\(\int \frac {(A+B x^2) (d+e x^2)^2}{(a+b x^2+c x^4)^{3/2}} \, dx\) [175]

Optimal result

Integrand size = 33, antiderivative size = 633 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {x \left (c \left (\frac {a B \left (b c d^2-4 a c d e+a b e^2\right )}{c}-A \left (b^2 d^2-2 a b d e-2 a \left (c d^2-a e^2\right )\right )\right )-\left (A c \left (b c d^2-4 a c d e+a b e^2\right )-a B \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )\right ) x^2\right )}{a c \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (A c \left (b c d^2-4 a c d e+a b e^2\right )-2 a B \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) x \sqrt {a+b x^2+c x^4}}{a c^{3/2} \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}+\frac {\left (A c \left (b c d^2-4 a c d e+a b e^2\right )-2 a B \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{a^{3/4} c^{7/4} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (A c^2 d^2+3 a^{3/2} B \sqrt {c} e^2-\sqrt {a} c^{3/2} d (B d+2 A e)+a e (2 B c d-2 b B e+A c e)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) c^{7/4} \sqrt {a+b x^2+c x^4}} \] Output:

-x*(c*(a*B*(a*b*e^2-4*a*c*d*e+b*c*d^2)/c-A*(b^2*d^2-2*a*b*d*e-2*a*(-a*e^2+ 
c*d^2)))-(A*c*(a*b*e^2-4*a*c*d*e+b*c*d^2)-a*B*(2*c^2*d^2+b^2*e^2-2*c*e*(a* 
e+b*d)))*x^2)/a/c/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(1/2)-(A*c*(a*b*e^2-4*a*c*d 
*e+b*c*d^2)-2*a*B*(c^2*d^2+b^2*e^2-c*e*(3*a*e+b*d)))*x*(c*x^4+b*x^2+a)^(1/ 
2)/a/c^(3/2)/(-4*a*c+b^2)/(a^(1/2)+c^(1/2)*x^2)+(A*c*(a*b*e^2-4*a*c*d*e+b* 
c*d^2)-2*a*B*(c^2*d^2+b^2*e^2-c*e*(3*a*e+b*d)))*(a^(1/2)+c^(1/2)*x^2)*((c* 
x^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*EllipticE(sin(2*arctan(c^(1/4) 
*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(3/4)/c^(7/4)/(-4*a*c+b^2) 
/(c*x^4+b*x^2+a)^(1/2)-1/2*(A*c^2*d^2+3*a^(3/2)*B*c^(1/2)*e^2-a^(1/2)*c^(3 
/2)*d*(2*A*e+B*d)+a*e*(A*c*e-2*B*b*e+2*B*c*d))*(a^(1/2)+c^(1/2)*x^2)*((c*x 
^4+b*x^2+a)/(a^(1/2)+c^(1/2)*x^2)^2)^(1/2)*InverseJacobiAM(2*arctan(c^(1/4 
)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(3/4)/(b-2*a^(1/2)*c^(1/2) 
)/c^(7/4)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 14.85 (sec) , antiderivative size = 766, normalized size of antiderivative = 1.21 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {-4 c \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x \left (-a B \left (a b e^2+2 c^2 d^2 x^2+b^2 e^2 x^2+b c d \left (d-2 e x^2\right )-2 a c e \left (2 d+e x^2\right )\right )+A c \left (b^2 d^2+2 a^2 e^2+b c d^2 x^2+a b e \left (-2 d+e x^2\right )-2 a c d \left (d+2 e x^2\right )\right )\right )-i \left (-b+\sqrt {b^2-4 a c}\right ) \left (-A c \left (b c d^2-4 a c d e+a b e^2\right )+2 a B \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+i \left (2 a B \left (b^2 \left (-b+\sqrt {b^2-4 a c}\right ) e^2+c^2 d \left (\sqrt {b^2-4 a c} d-4 a e\right )+c e \left (b^2 d-b \sqrt {b^2-4 a c} d+4 a b e-3 a \sqrt {b^2-4 a c} e\right )\right )+A c \left (b^2 \left (c d^2+a e^2\right )-b \sqrt {b^2-4 a c} \left (c d^2+a e^2\right )-4 a c \left (c d^2-\sqrt {b^2-4 a c} d e+a e^2\right )\right )\right ) \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {\frac {2 b-2 \sqrt {b^2-4 a c}+4 c x^2}{b-\sqrt {b^2-4 a c}}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )}{4 a c^2 \left (-b^2+4 a c\right ) \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} \sqrt {a+b x^2+c x^4}} \] Input:

Integrate[((A + B*x^2)*(d + e*x^2)^2)/(a + b*x^2 + c*x^4)^(3/2),x]
 

Output:

(-4*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x*(-(a*B*(a*b*e^2 + 2*c^2*d^2*x^2 + 
b^2*e^2*x^2 + b*c*d*(d - 2*e*x^2) - 2*a*c*e*(2*d + e*x^2))) + A*c*(b^2*d^2 
 + 2*a^2*e^2 + b*c*d^2*x^2 + a*b*e*(-2*d + e*x^2) - 2*a*c*d*(d + 2*e*x^2)) 
) - I*(-b + Sqrt[b^2 - 4*a*c])*(-(A*c*(b*c*d^2 - 4*a*c*d*e + a*b*e^2)) + 2 
*a*B*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)))*Sqrt[(b + Sqrt[b^2 - 4*a*c] 
+ 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c* 
x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt 
[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + I*( 
2*a*B*(b^2*(-b + Sqrt[b^2 - 4*a*c])*e^2 + c^2*d*(Sqrt[b^2 - 4*a*c]*d - 4*a 
*e) + c*e*(b^2*d - b*Sqrt[b^2 - 4*a*c]*d + 4*a*b*e - 3*a*Sqrt[b^2 - 4*a*c] 
*e)) + A*c*(b^2*(c*d^2 + a*e^2) - b*Sqrt[b^2 - 4*a*c]*(c*d^2 + a*e^2) - 4* 
a*c*(c*d^2 - Sqrt[b^2 - 4*a*c]*d*e + a*e^2)))*Sqrt[(b + Sqrt[b^2 - 4*a*c] 
+ 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(2*b - 2*Sqrt[b^2 - 4*a*c] + 4*c* 
x^2)/(b - Sqrt[b^2 - 4*a*c])]*EllipticF[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqrt 
[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])])/(4*a 
*c^2*(-b^2 + 4*a*c)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[a + b*x^2 + c*x^4 
])
 

Rubi [A] (verified)

Time = 0.93 (sec) , antiderivative size = 576, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2206, 27, 1511, 27, 1416, 1509}

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.

\(\displaystyle \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 2206

\(\displaystyle -\frac {\int \frac {\left (A c \left (b c d^2-4 a c e d+a b e^2\right )-2 a B \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) x^2+a (a e (4 B c d-b B e+2 A c e)+c d (2 A c d-b (B d+2 A e)))}{c \sqrt {c x^4+b x^2+a}}dx}{a \left (b^2-4 a c\right )}-\frac {x \left (c \left (\frac {a B \left (a b e^2-4 a c d e+b c d^2\right )}{c}-A \left (-2 a b d e-2 a \left (c d^2-a e^2\right )+b^2 d^2\right )\right )-x^2 \left (A c \left (a b e^2-4 a c d e+b c d^2\right )-a B \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )\right )\right )}{a c \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\left (A c \left (b c d^2-4 a c e d+a b e^2\right )-2 a B \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )\right ) x^2+a (a e (4 B c d-b B e+2 A c e)+c d (2 A c d-b (B d+2 A e)))}{\sqrt {c x^4+b x^2+a}}dx}{a c \left (b^2-4 a c\right )}-\frac {x \left (c \left (\frac {a B \left (a b e^2-4 a c d e+b c d^2\right )}{c}-A \left (-2 a b d e-2 a \left (c d^2-a e^2\right )+b^2 d^2\right )\right )-x^2 \left (A c \left (a b e^2-4 a c d e+b c d^2\right )-a B \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )\right )\right )}{a c \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\)

