3.43 \(\int \frac{(e x)^m (A+B x^2)}{(a+b x^2)^3 (c+d x^2)^3} \, dx\)

Optimal. Leaf size=665 \[ \frac{b^2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) \left (A b \left (a^2 d^2 \left (m^2-16 m+63\right )-2 a b c d \left (m^2-10 m+9\right )+b^2 c^2 \left (m^2-4 m+3\right )\right )+a B \left (-a^2 d^2 \left (m^2-12 m+35\right )-2 a b c d \left (-m^2+6 m+7\right )+b^2 c^2 \left (1-m^2\right )\right )\right )}{8 a^3 e (m+1) (b c-a d)^5}+\frac{d^2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) \left (-a^2 d^2 (1-m) (A d (3-m)+B c (m+1))+2 a b c d \left (A d \left (m^2-10 m+9\right )+B c \left (-m^2+6 m+7\right )\right )+b^2 c^2 (7-m) (B c (5-m)-A d (9-m))\right )}{8 c^3 e (m+1) (b c-a d)^5}+\frac{d (e x)^{m+1} \left (A (a d+b c) \left (a^2 d^2 (3-m)-2 a b c d (9-m)+b^2 c^2 (3-m)\right )+a B c \left (a^2 d^2 (m+1)+2 a b c d (11-m)+b^2 c^2 (m+1)\right )\right )}{8 a^2 c^2 e \left (c+d x^2\right ) (b c-a d)^4}-\frac{d (e x)^{m+1} \left (A \left (2 a^2 d^2+a b c d (13-m)-b^2 c^2 (3-m)\right )-a B c (a d (11-m)+b c (m+1))\right )}{8 a^2 c e \left (c+d x^2\right )^2 (b c-a d)^3}+\frac{(e x)^{m+1} (A b (b c (3-m)-a d (11-m))+a B (a d (7-m)+b c (m+1)))}{8 a^2 e \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 (b c-a d)} \]

[Out]

-(d*(A*(2*a^2*d^2 - b^2*c^2*(3 - m) + a*b*c*d*(13 - m)) - a*B*c*(a*d*(11 - m) + b*c*(1 + m)))*(e*x)^(1 + m))/(
8*a^2*c*(b*c - a*d)^3*e*(c + d*x^2)^2) + ((A*b - a*B)*(e*x)^(1 + m))/(4*a*(b*c - a*d)*e*(a + b*x^2)^2*(c + d*x
^2)^2) + ((A*b*(b*c*(3 - m) - a*d*(11 - m)) + a*B*(a*d*(7 - m) + b*c*(1 + m)))*(e*x)^(1 + m))/(8*a^2*(b*c - a*
d)^2*e*(a + b*x^2)*(c + d*x^2)^2) + (d*(A*(b*c + a*d)*(b^2*c^2*(3 - m) + a^2*d^2*(3 - m) - 2*a*b*c*d*(9 - m))
+ a*B*c*(2*a*b*c*d*(11 - m) + b^2*c^2*(1 + m) + a^2*d^2*(1 + m)))*(e*x)^(1 + m))/(8*a^2*c^2*(b*c - a*d)^4*e*(c
+ d*x^2)) + (b^2*(a*B*(b^2*c^2*(1 - m^2) - 2*a*b*c*d*(7 + 6*m - m^2) - a^2*d^2*(35 - 12*m + m^2)) + A*b*(a^2*
d^2*(63 - 16*m + m^2) - 2*a*b*c*d*(9 - 10*m + m^2) + b^2*c^2*(3 - 4*m + m^2)))*(e*x)^(1 + m)*Hypergeometric2F1
[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(8*a^3*(b*c - a*d)^5*e*(1 + m)) + (d^2*(b^2*c^2*(B*c*(5 - m) - A*d*(9
- m))*(7 - m) - a^2*d^2*(1 - m)*(A*d*(3 - m) + B*c*(1 + m)) + 2*a*b*c*d*(B*c*(7 + 6*m - m^2) + A*d*(9 - 10*m
+ m^2)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(8*c^3*(b*c - a*d)^5*e*(1 + m
))

