\(\int \frac {A+B x^2}{(d+e x^2)^2 \sqrt {a+b x^2+c x^4}} \, dx\) [172]

Optimal result

Integrand size = 33, antiderivative size = 782 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \sqrt {a+b x^2+c x^4}} \, dx=\frac {\sqrt {c} (B d-A e) x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {e (B d-A e) x \sqrt {a+b x^2+c x^4}}{2 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )}-\frac {\left (B \left (c d^3-a d e^2\right )-A e \left (3 c d^2-e (2 b d-a e)\right )\right ) \arctan \left (\frac {\sqrt {c d^2-b d e+a e^2} x}{\sqrt {d} \sqrt {e} \sqrt {a+b x^2+c x^4}}\right )}{4 d^{3/2} \sqrt {e} \left (c d^2-b d e+a e^2\right )^{3/2}}-\frac {\sqrt [4]{a} \sqrt [4]{c} (B d-A e) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 d \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}}+\frac {A \sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{2 \sqrt [4]{a} d \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {a+b x^2+c x^4}}+\frac {\left (\sqrt {c} d+\sqrt {a} e\right ) \left (B \left (c d^3-a d e^2\right )-A e \left (3 c d^2-e (2 b d-a e)\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticPi}\left (-\frac {\left (\sqrt {c} d-\sqrt {a} e\right )^2}{4 \sqrt {a} \sqrt {c} d e},2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{8 \sqrt [4]{a} \sqrt [4]{c} d^2 e \left (\sqrt {c} d-\sqrt {a} e\right ) \left (c d^2-b d e+a e^2\right ) \sqrt {a+b x^2+c x^4}} \] Output:

1/2*c^(1/2)*(-A*e+B*d)*x*(c*x^4+b*x^2+a)^(1/2)/d/(a*e^2-b*d*e+c*d^2)/(a^(1 
/2)+c^(1/2)*x^2)-1/2*e*(-A*e+B*d)*x*(c*x^4+b*x^2+a)^(1/2)/d/(a*e^2-b*d*e+c 
*d^2)/(e*x^2+d)-1/4*(B*(-a*d*e^2+c*d^3)-A*e*(3*c*d^2-e*(-a*e+2*b*d)))*arct 
an((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^(3 
/2)/e^(1/2)/(a*e^2-b*d*e+c*d^2)^(3/2)-1/2*a^(1/4)*c^(1/4)*(-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))/d/(a*e^ 
2-b*d*e+c*d^2)/(c*x^4+b*x^2+a)^(1/2)+1/2*A*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)*InverseJacobiAM(2*arctan(c^ 
(1/4)*x/a^(1/4)),1/2*(2-b/a^(1/2)/c^(1/2))^(1/2))/a^(1/4)/d/(c^(1/2)*d-a^( 
1/2)*e)/(c*x^4+b*x^2+a)^(1/2)+1/8*(c^(1/2)*d+a^(1/2)*e)*(B*(-a*d*e^2+c*d^3 
)-A*e*(3*c*d^2-e*(-a*e+2*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^2/e/(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 = 13.95 (sec) , antiderivative size = 1853, normalized size of antiderivative = 2.37 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^2 \sqrt {a+b x^2+c x^4}} \, dx =\text {Too large to display} \] Input:

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

Output:

((-1/8*I)*((-4*I)*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*d*e^2*(B*d - A*e)*x*(a + 
 b*x^2 + c*x^4) + Sqrt[2]*B*(b - Sqrt[b^2 - 4*a*c])*d^2*e*Sqrt[(b + Sqrt[b 
^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sq 
rt[b^2 - 4*a*c])]*(d + e*x^2)*(EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sqr 
t[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] - El 
lipticF[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])]) + Sqrt[2]*A*(-b + Sqrt[b^2 - 4*a*c]) 
*d*e^2*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqr 
t[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(d + e*x^2)*(EllipticE[I*ArcSinh[ 
Sqrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - S 
qrt[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*Sqrt[2]*B 
*c*d^3*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqr 
t[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]*(d + e*x^2)*EllipticF[I*ArcSinh[S 
qrt[2]*Sqrt[c/(b + Sqrt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sq 
rt[b^2 - 4*a*c])] - 2*Sqrt[2]*A*c*d^2*e*Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c* 
x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (2*c*x^2)/(b - Sqrt[b^2 - 4*a*c])]* 
(d + e*x^2)*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*Sqrt[2]*B*c*d^3*Sqr 
t[(b + Sqrt[b^2 - 4*a*c] + 2*c*x^2)/(b + Sqrt[b^2 - 4*a*c])]*Sqrt[1 + (...
 

Rubi [A] (verified)

Time = 1.59 (sec) , antiderivative size = 727, normalized size of antiderivative = 0.93, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2210, 25, 2232, 27, 1509, 2226, 27, 1416, 2220}

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.

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

\(\Big \downarrow \) 2210

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2232

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1509

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

\(\Big \downarrow \) 2226

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 2220

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

Input:

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

Output:

-1/2*(e*(B*d - A*e)*x*Sqrt[a + b*x^2 + c*x^4])/(d*(c*d^2 - b*d*e + a*e^2)* 
(d + e*x^2)) + (-(Sqrt[c]*(B*d - 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]))) + (A*c^( 
1/4)*(c*d^2 - b*d*e + a*e^2)*(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])/(a^(1/4)*(Sqrt[c]*d - Sqrt[a]*e)*Sqrt[a + b*x^ 
2 + c*x^4]) + ((B*(c*d^3 - a*d*e^2) - A*e*(3*c*d^2 - e*(2*b*d - a*e)))*(-1 
/2*((Sqrt[c]*d - Sqrt[a]*e)*ArcTan[(Sqrt[c*d^2 - b*d*e + a*e^2]*x)/(Sqrt[d 
]*Sqrt[e]*Sqrt[a + b*x^2 + c*x^4])])/(Sqrt[d]*Sqrt[e]*Sqrt[c*d^2 - b*d*e + 
 a*e^2]) + ((Sqrt[c]*d + Sqrt[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[a]*((Sqrt[c]*d 
)/Sqrt[a] - e)^2)/(Sqrt[c]*d*e), 2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sq 
rt[a]*Sqrt[c]))/4])/(4*a^(1/4)*c^(1/4)*d*e*Sqrt[a + b*x^2 + c*x^4])))/(Sqr 
t[c]*d - Sqrt[a]*e))/(2*d*(c*d^2 - b*d*e + a*e^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 2210

