\(\int \frac {(A+B x^2) (d+e x^2)^3}{(a+b x^2+c x^4)^{3/2}} \, dx\) [174]

Optimal result

Integrand size = 33, antiderivative size = 859 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\frac {x \left (A c \left (b^2 c d^3-2 a c d \left (c d^2-3 a e^2\right )-a b e \left (3 c d^2+a e^2\right )\right )+a B \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )-\left (a B (2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )+A c \left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}+\frac {B e^3 x \sqrt {a+b x^2+c x^4}}{3 c^2}+\frac {\left (a B \left (6 c^3 d^3-8 b^3 e^3-9 c^2 d e (b d+6 a e)+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c e \left (c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) x \sqrt {a+b x^2+c x^4}}{3 a c^{5/2} \left (b^2-4 a c\right ) \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\left (a B \left (6 c^3 d^3-8 b^3 e^3-9 c^2 d e (b d+6 a e)+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c e \left (c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{3 a^{3/4} c^{11/4} \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\left (3 A c^3 d^3-5 a^2 B c e^3-3 \sqrt {a} c^{5/2} d^2 (B d+3 A e)+a e (3 c d-2 b e) (3 B c d-4 b B e+3 A c e)+3 a^{3/2} \sqrt {c} e^2 (9 B c d-4 b B e+3 A c e)\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{6 a^{3/4} \left (b-2 \sqrt {a} \sqrt {c}\right ) c^{11/4} \sqrt {a+b x^2+c x^4}} \] Output:

x*(A*c*(b^2*c*d^3-2*a*c*d*(-3*a*e^2+c*d^2)-a*b*e*(a*e^2+3*c*d^2))+a*B*(a*b 
^2*e^3+2*a*c*e*(-a*e^2+3*c*d^2)-b*c*d*(3*a*e^2+c*d^2))-(a*B*(-b*e+2*c*d)*( 
c^2*d^2+b^2*e^2-c*e*(3*a*e+b*d))+A*c*(a*b^2*e^3+2*a*c*e*(-a*e^2+3*c*d^2)-b 
*c*d*(3*a*e^2+c*d^2)))*x^2)/a/c^2/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(1/2)+1/3*B 
*e^3*x*(c*x^4+b*x^2+a)^(1/2)/c^2+1/3*(a*B*(6*c^3*d^3-8*b^3*e^3-9*c^2*d*e*( 
6*a*e+b*d)+b*c*e^2*(29*a*e+18*b*d))+3*A*c*(2*a*b^2*e^3+6*a*c*e*(-a*e^2+c*d 
^2)-b*c*d*(3*a*e^2+c*d^2)))*x*(c*x^4+b*x^2+a)^(1/2)/a/c^(5/2)/(-4*a*c+b^2) 
/(a^(1/2)+c^(1/2)*x^2)-1/3*(a*B*(6*c^3*d^3-8*b^3*e^3-9*c^2*d*e*(6*a*e+b*d) 
+b*c*e^2*(29*a*e+18*b*d))+3*A*c*(2*a*b^2*e^3+6*a*c*e*(-a*e^2+c*d^2)-b*c*d* 
(3*a*e^2+c*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)*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^(11/4)/(-4*a*c+b^2)/(c*x^4+b*x^2+a)^(1/2)-1/6*(3 
*A*c^3*d^3-5*a^2*B*c*e^3-3*a^(1/2)*c^(5/2)*d^2*(3*A*e+B*d)+a*e*(-2*b*e+3*c 
*d)*(3*A*c*e-4*B*b*e+3*B*c*d)+3*a^(3/2)*c^(1/2)*e^2*(3*A*c*e-4*B*b*e+9*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^(11/4)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 16.63 (sec) , antiderivative size = 5432, normalized size of antiderivative = 6.32 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Result too large to show
 

Rubi [A] (verified)

Time = 1.76 (sec) , antiderivative size = 765, normalized size of antiderivative = 0.89, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {2206, 2207, 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.

