\(\int \frac {A+B x^2}{(d+e x^2)^3 \sqrt {a+b x^2+c x^4}} \, dx\) [173]

Optimal result

Integrand size = 33, antiderivative size = 1125 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^3 \sqrt {a+b x^2+c x^4}} \, dx =\text {Too large to display} \] Output:

-1/8*c^(1/2)*(3*A*e*(3*c*d^2-e*(-a*e+2*b*d))-B*d*(5*c*d^2-e*(a*e+2*b*d)))* 
x*(c*x^4+b*x^2+a)^(1/2)/d^2/(a*e^2-b*d*e+c*d^2)^2/(a^(1/2)+c^(1/2)*x^2)-1/ 
4*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)^2+1 
/8*e*(3*A*e*(3*c*d^2-e*(-a*e+2*b*d))-B*d*(5*c*d^2-e*(a*e+2*b*d)))*x*(c*x^4 
+b*x^2+a)^(1/2)/d^2/(a*e^2-b*d*e+c*d^2)^2/(e*x^2+d)-1/16*(B*d*(3*c^2*d^4-1 
0*a*c*d^2*e^2+a*e^3*(-a*e+4*b*d))-A*e*(15*c^2*d^4-2*c*d^2*e*(-3*a*e+10*b*d 
)+e^2*(3*a^2*e^2-8*a*b*d*e+8*b^2*d^2)))*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^(5/2)/e^(1/2)/(a*e^2-b*d*e+c*d^2 
)^(5/2)+1/8*a^(1/4)*c^(1/4)*(3*A*e*(3*c*d^2-e*(-a*e+2*b*d))-B*d*(5*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)*EllipticE(sin(2*arctan(c^(1/4)*x/a^(1/4))),1/2*(2-b/a^(1/2)/c^( 
1/2))^(1/2))/d^2/(a*e^2-b*d*e+c*d^2)^2/(c*x^4+b*x^2+a)^(1/2)+1/8*c^(1/4)*( 
a^(1/2)*c^(1/2)*d*(-A*e+B*d)+a*e*(3*A*e+B*d)+4*A*d*(-b*e+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^(1/4)/d^2/ 
(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)+1/32*(c^(1 
/2)*d+a^(1/2)*e)*(B*d*(3*c^2*d^4-10*a*c*d^2*e^2+a*e^3*(-a*e+4*b*d))-A*e*(1 
5*c^2*d^4-2*c*d^2*e*(-3*a*e+10*b*d)+e^2*(3*a^2*e^2-8*a*b*d*e+8*b^2*d^2)))* 
(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)*Elli 
pticPi(sin(2*arctan(c^(1/4)*x/a^(1/4))),-1/4*(c^(1/2)*d-a^(1/2)*e)^2/a^...
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 16.09 (sec) , antiderivative size = 781, normalized size of antiderivative = 0.69 \[ \int \frac {A+B x^2}{\left (d+e x^2\right )^3 \sqrt {a+b x^2+c x^4}} \, dx=\frac {-\frac {4 d e^2 x \left (a+b x^2+c x^4\right ) \left (2 d (B d-A e) \left (c d^2+e (-b d+a e)\right )+\left (-3 A e \left (3 c d^2+e (-2 b d+a e)\right )+B \left (5 c d^3-d e (2 b d+a e)\right )\right ) \left (d+e x^2\right )\right )}{\left (d+e x^2\right )^2}-\frac {i \sqrt {2} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^2}{b+\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^2}{b-\sqrt {b^2-4 a c}}} \left (\left (-b+\sqrt {b^2-4 a c}\right ) d e \left (3 A e \left (3 c d^2+e (-2 b d+a e)\right )+B \left (-5 c d^3+d e (2 b d+a e)\right )\right ) E\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right )|\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+d \left (B d \left (6 c^2 d^3+c d e \left (-5 b d+5 \sqrt {b^2-4 a c} d-6 a e\right )-\left (-b+\sqrt {b^2-4 a c}\right ) e^2 (2 b d+a e)\right )-A e \left (14 c^2 d^3-3 \left (-b+\sqrt {b^2-4 a c}\right ) e^2 (2 b d-a e)+c d e \left (-17 b d+9 \sqrt {b^2-4 a c} d+2 a e\right )\right )\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )+2 \left (B \left (-3 c^2 d^5+10 a c d^3 e^2+a d e^3 (-4 b d+a e)\right )+A e \left (15 c^2 d^4+2 c d^2 e (-10 b d+3 a e)+e^2 \left (8 b^2 d^2-8 a b d e+3 a^2 e^2\right )\right )\right ) \operatorname {EllipticPi}\left (\frac {\left (b+\sqrt {b^2-4 a c}\right ) e}{2 c d},i \text {arcsinh}\left (\sqrt {2} \sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}} x\right ),\frac {b+\sqrt {b^2-4 a c}}{b-\sqrt {b^2-4 a c}}\right )\right )}{\sqrt {\frac {c}{b+\sqrt {b^2-4 a c}}}}}{32 d^3 e \left (c d^2+e (-b d+a e)\right )^2 \sqrt {a+b x^2+c x^4}} \] Input:

