3.44 \(\int (e x)^m (a+b x^2)^p (A+B x^2) (c+d x^2)^3 \, dx\)

Optimal. Leaf size=1059 \[ \text{result too large to display} \]

[Out]

-(((a^3*B*d^3*(105 + 71*m + 15*m^2 + m^3) - a^2*b*d^2*(5 + m)*(A*d*(3 + m)*(9 + m + 2*p) + 2*B*c*(30 + 13*m +
m^2 + 2*p + 2*m*p)) + a*b^2*c*d*(2*A*d*(216 + m^3 + 84*p + 8*p^2 + 4*m^2*(5 + p) + m*(123 + 44*p + 4*p^2)) + B
*c*(267 + m^3 + 40*p + 4*p^2 + m^2*(21 + 4*p) + m*(143 + 44*p + 4*p^2))) - b^3*c^2*(48*B*c + A*d*(513 + m^3 +
366*p + 92*p^2 + 8*p^3 + m^2*(23 + 6*p) + m*(183 + 92*p + 12*p^2))))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p))/(b^4*e
*(3 + m + 2*p)*(5 + m + 2*p)*(7 + m + 2*p)*(9 + m + 2*p))) + ((a^2*B*d^2*(35 + 12*m + m^2) + b^2*c*(24*B*c + A
*d*(99 + m^2 + 40*p + 4*p^2 + 4*m*(5 + p))) - a*b*d*(A*d*(5 + m)*(9 + m + 2*p) + B*c*(65 + m^2 + 2*p + 2*m*(9
+ p))))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^2))/(b^3*e*(5 + m + 2*p)*(7 + m + 2*p)*(9 + m + 2*p)) - ((a
*B*d*(7 + m) - b*(6*B*c + A*d*(9 + m + 2*p)))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^2)^2)/(b^2*e*(7 + m +
2*p)*(9 + m + 2*p)) + (B*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^2)^3)/(b*e*(9 + m + 2*p)) + ((a*(1 + m)*(
a^3*B*d^3*(105 + 71*m + 15*m^2 + m^3) - a^2*b*d^2*(5 + m)*(A*d*(3 + m)*(9 + m + 2*p) + 2*B*c*(30 + 13*m + m^2
+ 2*p + 2*m*p)) + a*b^2*c*d*(2*A*d*(216 + m^3 + 84*p + 8*p^2 + 4*m^2*(5 + p) + m*(123 + 44*p + 4*p^2)) + B*c*(
267 + m^3 + 40*p + 4*p^2 + m^2*(21 + 4*p) + m*(143 + 44*p + 4*p^2))) - b^3*c^2*(48*B*c + A*d*(513 + m^3 + 366*
p + 92*p^2 + 8*p^3 + m^2*(23 + 6*p) + m*(183 + 92*p + 12*p^2)))) - b*c*(3 + m + 2*p)*(2*b*c*(2 + p)*(2*b*c*(3
+ p)*(a*B*(1 + m) - A*b*(9 + m + 2*p)) + (b*c - a*d)*(1 + m)*(a*B*(7 + m) - A*b*(9 + m + 2*p))) + (1 + m)*(b*c
*(2*b*c*(3 + p)*(a*B*(1 + m) - A*b*(9 + m + 2*p)) + (b*c - a*d)*(1 + m)*(a*B*(7 + m) - A*b*(9 + m + 2*p))) - a
*(2*b*c*d*(3 + p)*(a*B*(1 + m) - A*b*(9 + m + 2*p)) + d*(b*c - a*d)*(1 + m)*(a*B*(7 + m) - A*b*(9 + m + 2*p))
+ 4*(b*c - a*d)*(a*B*d*(7 + m) - b*(6*B*c + A*d*(9 + m + 2*p)))))))*(e*x)^(1 + m)*(a + b*x^2)^p*Hypergeometric
2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(b^4*e*(1 + m)*(3 + m + 2*p)*(5 + m + 2*p)*(7 + m + 2*p)*(9 + m +
2*p)*(1 + (b*x^2)/a)^p)

