\(\int (A+B x^2+C x^4) \sqrt {a c+(b c+a d) x^2+b d x^4} \, dx\) [6]

Optimal result

Integrand size = 38, antiderivative size = 563 \[ \int \left (A+B x^2+C x^4\right ) \sqrt {a c+(b c+a d) x^2+b d x^4} \, dx=-\frac {\left (5 b d (b c+a d) (a c C-7 A b d)+6 a b c d (4 b c C-7 b B d+4 a C d)-2 (b c+a d)^2 (4 b c C-7 b B d+4 a C d)\right ) x \left (c+d x^2\right )}{105 b^2 d^3 \sqrt {a c+(b c+a d) x^2+b d x^4}}-\frac {x \left (5 b d (a c C-7 A b d)+(b c+a d) (4 b c C-7 b B d+4 a C d)+3 b d (4 b c C-7 b B d+4 a C d) x^2\right ) \sqrt {a c+(b c+a d) x^2+b d x^4}}{105 b^2 d^2}+\frac {C x \left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}}{7 b d}+\frac {c \left (5 b d (b c+a d) (a c C-7 A b d)+6 a b c d (4 b c C-7 b B d+4 a C d)-2 (b c+a d)^2 (4 b c C-7 b B d+4 a C d)\right ) \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{105 \sqrt {a} b^{5/2} d^3 \sqrt {a c+(b c+a d) x^2+b d x^4}}-\frac {\sqrt {a} c (10 b d (a c C-7 A b d)-(b c+a d) (4 b c C-7 b B d+4 a C d)) \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{105 b^{5/2} d^2 \sqrt {a c+(b c+a d) x^2+b d x^4}} \] Output:

-1/105*(5*b*d*(a*d+b*c)*(-7*A*b*d+C*a*c)+6*a*b*c*d*(-7*B*b*d+4*C*a*d+4*C*b 
*c)-2*(a*d+b*c)^2*(-7*B*b*d+4*C*a*d+4*C*b*c))*x*(d*x^2+c)/b^2/d^3/(a*c+(a* 
d+b*c)*x^2+b*d*x^4)^(1/2)-1/105*x*(5*b*d*(-7*A*b*d+C*a*c)+(a*d+b*c)*(-7*B* 
b*d+4*C*a*d+4*C*b*c)+3*b*d*(-7*B*b*d+4*C*a*d+4*C*b*c)*x^2)*(a*c+(a*d+b*c)* 
x^2+b*d*x^4)^(1/2)/b^2/d^2+1/7*C*x*(a*c+(a*d+b*c)*x^2+b*d*x^4)^(3/2)/b/d+1 
/105*c*(5*b*d*(a*d+b*c)*(-7*A*b*d+C*a*c)+6*a*b*c*d*(-7*B*b*d+4*C*a*d+4*C*b 
*c)-2*(a*d+b*c)^2*(-7*B*b*d+4*C*a*d+4*C*b*c))*(b*x^2+a)*(a*(d*x^2+c)/c/(b* 
x^2+a))^(1/2)*EllipticE(b^(1/2)*x/a^(1/2)/(1+b*x^2/a)^(1/2),(1-a*d/b/c)^(1 
/2))/a^(1/2)/b^(5/2)/d^3/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)-1/105*a^(1/2)*c 
*(10*b*d*(-7*A*b*d+C*a*c)-(a*d+b*c)*(-7*B*b*d+4*C*a*d+4*C*b*c))*(b*x^2+a)* 
(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)*InverseJacobiAM(arctan(b^(1/2)*x/a^(1/2)), 
(1-a*d/b/c)^(1/2))/b^(5/2)/d^2/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.99 (sec) , antiderivative size = 388, normalized size of antiderivative = 0.69 \[ \int \left (A+B x^2+C x^4\right ) \sqrt {a c+(b c+a d) x^2+b d x^4} \, dx=\frac {-\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (c+d x^2\right ) \left (4 a^2 C d^2-a b d \left (2 c C+7 B d+3 C d x^2\right )+b^2 \left (4 c^2 C-c d \left (7 B+3 C x^2\right )-d^2 \left (35 A+21 B x^2+15 C x^4\right )\right )\right )-i c \left (8 a^3 C d^3-a^2 b d^2 (5 c C+14 B d)+b^3 c \left (8 c^2 C-14 B c d+35 A d^2\right )+a b^2 d \left (-5 c^2 C+14 B c d+35 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c (-b c+a d) \left (4 a^2 C d^2+a b d (c C-7 B d)+b^2 \left (-8 c^2 C+14 B c d-35 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )}{105 a^2 \left (\frac {b}{a}\right )^{5/2} d^3 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \] Input:

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

Output:

(-(Sqrt[b/a]*d*x*(a + b*x^2)*(c + d*x^2)*(4*a^2*C*d^2 - a*b*d*(2*c*C + 7*B 
*d + 3*C*d*x^2) + b^2*(4*c^2*C - c*d*(7*B + 3*C*x^2) - d^2*(35*A + 21*B*x^ 
2 + 15*C*x^4)))) - I*c*(8*a^3*C*d^3 - a^2*b*d^2*(5*c*C + 14*B*d) + b^3*c*( 
8*c^2*C - 14*B*c*d + 35*A*d^2) + a*b^2*d*(-5*c^2*C + 14*B*c*d + 35*A*d^2)) 
*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[Sqrt[b/a]*x], 
 (a*d)/(b*c)] + I*c*(-(b*c) + a*d)*(4*a^2*C*d^2 + a*b*d*(c*C - 7*B*d) + b^ 
2*(-8*c^2*C + 14*B*c*d - 35*A*d^2))*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c 
]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)])/(105*a^2*(b/a)^(5/2)*d^3 
*Sqrt[(a + b*x^2)*(c + d*x^2)])
 

Rubi [A] (verified)

Time = 1.05 (sec) , antiderivative size = 801, normalized size of antiderivative = 1.42, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {2207, 25, 1490, 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.

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

\(\Big \downarrow \) 2207

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1490

\(\displaystyle \frac {C x \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}{7 b d}-\frac {\frac {\int \frac {\left (-2 (4 b c C+4 a d C-7 b B d) (b c+a d)^2+5 b d (a c C-7 A b d) (b c+a d)+6 a b c d (4 b c C+4 a d C-7 b B d)\right ) x^2+a c (10 b d (a c C-7 A b d)-(b c+a d) (4 b c C+4 a d C-7 b B d))}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{15 b d}+\frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4} \left (5 b d (a c C-7 A b d)+3 b d x^2 (4 a C d-7 b B d+4 b c C)+(a d+b c) (4 a C d-7 b B d+4 b c C)\right )}{15 b d}}{7 b d}\)

\(\Big \downarrow \) 1511

\(\displaystyle \frac {C x \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}{7 b d}-\frac {\frac {\frac {\sqrt {a} \sqrt {c} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {c}\right )^2 \left (12 a^{3/2} \sqrt {b} \sqrt {c} C d^{3/2}-8 a^2 C d^2+3 \sqrt {a} b^{3/2} \sqrt {c} \sqrt {d} (4 c C-7 B d)-a b d (11 c C-14 B d)-\left (b^2 \left (35 A d^2-14 B c d+8 c^2 C\right )\right )\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}-\frac {\sqrt {a} \sqrt {c} \left (5 b d (a d+b c) (a c C-7 A b d)-2 (a d+b c)^2 (4 a C d-7 b B d+4 b c C)+6 a b c d (4 a C d-7 b B d+4 b c C)\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {a} \sqrt {c} \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{15 b d}+\frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4} \left (5 b d (a c C-7 A b d)+3 b d x^2 (4 a C d-7 b B d+4 b c C)+(a d+b c) (4 a C d-7 b B d+4 b c C)\right )}{15 b d}}{7 b d}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {C x \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}{7 b d}-\frac {\frac {\frac {\sqrt {a} \sqrt {c} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {c}\right )^2 \left (12 a^{3/2} \sqrt {b} \sqrt {c} C d^{3/2}-8 a^2 C d^2+3 \sqrt {a} b^{3/2} \sqrt {c} \sqrt {d} (4 c C-7 B d)-a b d (11 c C-14 B d)-\left (b^2 \left (35 A d^2-14 B c d+8 c^2 C\right )\right )\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}-\frac {\left (5 b d (a d+b c) (a c C-7 A b d)-2 (a d+b c)^2 (4 a C d-7 b B d+4 b c C)+6 a b c d (4 a C d-7 b B d+4 b c C)\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{15 b d}+\frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4} \left (5 b d (a c C-7 A b d)+3 b d x^2 (4 a C d-7 b B d+4 b c C)+(a d+b c) (4 a C d-7 b B d+4 b c C)\right )}{15 b d}}{7 b d}\)

\(\Big \downarrow \) 1416

\(\displaystyle \frac {C x \left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}}{7 b d}-\frac {\frac {\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {c}\right )^2 \left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right ) \sqrt {\frac {x^2 (a d+b c)+a c+b d x^4}{\left (\sqrt {a} \sqrt {c}+\sqrt {b} \sqrt {d} x^2\right )^2}} \left (12 a^{3/2} \sqrt {b} \sqrt {c} C d^{3/2}-8 a^2 C d^2+3 \sqrt {a} b^{3/2} \sqrt {c} \sqrt {d} (4 c C-7 B d)-a b d (11 c C-14 B d)-\left (b^2 \left (35 A d^2-14 B c d+8 c^2 C\right )\right )\right ) \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 b^{3/4} d^{3/4} \sqrt {x^2 (a d+b c)+a c+b d x^4}}-\frac {\left (5 b d (a d+b c) (a c C-7 A b d)-2 (a d+b c)^2 (4 a C d-7 b B d+4 b c C)+6 a b c d (4 a C d-7 b B d+4 b c C)\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{15 b d}+\frac {x \sqrt {x^2 (a d+b c)+a c+b d x^4} \left (5 b d (a c C-7 A b d)+3 b d x^2 (4 a C d-7 b B d+4 b c C)+(a d+b c) (4 a C d-7 b B d+4 b c C)\right )}{15 b d}}{7 b d}\)