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

Output:

((-4*d*e^2*x*(a + b*x^2 + c*x^4)*(2*d*(B*d - A*e)*(c*d^2 + e*(-(b*d) + a*e 
)) + (-3*A*e*(3*c*d^2 + e*(-2*b*d + a*e)) + B*(5*c*d^3 - d*e*(2*b*d + a*e) 
))*(d + e*x^2)))/(d + e*x^2)^2 - (I*Sqrt[2]*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 
])]*((-b + Sqrt[b^2 - 4*a*c])*d*e*(3*A*e*(3*c*d^2 + e*(-2*b*d + a*e)) + B* 
(-5*c*d^3 + d*e*(2*b*d + a*e)))*EllipticE[I*ArcSinh[Sqrt[2]*Sqrt[c/(b + Sq 
rt[b^2 - 4*a*c])]*x], (b + Sqrt[b^2 - 4*a*c])/(b - Sqrt[b^2 - 4*a*c])] + d 
*(B*d*(6*c^2*d^3 + c*d*e*(-5*b*d + 5*Sqrt[b^2 - 4*a*c]*d - 6*a*e) - (-b + 
Sqrt[b^2 - 4*a*c])*e^2*(2*b*d + a*e)) - A*e*(14*c^2*d^3 - 3*(-b + Sqrt[b^2 
 - 4*a*c])*e^2*(2*b*d - a*e) + c*d*e*(-17*b*d + 9*Sqrt[b^2 - 4*a*c]*d + 2* 
a*e)))*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*(B*(-3*c^2*d^5 + 10*a*c* 
d^3*e^2 + a*d*e^3*(-4*b*d + a*e)) + A*e*(15*c^2*d^4 + 2*c*d^2*e*(-10*b*d + 
 3*a*e) + e^2*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2)))*EllipticPi[((b + Sqrt[ 
b^2 - 4*a*c])*e)/(2*c*d), 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[c/(b + Sqrt[ 
b^2 - 4*a*c])])/(32*d^3*e*(c*d^2 + e*(-(b*d) + a*e))^2*Sqrt[a + b*x^2 + c* 
x^4])
 

Rubi [A] (verified)

Time = 3.59 (sec) , antiderivative size = 985, normalized size of antiderivative = 0.88, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2210, 25, 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.

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

\(\Big \downarrow \) 2210

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 2210

\(\displaystyle \frac {\frac {e x \sqrt {a+b x^2+c x^4} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (a e+2 b d)\right )\right )}{2 d \left (d+e x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac {\int -\frac {-c e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) x^4-2 c d \left (A e \left (8 c d^2-e (5 b d-2 a e)\right )-B \left (4 c d^3-d e (b d+2 a e)\right )\right ) x^2+a d e (B d-A e) (5 c d-2 b e)+\left (4 A c d^2+a B e d-A e (4 b d-3 a e)\right ) \left (2 c d^2-e (2 b d-a e)\right )}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx}{2 d \left (a e^2-b d e+c d^2\right )}}{4 d \left (a e^2-b d e+c d^2\right )}-\frac {e x \sqrt {a+b x^2+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {-c e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) x^4-2 c d \left (A e \left (8 c d^2-e (5 b d-2 a e)\right )-B \left (4 c d^3-d e (b d+2 a e)\right )\right ) x^2+a d e (B d-A e) (5 c d-2 b e)+\left (4 A c d^2+a B e d-A e (4 b d-3 a e)\right ) \left (2 c d^2-e (2 b d-a e)\right )}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx}{2 d \left (a e^2-b d e+c d^2\right )}+\frac {e x \sqrt {a+b x^2+c x^4} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (a e+2 b d)\right )\right )}{2 d \left (d+e x^2\right ) \left (a e^2-b d e+c d^2\right )}}{4 d \left (a e^2-b d e+c d^2\right )}-\frac {e x \sqrt {a+b x^2+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 2232