________________________________________________________________________________________

Rubi [A]  time = 2.51227, antiderivative size = 1047, normalized size of antiderivative = 0.99, number of steps used = 6, number of rules used = 4, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.129, Rules used = {581, 459, 365, 364} \[ -\frac{\left (-c^2 \left (48 B c+A d \left (m^3+(6 p+23) m^2+\left (12 p^2+92 p+183\right ) m+8 p^3+92 p^2+366 p+513\right )\right ) b^3+a c d \left (2 A d \left (m^3+4 (p+5) m^2+\left (4 p^2+44 p+123\right ) m+8 p^2+84 p+216\right )+B c \left (m^3+(4 p+21) m^2+\left (4 p^2+44 p+143\right ) m+4 p^2+40 p+267\right )\right ) b^2-a^2 d^2 (m+5) \left (A d (m+3) (m+2 p+9)+2 B c \left (m^2+2 p m+13 m+2 p+30\right )\right ) b+a^3 B d^3 \left (m^3+15 m^2+71 m+105\right )\right ) \left (b x^2+a\right )^{p+1} (e x)^{m+1}}{b^4 e (m+2 p+3) (m+2 p+5) (m+2 p+7) (m+2 p+9)}+\frac{B \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^3 (e x)^{m+1}}{b e (m+2 p+9)}+\frac{(6 b B c-a B d (m+7)+A b d (m+2 p+9)) \left (b x^2+a\right )^{p+1} \left (d x^2+c\right )^2 (e x)^{m+1}}{b^2 e (m+2 p+7) (m+2 p+9)}+\frac{\left (c \left (24 B c+A d \left (m^2+4 (p+5) m+4 p^2+40 p+99\right )\right ) b^2-a d \left (A d (m+5) (m+2 p+9)+B c \left (m^2+2 (p+9) m+2 p+65\right )\right ) b+a^2 B d^2 \left (m^2+12 m+35\right )\right ) \left (b x^2+a\right )^{p+1} \left (d x^2+c\right ) (e x)^{m+1}}{b^3 e (m+2 p+5) (m+2 p+7) (m+2 p+9)}-\frac{\left (c \left (2 b^2 (p+3) (a B (m+1)-A b (m+2 p+9)) c^2-2 a b d (p+3) (a B (m+1)-A b (m+2 p+9)) c+b (b c-a d) (m+1) (a B (m+7)-A b (m+2 p+9)) c+\frac{2 b (p+2) (2 b c (p+3) (a B (m+1)-A b (m+2 p+9))+(b c-a d) (m+1) (a B (m+7)-A b (m+2 p+9))) c}{m+1}-a d (b c-a d) (m+1) (a B (m+7)-A b (m+2 p+9))+4 a (b c-a d) (6 b B c-a B d (m+7)+A b d (m+2 p+9))\right )-\frac{a \left (-c^2 \left (48 B c+A d \left (m^3+(6 p+23) m^2+\left (12 p^2+92 p+183\right ) m+8 p^3+92 p^2+366 p+513\right )\right ) b^3+a c d \left (2 A d \left (m^3+4 (p+5) m^2+\left (4 p^2+44 p+123\right ) m+8 p^2+84 p+216\right )+B c \left (m^3+(4 p+21) m^2+\left (4 p^2+44 p+143\right ) m+4 p^2+40 p+267\right )\right ) b^2-a^2 d^2 (m+5) \left (A d (m+3) (m+2 p+9)+2 B c \left (m^2+2 p m+13 m+2 p+30\right )\right ) b+a^3 B d^3 \left (m^3+15 m^2+71 m+105\right )\right )}{b (m+2 p+3)}\right ) \left (b x^2+a\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right ) (e x)^{m+1}}{b^3 e (m+2 p+5) (m+2 p+7) (m+2 p+9)} \]

Antiderivative was successfully verified.