Int[((P4x_)*((d_) + (e_.)*(x_)^2)^(q_))/Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x 
_)^4], x_Symbol] :> With[{A = Coeff[P4x, x, 0], B = Coeff[P4x, x, 2], C = C 
oeff[P4x, x, 4]}, Simp[(-(C*d^2 - B*d*e + A*e^2))*x*(d + e*x^2)^(q + 1)*(Sq 
rt[a + b*x^2 + c*x^4]/(2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/( 
2*d*(q + 1)*(c*d^2 - b*d*e + a*e^2))   Int[((d + e*x^2)^(q + 1)/Sqrt[a + b* 
x^2 + c*x^4])*Simp[a*d*(C*d - B*e) + A*(a*e^2*(2*q + 3) + 2*d*(c*d - b*e)*( 
q + 1)) - 2*((B*d - A*e)*(b*e*(q + 2) - c*d*(q + 1)) - C*d*(b*d + a*e*(q + 
1)))*x^2 + c*(C*d^2 - B*d*e + A*e^2)*(2*q + 5)*x^4, x], x], x]] /; FreeQ[{a 
, b, c, d, e}, x] && PolyQ[P4x, x^2] && LeQ[Expon[P4x, x], 4] && ILtQ[q, -1 
]
 

rule 2220

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

rule 2226

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

rule 2232

Int[(P4x_)/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4]) 
, x_Symbol] :> With[{q = Rt[c/a, 2], A = Coeff[P4x, x, 0], B = Coeff[P4x, x 
, 2], C = Coeff[P4x, x, 4]}, Simp[-C/(e*q)   Int[(1 - q*x^2)/Sqrt[a + b*x^2 
 + c*x^4], x], x] + Simp[1/(c*e)   Int[(A*c*e + a*C*d*q + (B*c*e - C*(c*d - 
 a*e*q))*x^2)/((d + e*x^2)*Sqrt[a + b*x^2 + c*x^4]), x], x]] /; FreeQ[{a, b 
, c, d, e}, x] && PolyQ[P4x, x^2, 2] && NeQ[c*d^2 - a*e^2, 0] && PosQ[c/a] 
&&  !GtQ[b^2 - 4*a*c, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1494\) vs. \(2(654)=1308\).

Time = 3.07 (sec) , antiderivative size = 1495, normalized size of antiderivative = 1.91

method	result	size
default	\(\text {Expression too large to display}\)	\(1495\)
elliptic	\(\text {Expression too large to display}\)	\(2301\)

Input:

int((B*x^2+A)/(e*x^2+d)^2/(c*x^4+b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

B/e/d*2^(1/2)/(-b/a+1/a*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b/a*x^2-1/2/a*x^2 
*(-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b/a*x^2+1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(1 
/2)/(c*x^4+b*x^2+a)^(1/2)*EllipticPi(1/2*x*2^(1/2)*((-b+(-4*a*c+b^2)^(1/2) 
)/a)^(1/2),-2/(-b+(-4*a*c+b^2)^(1/2))*a/d*e,(-1/2*(b+(-4*a*c+b^2)^(1/2))/a 
)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))/a)^(1/2))+(A*e-B*d)/e*(1/2*e^2/(a 
*e^2-b*d*e+c*d^2)/d*x*(c*x^4+b*x^2+a)^(1/2)/(e*x^2+d)-1/8*c/(a*e^2-b*d*e+c 
*d^2)*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))+1/4*c*e/(a*e^2-b*d*e+c*d^ 
2)/d*a*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)/(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))-1 
/4*c*e/(a*e^2-b*d*e+c*d^2)/d*a*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)/(b+(-4*a*c+b^2)^(1/2))*Elliptic 
E(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/(a*e^2-b*d*e+c*d^2)/d^2*e^2*2^(1/2)/(-b/a+1/a*( 
-4*a*c+b^2)^(1/2))^(1/2)*(1+1/2*b/a*x^2-1/2/a*x^2*(-4*a*c+b^2)^(1/2))^(...
 

Fricas [F]

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

integrate((B*x^2+A)/(e*x^2+d)^2/(c*x^4+b*x^2+a)^(1/2),x, algorithm="fricas 
")
 

Output:

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

Sympy [F]

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

integrate((B*x**2+A)/(e*x**2+d)**2/(c*x**4+b*x**2+a)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((B*x^2+A)/(e*x^2+d)^2/(c*x^4+b*x^2+a)^(1/2),x, algorithm="maxima 
")
 

Output:

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

Giac [F]

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

integrate((B*x^2+A)/(e*x^2+d)^2/(c*x^4+b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((A + B*x^2)/((d + e*x^2)^2*(a + b*x^2 + c*x^4)^(1/2)),x)
 

Output:

int((A + B*x^2)/((d + e*x^2)^2*(a + b*x^2 + c*x^4)^(1/2)), x)
 

Reduce [F]

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

int((B*x^2+A)/(e*x^2+d)^2/(c*x^4+b*x^2+a)^(1/2),x)
 

Output:

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