________________________________________________________________________________________

Rubi [A]  time = 2.17864, antiderivative size = 665, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 3, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.097, Rules used = {579, 584, 364} \[ \frac{b^2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) \left (A b \left (a^2 d^2 \left (m^2-16 m+63\right )-2 a b c d \left (m^2-10 m+9\right )+b^2 c^2 \left (m^2-4 m+3\right )\right )+a B \left (-a^2 d^2 \left (m^2-12 m+35\right )-2 a b c d \left (-m^2+6 m+7\right )+b^2 c^2 \left (1-m^2\right )\right )\right )}{8 a^3 e (m+1) (b c-a d)^5}+\frac{d^2 (e x)^{m+1} \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) \left (-a^2 d^2 (1-m) (A d (3-m)+B c (m+1))+2 a b c d \left (A d \left (m^2-10 m+9\right )+B c \left (-m^2+6 m+7\right )\right )+b^2 c^2 (7-m) (B c (5-m)-A d (9-m))\right )}{8 c^3 e (m+1) (b c-a d)^5}+\frac{d (e x)^{m+1} \left (A (a d+b c) \left (a^2 d^2 (3-m)-2 a b c d (9-m)+b^2 c^2 (3-m)\right )+a B c \left (a^2 d^2 (m+1)+2 a b c d (11-m)+b^2 c^2 (m+1)\right )\right )}{8 a^2 c^2 e \left (c+d x^2\right ) (b c-a d)^4}-\frac{d (e x)^{m+1} \left (A \left (2 a^2 d^2+a b c d (13-m)-b^2 c^2 (3-m)\right )-a B c (a d (11-m)+b c (m+1))\right )}{8 a^2 c e \left (c+d x^2\right )^2 (b c-a d)^3}+\frac{(e x)^{m+1} (A b (b c (3-m)-a d (11-m))+a B (a d (7-m)+b c (m+1)))}{8 a^2 e \left (a+b x^2\right ) \left (c+d x^2\right )^2 (b c-a d)^2}+\frac{(e x)^{m+1} (A b-a B)}{4 a e \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*(A*(2*a^2*d^2 - b^2*c^2*(3 - m) + a*b*c*d*(13 - m)) - a*B*c*(a*d*(11 - m) + b*c*(1 + m)))*(e*x)^(1 + m))/(
8*a^2*c*(b*c - a*d)^3*e*(c + d*x^2)^2) + ((A*b - a*B)*(e*x)^(1 + m))/(4*a*(b*c - a*d)*e*(a + b*x^2)^2*(c + d*x
^2)^2) + ((A*b*(b*c*(3 - m) - a*d*(11 - m)) + a*B*(a*d*(7 - m) + b*c*(1 + m)))*(e*x)^(1 + m))/(8*a^2*(b*c - a*
d)^2*e*(a + b*x^2)*(c + d*x^2)^2) + (d*(A*(b*c + a*d)*(b^2*c^2*(3 - m) + a^2*d^2*(3 - m) - 2*a*b*c*d*(9 - m))
+ a*B*c*(2*a*b*c*d*(11 - m) + b^2*c^2*(1 + m) + a^2*d^2*(1 + m)))*(e*x)^(1 + m))/(8*a^2*c^2*(b*c - a*d)^4*e*(c
+ d*x^2)) + (b^2*(a*B*(b^2*c^2*(1 - m^2) - 2*a*b*c*d*(7 + 6*m - m^2) - a^2*d^2*(35 - 12*m + m^2)) + A*b*(a^2*
d^2*(63 - 16*m + m^2) - 2*a*b*c*d*(9 - 10*m + m^2) + b^2*c^2*(3 - 4*m + m^2)))*(e*x)^(1 + m)*Hypergeometric2F1
[1, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)])/(8*a^3*(b*c - a*d)^5*e*(1 + m)) + (d^2*(b^2*c^2*(B*c*(5 - m) - A*d*(9
- m))*(7 - m) - a^2*d^2*(1 - m)*(A*d*(3 - m) + B*c*(1 + m)) + 2*a*b*c*d*(B*c*(7 + 6*m - m^2) + A*d*(9 - 10*m
+ m^2)))*(e*x)^(1 + m)*Hypergeometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])/(8*c^3*(b*c - a*d)^5*e*(1 + m
))

Rule 579

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_)*((e_) + (f_.)*(x_)^(n_)), x
_Symbol] :> -Simp[((b*e - a*f)*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q + 1))/(a*g*n*(b*c - a*d)*(p +
1)), x] + Dist[1/(a*n*(b*c - a*d)*(p + 1)), Int[(g*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f)*(
m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d,
e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 584

