\(\int \frac {(A+B x^2) (d+e x^2)^3}{\sqrt {a+b x^2+c x^4}} \, dx\) [167]

Optimal result

Integrand size = 33, antiderivative size = 755 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\sqrt {a+b x^2+c x^4}} \, dx=\frac {e \left (7 A c e (15 c d-4 b e)+B \left (105 c^2 d^2+24 b^2 e^2-c e (84 b d+25 a e)\right )\right ) x \sqrt {a+b x^2+c x^4}}{105 c^3}+\frac {e^2 (21 B c d-6 b B e+7 A c e) x^3 \sqrt {a+b x^2+c x^4}}{35 c^2}+\frac {B e^3 x^5 \sqrt {a+b x^2+c x^4}}{7 c}+\frac {\left (7 A c e \left (45 c^2 d^2+8 b^2 e^2-3 c e (10 b d+3 a e)\right )+B \left (105 c^3 d^3-48 b^3 e^3-21 c^2 d e (10 b d+9 a e)+8 b c e^2 (21 b d+13 a e)\right )\right ) x \sqrt {a+b x^2+c x^4}}{105 c^{7/2} \left (\sqrt {a}+\sqrt {c} x^2\right )}-\frac {\sqrt [4]{a} \left (7 A c e \left (45 c^2 d^2+8 b^2 e^2-3 c e (10 b d+3 a e)\right )+B \left (105 c^3 d^3-48 b^3 e^3-21 c^2 d e (10 b d+9 a e)+8 b c e^2 (21 b d+13 a e)\right )\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{105 c^{15/4} \sqrt {a+b x^2+c x^4}}+\frac {\left (7 A c \left (15 c^2 d^3-15 a c d e^2+4 a b e^3\right )-a B e \left (105 c^2 d^2+24 b^2 e^2-c e (84 b d+25 a e)\right )+\frac {\sqrt {a} \left (7 A c e \left (45 c^2 d^2+8 b^2 e^2-3 c e (10 b d+3 a e)\right )+B \left (105 c^3 d^3-48 b^3 e^3-21 c^2 d e (10 b d+9 a e)+8 b c e^2 (21 b d+13 a e)\right )\right )}{\sqrt {c}}\right ) \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{210 \sqrt [4]{a} c^{13/4} \sqrt {a+b x^2+c x^4}} \] Output:

1/105*e*(7*A*c*e*(-4*b*e+15*c*d)+B*(105*c^2*d^2+24*b^2*e^2-c*e*(25*a*e+84* 
b*d)))*x*(c*x^4+b*x^2+a)^(1/2)/c^3+1/35*e^2*(7*A*c*e-6*B*b*e+21*B*c*d)*x^3 
*(c*x^4+b*x^2+a)^(1/2)/c^2+1/7*B*e^3*x^5*(c*x^4+b*x^2+a)^(1/2)/c+1/105*(7* 
A*c*e*(45*c^2*d^2+8*b^2*e^2-3*c*e*(3*a*e+10*b*d))+B*(105*c^3*d^3-48*b^3*e^ 
3-21*c^2*d*e*(9*a*e+10*b*d)+8*b*c*e^2*(13*a*e+21*b*d)))*x*(c*x^4+b*x^2+a)^ 
(1/2)/c^(7/2)/(a^(1/2)+c^(1/2)*x^2)-1/105*a^(1/4)*(7*A*c*e*(45*c^2*d^2+8*b 
^2*e^2-3*c*e*(3*a*e+10*b*d))+B*(105*c^3*d^3-48*b^3*e^3-21*c^2*d*e*(9*a*e+1 
0*b*d)+8*b*c*e^2*(13*a*e+21*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))/c^(15/4)/(c*x^4+b*x^2+a)^(1/2)+1/210*(7*A 
*c*(4*a*b*e^3-15*a*c*d*e^2+15*c^2*d^3)-a*B*e*(105*c^2*d^2+24*b^2*e^2-c*e*( 
25*a*e+84*b*d))+a^(1/2)*(7*A*c*e*(45*c^2*d^2+8*b^2*e^2-3*c*e*(3*a*e+10*b*d 
))+B*(105*c^3*d^3-48*b^3*e^3-21*c^2*d*e*(9*a*e+10*b*d)+8*b*c*e^2*(13*a*e+2 
1*b*d)))/c^(1/2))*(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)*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)/c^(13/4)/(c*x^4+b*x^2+a)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 16.53 (sec) , antiderivative size = 4473, normalized size of antiderivative = 5.92 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\sqrt {a+b x^2+c x^4}} \, dx=\text {Result too large to show} \] Input:

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

Output:

Sqrt[a + b*x^2 + c*x^4]*(-1/105*(e*(-105*B*c^2*d^2 + 84*b*B*c*d*e - 105*A* 
c^2*d*e - 24*b^2*B*e^2 + 28*A*b*c*e^2 + 25*a*B*c*e^2)*x)/c^3 + (e^2*(21*B* 
c*d - 6*b*B*e + 7*A*c*e)*x^3)/(35*c^2) + (B*e^3*x^5)/(7*c)) + ((((105*I)/2 
)*B*c^2*(-b + Sqrt[b^2 - 4*a*c])*d^3*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[b^2 - 4 
*a*c])]*Sqrt[1 - (2*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])] - 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])]))/ 
(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4]) - (( 
105*I)*b*B*c*(-b + Sqrt[b^2 - 4*a*c])*d^2*e*Sqrt[1 - (2*c*x^2)/(-b - Sqrt[ 
b^2 - 4*a*c])]*Sqrt[1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]*(EllipticE[I*A 
rcSinh[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])] - 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])]))/(Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*Sqrt[a + b*x^2 + c*x^4 
]) + (((315*I)/2)*A*c^2*(-b + Sqrt[b^2 - 4*a*c])*d^2*e*Sqrt[1 - (2*c*x^2)/ 
(-b - Sqrt[b^2 - 4*a*c])]*Sqrt[1 - (2*c*x^2)/(-b + Sqrt[b^2 - 4*a*c])]*(El 
lipticE[I*ArcSinh[Sqrt[2]*Sqrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*x], (-b - Sq 
rt[b^2 - 4*a*c])/(-b + Sqrt[b^2 - 4*a*c])] - EllipticF[I*ArcSinh[Sqrt[2]*S 
qrt[-(c/(-b - Sqrt[b^2 - 4*a*c]))]*x], (-b - Sqrt[b^2 - 4*a*c])/(-b + S...
 

Rubi [A] (verified)

Time = 1.61 (sec) , antiderivative size = 690, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {2207, 2207, 2207, 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.

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

\(\Big \downarrow \) 2207

\(\displaystyle \frac {\int \frac {e^2 (21 B c d-6 b B e+7 A c e) x^6+e \left (21 B c d^2+21 A c e d-5 a B e^2\right ) x^4+7 c d^2 (B d+3 A e) x^2+7 A c d^3}{\sqrt {c x^4+b x^2+a}}dx}{7 c}+\frac {B e^3 x^5 \sqrt {a+b x^2+c x^4}}{7 c}\)

\(\Big \downarrow \) 2207

\(\displaystyle \frac {\frac {\int \frac {e \left (7 A c e (15 c d-4 b e)+B \left (105 c^2 d^2+24 b^2 e^2-c e (84 b d+25 a e)\right )\right ) x^4+\left (21 A c e \left (5 c d^2-a e^2\right )+B \left (35 c^2 d^3-63 a c e^2 d+18 a b e^3\right )\right ) x^2+35 A c^2 d^3}{\sqrt {c x^4+b x^2+a}}dx}{5 c}+\frac {e^2 x^3 \sqrt {a+b x^2+c x^4} (7 A c e-6 b B e+21 B c d)}{5 c}}{7 c}+\frac {B e^3 x^5 \sqrt {a+b x^2+c x^4}}{7 c}\)

