2 p+5)}-\frac {\left (c+d x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1} (-a B d (m+5)+A b d (m+2 p+7)+4 b B c)}{b e (m+2 p+5)}}{b (m+2 p+7)}\)

\(\Big \downarrow \) 279

\(\displaystyle \frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)}-\frac {\frac {\left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (\frac {a (m+1) \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b (m+2 p+3)}+c ((m+1) (b c-a d) (a B (m+5)-A b (m+2 p+7))+2 b c (p+2) (a B (m+1)-A b (m+2 p+7)))\right ) \int (e x)^m \left (\frac {b x^2}{a}+1\right )^pdx-\frac {(e x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b e (m+2 p+3)}}{b (m+2 p+5)}-\frac {\left (c+d x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1} (-a B d (m+5)+A b d (m+2 p+7)+4 b B c)}{b e (m+2 p+5)}}{b (m+2 p+7)}\)

\(\Big \downarrow \) 278

\(\displaystyle \frac {B \left (c+d x^2\right )^2 (e x)^{m+1} \left (a+b x^2\right )^{p+1}}{b e (m+2 p+7)}-\frac {\frac {\frac {(e x)^{m+1} \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {m+1}{2},-p,\frac {m+3}{2},-\frac {b x^2}{a}\right ) \left (\frac {a (m+1) \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b (m+2 p+3)}+c ((m+1) (b c-a d) (a B (m+5)-A b (m+2 p+7))+2 b c (p+2) (a B (m+1)-A b (m+2 p+7)))\right )}{e (m+1)}-\frac {(e x)^{m+1} \left (a+b x^2\right )^{p+1} \left (a^2 B d^2 \left (m^2+8 m+15\right )-a b d \left (A d (m+3) (m+2 p+7)+B c \left (m^2+2 m (p+6)+2 p+27\right )\right )+b^2 c \left (A d (m+2 p+7)^2+8 B c\right )\right )}{b e (m+2 p+3)}}{b (m+2 p+5)}-\frac {\left (c+d x^2\right ) (e x)^{m+1} \left (a+b x^2\right )^{p+1} (-a B d (m+5)+A b d (m+2 p+7)+4 b B c)}{b e (m+2 p+5)}}{b (m+2 p+7)}\)

Input:

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

Output:

(B*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^2)^2)/(b*e*(7 + m + 2*p)) - 
(-(((4*b*B*c - a*B*d*(5 + m) + A*b*d*(7 + m + 2*p))*(e*x)^(1 + m)*(a + b*x 
^2)^(1 + p)*(c + d*x^2))/(b*e*(5 + m + 2*p))) + (-(((a^2*B*d^2*(15 + 8*m + 
 m^2) + b^2*c*(8*B*c + A*d*(7 + m + 2*p)^2) - a*b*d*(A*d*(3 + m)*(7 + m + 
2*p) + B*c*(27 + m^2 + 2*p + 2*m*(6 + p))))*(e*x)^(1 + m)*(a + b*x^2)^(1 + 
 p))/(b*e*(3 + m + 2*p))) + ((c*(2*b*c*(2 + p)*(a*B*(1 + m) - A*b*(7 + m + 
 2*p)) + (b*c - a*d)*(1 + m)*(a*B*(5 + m) - A*b*(7 + m + 2*p))) + (a*(1 + 
m)*(a^2*B*d^2*(15 + 8*m + m^2) + b^2*c*(8*B*c + A*d*(7 + m + 2*p)^2) - a*b 
*d*(A*d*(3 + m)*(7 + m + 2*p) + B*c*(27 + m^2 + 2*p + 2*m*(6 + 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 
)))/(b*(7 + 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]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

integral((B*d^2*x^6 + (2*B*c*d + A*d^2)*x^4 + A*c^2 + (B*c^2 + 2*A*c*d)*x^ 
2)*(b*x^2 + a)^p*(e*x)^m, x)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(e**m*( - 4*x**m*(a + b*x**2)**p*a**3*d**2*m*p**2*x - 4*x**m*(a + b*x**2)* 
*p*a**3*d**2*m*p*x - 12*x**m*(a + b*x**2)**p*a**3*d**2*p**2*x - 12*x**m*(a 
 + b*x**2)**p*a**3*d**2*p*x + 8*x**m*(a + b*x**2)**p*a**2*b*c*d*m*p**2*x + 
 8*x**m*(a + b*x**2)**p*a**2*b*c*d*m*p*x + 16*x**m*(a + b*x**2)**p*a**2*b* 
c*d*p**3*x + 72*x**m*(a + b*x**2)**p*a**2*b*c*d*p**2*x + 56*x**m*(a + b*x* 
*2)**p*a**2*b*c*d*p*x + 4*x**m*(a + b*x**2)**p*a**2*b*d**2*m*p**2*x**3 + 4 
*x**m*(a + b*x**2)**p*a**2*b*d**2*m*p*x**3 + 8*x**m*(a + b*x**2)**p*a**2*b 
*d**2*p**3*x**3 + 12*x**m*(a + b*x**2)**p*a**2*b*d**2*p**2*x**3 + 4*x**m*( 
a + b*x**2)**p*a**2*b*d**2*p*x**3 + x**m*(a + b*x**2)**p*a*b**2*c**2*m**3* 
x + 8*x**m*(a + b*x**2)**p*a*b**2*c**2*m**2*p*x + 15*x**m*(a + b*x**2)**p* 
a*b**2*c**2*m**2*x + 20*x**m*(a + b*x**2)**p*a*b**2*c**2*m*p**2*x + 84*x** 
m*(a + b*x**2)**p*a*b**2*c**2*m*p*x + 71*x**m*(a + b*x**2)**p*a*b**2*c**2* 
m*x + 16*x**m*(a + b*x**2)**p*a*b**2*c**2*p**3*x + 108*x**m*(a + b*x**2)** 
p*a*b**2*c**2*p**2*x + 212*x**m*(a + b*x**2)**p*a*b**2*c**2*p*x + 105*x**m 
*(a + b*x**2)**p*a*b**2*c**2*x + 2*x**m*(a + b*x**2)**p*a*b**2*c*d*m**3*x* 
*3 + 16*x**m*(a + b*x**2)**p*a*b**2*c*d*m**2*p*x**3 + 26*x**m*(a + b*x**2) 
**p*a*b**2*c*d*m**2*x**3 + 40*x**m*(a + b*x**2)**p*a*b**2*c*d*m*p**2*x**3 
+ 136*x**m*(a + b*x**2)**p*a*b**2*c*d*m*p*x**3 + 94*x**m*(a + b*x**2)**p*a 
*b**2*c*d*m*x**3 + 32*x**m*(a + b*x**2)**p*a*b**2*c*d*p**3*x**3 + 168*x**m 
*(a + b*x**2)**p*a*b**2*c*d*p**2*x**3 + 216*x**m*(a + b*x**2)**p*a*b**2...