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n_)))/((c_) + (d_.)*(x_)^(n_)), x_Sy
mbol] :> Int[ExpandIntegrand[((g*x)^m*(a + b*x^n)^p*(e + f*x^n))/(c + d*x^n), x], x] /; FreeQ[{a, b, c, d, e,
f, g, m, p}, x] && IGtQ[n, 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 \frac{(e x)^m \left (A+B x^2\right )}{\left (a+b x^2\right )^3 \left (c+d x^2\right )^3} \, dx &=\frac{(A b-a B) (e x)^{1+m}}{4 a (b c-a d) e \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}-\frac{\int \frac{(e x)^m \left (4 a A d-A b c (3-m)-a B c (1+m)-(A b-a B) d (7-m) x^2\right )}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^3} \, dx}{4 a (b c-a d)}\\ &=\frac{(A b-a B) (e x)^{1+m}}{4 a (b c-a d) e \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac{(A b (b c (3-m)-a d (11-m))+a B (a d (7-m)+b c (1+m))) (e x)^{1+m}}{8 a^2 (b c-a d)^2 e \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{\int \frac{(e x)^m \left (-a B c (1+m) (a d (9-m)-b (c-c m))+A \left (8 a^2 d^2-a b c d \left (3-12 m+m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )\right )+d (5-m) (A b (b c (3-m)-a d (11-m))+a B (a d (7-m)+b c (1+m))) x^2\right )}{\left (a+b x^2\right ) \left (c+d x^2\right )^3} \, dx}{8 a^2 (b c-a d)^2}\\ &=-\frac{d \left (A \left (2 a^2 d^2-b^2 c^2 (3-m)+a b c d (13-m)\right )-a B c (a d (11-m)+b c (1+m))\right ) (e x)^{1+m}}{8 a^2 c (b c-a d)^3 e \left (c+d x^2\right )^2}+\frac{(A b-a B) (e x)^{1+m}}{4 a (b c-a d) e \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac{(A b (b c (3-m)-a d (11-m))+a B (a d (7-m)+b c (1+m))) (e x)^{1+m}}{8 a^2 (b c-a d)^2 e \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{\int \frac{(e x)^m \left (-4 \left (a B c \left (2 a^2 d^2-b^2 c^2 (1-m)+a b c d (11-m)\right ) (1+m)-A \left (24 a^2 b c d^2-2 a^3 d^3 (3-m)-a b^2 c^2 d \left (9-14 m+m^2\right )+b^3 c^3 \left (3-4 m+m^2\right )\right )\right )-4 b d (3-m) \left (A \left (2 a^2 d^2-b^2 c^2 (3-m)+a b c d (13-m)\right )-a B c (a d (11-m)+b c (1+m))\right ) x^2\right )}{\left (a+b x^2\right ) \left (c+d x^2\right )^2} \, dx}{32 a^2 c (b c-a d)^3}\\ &=-\frac{d \left (A \left (2 a^2 d^2-b^2 c^2 (3-m)+a b c d (13-m)\right )-a B c (a d (11-m)+b c (1+m))\right ) (e x)^{1+m}}{8 a^2 c (b c-a d)^3 e \left (c+d x^2\right )^2}+\frac{(A b-a B) (e x)^{1+m}}{4 a (b c-a d) e \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac{(A b (b c (3-m)-a d (11-m))+a B (a d (7-m)+b c (1+m))) (e x)^{1+m}}{8 a^2 (b c-a d)^2 e \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (A (b c+a d) \left (b^2 c^2 (3-m)+a^2 d^2 (3-m)-2 a b c d (9-m)\right )+a B c \left (2 a b c d (11-m)+b^2 c^2 (1+m)+a^2 d^2 (1+m)\right )\right ) (e x)^{1+m}}{8 a^2 c^2 (b c-a d)^4 e \left (c+d x^2\right )}+\frac{\int \frac{(e x)^m \left (8 \left (a B c (b c+a d) \left (b^2 c^2 (1-m)+a^2 d^2 (1-m)-2 a b c d (7-m)\right ) (1+m)+A \left (48 a^2 b^2 c^2 d^2-a b^3 c^3 d \left (15-16 m+m^2\right )-a^3 b c d^3 \left (15-16 m+m^2\right )+b^4 c^4 \left (3-4 m+m^2\right )+a^4 d^4 \left (3-4 m+m^2\right )\right )\right )+8 b d (1-m) \left (A (b c+a d) \left (b^2 c^2 (3-m)+a^2 d^2 (3-m)-2 a b c d (9-m)\right )+a B c \left (2 a b c d (11-m)+b^2 c^2 (1+m)+a^2 d^2 (1+m)\right )\right ) x^2\right )}{\left (a+b x^2\right ) \left (c+d x^2\right )} \, dx}{64 a^2 c^2 (b c-a d)^4}\\ &=-\frac{d \left (A \left (2 a^2 d^2-b^2 c^2 (3-m)+a b c d (13-m)\right )-a B c (a d (11-m)+b c (1+m))\right ) (e x)^{1+m}}{8 a^2 c (b c-a d)^3 e \left (c+d x^2\right )^2}+\frac{(A b-a B) (e x)^{1+m}}{4 a (b c-a d) e \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac{(A