\(\Big \downarrow \) 1511

\(\displaystyle -\frac {\frac {\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (3 a^{3/2} B \sqrt {c} e^2+a e (A c e-2 b B e+2 B c d)-\sqrt {a} c^{3/2} d (2 A e+B d)+A c^2 d^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\sqrt {a} \left (A c \left (a b e^2-4 a c d e+b c d^2\right )-2 a B \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{a c \left (b^2-4 a c\right )}-\frac {x \left (c \left (\frac {a B \left (a b e^2-4 a c d e+b c d^2\right )}{c}-A \left (-2 a b d e-2 a \left (c d^2-a e^2\right )+b^2 d^2\right )\right )-x^2 \left (A c \left (a b e^2-4 a c d e+b c d^2\right )-a B \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )\right )\right )}{a c \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (3 a^{3/2} B \sqrt {c} e^2+a e (A c e-2 b B e+2 B c d)-\sqrt {a} c^{3/2} d (2 A e+B d)+A c^2 d^2\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\left (A c \left (a b e^2-4 a c d e+b c d^2\right )-2 a B \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{a c \left (b^2-4 a c\right )}-\frac {x \left (c \left (\frac {a B \left (a b e^2-4 a c d e+b c d^2\right )}{c}-A \left (-2 a b d e-2 a \left (c d^2-a e^2\right )+b^2 d^2\right )\right )-x^2 \left (A c \left (a b e^2-4 a c d e+b c d^2\right )-a B \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )\right )\right )}{a c \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\)

\(\Big \downarrow \) 1416

\(\displaystyle -\frac {\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (3 a^{3/2} B \sqrt {c} e^2+a e (A c e-2 b B e+2 B c d)-\sqrt {a} c^{3/2} d (2 A e+B d)+A c^2 d^2\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (A c \left (a b e^2-4 a c d e+b c d^2\right )-2 a B \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{a c \left (b^2-4 a c\right )}-\frac {x \left (c \left (\frac {a B \left (a b e^2-4 a c d e+b c d^2\right )}{c}-A \left (-2 a b d e-2 a \left (c d^2-a e^2\right )+b^2 d^2\right )\right )-x^2 \left (A c \left (a b e^2-4 a c d e+b c d^2\right )-a B \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )\right )\right )}{a c \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\)

\(\Big \downarrow \) 1509

\(\displaystyle -\frac {\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (3 a^{3/2} B \sqrt {c} e^2+a e (A c e-2 b B e+2 B c d)-\sqrt {a} c^{3/2} d (2 A e+B d)+A c^2 d^2\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (A c \left (a b e^2-4 a c d e+b c d^2\right )-2 a B \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right )}{\sqrt {c}}}{a c \left (b^2-4 a c\right )}-\frac {x \left (c \left (\frac {a B \left (a b e^2-4 a c d e+b c d^2\right )}{c}-A \left (-2 a b d e-2 a \left (c d^2-a e^2\right )+b^2 d^2\right )\right )-x^2 \left (A c \left (a b e^2-4 a c d e+b c d^2\right )-a B \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right )\right )\right )}{a c \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}\)

Input:

Int[((A + B*x^2)*(d + e*x^2)^2)/(a + b*x^2 + c*x^4)^(3/2),x]
 

Output:

-((x*(c*((a*B*(b*c*d^2 - 4*a*c*d*e + a*b*e^2))/c - A*(b^2*d^2 - 2*a*b*d*e 
- 2*a*(c*d^2 - a*e^2))) - (A*c*(b*c*d^2 - 4*a*c*d*e + a*b*e^2) - a*B*(2*c^ 
2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e)))*x^2))/(a*c*(b^2 - 4*a*c)*Sqrt[a + b* 
x^2 + c*x^4])) - (-(((A*c*(b*c*d^2 - 4*a*c*d*e + a*b*e^2) - 2*a*B*(c^2*d^2 
 + b^2*e^2 - c*e*(b*d + 3*a*e)))*(-((x*Sqrt[a + b*x^2 + c*x^4])/(Sqrt[a] + 
 Sqrt[c]*x^2)) + (a^(1/4)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4) 
/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - 
b/(Sqrt[a]*Sqrt[c]))/4])/(c^(1/4)*Sqrt[a + b*x^2 + c*x^4])))/Sqrt[c]) + (a 
^(1/4)*(b + 2*Sqrt[a]*Sqrt[c])*(A*c^2*d^2 + 3*a^(3/2)*B*Sqrt[c]*e^2 - Sqrt 
[a]*c^(3/2)*d*(B*d + 2*A*e) + a*e*(2*B*c*d - 2*b*B*e + A*c*e))*(Sqrt[a] + 
Sqrt[c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF 
[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*c^(3/4)*S 
qrt[a + b*x^2 + c*x^4]))/(a*c*(b^2 - 4*a*c))
 