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

\(\Big \downarrow \) 2206

\(\displaystyle \frac {x \left (-\left (x^2 \left (A c \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )+a B (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right )\right )+A c \left (-a b e \left (a e^2+3 c d^2\right )-2 a c d \left (c d^2-3 a e^2\right )+b^2 c d^3\right )+a B \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\int \frac {a B \left (4 a-\frac {b^2}{c}\right ) e^3 x^4-\frac {\left (a B \left (2 c^3 d^3-3 c^2 e (b d+6 a e) d-2 b^3 e^3+b c e^2 (6 b d+7 a e)\right )+A c \left (2 a b^2 e^3+6 a c \left (c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2}{c^2}+\frac {a \left (a b^2 B e^3-b c \left (B c d^3+3 A c e d^2+3 a B e^2 d+a A e^3\right )+2 c \left (a B e \left (3 c d^2-a e^2\right )+A c d \left (c d^2+3 a e^2\right )\right )\right )}{c^2}}{\sqrt {c x^4+b x^2+a}}dx}{a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 2207

\(\displaystyle \frac {x \left (-\left (x^2 \left (A c \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )+a B (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right )\right )+A c \left (-a b e \left (a e^2+3 c d^2\right )-2 a c d \left (c d^2-3 a e^2\right )+b^2 c d^3\right )+a B \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\frac {\int \frac {a \left (4 a b^2 B e^3-3 b c \left (B c d^3+3 A c e d^2+3 a B e^2 d+a A e^3\right )+2 c \left (a B e \left (9 c d^2-5 a e^2\right )+3 A c d \left (c d^2+3 a e^2\right )\right )\right )-\left (a B \left (6 c^3 d^3-9 c^2 e (b d+6 a e) d-8 b^3 e^3+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c \left (c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2}{c \sqrt {c x^4+b x^2+a}}dx}{3 c}-\frac {a B e^3 x \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}{3 c^2}}{a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \left (-\left (x^2 \left (A c \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )+a B (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right )\right )+A c \left (-a b e \left (a e^2+3 c d^2\right )-2 a c d \left (c d^2-3 a e^2\right )+b^2 c d^3\right )+a B \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\frac {\int \frac {a \left (4 a b^2 B e^3-3 b c \left (B c d^3+3 A c e d^2+3 a B e^2 d+a A e^3\right )+2 c \left (a B e \left (9 c d^2-5 a e^2\right )+3 A c d \left (c d^2+3 a e^2\right )\right )\right )-\left (a B \left (6 c^3 d^3-9 c^2 e (b d+6 a e) d-8 b^3 e^3+b c e^2 (18 b d+29 a e)\right )+3 A c \left (2 a b^2 e^3+6 a c \left (c d^2-a e^2\right ) e-b c d \left (c d^2+3 a e^2\right )\right )\right ) x^2}{\sqrt {c x^4+b x^2+a}}dx}{3 c^2}-\frac {a B e^3 x \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}{3 c^2}}{a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {x \left (-\left (x^2 \left (A c \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )+a B (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right )\right )+A c \left (-a b e \left (a e^2+3 c d^2\right )-2 a c d \left (c d^2-3 a e^2\right )+b^2 c d^3\right )+a B \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\frac {\frac {\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (3 a^{3/2} \sqrt {c} e^2 (3 A c e-4 b B e+9 B c d)-5 a^2 B c e^3+a e (3 c d-2 b e) (3 A c e-4 b B e+3 B c d)-3 \sqrt {a} c^{5/2} d^2 (3 A e+B d)+3 A c^3 d^3\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}+\frac {\sqrt {a} \left (3 A c \left (2 a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+6 a c e \left (c d^2-a e^2\right )\right )+a B \left (-9 c^2 d e (6 a e+b d)+b c e^2 (29 a e+18 b d)-8 b^3 e^3+6 c^3 d^3\right )\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{3 c^2}-\frac {a B e^3 x \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}{3 c^2}}{a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {x \left (-\left (x^2 \left (A c \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )+a B (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right )\right )+A c \left (-a b e \left (a e^2+3 c d^2\right )-2 a c d \left (c d^2-3 a e^2\right )+b^2 c d^3\right )+a B \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\frac {\frac {\sqrt {a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (3 a^{3/2} \sqrt {c} e^2 (3 A c e-4 b B e+9 B c d)-5 a^2 B c e^3+a e (3 c d-2 b e) (3 A c e-4 b B e+3 B c d)-3 \sqrt {a} c^{5/2} d^2 (3 A e+B d)+3 A c^3 d^3\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}+\frac {\left (3 A c \left (2 a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+6 a c e \left (c d^2-a e^2\right )\right )+a B \left (-9 c^2 d e (6 a e+b d)+b c e^2 (29 a e+18 b d)-8 b^3 e^3+6 c^3 d^3\right )\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{3 c^2}-\frac {a B e^3 x \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}{3 c^2}}{a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {x \left (-\left (x^2 \left (A c \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )+a B (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right )\right )+A c \left (-a b e \left (a e^2+3 c d^2\right )-2 a c d \left (c d^2-3 a e^2\right )+b^2 c d^3\right )+a B \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\frac {\frac {\left (3 A c \left (2 a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+6 a c e \left (c d^2-a e^2\right )\right )+a B \left (-9 c^2 d e (6 a e+b d)+b c e^2 (29 a e+18 b d)-8 b^3 e^3+6 c^3 d^3\right )\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}+\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (3 a^{3/2} \sqrt {c} e^2 (3 A c e-4 b B e+9 B c d)-5 a^2 B c e^3+a e (3 c d-2 b e) (3 A c e-4 b B e+3 B c d)-3 \sqrt {a} c^{5/2} d^2 (3 A e+B d)+3 A c^3 d^3\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}}{3 c^2}-\frac {a B e^3 x \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}{3 c^2}}{a \left (b^2-4 a c\right )}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {x \left (-\left (x^2 \left (A c \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )+a B (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )\right )\right )+A c \left (-a b e \left (a e^2+3 c d^2\right )-2 a c d \left (c d^2-3 a e^2\right )+b^2 c d^3\right )+a B \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{a c^2 \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}-\frac {\frac {\frac {\sqrt [4]{a} \left (2 \sqrt {a} \sqrt {c}+b\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (3 a^{3/2} \sqrt {c} e^2 (3 A c e-4 b B e+9 B c d)-5 a^2 B c e^3+a e (3 c d-2 b e) (3 A c e-4 b B e+3 B c d)-3 \sqrt {a} c^{5/2} d^2 (3 A e+B d)+3 A c^3 d^3\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}+\frac {\left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (3 A c \left (2 a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+6 a c e \left (c d^2-a e^2\right )\right )+a B \left (-9 c^2 d e (6 a e+b d)+b c e^2 (29 a e+18 b d)-8 b^3 e^3+6 c^3 d^3\right )\right )}{\sqrt {c}}}{3 c^2}-\frac {a B e^3 x \left (b^2-4 a c\right ) \sqrt {a+b x^2+c x^4}}{3 c^2}}{a \left (b^2-4 a c\right )}\)