[In]

Int[(e*x)^m*(a + b*x^2)^p*(A + B*x^2)*(c + d*x^2)^3,x]

[Out]

-(((a^3*B*d^3*(105 + 71*m + 15*m^2 + m^3) - a^2*b*d^2*(5 + m)*(A*d*(3 + m)*(9 + m + 2*p) + 2*B*c*(30 + 13*m +
m^2 + 2*p + 2*m*p)) + a*b^2*c*d*(2*A*d*(216 + m^3 + 84*p + 8*p^2 + 4*m^2*(5 + p) + m*(123 + 44*p + 4*p^2)) + B
*c*(267 + m^3 + 40*p + 4*p^2 + m^2*(21 + 4*p) + m*(143 + 44*p + 4*p^2))) - b^3*c^2*(48*B*c + A*d*(513 + m^3 +
366*p + 92*p^2 + 8*p^3 + m^2*(23 + 6*p) + m*(183 + 92*p + 12*p^2))))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p))/(b^4*e
*(3 + m + 2*p)*(5 + m + 2*p)*(7 + m + 2*p)*(9 + m + 2*p))) + ((a^2*B*d^2*(35 + 12*m + m^2) + b^2*c*(24*B*c + A
*d*(99 + m^2 + 40*p + 4*p^2 + 4*m*(5 + p))) - a*b*d*(A*d*(5 + m)*(9 + m + 2*p) + B*c*(65 + m^2 + 2*p + 2*m*(9
+ p))))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^2))/(b^3*e*(5 + m + 2*p)*(7 + m + 2*p)*(9 + m + 2*p)) + ((6
*b*B*c - a*B*d*(7 + m) + A*b*d*(9 + m + 2*p))*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^2)^2)/(b^2*e*(7 + m +
2*p)*(9 + m + 2*p)) + (B*(e*x)^(1 + m)*(a + b*x^2)^(1 + p)*(c + d*x^2)^3)/(b*e*(9 + m + 2*p)) - ((c*(2*b^2*c^
2*(3 + p)*(a*B*(1 + m) - A*b*(9 + m + 2*p)) - 2*a*b*c*d*(3 + p)*(a*B*(1 + m) - A*b*(9 + m + 2*p)) + b*c*(b*c -
a*d)*(1 + m)*(a*B*(7 + m) - A*b*(9 + m + 2*p)) - a*d*(b*c - a*d)*(1 + m)*(a*B*(7 + m) - A*b*(9 + m + 2*p)) +
4*a*(b*c - a*d)*(6*b*B*c - a*B*d*(7 + m) + A*b*d*(9 + m + 2*p)) + (2*b*c*(2 + p)*(2*b*c*(3 + p)*(a*B*(1 + m) -
A*b*(9 + m + 2*p)) + (b*c - a*d)*(1 + m)*(a*B*(7 + m) - A*b*(9 + m + 2*p))))/(1 + m)) - (a*(a^3*B*d^3*(105 +
71*m + 15*m^2 + m^3) - a^2*b*d^2*(5 + m)*(A*d*(3 + m)*(9 + m + 2*p) + 2*B*c*(30 + 13*m + m^2 + 2*p + 2*m*p)) +
a*b^2*c*d*(2*A*d*(216 + m^3 + 84*p + 8*p^2 + 4*m^2*(5 + p) + m*(123 + 44*p + 4*p^2)) + B*c*(267 + m^3 + 40*p
+ 4*p^2 + m^2*(21 + 4*p) + m*(143 + 44*p + 4*p^2))) - b^3*c^2*(48*B*c + A*d*(513 + m^3 + 366*p + 92*p^2 + 8*p^
3 + m^2*(23 + 6*p) + m*(183 + 92*p + 12*p^2)))))/(b*(3 + m + 2*p)))*(e*x)^(1 + m)*(a + b*x^2)^p*Hypergeometric
2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(b^3*e*(5 + m + 2*p)*(7 + m + 2*p)*(9 + m + 2*p)*(1 + (b*x^2)/a)^
p)