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 1416

Int[1/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbol] :> With[{q = Rt[c 
/a, 4]}, Simp[(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2*x^2)^2)]/ 
(2*q*Sqrt[a + b*x^2 + c*x^4]))*EllipticF[2*ArcTan[q*x], 1/2 - b*(q^2/(4*c)) 
], x]] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && PosQ[c/a]
 

rule 1509

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 4]}, Simp[(-d)*x*(Sqrt[a + b*x^2 + c*x^4]/(a*(1 + q 
^2*x^2))), x] + Simp[d*(1 + q^2*x^2)*(Sqrt[(a + b*x^2 + c*x^4)/(a*(1 + q^2* 
x^2)^2)]/(q*Sqrt[a + b*x^2 + c*x^4]))*EllipticE[2*ArcTan[q*x], 1/2 - b*(q^2 
/(4*c))], x] /; EqQ[e + d*q^2, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 
- 4*a*c, 0] && PosQ[c/a]
 

rule 1511

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4], x_Symbo 
l] :> With[{q = Rt[c/a, 2]}, Simp[(e + d*q)/q   Int[1/Sqrt[a + b*x^2 + c*x^ 
4], x], x] - Simp[e/q   Int[(1 - q*x^2)/Sqrt[a + b*x^2 + c*x^4], x], x] /; 
NeQ[e + d*q, 0]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && Pos 
Q[c/a]
 

rule 2206

Int[(Px_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> With[{d = 
 Coeff[PolynomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 0], e = Coeff[Poly 
nomialRemainder[Px, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^ 
4)^(p + 1)*((a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*x^2)/(2*a*(p + 1)*(b 
^2 - 4*a*c))), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(a + b*x^2 + c 
*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[Px, 
a + b*x^2 + c*x^4, x] + b^2*d*(2*p + 3) - 2*a*c*d*(4*p + 5) - a*b*e + c*(4* 
p + 7)*(b*d - 2*a*e)*x^2, x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x 
^2] && Expon[Px, x^2] > 1 && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1]
 

Maple [A] (verified)

Time = 4.23 (sec) , antiderivative size = 799, normalized size of antiderivative = 1.26

method	result	size
elliptic	\(-\frac {2 c \left (\frac {\left (A a b c \,e^{2}-4 A a \,c^{2} d e +A b \,c^{2} d^{2}+2 B \,a^{2} c \,e^{2}-B a \,b^{2} e^{2}+2 B b c d e a -2 B a \,c^{2} d^{2}\right ) x^{3}}{2 c^{2} a \left (4 a c -b^{2}\right )}+\frac {\left (2 A \,a^{2} c \,e^{2}-2 A a b c d e -2 A \,c^{2} d^{2} a +A \,b^{2} c \,d^{2}-B \,a^{2} b \,e^{2}+4 B \,a^{2} c d e -B a b c \,d^{2}\right ) x}{2 c^{2} \left (4 a c -b^{2}\right ) a}\right )}{\sqrt {\left (x^{4}+\frac {b \,x^{2}}{c}+\frac {a}{c}\right ) c}}+\frac {\left (\frac {e \left (A c e -B b e +2 B c d \right )}{c^{2}}-\frac {A a c \,e^{2}-A \,c^{2} d^{2}-B a b \,e^{2}+2 B a c d e}{c^{2} a}+\frac {2 A \,a^{2} c \,e^{2}-2 A a b c d e -2 A \,c^{2} d^{2} a +A \,b^{2} c \,d^{2}-B \,a^{2} b \,e^{2}+4 B \,a^{2} c d e -B a b c \,d^{2}}{a \left (4 a c -b^{2}\right ) c}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (\frac {B \,e^{2}}{c}+\frac {A a b c \,e^{2}-4 A a \,c^{2} d e +A b \,c^{2} d^{2}+2 B \,a^{2} c \,e^{2}-B a \,b^{2} e^{2}+2 B b c d e a -2 B a \,c^{2} d^{2}}{a \left (4 a c -b^{2}\right ) c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(799\)
default	\(\text {Expression too large to display}\)	\(1887\)

Input:

int((B*x^2+A)*(e*x^2+d)^2/(c*x^4+b*x^2+a)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

