\(\int \frac {A+B x^2}{(d+e x^2) (a+b x^2+c x^4)^{3/2}} \, dx\) [178]

Optimal result

Integrand size = 33, antiderivative size = 882 \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {x \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 )}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {\sqrt {c} \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c e\right )\right ) x \sqrt {a+b x^2+c x^4}}{a \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e^{3/2} (B d-A e) \arctan \left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{2 \sqrt {d} \left (c d^2-b d e+a e^2\right )^{3/2}}-\frac {\sqrt [4]{c} \left (a B (2 c d-b e)-A \left (b c d-b^2 e+2 a c 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} \left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (\sqrt {a} B-A \sqrt {c}\right ) \sqrt [4]{c} \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 ) \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {a+b x^2+c x^4}}+\frac {e \left (\sqrt {c} d+\sqrt {a} e\right ) (B d-A e) \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 {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},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 )}{4 \sqrt [4]{a} \sqrt [4]{c} d \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}} \] Output:

-x*(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^(1/2)*(a*B*(-b*e+2*c*d)-A*(2*a*c*e-b^2*e+b*c* 
d))*x*(c*x^4+b*x^2+a)^(1/2)/a/(-4*a*c+b^2)/(a*e^2-b*d*e+c*d^2)/(a^(1/2)+c^ 
(1/2)*x^2)-1/2*e^(3/2)*(-A*e+B*d)*arctan((a*e^2-b*d*e+c*d^2)^(1/2)*x/d^(1/ 
2)/e^(1/2)/(c*x^4+b*x^2+a)^(1/2))/d^(1/2)/(a*e^2-b*d*e+c*d^2)^(3/2)-c^(1/4 
)*(a*B*(-b*e+2*c*d)-A*(2*a*c*e-b^2*e+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)*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)/(-4*a*c+b^2)/(a*e^2-b*d 
*e+c*d^2)/(c*x^4+b*x^2+a)^(1/2)+1/2*(a^(1/2)*B-A*c^(1/2))*c^(1/4)*(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)*InverseJacob 
iAM(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^(1/2)*d-a^(1/2)*e)/(c*x^4+b*x^2+a)^(1/2)+1/4*e*(c^ 
(1/2)*d+a^(1/2)*e)*(-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)*EllipticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4 
*(c^(1/2)*d-a^(1/2)*e)^2/a^(1/2)/c^(1/2)/d/e,1/2*(2-b/a^(1/2)/c^(1/2))^(1/ 
2))/a^(1/4)/c^(1/4)/d/(c^(1/2)*d-a^(1/2)*e)/(a*e^2-b*d*e+c*d^2)/(c*x^4+b*x 
^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 15.46 (sec) , antiderivative size = 1736, normalized size of antiderivative = 1.97 \[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(-4*A*b^2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d^2*x + 4*a*b*B*c*Sqrt[c/(b + 
Sqrt[b^2 - 4*a*c])]*d^2*x + 8*a*A*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d^2* 
x + 4*A*b^3*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*x - 4*a*b^2*B*Sqrt[c/(b + 
Sqrt[b^2 - 4*a*c])]*d*e*x - 12*a*A*b*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e 
*x + 8*a^2*B*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*x - 4*A*b*c^2*Sqrt[c/(b 
 + Sqrt[b^2 - 4*a*c])]*d^2*x^3 + 8*a*B*c^2*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])] 
*d^2*x^3 + 4*A*b^2*c*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*x^3 - 4*a*b*B*c*S 
qrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e*x^3 - 8*a*A*c^2*Sqrt[c/(b + Sqrt[b^2 - 
4*a*c])]*d*e*x^3 - I*(-b + Sqrt[b^2 - 4*a*c])*d*(a*B*(2*c*d - b*e) + A*(-( 
b*c*d) + b^2*e - 2*a*c*e))*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqr 
t[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*d*(a*B*(b*(b - Sqrt[ 
b^2 - 4*a*c])*e + 2*c*(Sqrt[b^2 - 4*a*c]*d - 2*a*e)) + A*(-(b^3*e) + b*c*( 
-(Sqrt[b^2 - 4*a*c]*d) + 4*a*e) + b^2*(c*d + Sqrt[b^2 - 4*a*c]*e) - 2*a*c* 
(2*c*d + Sqrt[b^2 - 4*a*c]*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])]*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])] - (2*I)*a*b^2*B*d*e 
*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[(...
 

Rubi [A] (verified)

Time = 1.63 (sec) , antiderivative size = 1045, normalized size of antiderivative = 1.18, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {2258, 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.

