\(\int \frac {(e x)^m (A+B x^2)}{(a+b x^2)^2 (c+d x^2)^2} \, dx\) [35]

Optimal result

Integrand size = 31, antiderivative size = 304 \[ \int \frac {(e x)^m \left (A+B x^2\right )}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {d (A b c-2 a B c+a A d) (e x)^{1+m}}{2 a c (b c-a d)^2 e \left (c+d x^2\right )}+\frac {(A b-a B) (e x)^{1+m}}{2 a (b c-a d) e \left (a+b x^2\right ) \left (c+d x^2\right )}+\frac {b (A b (b c (1-m)-a d (5-m))+a B (a d (3-m)+b c (1+m))) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )}{2 a^2 (b c-a d)^3 e (1+m)}-\frac {d (b c (B c (3-m)-A d (5-m))+a d (A d (1-m)+B c (1+m))) (e x)^{1+m} \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {d x^2}{c}\right )}{2 c^2 (b c-a d)^3 e (1+m)} \] Output:

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

Mathematica [A] (verified)

Time = 0.88 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.68 \[ \int \frac {(e x)^m \left (A+B x^2\right )}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=\frac {x (e x)^m \left (-a b c^2 (b B c-2 A b d+a B d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )+a^2 c d (b B c-2 A b d+a B d) \operatorname {Hypergeometric2F1}\left (1,\frac {1+m}{2},\frac {3+m}{2},-\frac {d x^2}{c}\right )-(b c-a d) \left (b (A b-a B) c^2 \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {b x^2}{a}\right )+a^2 d (-B c+A d) \operatorname {Hypergeometric2F1}\left (2,\frac {1+m}{2},\frac {3+m}{2},-\frac {d x^2}{c}\right )\right )\right )}{a^2 c^2 (-b c+a d)^3 (1+m)} \] Input:

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

Output:

(x*(e*x)^m*(-(a*b*c^2*(b*B*c - 2*A*b*d + a*B*d)*Hypergeometric2F1[1, (1 + 
m)/2, (3 + m)/2, -((b*x^2)/a)]) + a^2*c*d*(b*B*c - 2*A*b*d + a*B*d)*Hyperg 
eometric2F1[1, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)] - (b*c - a*d)*(b*(A*b - 
 a*B)*c^2*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((b*x^2)/a)] + a^2*d 
*(-(B*c) + A*d)*Hypergeometric2F1[2, (1 + m)/2, (3 + m)/2, -((d*x^2)/c)])) 
)/(a^2*c^2*(-(b*c) + a*d)^3*(1 + m))
 

Rubi [A] (verified)

Time = 0.74 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.08, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {441, 441, 27, 446, 2009}

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.

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

\(\Big \downarrow \) 441

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

\(\Big \downarrow \) 441

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 446

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 441

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

rule 446

Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((e_) + (f_.)*(x_)^2))/( 
(c_) + (d_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^2)^ 
p*((e + f*x^2)/(c + d*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x 
]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [F]

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

Input:

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

Output:

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

Fricas [F]

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

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

Output:

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

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(e x)^m \left (A+B x^2\right )}{\left (a+b x^2\right )^2 \left (c+d x^2\right )^2} \, dx=e^{m} \left (\int \frac {x^{m}}{b \,d^{2} x^{6}+a \,d^{2} x^{4}+2 b c d \,x^{4}+2 a c d \,x^{2}+b \,c^{2} x^{2}+a \,c^{2}}d x \right ) \] Input:

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

Output:

e**m*int(x**m/(a*c**2 + 2*a*c*d*x**2 + a*d**2*x**4 + b*c**2*x**2 + 2*b*c*d 
*x**4 + b*d**2*x**6),x)