-2*c*(1/2/c^2*(A*a*b*c*e^2-4*A*a*c^2*d*e+A*b*c^2*d^2+2*B*a^2*c*e^2-B*a*b^2 
*e^2+2*B*a*b*c*d*e-2*B*a*c^2*d^2)/a/(4*a*c-b^2)*x^3+1/2*(2*A*a^2*c*e^2-2*A 
*a*b*c*d*e-2*A*a*c^2*d^2+A*b^2*c*d^2-B*a^2*b*e^2+4*B*a^2*c*d*e-B*a*b*c*d^2 
)/c^2/(4*a*c-b^2)/a*x)/((x^4+b/c*x^2+a/c)*c)^(1/2)+1/4*(e*(A*c*e-B*b*e+2*B 
*c*d)/c^2-1/c^2*(A*a*c*e^2-A*c^2*d^2-B*a*b*e^2+2*B*a*c*d*e)/a+1/a*(2*A*a^2 
*c*e^2-2*A*a*b*c*d*e-2*A*a*c^2*d^2+A*b^2*c*d^2-B*a^2*b*e^2+4*B*a^2*c*d*e-B 
*a*b*c*d^2)/(4*a*c-b^2)/c)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2)*(4-2* 
(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1 
/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticF(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2)) 
/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2))-1/2*(B*e^2/c+1/a* 
(A*a*b*c*e^2-4*A*a*c^2*d*e+A*b*c^2*d^2+2*B*a^2*c*e^2-B*a*b^2*e^2+2*B*a*b*c 
*d*e-2*B*a*c^2*d^2)/(4*a*c-b^2)/c)*a*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^( 
1/2)*(4-2*(-b+(-4*a*c+b^2)^(1/2))/a*x^2)^(1/2)*(4+2*(b+(-4*a*c+b^2)^(1/2)) 
/a*x^2)^(1/2)/(c*x^4+b*x^2+a)^(1/2)/(b+(-4*a*c+b^2)^(1/2))*(EllipticF(1/2* 
x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2),1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1 
/2))/a/c)^(1/2))-EllipticE(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2) 
,1/2*(-4+2*b*(b+(-4*a*c+b^2)^(1/2))/a/c)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1669 vs. \(2 (564) = 1128\).

Time = 0.10 (sec) , antiderivative size = 1669, normalized size of antiderivative = 2.64 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

integrate((B*x^2+A)*(e*x^2+d)^2/(c*x^4+b*x^2+a)^(3/2),x, algorithm="fricas 
")
 

Output:

-1/2*(sqrt(1/2)*(((2*B*a*b - A*b^2)*c^3*d^2 - 2*(B*a*b^2*c^2 - 2*A*a*b*c^3 
)*d*e + (2*B*a*b^3*c - (6*B*a^2*b + A*a*b^2)*c^2)*e^2)*x^5 + ((2*B*a*b^2 - 
 A*b^3)*c^2*d^2 - 2*(B*a*b^3*c - 2*A*a*b^2*c^2)*d*e + (2*B*a*b^4 - (6*B*a^ 
2*b^2 + A*a*b^3)*c)*e^2)*x^3 + ((2*B*a^2*b - A*a*b^2)*c^2*d^2 - 2*(B*a^2*b 
^2*c - 2*A*a^2*b*c^2)*d*e + (2*B*a^2*b^3 - (6*B*a^3*b + A*a^2*b^2)*c)*e^2) 
*x - (((2*B*a - A*b)*c^4*d^2 - 2*(B*a*b*c^3 - 2*A*a*c^4)*d*e + (2*B*a*b^2* 
c^2 - (6*B*a^2 + A*a*b)*c^3)*e^2)*x^5 + ((2*B*a*b - A*b^2)*c^3*d^2 - 2*(B* 
a*b^2*c^2 - 2*A*a*b*c^3)*d*e + (2*B*a*b^3*c - (6*B*a^2*b + A*a*b^2)*c^2)*e 
^2)*x^3 + ((2*B*a^2 - A*a*b)*c^3*d^2 - 2*(B*a^2*b*c^2 - 2*A*a^2*c^3)*d*e + 
 (2*B*a^2*b^2*c - (6*B*a^3 + A*a^2*b)*c^2)*e^2)*x)*sqrt((b^2 - 4*a*c)/c^2) 
)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)*elliptic_e(arcsin(sqrt(1 
/2)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b)/c)/x), 1/2*(b*c*sqrt((b^2 - 4*a*c 
)/c^2) + b^2 - 2*a*c)/(a*c)) + sqrt(1/2)*(((2*A*b*c^4 - (2*B*a*b - (A - B) 
*b^2)*c^3)*d^2 + 2*(B*a*b^2*c^2 - (2*(A - B)*a*b + A*b^2)*c^3)*d*e - (2*B* 
a*b^3*c - 2*A*a*b*c^3 - (6*B*a^2*b + (A - B)*a*b^2)*c^2)*e^2)*x^5 + ((2*A* 
b^2*c^3 - (2*B*a*b^2 - (A - B)*b^3)*c^2)*d^2 + 2*(B*a*b^3*c - (2*(A - B)*a 
*b^2 + A*b^3)*c^2)*d*e - (2*B*a*b^4 - 2*A*a*b^2*c^2 - (6*B*a^2*b^2 + (A - 
B)*a*b^3)*c)*e^2)*x^3 + ((2*A*a*b*c^3 - (2*B*a^2*b - (A - B)*a*b^2)*c^2)*d 
^2 + 2*(B*a^2*b^2*c - (2*(A - B)*a^2*b + A*a*b^2)*c^2)*d*e - (2*B*a^2*b^3 
- 2*A*a^2*b*c^2 - (6*B*a^3*b + (A - B)*a^2*b^2)*c)*e^2)*x + (((2*A*c^5 ...
 