Input:

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

Output:

(x*(A*c*(b^2*c*d^3 - 2*a*c*d*(c*d^2 - 3*a*e^2) - a*b*e*(3*c*d^2 + a*e^2)) 
+ a*B*(a*b^2*e^3 + 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2)) - 
(a*B*(2*c*d - b*e)*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*e)) + A*c*(a*b^2*e^ 
3 + 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2)))*x^2))/(a*c^2*(b^ 
2 - 4*a*c)*Sqrt[a + b*x^2 + c*x^4]) - (-1/3*(a*B*(b^2 - 4*a*c)*e^3*x*Sqrt[ 
a + b*x^2 + c*x^4])/c^2 + (((a*B*(6*c^3*d^3 - 8*b^3*e^3 - 9*c^2*d*e*(b*d + 
 6*a*e) + b*c*e^2*(18*b*d + 29*a*e)) + 3*A*c*(2*a*b^2*e^3 + 6*a*c*e*(c*d^2 
 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2)))*(-((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])*(3*A*c^3*d^3 - 5*a^2*B*c*e^3 - 3*Sqrt[a 
]*c^(5/2)*d^2*(B*d + 3*A*e) + a*e*(3*c*d - 2*b*e)*(3*B*c*d - 4*b*B*e + 3*A 
*c*e) + 3*a^(3/2)*Sqrt[c]*e^2*(9*B*c*d - 4*b*B*e + 3*A*c*e))*(Sqrt[a] + Sq 
rt[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)*Sqr 
t[a + b*x^2 + c*x^4]))/(3*c^2))/(a*(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]
 

rule 2207

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

Maple [A] (verified)

Time = 12.68 (sec) , antiderivative size = 1141, normalized size of antiderivative = 1.33

method	result	size
elliptic	\(\text {Expression too large to display}\)	\(1141\)
default	\(\text {Expression too large to display}\)	\(2436\)
risch	\(\text {Expression too large to display}\)	\(2472\)

Input:

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

Output:

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2623 vs. \(2 (780) = 1560\).

