\(\int \frac {(A+B x^2) (d+e x^2)^q}{(a+b x^2+c x^4)^2} \, dx\) [258]

Optimal result

Integrand size = 31, antiderivative size = 857 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^q}{\left (a+b x^2+c x^4\right )^2} \, dx=-\frac {x \left (d+e x^2\right )^{1+q} \left (a B \left (b c d-b^2 e+2 a c e\right )-A \left (b^2 c d-2 a c^2 d-b^3 e+3 a b c e\right )+c \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )\right ) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (a+b x^2+c x^4\right )}-\frac {c \left (2 a B \left (c d^2+e (a e (1-2 q)-b d (1-q))\right )+A \left (b^2 d e-b \left (c d^2+a e^2 (1-2 q)\right )-4 a c d e q\right )+\frac {A \left (b^3 d e-12 a b c d e-b^2 \left (c d^2+a e^2 (1-2 q)\right )+4 a c \left (3 c d^2+a e^2 (3-2 q)\right )\right )-2 a B \left (2 b \left (c d^2+a e^2\right )-b^2 d e (2-q)-4 a c d e q\right )}{\sqrt {b^2-4 a c}}\right ) x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-q,1,\frac {3}{2},-\frac {e x^2}{d},-\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}\right )}{2 a \left (b^2-4 a c\right ) \left (b-\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )}-\frac {c \left (2 a B \left (c d^2+e (a e (1-2 q)-b d (1-q))\right )+A \left (b^2 d e-b \left (c d^2+a e^2 (1-2 q)\right )-4 a c d e q\right )-\frac {A \left (b^3 d e-12 a b c d e-b^2 \left (c d^2+a e^2 (1-2 q)\right )+4 a c \left (3 c d^2+a e^2 (3-2 q)\right )\right )-2 a B \left (2 b \left (c d^2+a e^2\right )-b^2 d e (2-q)-4 a c d e q\right )}{\sqrt {b^2-4 a c}}\right ) x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \operatorname {AppellF1}\left (\frac {1}{2},-q,1,\frac {3}{2},-\frac {e x^2}{d},-\frac {2 c x^2}{b+\sqrt {b^2-4 a c}}\right )}{2 a \left (b^2-4 a c\right ) \left (b+\sqrt {b^2-4 a c}\right ) \left (c d^2-b d e+a e^2\right )}+\frac {e \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )\right ) (1+2 q) x \left (d+e x^2\right )^q \left (1+\frac {e x^2}{d}\right )^{-q} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-q,\frac {3}{2},-\frac {e x^2}{d}\right )}{2 a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )} \] Output:

-1/2*x*(e*x^2+d)^(1+q)*(a*B*(2*a*c*e-b^2*e+b*c*d)-A*(3*a*b*c*e-2*a*c^2*d-b 
^3*e+b^2*c*d)+c*(a*B*(-b*e+2*c*d)-A*(2*a*c*e-b^2*e+b*c*d))*x^2)/a/(-4*a*c+ 
b^2)/(a*e^2-b*d*e+c*d^2)/(c*x^4+b*x^2+a)-1/2*c*(2*a*B*(c*d^2+e*(a*e*(1-2*q 
)-b*d*(1-q)))+A*(b^2*d*e-b*(c*d^2+a*e^2*(1-2*q))-4*a*c*d*e*q)+(A*(b^3*d*e- 
12*a*b*c*d*e-b^2*(c*d^2+a*e^2*(1-2*q))+4*a*c*(3*c*d^2+a*e^2*(3-2*q)))-2*a* 
B*(2*b*(a*e^2+c*d^2)-b^2*d*e*(2-q)-4*a*c*d*e*q))/(-4*a*c+b^2)^(1/2))*x*(e* 
x^2+d)^q*AppellF1(1/2,1,-q,3/2,-2*c*x^2/(b-(-4*a*c+b^2)^(1/2)),-e*x^2/d)/a 
/(-4*a*c+b^2)/(b-(-4*a*c+b^2)^(1/2))/(a*e^2-b*d*e+c*d^2)/((1+e*x^2/d)^q)-1 
/2*c*(2*a*B*(c*d^2+e*(a*e*(1-2*q)-b*d*(1-q)))+A*(b^2*d*e-b*(c*d^2+a*e^2*(1 
-2*q))-4*a*c*d*e*q)-(A*(b^3*d*e-12*a*b*c*d*e-b^2*(c*d^2+a*e^2*(1-2*q))+4*a 
*c*(3*c*d^2+a*e^2*(3-2*q)))-2*a*B*(2*b*(a*e^2+c*d^2)-b^2*d*e*(2-q)-4*a*c*d 
*e*q))/(-4*a*c+b^2)^(1/2))*x*(e*x^2+d)^q*AppellF1(1/2,1,-q,3/2,-2*c*x^2/(b 
+(-4*a*c+b^2)^(1/2)),-e*x^2/d)/a/(-4*a*c+b^2)/(b+(-4*a*c+b^2)^(1/2))/(a*e^ 
2-b*d*e+c*d^2)/((1+e*x^2/d)^q)+1/2*e*(a*B*(-b*e+2*c*d)-A*(2*a*c*e-b^2*e+b* 
c*d))*(1+2*q)*x*(e*x^2+d)^q*hypergeom([1/2, -q],[3/2],-e*x^2/d)/a/(-4*a*c+ 
b^2)/(a*e^2-b*d*e+c*d^2)/((1+e*x^2/d)^q)
 