\(\Big \downarrow \) 2207

\(\displaystyle \frac {\frac {\frac {\int \frac {\left (7 A c e \left (45 c^2 d^2+8 b^2 e^2-3 c e (10 b d+3 a e)\right )+B \left (105 c^3 d^3-21 c^2 e (10 b d+9 a e) d-48 b^3 e^3+8 b c e^2 (21 b d+13 a e)\right )\right ) x^2+7 A c \left (15 c^2 d^3-15 a c e^2 d+4 a b e^3\right )-a B e \left (105 c^2 d^2+24 b^2 e^2-c e (84 b d+25 a e)\right )}{\sqrt {c x^4+b x^2+a}}dx}{3 c}+\frac {e x \sqrt {a+b x^2+c x^4} \left (B \left (-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right )+7 A c e (15 c d-4 b e)\right )}{3 c}}{5 c}+\frac {e^2 x^3 \sqrt {a+b x^2+c x^4} (7 A c e-6 b B e+21 B c d)}{5 c}}{7 c}+\frac {B e^3 x^5 \sqrt {a+b x^2+c x^4}}{7 c}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {\frac {\frac {\frac {\sqrt {a} \left (\frac {\sqrt {c} \left (7 A c \left (4 a b e^3-15 a c d e^2+15 c^2 d^3\right )-a B e \left (-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right )\right )}{\sqrt {a}}+7 A c e \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )+B \left (-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right )\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\sqrt {a} \left (7 A c e \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )+B \left (-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right )\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {a} \sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{3 c}+\frac {e x \sqrt {a+b x^2+c x^4} \left (B \left (-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right )+7 A c e (15 c d-4 b e)\right )}{3 c}}{5 c}+\frac {e^2 x^3 \sqrt {a+b x^2+c x^4} (7 A c e-6 b B e+21 B c d)}{5 c}}{7 c}+\frac {B e^3 x^5 \sqrt {a+b x^2+c x^4}}{7 c}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\frac {\frac {\sqrt {a} \left (\frac {\sqrt {c} \left (7 A c \left (4 a b e^3-15 a c d e^2+15 c^2 d^3\right )-a B e \left (-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right )\right )}{\sqrt {a}}+7 A c e \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )+B \left (-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right )\right ) \int \frac {1}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}-\frac {\left (7 A c e \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )+B \left (-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right )\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{3 c}+\frac {e x \sqrt {a+b x^2+c x^4} \left (B \left (-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right )+7 A c e (15 c d-4 b e)\right )}{3 c}}{5 c}+\frac {e^2 x^3 \sqrt {a+b x^2+c x^4} (7 A c e-6 b B e+21 B c d)}{5 c}}{7 c}+\frac {B e^3 x^5 \sqrt {a+b x^2+c x^4}}{7 c}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {\frac {\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\frac {\sqrt {c} \left (7 A c \left (4 a b e^3-15 a c d e^2+15 c^2 d^3\right )-a B e \left (-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right )\right )}{\sqrt {a}}+7 A c e \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )+B \left (-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (7 A c e \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )+B \left (-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right )\right ) \int \frac {\sqrt {a}-\sqrt {c} x^2}{\sqrt {c x^4+b x^2+a}}dx}{\sqrt {c}}}{3 c}+\frac {e x \sqrt {a+b x^2+c x^4} \left (B \left (-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right )+7 A c e (15 c d-4 b e)\right )}{3 c}}{5 c}+\frac {e^2 x^3 \sqrt {a+b x^2+c x^4} (7 A c e-6 b B e+21 B c d)}{5 c}}{7 c}+\frac {B e^3 x^5 \sqrt {a+b x^2+c x^4}}{7 c}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {\frac {\frac {\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right ),\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right ) \left (\frac {\sqrt {c} \left (7 A c \left (4 a b e^3-15 a c d e^2+15 c^2 d^3\right )-a B e \left (-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right )\right )}{\sqrt {a}}+7 A c e \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )+B \left (-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right )\right )}{2 c^{3/4} \sqrt {a+b x^2+c x^4}}-\frac {\left (\frac {\sqrt [4]{a} \left (\sqrt {a}+\sqrt {c} x^2\right ) \sqrt {\frac {a+b x^2+c x^4}{\left (\sqrt {a}+\sqrt {c} x^2\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{c} x}{\sqrt [4]{a}}\right )|\frac {1}{4} \left (2-\frac {b}{\sqrt {a} \sqrt {c}}\right )\right )}{\sqrt [4]{c} \sqrt {a+b x^2+c x^4}}-\frac {x \sqrt {a+b x^2+c x^4}}{\sqrt {a}+\sqrt {c} x^2}\right ) \left (7 A c e \left (-3 c e (3 a e+10 b d)+8 b^2 e^2+45 c^2 d^2\right )+B \left (-21 c^2 d e (9 a e+10 b d)+8 b c e^2 (13 a e+21 b d)-48 b^3 e^3+105 c^3 d^3\right )\right )}{\sqrt {c}}}{3 c}+\frac {e x \sqrt {a+b x^2+c x^4} \left (B \left (-c e (25 a e+84 b d)+24 b^2 e^2+105 c^2 d^2\right )+7 A c e (15 c d-4 b e)\right )}{3 c}}{5 c}+\frac {e^2 x^3 \sqrt {a+b x^2+c x^4} (7 A c e-6 b B e+21 B c d)}{5 c}}{7 c}+\frac {B e^3 x^5 \sqrt {a+b x^2+c x^4}}{7 c}\)