Rule 581

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

Rule 459

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

Rule 365

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

Rule 364

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

Rubi steps

\begin{align*} \int (e x)^m \left (a+b x^2\right )^p \left (A+B x^2\right ) \left (c+d x^2\right )^3 \, dx &=\frac{B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^3}{b e (9+m+2 p)}+\frac{\int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right )^2 \left (-c (a B (1+m)-A b (9+m+2 p))+(6 b B c-a B d (7+m)+A b d (9+m+2 p)) x^2\right ) \, dx}{b (9+m+2 p)}\\ &=\frac{(6 b B c-a B d (7+m)+A b d (9+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b^2 e (7+m+2 p) (9+m+2 p)}+\frac{B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^3}{b e (9+m+2 p)}+\frac{\int (e x)^m \left (a+b x^2\right )^p \left (c+d x^2\right ) \left (-c (2 b c (3+p) (a B (1+m)-A b (9+m+2 p))+(b c-a d) (1+m) (a B (7+m)-A b (9+m+2 p)))+(-2 b c d (3+p) (a B (1+m)-A b (9+m+2 p))-d (b c-a d) (1+m) (a B (7+m)-A b (9+m+2 p))+4 (b c-a d) (6 b B c-a B d (7+m)+A b d (9+m+2 p))) x^2\right ) \, dx}{b^2 (7+m+2 p) (9+m+2 p)}\\ &=\frac{\left (a^2 B d^2 \left (35+12 m+m^2\right )+b^2 c \left (24 B c+A d \left (99+m^2+40 p+4 p^2+4 m (5+p)\right )\right )-a b d \left (A d (5+m) (9+m+2 p)+B c \left (65+m^2+2 p+2 m (9+p)\right )\right )\right ) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b^3 e (5+m+2 p) (7+m+2 p) (9+m+2 p)}+\frac{(6 b B c-a B d (7+m)+A b d (9+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b^2 e (7+m+2 p) (9+m+2 p)}+\frac{B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^3}{b e (9+m+2 p)}+\frac{\int (e x)^m \left (a+b x^2\right )^p \left (-c \left (2 b c (2+p) (2 b c (3+p) (a B (1+m)-A b (9+m+2 p))+(b c-a d) (1+m) (a B (7+m)-A b (9+m+2 p)))+(1+m) \left (2 b^2 c^2 (3+p) (a B (1+m)-A b (9+m+2 p))-2 a b c d (3+p) (a B (1+m)-A b (9+m+2 p))+b c (b c-a d) (1+m) (a B (7+m)-A b (9+m+2 p))-a d (b c-a d) (1+m) (a B (7+m)-A b (9+m+2 p))+4 a (b c-a d) (6 b B c-a B d (7+m)+A b d (9+m+2 p))\right )\right )-\left (a^3 B d^3 \left (105+71 m+15 m^2+m^3\right )-a^2 b d^2 (5+m) \left (A d (3+m) (9+m+2 p)+2 B c \left (30+13 m+m^2+2 p+2 m p\right )\right )+a b^2 c d \left (2 A d \left (216+m^3+84 p+8 p^2+4 m^2 (5+p)+m \left (123+44 p+4 p^2\right )\right )+B c \left (267+m^3+40 p+4 p^2+m^2 (21+4 p)+m \left (143+44 p+4 p^2\right )\right )\right )-b^3 c^2 \left (48 B c+A d \left (513+m^3+366 p+92 p^2+8 p^3+m^2 (23+6 p)+m \left (183+92 p+12 p^2\right )\right )\right )\right ) x^2\right ) \, dx}{b^3 (5+m+2 p) (7+m+2 p) (9+m+2 p)}\\ &=-\frac{\left (a^3 B d^3 \left (105+71 m+15 m^2+m^3\right )-a^2 b d^2 (5+m) \left (A d (3+m) (9+m+2 p)+2 B c \left (30+13 m+m^2+2 p+2 m p\right )\right )+a b^2 c d \left (2 A d \left (216+m^3+84 p+8 p^2+4 m^2 (5+p)+m \left (123+44 p+4 p^2\right )\right )+B c \left (267+m^3+40 p+4 p^2+m^2 (21+4 p)+m \left (143+44 p+4 p^2\right )\right )\right )-b^3 c^2 \left (48 