\(\displaystyle \frac {\frac {\sqrt {a} \sqrt {c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (a e+2 b d)\right )\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx+\frac {\int \frac {c e \left (\left (\left (c d-\sqrt {a} \sqrt {c} e\right ) \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right )-2 c d \left (A e \left (8 c d^2-e (5 b d-2 a e)\right )-B \left (4 c d^3-d e (b d+2 a e)\right )\right )\right ) x^2+a d e (B d-A e) (5 c d-2 b e)+\left (4 A c d^2+a B e d-A e (4 b d-3 a e)\right ) \left (2 c d^2-e (2 b d-a e)\right )-\sqrt {a} \sqrt {c} d \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right )\right )}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx}{c e}}{2 d \left (a e^2-b d e+c d^2\right )}+\frac {e x \sqrt {a+b x^2+c x^4} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (a e+2 b d)\right )\right )}{2 d \left (d+e x^2\right ) \left (a e^2-b d e+c d^2\right )}}{4 d \left (a e^2-b d e+c d^2\right )}-\frac {e x \sqrt {a+b x^2+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\sqrt {c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (a e+2 b d)\right )\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx+\int \frac {\left (\left (c d-\sqrt {a} \sqrt {c} e\right ) \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right )-2 c d \left (A e \left (8 c d^2-e (5 b d-2 a e)\right )-B \left (4 c d^3-d e (b d+2 a e)\right )\right )\right ) x^2+a d e (B d-A e) (5 c d-2 b e)+\left (4 A c d^2+a B e d-A e (4 b d-3 a e)\right ) \left (2 c d^2-e (2 b d-a e)\right )-\sqrt {a} \sqrt {c} d \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right )}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx}{2 d \left (a e^2-b d e+c d^2\right )}+\frac {e x \sqrt {a+b x^2+c x^4} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (a e+2 b d)\right )\right )}{2 d \left (d+e x^2\right ) \left (a e^2-b d e+c d^2\right )}}{4 d \left (a e^2-b d e+c d^2\right )}-\frac {e x \sqrt {a+b x^2+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {\frac {\int \frac {\left (\left (c d-\sqrt {a} \sqrt {c} e\right ) \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right )-2 c d \left (A e \left (8 c d^2-e (5 b d-2 a e)\right )-B \left (4 c d^3-d e (b d+2 a e)\right )\right )\right ) x^2+a d e (B d-A e) (5 c d-2 b e)+\left (4 A c d^2+a B e d-A e (4 b d-3 a e)\right ) \left (2 c d^2-e (2 b d-a e)\right )-\sqrt {a} \sqrt {c} d \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right )}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx+\sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (a e+2 b d)\right )\right )}{2 d \left (a e^2-b d e+c d^2\right )}+\frac {e x \sqrt {a+b x^2+c x^4} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (a e+2 b d)\right )\right )}{2 d \left (d+e x^2\right ) \left (a e^2-b d e+c d^2\right )}}{4 d \left (a e^2-b d e+c d^2\right )}-\frac {e x \sqrt {a+b x^2+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 2226