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

\(\Big \downarrow \) 2258

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

\(\Big \downarrow \) 2009

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

Input:

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

Output:

-((x*(a*b*c*(B*d - A*e) - (b^2 - 2*a*c)*(A*c*d - A*b*e + a*B*e) + c*(a*B*( 
2*c*d - b*e) - A*(b*c*d - b^2*e + 2*a*c*e))*x^2))/(a*(b^2 - 4*a*c)*(c*d^2 
- b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4])) + (Sqrt[c]*(a*B*(2*c*d - b*e) - 
 A*(b*c*d - b^2*e + 2*a*c*e))*x*Sqrt[a + b*x^2 + c*x^4])/(a*(b^2 - 4*a*c)* 
(c*d^2 - b*d*e + a*e^2)*(Sqrt[a] + Sqrt[c]*x^2)) - (e^(3/2)*(B*d - A*e)*Ar 
cTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d]*Sqrt[e]*Sqrt[a + b*x^2 + c*x 
^4])])/(2*Sqrt[d]*(c*d^2 - b*d*e + a*e^2)^(3/2)) - (c^(1/4)*(a*B*(2*c*d - 
b*e) - A*(b*c*d - b^2*e + 2*a*c*e))*(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])/(a^(3/4)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + 
 a*e^2)*Sqrt[a + b*x^2 + c*x^4]) - (c^(1/4)*e*(B*d - A*e)*(Sqrt[a] + Sqrt[ 
c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*Ar 
cTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(1/4)*(Sqrt[ 
c]*d - Sqrt[a]*e)*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) - (c^(1 
/4)*(a*B*e - Sqrt[a]*Sqrt[c]*(B*d - A*e) + A*(c*d - b*e))*(Sqrt[a] + Sqrt[ 
c]*x^2)*Sqrt[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticF[2*Ar 
cTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqrt[c]))/4])/(2*a^(3/4)*(b - 2 
*Sqrt[a]*Sqrt[c])*(c*d^2 - b*d*e + a*e^2)*Sqrt[a + b*x^2 + c*x^4]) + (a^(3 
/4)*e*((Sqrt[c]*d)/Sqrt[a] + e)^2*(B*d - A*e)*(Sqrt[a] + Sqrt[c]*x^2)*Sqrt 
[(a + b*x^2 + c*x^4)/(Sqrt[a] + Sqrt[c]*x^2)^2]*EllipticPi[-1/4*(Sqrt[c...
 

Defintions of rubi rules used

rule 2009

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

rule 2258

Int[(Px_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^ 
(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + b*x^2 + c*x^4], Px*(d + e 
*x^2)^q*(a + b*x^2 + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d, e}, x] 
&& PolyQ[Px, x] && IntegerQ[p + 1/2] && IntegerQ[q]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3240\) vs. \(2(752)=1504\).

Time = 1.16 (sec) , antiderivative size = 3241, normalized size of antiderivative = 3.67

method	result	size
default	\(\text {Expression too large to display}\)	\(3241\)
elliptic	\(\text {Expression too large to display}\)	\(4597\)

Input:

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

Output:

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

Fricas [F(-1)]

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

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

Output:

Timed out
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {A+B x^2}{\left (d+e x^2\right ) \left (a+b x^2+c x^4\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}}{c^{2} e \,x^{10}+2 b c e \,x^{8}+c^{2} d \,x^{8}+2 a c e \,x^{6}+b^{2} e \,x^{6}+2 b c d \,x^{6}+2 a b e \,x^{4}+2 a c d \,x^{4}+b^{2} d \,x^{4}+a^{2} e \,x^{2}+2 a b d \,x^{2}+a^{2} d}d x \right ) a +\left (\int \frac {\sqrt {c \,x^{4}+b \,x^{2}+a}\, x^{2}}{c^{2} e \,x^{10}+2 b c e \,x^{8}+c^{2} d \,x^{8}+2 a c e \,x^{6}+b^{2} e \,x^{6}+2 b c d \,x^{6}+2 a b e \,x^{4}+2 a c d \,x^{4}+b^{2} d \,x^{4}+a^{2} e \,x^{2}+2 a b d \,x^{2}+a^{2} d}d x \right ) b \] Input:

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

Output:

int(sqrt(a + b*x**2 + c*x**4)/(a**2*d + a**2*e*x**2 + 2*a*b*d*x**2 + 2*a*b 
*e*x**4 + 2*a*c*d*x**4 + 2*a*c*e*x**6 + b**2*d*x**4 + b**2*e*x**6 + 2*b*c* 
d*x**6 + 2*b*c*e*x**8 + c**2*d*x**8 + c**2*e*x**10),x)*a + int((sqrt(a + b 
*x**2 + c*x**4)*x**2)/(a**2*d + a**2*e*x**2 + 2*a*b*d*x**2 + 2*a*b*e*x**4 
+ 2*a*c*d*x**4 + 2*a*c*e*x**6 + b**2*d*x**4 + b**2*e*x**6 + 2*b*c*d*x**6 + 
 2*b*c*e*x**8 + c**2*d*x**8 + c**2*e*x**10),x)*b