B c+A d \left (513+m^3+366 p+92 p^2+8 p^3+m^2 (23+6 p)+m \left (183+92 p+12 p^2\right )\right )\right )\right ) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{b^4 e (3+m+2 p) (5+m+2 p) (7+m+2 p) (9+m+2 p)}+\frac{\left (a^2 B d^2 \left (35+12 m+m^2\right )+b^2 c \left (24 B c+A d \left (99+m^2+40 p+4 p^2+4 m (5+p)\right )\right )-a b d \left (A d (5+m) (9+m+2 p)+B c \left (65+m^2+2 p+2 m (9+p)\right )\right )\right ) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b^3 e (5+m+2 p) (7+m+2 p) (9+m+2 p)}+\frac{(6 b B c-a B d (7+m)+A b d (9+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b^2 e (7+m+2 p) (9+m+2 p)}+\frac{B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^3}{b e (9+m+2 p)}-\frac{\left (c \left (2 b c (2+p) (2 b c (3+p) (a B (1+m)-A b (9+m+2 p))+(b c-a d) (1+m) (a B (7+m)-A b (9+m+2 p)))+(1+m) \left (2 b^2 c^2 (3+p) (a B (1+m)-A b (9+m+2 p))-2 a b c d (3+p) (a B (1+m)-A b (9+m+2 p))+b c (b c-a d) (1+m) (a B (7+m)-A b (9+m+2 p))-a d (b c-a d) (1+m) (a B (7+m)-A b (9+m+2 p))+4 a (b c-a d) (6 b B c-a B d (7+m)+A b d (9+m+2 p))\right )\right )-\frac{a (1+m) \left (a^3 B d^3 \left (105+71 m+15 m^2+m^3\right )-a^2 b d^2 (5+m) \left (A d (3+m) (9+m+2 p)+2 B c \left (30+13 m+m^2+2 p+2 m p\right )\right )+a b^2 c d \left (2 A d \left (216+m^3+84 p+8 p^2+4 m^2 (5+p)+m \left (123+44 p+4 p^2\right )\right )+B c \left (267+m^3+40 p+4 p^2+m^2 (21+4 p)+m \left (143+44 p+4 p^2\right )\right )\right )-b^3 c^2 \left (48 B c+A d \left (513+m^3+366 p+92 p^2+8 p^3+m^2 (23+6 p)+m \left (183+92 p+12 p^2\right )\right )\right )\right )}{b (3+m+2 p)}\right ) \int (e x)^m \left (a+b x^2\right )^p \, dx}{b^3 (5+m+2 p) (7+m+2 p) (9+m+2 p)}\\ &=-\frac{\left (a^3 B d^3 \left (105+71 m+15 m^2+m^3\right )-a^2 b d^2 (5+m) \left (A d (3+m) (9+m+2 p)+2 B c \left (30+13 m+m^2+2 p+2 m p\right )\right )+a b^2 c d \left (2 A d \left (216+m^3+84 p+8 p^2+4 m^2 (5+p)+m \left (123+44 p+4 p^2\right )\right )+B c \left (267+m^3+40 p+4 p^2+m^2 (21+4 p)+m \left (143+44 p+4 p^2\right )\right )\right )-b^3 c^2 \left (48 B c+A d \left (513+m^3+366 p+92 p^2+8 p^3+m^2 (23+6 p)+m \left (183+92 p+12 p^2\right )\right )\right )\right ) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{b^4 e (3+m+2 p) (5+m+2 p) (7+m+2 p) (9+m+2 p)}+\frac{\left (a^2 B d^2 \left (35+12 m+m^2\right )+b^2 c \left (24 B c+A d \left (99+m^2+40 p+4 p^2+4 m (5+p)\right )\right )-a b d \left (A d (5+m) (9+m+2 p)+B c \left (65+m^2+2 p+2 m (9+p)\right )\right )\right ) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b^3 e (5+m+2 p) (7+m+2 p) (9+m+2 p)}+\frac{(6 b B c-a B d (7+m)+A b d (9+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b^2 e (7+m+2 p) (9+m+2 p)}+\frac{B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^3}{b e (9+m+2 p)}-\frac{\left (\left (c \left (2 b c (2+p) (2 b c (3+p) (a B (1+m)-A b (9+m+2 p))+(b c-a d) (1+m) (a B (7+m)-A b (9+m+2 p)))+(1+m) \left (2 b^2 c^2 (3+p) (a B (1+m)-A b (9+m+2 p))-2 a b c d (3+p) (a B (1+m)-A b (9+m+2 p))+b c (b c-a