\(\displaystyle \frac {\frac {\frac {\sqrt {a} \left (B \left (a d e^3 (4 b d-a e)-10 a c d^3 e^2+3 c^2 d^5\right )-A e \left (e^2 \left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right )-2 c d^2 e (10 b d-3 a e)+15 c^2 d^4\right )\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\sqrt {a} \left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c} d-\sqrt {a} e}+\frac {2 \sqrt {c} \left (a e^2-b d e+c d^2\right ) \left (\sqrt {a} \sqrt {c} d (B d-A e)+a e (3 A e+B d)+4 A d (c d-b e)\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c} d-\sqrt {a} e}+\sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (a e+2 b d)\right )\right )}{2 d \left (a e^2-b d e+c d^2\right )}+\frac {e x \sqrt {a+b x^2+c x^4} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (a e+2 b d)\right )\right )}{2 d \left (d+e x^2\right ) \left (a e^2-b d e+c d^2\right )}}{4 d \left (a e^2-b d e+c d^2\right )}-\frac {e x \sqrt {a+b x^2+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\left (B \left (a d e^3 (4 b d-a e)-10 a c d^3 e^2+3 c^2 d^5\right )-A e \left (e^2 \left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right )-2 c d^2 e (10 b d-3 a e)+15 c^2 d^4\right )\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c} d-\sqrt {a} e}+\frac {2 \sqrt {c} \left (a e^2-b d e+c d^2\right ) \left (\sqrt {a} \sqrt {c} d (B d-A e)+a e (3 A e+B d)+4 A d (c d-b e)\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c} d-\sqrt {a} e}+\sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (a e+2 b d)\right )\right )}{2 d \left (a e^2-b d e+c d^2\right )}+\frac {e x \sqrt {a+b x^2+c x^4} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (a e+2 b d)\right )\right )}{2 d \left (d+e x^2\right ) \left (a e^2-b d e+c d^2\right )}}{4 d \left (a e^2-b d e+c d^2\right )}-\frac {e x \sqrt {a+b x^2+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\frac {\frac {\left (B \left (a d e^3 (4 b d-a e)-10 a c d^3 e^2+3 c^2 d^5\right )-A e \left (e^2 \left (3 a^2 e^2-8 a b d e+8 b^2 d^2\right )-2 c d^2 e (10 b d-3 a e)+15 c^2 d^4\right )\right ) \int \frac {\sqrt {c} x^2+\sqrt {a}}{\left (e x^2+d\right ) \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c} d-\sqrt {a} e}+\frac {\sqrt [4]{c} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \left (a e^2-b d e+c d^2\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\sqrt {a} \sqrt {c} d (B d-A e)+a e (3 A e+B d)+4 A d (c d-b e)\right )}{\sqrt [4]{a} \sqrt {a+b x^2+c x^4} \left (\sqrt {c} d-\sqrt {a} e\right )}+\sqrt {c} \left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (a e+2 b d)\right )\right )}{2 d \left (a e^2-b d e+c d^2\right )}+\frac {e x \sqrt {a+b x^2+c x^4} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (a e+2 b d)\right )\right )}{2 d \left (d+e x^2\right ) \left (a e^2-b d e+c d^2\right )}}{4 d \left (a e^2-b d e+c d^2\right )}-\frac {e x \sqrt {a+b x^2+c x^4} (B d-A e)}{4 d \left (d+e x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\)

\(\Big \downarrow \) 2220

\(\displaystyle \frac {\frac {e \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) \sqrt {c x^4+b x^2+a} x}{2 d \left (c d^2-b e d+a e^2\right ) \left (e x^2+d\right )}+\frac {\sqrt {c} \left (3 A e \left (3 c d^2-e (2 b d-a e)\right )-B \left (5 c d^3-d e (2 b d+a e)\right )\right ) \left (\frac {\sqrt [4]{a} \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 )}{\sqrt [4]{c} \sqrt {c x^4+b x^2+a}}-\frac {x \sqrt {c x^4+b x^2+a}}{\sqrt {c} x^2+\sqrt {a}}\right )+\frac {\sqrt [4]{c} \left (c d^2-b e d+a e^2\right ) \left (\sqrt {a} \sqrt {c} d (B d-A e)+a e (B d+3 A e)+4 A d (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 )}{\sqrt [4]{a} \left (\sqrt {c} d-\sqrt {a} e\right ) \sqrt {c x^4+b x^2+a}}+\frac {\left (B \left (3 c^2 d^5-10 a c e^2 d^3+a e^3 (4 b d-a e) d\right )-A e \left (15 c^2 d^4-2 c e (10 b d-3 a e) d^2+e^2 \left (8 b^2 d^2-8 a b e d+3 a^2 e^2\right )\right )\right ) \left (\frac {\left (\sqrt {c} d+\sqrt {a} 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 {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 {c x^4+b x^2+a}}-\frac {\left (\sqrt {c} d-\sqrt {a} e\right ) \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} \sqrt {e} \sqrt {c d^2-b e d+a e^2}}\right )}{\sqrt {c} d-\sqrt {a} e}}{2 d \left (c d^2-b e d+a e^2\right )}}{4 d \left (c d^2-b e d+a e^2\right )}-\frac {e (B d-A e) x \sqrt {c x^4+b x^2+a}}{4 d \left (c d^2-b e d+a e^2\right ) \left (e x^2+d\right )^2}\)