b (b c (3-m)-a d (11-m))+a B (a d (7-m)+b c (1+m))) (e x)^{1+m}}{8 a^2 (b c-a d)^2 e \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (A (b c+a d) \left (b^2 c^2 (3-m)+a^2 d^2 (3-m)-2 a b c d (9-m)\right )+a B c \left (2 a b c d (11-m)+b^2 c^2 (1+m)+a^2 d^2 (1+m)\right )\right ) (e x)^{1+m}}{8 a^2 c^2 (b c-a d)^4 e \left (c+d x^2\right )}+\frac{\int \left (\frac{8 b^2 c^2 \left (a B \left (b^2 c^2 \left (1-m^2\right )-2 a b c d \left (7+6 m-m^2\right )-a^2 d^2 \left (35-12 m+m^2\right )\right )+A b \left (a^2 d^2 \left (63-16 m+m^2\right )-2 a b c d \left (9-10 m+m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )\right )\right ) (e x)^m}{(b c-a d) \left (a+b x^2\right )}+\frac{8 a^2 d^2 \left (b^2 c^2 (B c (5-m)-A d (9-m)) (7-m)-a^2 d^2 (1-m) (A d (3-m)+B c (1+m))+2 a b c d \left (B c \left (7+6 m-m^2\right )+A d \left (9-10 m+m^2\right )\right )\right ) (e x)^m}{(b c-a d) \left (c+d x^2\right )}\right ) \, dx}{64 a^2 c^2 (b c-a d)^4}\\ &=-\frac{d \left (A \left (2 a^2 d^2-b^2 c^2 (3-m)+a b c d (13-m)\right )-a B c (a d (11-m)+b c (1+m))\right ) (e x)^{1+m}}{8 a^2 c (b c-a d)^3 e \left (c+d x^2\right )^2}+\frac{(A b-a B) (e x)^{1+m}}{4 a (b c-a d) e \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac{(A b (b c (3-m)-a d (11-m))+a B (a d (7-m)+b c (1+m))) (e x)^{1+m}}{8 a^2 (b c-a d)^2 e \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (A (b c+a d) \left (b^2 c^2 (3-m)+a^2 d^2 (3-m)-2 a b c d (9-m)\right )+a B c \left (2 a b c d (11-m)+b^2 c^2 (1+m)+a^2 d^2 (1+m)\right )\right ) (e x)^{1+m}}{8 a^2 c^2 (b c-a d)^4 e \left (c+d x^2\right )}+\frac{\left (d^2 \left (b^2 c^2 (B c (5-m)-A d (9-m)) (7-m)-a^2 d^2 (1-m) (A d (3-m)+B c (1+m))+2 a b c d \left (B c \left (7+6 m-m^2\right )+A d \left (9-10 m+m^2\right )\right )\right )\right ) \int \frac{(e x)^m}{c+d x^2} \, dx}{8 c^2 (b c-a d)^5}+\frac{\left (b^2 \left (a B \left (b^2 c^2 \left (1-m^2\right )-2 a b c d \left (7+6 m-m^2\right )-a^2 d^2 \left (35-12 m+m^2\right )\right )+A b \left (a^2 d^2 \left (63-16 m+m^2\right )-2 a b c d \left (9-10 m+m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )\right )\right )\right ) \int \frac{(e x)^m}{a+b x^2} \, dx}{8 a^2 (b c-a d)^5}\\ &=-\frac{d \left (A \left (2 a^2 d^2-b^2 c^2 (3-m)+a b c d (13-m)\right )-a B c (a d (11-m)+b c (1+m))\right ) (e x)^{1+m}}{8 a^2 c (b c-a d)^3 e \left (c+d x^2\right )^2}+\frac{(A b-a B) (e x)^{1+m}}{4 a (b c-a d) e \left (a+b x^2\right )^2 \left (c+d x^2\right )^2}+\frac{(A b (b c (3-m)-a d (11-m))+a B (a d (7-m)+b c (1+m))) (e x)^{1+m}}{8 a^2 (b c-a d)^2 e \left (a+b x^2\right ) \left (c+d x^2\right )^2}+\frac{d \left (A (b c+a d) \left (b^2 c^2 (3-m)+a^2 d^2 (3-m)-2 a b c d (9-m)\right )+a B c \left (2 a b c d (11-m)+b^2 c^2 (1+m)+a^2 d^2 (1+m)\right )\right ) (e x)^{1+m}}{8 a^2 c^2 (b c-a d)^4 e \left (c+d x^2\right )}+\frac{b^2 \left (a B \left (b^2 c^2 \left (1-m^2\right )-2 a b c d \left (7+6 m-m^2\right )-a^2 d^2 \left (35-12 m+m^2\right )\right )+A b \left (a^2 d^2 \left (63-16 m+m^2\right )-2 a b c d \left (9-10 m+m^2\right )+b^2 c^2 \left (3-4 m+m^2\right )\right )\right ) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{b x^2}{a}\right )}{8 a^3 (b c-a d)^5 e (1+m)}+\frac{d^2 \left (b^2 c^2 (B c (5-m)-A d (9-m)) (7-m)-a^2 d^2 (1-m) (A d (3-m)+B c (1+m))+2 a b c d \left (B c \left (7+6 m-m^2\right )+A d \left (9-10 m+m^2\right )\right )\right ) (e x)^{1+m} \, _2F_1\left (1,\frac{1+m}{2};\frac{3+m}{2};-\frac{d x^2}{c}\right )}{8 c^3 (b c-a d)^5 e (1+m)}\\ \end{align*}