d) (1+m) (a B (7+m)-A b (9+m+2 p))-a d (b c-a d) (1+m) (a B (7+m)-A b (9+m+2 p))+4 a (b c-a d) (6 b B c-a B d (7+m)+A b d (9+m+2 p))\right )\right )-\frac{a (1+m) \left (a^3 B d^3 \left (105+71 m+15 m^2+m^3\right )-a^2 b d^2 (5+m) \left (A d (3+m) (9+m+2 p)+2 B c \left (30+13 m+m^2+2 p+2 m p\right )\right )+a b^2 c d \left (2 A d \left (216+m^3+84 p+8 p^2+4 m^2 (5+p)+m \left (123+44 p+4 p^2\right )\right )+B c \left (267+m^3+40 p+4 p^2+m^2 (21+4 p)+m \left (143+44 p+4 p^2\right )\right )\right )-b^3 c^2 \left (48 B c+A d \left (513+m^3+366 p+92 p^2+8 p^3+m^2 (23+6 p)+m \left (183+92 p+12 p^2\right )\right )\right )\right )}{b (3+m+2 p)}\right ) \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p}\right ) \int (e x)^m \left (1+\frac{b x^2}{a}\right )^p \, dx}{b^3 (5+m+2 p) (7+m+2 p) (9+m+2 p)}\\ &=-\frac{\left (a^3 B d^3 \left (105+71 m+15 m^2+m^3\right )-a^2 b d^2 (5+m) \left (A d (3+m) (9+m+2 p)+2 B c \left (30+13 m+m^2+2 p+2 m p\right )\right )+a b^2 c d \left (2 A d \left (216+m^3+84 p+8 p^2+4 m^2 (5+p)+m \left (123+44 p+4 p^2\right )\right )+B c \left (267+m^3+40 p+4 p^2+m^2 (21+4 p)+m \left (143+44 p+4 p^2\right )\right )\right )-b^3 c^2 \left (48 B c+A d \left (513+m^3+366 p+92 p^2+8 p^3+m^2 (23+6 p)+m \left (183+92 p+12 p^2\right )\right )\right )\right ) (e x)^{1+m} \left (a+b x^2\right )^{1+p}}{b^4 e (3+m+2 p) (5+m+2 p) (7+m+2 p) (9+m+2 p)}+\frac{\left (a^2 B d^2 \left (35+12 m+m^2\right )+b^2 c \left (24 B c+A d \left (99+m^2+40 p+4 p^2+4 m (5+p)\right )\right )-a b d \left (A d (5+m) (9+m+2 p)+B c \left (65+m^2+2 p+2 m (9+p)\right )\right )\right ) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )}{b^3 e (5+m+2 p) (7+m+2 p) (9+m+2 p)}+\frac{(6 b B c-a B d (7+m)+A b d (9+m+2 p)) (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^2}{b^2 e (7+m+2 p) (9+m+2 p)}+\frac{B (e x)^{1+m} \left (a+b x^2\right )^{1+p} \left (c+d x^2\right )^3}{b e (9+m+2 p)}-\frac{\left (c \left (2 b c (2+p) (2 b c (3+p) (a B (1+m)-A b (9+m+2 p))+(b c-a d) (1+m) (a B (7+m)-A b (9+m+2 p)))+(1+m) \left (2 b^2 c^2 (3+p) (a B (1+m)-A b (9+m+2 p))-2 a b c d (3+p) (a B (1+m)-A b (9+m+2 p))+b c (b c-a d) (1+m) (a B (7+m)-A b (9+m+2 p))-a d (b c-a d) (1+m) (a B (7+m)-A b (9+m+2 p))+4 a (b c-a d) (6 b B c-a B d (7+m)+A b d (9+m+2 p))\right )\right )-\frac{a (1+m) \left (a^3 B d^3 \left (105+71 m+15 m^2+m^3\right )-a^2 b d^2 (5+m) \left (A d (3+m) (9+m+2 p)+2 B c \left (30+13 m+m^2+2 p+2 m p\right )\right )+a b^2 c d \left (2 A d \left (216+m^3+84 p+8 p^2+4 m^2 (5+p)+m \left (123+44 p+4 p^2\right )\right )+B c \left (267+m^3+40 p+4 p^2+m^2 (21+4 p)+m \left (143+44 p+4 p^2\right )\right )\right )-b^3 c^2 \left (48 B c+A d \left (513+m^3+366 p+92 p^2+8 p^3+m^2 (23+6 p)+m \left (183+92 p+12 p^2\right )\right )\right )\right )}{b (3+m+2 p)}\right ) (e x)^{1+m} \left (a+b x^2\right )^p \left (1+\frac{b x^2}{a}\right )^{-p} \, _2F_1\left (\frac{1+m}{2},-p;\frac{3+m}{2};-\frac{b x^2}{a}\right )}{b^3 e (1+m) (5+m+2 p) (7+m+2 p) (9+m+2 p)}\\ \end{align*}