Input:

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

Output:

-1/4*(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)^2) + ((e*(3*A*e*(3*c*d^2 - e*(2*b*d - a*e)) - B*(5*c*d^3 - d*e 
*(2*b*d + a*e)))*x*Sqrt[a + b*x^2 + c*x^4])/(2*d*(c*d^2 - b*d*e + a*e^2)*( 
d + e*x^2)) + (Sqrt[c]*(3*A*e*(3*c*d^2 - e*(2*b*d - a*e)) - B*(5*c*d^3 - d 
*e*(2*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] + Sq 
rt[c]*x^2)^2]*EllipticE[2*ArcTan[(c^(1/4)*x)/a^(1/4)], (2 - b/(Sqrt[a]*Sqr 
t[c]))/4])/(c^(1/4)*Sqrt[a + b*x^2 + c*x^4])) + (c^(1/4)*(c*d^2 - b*d*e + 
a*e^2)*(Sqrt[a]*Sqrt[c]*d*(B*d - A*e) + a*e*(B*d + 3*A*e) + 4*A*d*(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*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*(3*c^2 
*d^5 - 10*a*c*d^3*e^2 + a*d*e^3*(4*b*d - a*e)) - A*e*(15*c^2*d^4 - 2*c*d^2 
*e*(10*b*d - 3*a*e) + e^2*(8*b^2*d^2 - 8*a*b*d*e + 3*a^2*e^2)))*(-1/2*((Sq 
rt[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/(Sqrt[a]*S 
qrt[c]))/4])/(4*a^(1/4)*c^(1/4)*d*e*Sqrt[a + b*x^2 + c*x^4])))/(Sqrt[c]...
 

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. \(4475\) vs. \(2(989)=1978\).

Time = 4.90 (sec) , antiderivative size = 4476, normalized size of antiderivative = 3.98

method	result	size
default	\(\text {Expression too large to display}\)	\(4476\)
elliptic	\(\text {Expression too large to display}\)	\(5671\)

Input:

int((B*x^2+A)/(e*x^2+d)^3/(c*x^4+b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

B/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))*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))+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))^(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-1/(a*e^2-b*d*e+c*d^2)/d*e*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...
 

Fricas [F(-1)]

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

integrate((B*x^2+A)/(e*x^2+d)^3/(c*x^4+b*x^2+a)^(1/2),x, algorithm="fricas 
")
 

Output:

Timed out
 

Sympy [F]

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

integrate((B*x**2+A)/(e*x**2+d)**3/(c*x**4+b*x**2+a)**(1/2),x)
 

Output:

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

Maxima [F]

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

integrate((B*x^2+A)/(e*x^2+d)^3/(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)^3), x)
 

Giac [F]

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

integrate((B*x^2+A)/(e*x^2+d)^3/(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)^3), x)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int(sqrt(a + b*x**2 + c*x**4)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d*e**2*x**4 
+ a*e**3*x**6 + b*d**3*x**2 + 3*b*d**2*e*x**4 + 3*b*d*e**2*x**6 + b*e**3*x 
**8 + c*d**3*x**4 + 3*c*d**2*e*x**6 + 3*c*d*e**2*x**8 + c*e**3*x**10),x)*a 
 + int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a*d**3 + 3*a*d**2*e*x**2 + 3*a*d* 
e**2*x**4 + a*e**3*x**6 + b*d**3*x**2 + 3*b*d**2*e*x**4 + 3*b*d*e**2*x**6 
+ b*e**3*x**8 + c*d**3*x**4 + 3*c*d**2*e*x**6 + 3*c*d*e**2*x**8 + c*e**3*x 
**10),x)*b