Input:

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

Output:

(B*e^3*x^5*Sqrt[a + b*x^2 + c*x^4])/(7*c) + ((e^2*(21*B*c*d - 6*b*B*e + 7* 
A*c*e)*x^3*Sqrt[a + b*x^2 + c*x^4])/(5*c) + ((e*(7*A*c*e*(15*c*d - 4*b*e) 
+ B*(105*c^2*d^2 + 24*b^2*e^2 - c*e*(84*b*d + 25*a*e)))*x*Sqrt[a + b*x^2 + 
 c*x^4])/(3*c) + (-(((7*A*c*e*(45*c^2*d^2 + 8*b^2*e^2 - 3*c*e*(10*b*d + 3* 
a*e)) + B*(105*c^3*d^3 - 48*b^3*e^3 - 21*c^2*d*e*(10*b*d + 9*a*e) + 8*b*c* 
e^2*(21*b*d + 13*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)*(7 
*A*c*e*(45*c^2*d^2 + 8*b^2*e^2 - 3*c*e*(10*b*d + 3*a*e)) + B*(105*c^3*d^3 
- 48*b^3*e^3 - 21*c^2*d*e*(10*b*d + 9*a*e) + 8*b*c*e^2*(21*b*d + 13*a*e)) 
+ (Sqrt[c]*(7*A*c*(15*c^2*d^3 - 15*a*c*d*e^2 + 4*a*b*e^3) - a*B*e*(105*c^2 
*d^2 + 24*b^2*e^2 - c*e*(84*b*d + 25*a*e))))/Sqrt[a])*(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)*Sqrt[a + b 
*x^2 + c*x^4]))/(3*c))/(5*c))/(7*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 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 = 10.95 (sec) , antiderivative size = 673, normalized size of antiderivative = 0.89

method	result	size
elliptic	\(\frac {B \,e^{3} x^{5} \sqrt {c \,x^{4}+b \,x^{2}+a}}{7 c}+\frac {\left (A \,e^{3}+3 B d \,e^{2}-\frac {6 e^{3} B b}{7 c}\right ) x^{3} \sqrt {c \,x^{4}+b \,x^{2}+a}}{5 c}+\frac {\left (3 A d \,e^{2}+3 B e \,d^{2}-\frac {5 e^{3} B a}{7 c}-\frac {4 \left (A \,e^{3}+3 B d \,e^{2}-\frac {6 e^{3} B b}{7 c}\right ) b}{5 c}\right ) x \sqrt {c \,x^{4}+b \,x^{2}+a}}{3 c}+\frac {\left (A \,d^{3}-\frac {\left (3 A d \,e^{2}+3 B e \,d^{2}-\frac {5 e^{3} B a}{7 c}-\frac {4 \left (A \,e^{3}+3 B d \,e^{2}-\frac {6 e^{3} B b}{7 c}\right ) b}{5 c}\right ) a}{3 c}\right ) \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )}{4 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}}-\frac {\left (3 A \,d^{2} e +B \,d^{3}-\frac {3 \left (A \,e^{3}+3 B d \,e^{2}-\frac {6 e^{3} B b}{7 c}\right ) a}{5 c}-\frac {2 \left (3 A d \,e^{2}+3 B e \,d^{2}-\frac {5 e^{3} B a}{7 c}-\frac {4 \left (A \,e^{3}+3 B d \,e^{2}-\frac {6 e^{3} B b}{7 c}\right ) b}{5 c}\right ) b}{3 c}\right ) a \sqrt {2}\, \sqrt {4-\frac {2 \left (-b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \sqrt {4+\frac {2 \left (b +\sqrt {-4 a c +b^{2}}\right ) x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )-\operatorname {EllipticE}\left (\frac {x \sqrt {2}\, \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}}{2}, \frac {\sqrt {-4+\frac {2 b \left (b +\sqrt {-4 a c +b^{2}}\right )}{a c}}}{2}\right )\right )}{2 \sqrt {\frac {-b +\sqrt {-4 a c +b^{2}}}{a}}\, \sqrt {c \,x^{4}+b \,x^{2}+a}\, \left (b +\sqrt {-4 a c +b^{2}}\right )}\)	\(673\)
risch	\(\text {Expression too large to display}\)	\(1499\)
default	\(\text {Expression too large to display}\)	\(1699\)