Mathematica [A]  time = 0.445544, size = 248, normalized size = 0.23 \[ x (e x)^m \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (\frac{c^2 x^2 (3 A d+B c) \, _2F_1\left (\frac{m+3}{2},-p;\frac{m+5}{2};-\frac{b x^2}{a}\right )}{m+3}+d x^4 \left (d x^2 \left (\frac{(A d+3 B c) \, _2F_1\left (\frac{m+7}{2},-p;\frac{m+9}{2};-\frac{b x^2}{a}\right )}{m+7}+\frac{B d x^2 \, _2F_1\left (\frac{m+9}{2},-p;\frac{m+11}{2};-\frac{b x^2}{a}\right )}{m+9}\right )+\frac{3 c (A d+B c) \, _2F_1\left (\frac{m+5}{2},-p;\frac{m+7}{2};-\frac{b x^2}{a}\right )}{m+5}\right )+\frac{A c^3 \, _2F_1\left (\frac{m+1}{2},-p;\frac{m+3}{2};-\frac{b x^2}{a}\right )}{m+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^m*(a + b*x^2)^p*(A + B*x^2)*(c + d*x^2)^3,x]

[Out]

(x*(e*x)^m*(a + b*x^2)^p*((A*c^3*Hypergeometric2F1[(1 + m)/2, -p, (3 + m)/2, -((b*x^2)/a)])/(1 + m) + (c^2*(B*
c + 3*A*d)*x^2*Hypergeometric2F1[(3 + m)/2, -p, (5 + m)/2, -((b*x^2)/a)])/(3 + m) + d*x^4*((3*c*(B*c + A*d)*Hy
pergeometric2F1[(5 + m)/2, -p, (7 + m)/2, -((b*x^2)/a)])/(5 + m) + d*x^2*(((3*B*c + A*d)*Hypergeometric2F1[(7
+ m)/2, -p, (9 + m)/2, -((b*x^2)/a)])/(7 + m) + (B*d*x^2*Hypergeometric2F1[(9 + m)/2, -p, (11 + m)/2, -((b*x^2
)/a)])/(9 + m)))))/(1 + (b*x^2)/a)^p

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Maple [F]  time = 0.072, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( b{x}^{2}+a \right ) ^{p} \left ( B{x}^{2}+A \right ) \left ( d{x}^{2}+c \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^3,x)

[Out]

int((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B x^{2} + A\right )}{\left (d x^{2} + c\right )}^{3}{\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^3,x, algorithm="maxima")

[Out]

integrate((B*x^2 + A)*(d*x^2 + c)^3*(b*x^2 + a)^p*(e*x)^m, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B d^{3} x^{8} +{\left (3 \, B c d^{2} + A d^{3}\right )} x^{6} + 3 \,{\left (B c^{2} d + A c d^{2}\right )} x^{4} + A c^{3} +{\left (B c^{3} + 3 \, A c^{2} d\right )} x^{2}\right )}{\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^3,x, algorithm="fricas")

[Out]

integral((B*d^3*x^8 + (3*B*c*d^2 + A*d^3)*x^6 + 3*(B*c^2*d + A*c*d^2)*x^4 + A*c^3 + (B*c^3 + 3*A*c^2*d)*x^2)*(
b*x^2 + a)^p*(e*x)^m, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(b*x**2+a)**p*(B*x**2+A)*(d*x**2+c)**3,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B x^{2} + A\right )}{\left (d x^{2} + c\right )}^{3}{\left (b x^{2} + a\right )}^{p} \left (e x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^m*(b*x^2+a)^p*(B*x^2+A)*(d*x^2+c)^3,x, algorithm="giac")

[Out]

integrate((B*x^2 + A)*(d*x^2 + c)^3*(b*x^2 + a)^p*(e*x)^m, x)