Time = 0.12 (sec) , antiderivative size = 2623, normalized size of antiderivative = 3.05 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^{3/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/6*(sqrt(1/2)*((3*(2*B*a*b - A*b^2)*c^4*d^3 - 9*(B*a*b^2*c^3 - 2*A*a*b*c 
^4)*d^2*e + 9*(2*B*a*b^3*c^2 - (6*B*a^2*b + A*a*b^2)*c^3)*d*e^2 - (8*B*a*b 
^4*c + 18*A*a^2*b*c^3 - (29*B*a^2*b^2 + 6*A*a*b^3)*c^2)*e^3)*x^5 + (3*(2*B 
*a*b^2 - A*b^3)*c^3*d^3 - 9*(B*a*b^3*c^2 - 2*A*a*b^2*c^3)*d^2*e + 9*(2*B*a 
*b^4*c - (6*B*a^2*b^2 + A*a*b^3)*c^2)*d*e^2 - (8*B*a*b^5 + 18*A*a^2*b^2*c^ 
2 - (29*B*a^2*b^3 + 6*A*a*b^4)*c)*e^3)*x^3 + (3*(2*B*a^2*b - A*a*b^2)*c^3* 
d^3 - 9*(B*a^2*b^2*c^2 - 2*A*a^2*b*c^3)*d^2*e + 9*(2*B*a^2*b^3*c - (6*B*a^ 
3*b + A*a^2*b^2)*c^2)*d*e^2 - (8*B*a^2*b^4 + 18*A*a^3*b*c^2 - (29*B*a^3*b^ 
2 + 6*A*a^2*b^3)*c)*e^3)*x - ((3*(2*B*a - A*b)*c^5*d^3 - 9*(B*a*b*c^4 - 2* 
A*a*c^5)*d^2*e + 9*(2*B*a*b^2*c^3 - (6*B*a^2 + A*a*b)*c^4)*d*e^2 - (8*B*a* 
b^3*c^2 + 18*A*a^2*c^4 - (29*B*a^2*b + 6*A*a*b^2)*c^3)*e^3)*x^5 + (3*(2*B* 
a*b - A*b^2)*c^4*d^3 - 9*(B*a*b^2*c^3 - 2*A*a*b*c^4)*d^2*e + 9*(2*B*a*b^3* 
c^2 - (6*B*a^2*b + A*a*b^2)*c^3)*d*e^2 - (8*B*a*b^4*c + 18*A*a^2*b*c^3 - ( 
29*B*a^2*b^2 + 6*A*a*b^3)*c^2)*e^3)*x^3 + (3*(2*B*a^2 - A*a*b)*c^4*d^3 - 9 
*(B*a^2*b*c^3 - 2*A*a^2*c^4)*d^2*e + 9*(2*B*a^2*b^2*c^2 - (6*B*a^3 + A*a^2 
*b)*c^3)*d*e^2 - (8*B*a^2*b^3*c + 18*A*a^3*c^3 - (29*B*a^3*b + 6*A*a^2*b^2 
)*c^2)*e^3)*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)*((3*(2*A*b*c^5 - (2*B*a*b - (A - B)*b^2)*c^4)*d^3 + 9*(B*a*b^2*c^3...
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(11*sqrt(a + b*x**2 + c*x**4)*a*b*c*e**3*x - 9*sqrt(a + b*x**2 + c*x**4)*a 
*c**2*d*e**2*x + 3*sqrt(a + b*x**2 + c*x**4)*a*c**2*e**3*x**3 - 8*sqrt(a + 
 b*x**2 + c*x**4)*b**3*e**3*x + 18*sqrt(a + b*x**2 + c*x**4)*b**2*c*d*e**2 
*x - 4*sqrt(a + b*x**2 + c*x**4)*b**2*c*e**3*x**3 - 9*sqrt(a + b*x**2 + c* 
x**4)*b*c**2*d**2*e*x + 9*sqrt(a + b*x**2 + c*x**4)*b*c**2*d*e**2*x**3 + s 
qrt(a + b*x**2 + c*x**4)*b*c**2*e**3*x**5 - 11*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*b*c*e**3 + 9*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**2*d*e**2 + 8*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**3*e**3 - 18*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*c*d*e**2 - 11*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*c*e**3*x**2 
 + 9*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**2*d**2*e + 9*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**2*d*e**2*x**2 - 11*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**2*e**3*x**4 + 3*int(sqrt(a + b*x**2 + c*x**4)/(a**2 + 2*a*b*x**2 ...