Input:

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

Output:

1/7*B*e^3*x^5*(c*x^4+b*x^2+a)^(1/2)/c+1/5*(A*e^3+3*B*d*e^2-6/7*e^3*B/c*b)/ 
c*x^3*(c*x^4+b*x^2+a)^(1/2)+1/3*(3*A*d*e^2+3*B*e*d^2-5/7*e^3*B/c*a-4/5*(A* 
e^3+3*B*d*e^2-6/7*e^3*B/c*b)/c*b)/c*x*(c*x^4+b*x^2+a)^(1/2)+1/4*(A*d^3-1/3 
*(3*A*d*e^2+3*B*e*d^2-5/7*e^3*B/c*a-4/5*(A*e^3+3*B*d*e^2-6/7*e^3*B/c*b)/c* 
b)/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)*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*(3*A*d^2*e+B*d^3-3/5*(A*e^3+3*B* 
d*e^2-6/7*e^3*B/c*b)/c*a-2/3*(3*A*d*e^2+3*B*e*d^2-5/7*e^3*B/c*a-4/5*(A*e^3 
+3*B*d*e^2-6/7*e^3*B/c*b)/c*b)/c*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)^(1/2)/(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 [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 1041, normalized size of antiderivative = 1.38 \[ \int \frac {\left (A+B x^2\right ) \left (d+e x^2\right )^3}{\sqrt {a+b x^2+c x^4}} \, dx=\text {Too large to display} \] Input:

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

Output:

1/210*(sqrt(1/2)*((105*B*a*c^4*d^3 - 105*(2*B*a*b*c^3 - 3*A*a*c^4)*d^2*e + 
 21*(8*B*a*b^2*c^2 - (9*B*a^2 + 10*A*a*b)*c^3)*d*e^2 - (48*B*a*b^3*c + 63* 
A*a^2*c^3 - 8*(13*B*a^2*b + 7*A*a*b^2)*c^2)*e^3)*x*sqrt((b^2 - 4*a*c)/c^2) 
 - (105*B*a*b*c^3*d^3 - 105*(2*B*a*b^2*c^2 - 3*A*a*b*c^3)*d^2*e + 21*(8*B* 
a*b^3*c - (9*B*a^2*b + 10*A*a*b^2)*c^2)*d*e^2 - (48*B*a*b^4 + 63*A*a^2*b*c 
^2 - 8*(13*B*a^2*b^2 + 7*A*a*b^3)*c)*e^3)*x)*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)*((105*(B*a*c^4 - A*c^5)*d^3 - 105*(2*B*a*b*c^3 - (3*A + B)*a*c^4 
)*d^2*e + 21*(8*B*a*b^2*c^2 + 5*A*a*c^4 - (9*B*a^2 + 2*(5*A + 2*B)*a*b)*c^ 
3)*d*e^2 - (48*B*a*b^3*c + ((63*A + 25*B)*a^2 + 28*A*a*b)*c^3 - 8*(13*B*a^ 
2*b + (7*A + 3*B)*a*b^2)*c^2)*e^3)*x*sqrt((b^2 - 4*a*c)/c^2) - (105*(B*a*b 
*c^3 + A*b*c^4)*d^3 - 105*(2*B*a*b^2*c^2 - (3*A - B)*a*b*c^3)*d^2*e + 21*( 
8*B*a*b^3*c - 5*A*a*b*c^3 - (9*B*a^2*b + 2*(5*A - 2*B)*a*b^2)*c^2)*d*e^2 - 
 (48*B*a*b^4 + ((63*A - 25*B)*a^2*b - 28*A*a*b^2)*c^2 - 8*(13*B*a^2*b^2 + 
(7*A - 3*B)*a*b^3)*c)*e^3)*x)*sqrt(c)*sqrt((c*sqrt((b^2 - 4*a*c)/c^2) - b) 
/c)*elliptic_f(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)) + 2*(15*B*a*c^4*e 
^3*x^6 + 105*B*a*c^4*d^3 + 3*(21*B*a*c^4*d*e^2 - (6*B*a*b*c^3 - 7*A*a*c^4) 
*e^3)*x^4 - 105*(2*B*a*b*c^3 - 3*A*a*c^4)*d^2*e + 21*(8*B*a*b^2*c^2 - (...
                                                                                    
                                                                                    
 

Sympy [F]

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

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

Giac [F]

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

Mupad [F(-1)]

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

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

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

Output:

( - 53*sqrt(a + b*x**2 + c*x**4)*a*b*c*e**3*x + 105*sqrt(a + b*x**2 + c*x* 
*4)*a*c**2*d*e**2*x + 21*sqrt(a + b*x**2 + c*x**4)*a*c**2*e**3*x**3 + 24*s 
qrt(a + b*x**2 + c*x**4)*b**3*e**3*x - 84*sqrt(a + b*x**2 + c*x**4)*b**2*c 
*d*e**2*x - 18*sqrt(a + b*x**2 + c*x**4)*b**2*c*e**3*x**3 + 105*sqrt(a + b 
*x**2 + c*x**4)*b*c**2*d**2*e*x + 63*sqrt(a + b*x**2 + c*x**4)*b*c**2*d*e* 
*2*x**3 + 15*sqrt(a + b*x**2 + c*x**4)*b*c**2*e**3*x**5 + 53*int(sqrt(a + 
b*x**2 + c*x**4)/(a + b*x**2 + c*x**4),x)*a**2*b*c*e**3 - 105*int(sqrt(a + 
 b*x**2 + c*x**4)/(a + b*x**2 + c*x**4),x)*a**2*c**2*d*e**2 - 24*int(sqrt( 
a + b*x**2 + c*x**4)/(a + b*x**2 + c*x**4),x)*a*b**3*e**3 + 84*int(sqrt(a 
+ b*x**2 + c*x**4)/(a + b*x**2 + c*x**4),x)*a*b**2*c*d*e**2 - 105*int(sqrt 
(a + b*x**2 + c*x**4)/(a + b*x**2 + c*x**4),x)*a*b*c**2*d**2*e + 105*int(s 
qrt(a + b*x**2 + c*x**4)/(a + b*x**2 + c*x**4),x)*a*c**3*d**3 - 63*int((sq 
rt(a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x**4),x)*a**2*c**2*e**3 + 16 
0*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x**4),x)*a*b**2*c*e 
**3 - 399*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x**4),x)*a* 
b*c**2*d*e**2 + 315*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b*x**2 + c*x 
**4),x)*a*c**3*d**2*e - 48*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b*x** 
2 + c*x**4),x)*b**4*e**3 + 168*int((sqrt(a + b*x**2 + c*x**4)*x**2)/(a + b 
*x**2 + c*x**4),x)*b**3*c*d*e**2 - 210*int((sqrt(a + b*x**2 + c*x**4)*x**2 
)/(a + b*x**2 + c*x**4),x)*b**2*c**2*d**2*e + 105*int((sqrt(a + b*x**2 ...