Sympy [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {\left (A + B x^{2}\right ) \left (d + e x^{2}\right )^{2}}{\left (a + b x^{2} + c x^{4}\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((B*x**2+A)*(e*x**2+d)**2/(c*x**4+b*x**2+a)**(3/2),x)
 

Output:

Integral((A + B*x**2)*(d + e*x**2)**2/(a + b*x**2 + c*x**4)**(3/2), x)
 

Maxima [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{2}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^2/(c*x^4+b*x^2+a)^(3/2),x, algorithm="maxima 
")
 

Output:

integrate((B*x^2 + A)*(e*x^2 + d)^2/(c*x^4 + b*x^2 + a)^(3/2), x)
 

Giac [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int { \frac {{\left (B x^{2} + A\right )} {\left (e x^{2} + d\right )}^{2}}{{\left (c x^{4} + b x^{2} + a\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((B*x^2+A)*(e*x^2+d)^2/(c*x^4+b*x^2+a)^(3/2),x, algorithm="giac")
 

Output:

integrate((B*x^2 + A)*(e*x^2 + d)^2/(c*x^4 + b*x^2 + a)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\int \frac {\left (B\,x^2+A\right )\,{\left (e\,x^2+d\right )}^2}{{\left (c\,x^4+b\,x^2+a\right )}^{3/2}} \,d x \] Input:

int(((A + B*x^2)*(d + e*x^2)^2)/(a + b*x^2 + c*x^4)^(3/2),x)
 

Output:

int(((A + B*x^2)*(d + e*x^2)^2)/(a + b*x^2 + c*x^4)^(3/2), x)
 

Reduce [F]

\[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^2}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

int((B*x^2+A)*(e*x^2+d)^2/(c*x^4+b*x^2+a)^(3/2),x)
 

Output:

( - sqrt(a + b*x**2 + c*x**4)*a*c*e**2*x + 2*sqrt(a + b*x**2 + c*x**4)*b** 
2*e**2*x - 2*sqrt(a + b*x**2 + c*x**4)*b*c*d*e*x + sqrt(a + b*x**2 + c*x** 
4)*b*c*e**2*x**3 + int(sqrt(a + b*x**2 + c*x**4)/(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**3*c*e**2 - 2*int(sqrt(a 
 + b*x**2 + c*x**4)/(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**2*b**2*e**2 + 2*int(sqrt(a + b*x**2 + c*x**4)/(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**2*b* 
c*d*e + int(sqrt(a + b*x**2 + c*x**4)/(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**2*b*c*e**2*x**2 + int(sqrt(a + b*x 
**2 + c*x**4)/(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**2*c**2*d**2 + int(sqrt(a + b*x**2 + c*x**4)/(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**2*c**2*e**2* 
x**4 - 2*int(sqrt(a + b*x**2 + c*x**4)/(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*b**3*e**2*x**2 + 2*int(sqrt(a + b* 
x**2 + c*x**4)/(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*b**2*c*d*e*x**2 - 2*int(sqrt(a + b*x**2 + c*x**4)/(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*b**2*c* 
e**2*x**4 + int(sqrt(a + b*x**2 + c*x**4)/(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*b*c**2*d**2*x**2 + 2*int(sqrt(a 
 + b*x**2 + c*x**4)/(a**2 + 2*a*b*x**2 + 2*a*c*x**4 + b**2*x**4 + 2*b*c...