Mathematica [F]

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

Integrate[((A + B*x^2)*(d + e*x^2)^q)/(a + b*x^2 + c*x^4)^2,x]
 

Output:

Integrate[((A + B*x^2)*(d + e*x^2)^q)/(a + b*x^2 + c*x^4)^2, x]
 

Rubi [F]

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.

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

\(\Big \downarrow \) 2256

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

\(\Big \downarrow \) 2009

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

Input:

Int[((A + B*x^2)*(d + e*x^2)^q)/(a + b*x^2 + c*x^4)^2,x]
 

Output:

$Aborted
 

Defintions of rubi rules used

rule 2009

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

rule 2256

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ 
(p_.), x_Symbol] :> Int[ExpandIntegrand[Px*(d + e*x^2)^q*(a + b*x^2 + c*x^4 
)^p, x], x] /; FreeQ[{a, b, c, d, e, q}, x] && PolyQ[Px, x] && IntegerQ[p]
 

Maple [F]

\[\int \frac {\left (B \,x^{2}+A \right ) \left (e \,x^{2}+d \right )^{q}}{\left (c \,x^{4}+b \,x^{2}+a \right )^{2}}d x\]

Input:

int((B*x^2+A)*(e*x^2+d)^q/(c*x^4+b*x^2+a)^2,x)
 

Output:

int((B*x^2+A)*(e*x^2+d)^q/(c*x^4+b*x^2+a)^2,x)
 

Fricas [F]

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

integrate((B*x^2+A)*(e*x^2+d)^q/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")
 

Output:

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

Sympy [F(-1)]

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

integrate((B*x**2+A)*(e*x**2+d)**q/(c*x**4+b*x**2+a)**2,x)
 

Output:

Timed out
 

Maxima [F]

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

integrate((B*x^2+A)*(e*x^2+d)^q/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")
 

Output:

integrate((B*x^2 + A)*(e*x^2 + d)^q/(c*x^4 + b*x^2 + a)^2, x)
 

Giac [F]

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

integrate((B*x^2+A)*(e*x^2+d)^q/(c*x^4+b*x^2+a)^2,x, algorithm="giac")
 

Output:

integrate((B*x^2 + A)*(e*x^2 + d)^q/(c*x^4 + b*x^2 + a)^2, x)
 

Mupad [F(-1)]

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

int(((A + B*x^2)*(d + e*x^2)^q)/(a + b*x^2 + c*x^4)^2,x)
 

Output:

int(((A + B*x^2)*(d + e*x^2)^q)/(a + b*x^2 + c*x^4)^2, x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^q}{\left (a+b x^2+c x^4\right )^2} \, dx=\left (\int \frac {\left (e \,x^{2}+d \right )^{q}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) a +\left (\int \frac {\left (e \,x^{2}+d \right )^{q} x^{2}}{c^{2} x^{8}+2 b c \,x^{6}+2 a c \,x^{4}+b^{2} x^{4}+2 a b \,x^{2}+a^{2}}d x \right ) b \] Input:

int((B*x^2+A)*(e*x^2+d)^q/(c*x^4+b*x^2+a)^2,x)
 

Output:

int((d + e*x**2)**q/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c*x* 
*6 + c**2*x**8),x)*a + int(((d + e*x**2)**q*x**2)/(a**2 + 2*a*b*x**2 + 2*a 
*c*x**4 + b**2*x**4 + 2*b*c*x**6 + c**2*x**8),x)*b