Mathematica [A]  time = 0.499161, size = 329, normalized size = 0.49 \[ \frac{x (e x)^m \left (\frac{b^2 (b c-a d) \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) (2 a B d-3 A b d+b B c)}{a^2}+\frac{b^2 (A b-a B) (b c-a d)^2 \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right )}{a^3}-\frac{3 b^2 d \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{b x^2}{a}\right ) (a B d-2 A b d+b B c)}{a}+\frac{d^2 (b c-a d) \, _2F_1\left (2,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) (a B d-3 A b d+2 b B c)}{c^2}+\frac{d^2 (b c-a d)^2 (B c-A d) \, _2F_1\left (3,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right )}{c^3}+\frac{3 b d^2 \, _2F_1\left (1,\frac{m+1}{2};\frac{m+3}{2};-\frac{d x^2}{c}\right ) (a B d-2 A b d+b B c)}{c}\right )}{(m+1) (b c-a d)^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.076, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( B{x}^{2}+A \right ) \left ( ex \right ) ^{m}}{ \left ( b{x}^{2}+a \right ) ^{3} \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)/(b*x^2+a)^3/(d*x^2+c)^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{b^{3} d^{3} x^{12} + 3 \,{\left (b^{3} c d^{2} + a b^{2} d^{3}\right )} x^{10} + 3 \,{\left (b^{3} c^{2} d + 3 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{8} +{\left (b^{3} c^{3} + 9 \, a b^{2} c^{2} d + 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{6} + a^{3} c^{3} + 3 \,{\left (a b^{2} c^{3} + 3 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} x^{4} + 3 \,{\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((B*x^2 + A)*(e*x)^m/(b^3*d^3*x^12 + 3*(b^3*c*d^2 + a*b^2*d^3)*x^10 + 3*(b^3*c^2*d + 3*a*b^2*c*d^2 + a
^2*b*d^3)*x^8 + (b^3*c^3 + 9*a*b^2*c^2*d + 9*a^2*b*c*d^2 + a^3*d^3)*x^6 + a^3*c^3 + 3*(a*b^2*c^3 + 3*a^2*b*c^2
*d + a^3*c*d^2)*x^4 + 3*(a^2*b*c^3 + a^3*c^2*d)*x^2), 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)/(b*x**2+a)**3/(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 \frac{{\left (B x^{2} + A\right )} \left (e x\right )^{m}}{{\left (b x^{2} + a\right )}^{3}{\left (d x^{2} + c\right )}^{3}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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