\(\Big \downarrow \) 1509

\(\displaystyle \frac {C x \left (b d x^4+(b c+a d) x^2+a c\right )^{3/2}}{7 b d}-\frac {\frac {x \sqrt {b d x^4+(b c+a d) x^2+a c} \left (3 b d (4 b c C+4 a d C-7 b B d) x^2+5 b d (a c C-7 A b d)+(b c+a d) (4 b c C+4 a d C-7 b B d)\right )}{15 b d}+\frac {\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {b} \sqrt {c}+\sqrt {a} \sqrt {d}\right )^2 \left (-\left (\left (8 C c^2-14 B d c+35 A d^2\right ) b^2\right )+3 \sqrt {a} \sqrt {c} \sqrt {d} (4 c C-7 B d) b^{3/2}-a d (11 c C-14 B d) b+12 a^{3/2} \sqrt {c} C d^{3/2} \sqrt {b}-8 a^2 C d^2\right ) \left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right ) \sqrt {\frac {b d x^4+(b c+a d) x^2+a c}{\left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right ),\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{2 b^{3/4} d^{3/4} \sqrt {b d x^4+(b c+a d) x^2+a c}}-\frac {\left (-2 (4 b c C+4 a d C-7 b B d) (b c+a d)^2+5 b d (a c C-7 A b d) (b c+a d)+6 a b c d (4 b c C+4 a d C-7 b B d)\right ) \left (\frac {\sqrt [4]{a} \sqrt [4]{c} \left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right ) \sqrt {\frac {b d x^4+(b c+a d) x^2+a c}{\left (\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{b} \sqrt [4]{d} x}{\sqrt [4]{a} \sqrt [4]{c}}\right )|\frac {1}{4} \left (2-\frac {b c+a d}{\sqrt {a} \sqrt {b} \sqrt {c} \sqrt {d}}\right )\right )}{\sqrt [4]{b} \sqrt [4]{d} \sqrt {b d x^4+(b c+a d) x^2+a c}}-\frac {x \sqrt {b d x^4+(b c+a d) x^2+a c}}{\sqrt {b} \sqrt {d} x^2+\sqrt {a} \sqrt {c}}\right )}{\sqrt {b} \sqrt {d}}}{15 b d}}{7 b d}\)

Input:

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

Output:

(C*x*(a*c + (b*c + a*d)*x^2 + b*d*x^4)^(3/2))/(7*b*d) - ((x*(5*b*d*(a*c*C 
- 7*A*b*d) + (b*c + a*d)*(4*b*c*C - 7*b*B*d + 4*a*C*d) + 3*b*d*(4*b*c*C - 
7*b*B*d + 4*a*C*d)*x^2)*Sqrt[a*c + (b*c + a*d)*x^2 + b*d*x^4])/(15*b*d) + 
(-(((5*b*d*(b*c + a*d)*(a*c*C - 7*A*b*d) + 6*a*b*c*d*(4*b*c*C - 7*b*B*d + 
4*a*C*d) - 2*(b*c + a*d)^2*(4*b*c*C - 7*b*B*d + 4*a*C*d))*(-((x*Sqrt[a*c + 
 (b*c + a*d)*x^2 + b*d*x^4])/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)) + (a 
^(1/4)*c^(1/4)*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)*Sqrt[(a*c + (b*c + 
a*d)*x^2 + b*d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^2)^2]*EllipticE[2 
*ArcTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 - (b*c + a*d)/(Sqrt[a]* 
Sqrt[b]*Sqrt[c]*Sqrt[d]))/4])/(b^(1/4)*d^(1/4)*Sqrt[a*c + (b*c + a*d)*x^2 
+ b*d*x^4])))/(Sqrt[b]*Sqrt[d])) + (a^(1/4)*c^(1/4)*(Sqrt[b]*Sqrt[c] + Sqr 
t[a]*Sqrt[d])^2*(12*a^(3/2)*Sqrt[b]*Sqrt[c]*C*d^(3/2) - 8*a^2*C*d^2 - a*b* 
d*(11*c*C - 14*B*d) + 3*Sqrt[a]*b^(3/2)*Sqrt[c]*Sqrt[d]*(4*c*C - 7*B*d) - 
b^2*(8*c^2*C - 14*B*c*d + 35*A*d^2))*(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[d]*x^ 
2)*Sqrt[(a*c + (b*c + a*d)*x^2 + b*d*x^4)/(Sqrt[a]*Sqrt[c] + Sqrt[b]*Sqrt[ 
d]*x^2)^2]*EllipticF[2*ArcTan[(b^(1/4)*d^(1/4)*x)/(a^(1/4)*c^(1/4))], (2 - 
 (b*c + a*d)/(Sqrt[a]*Sqrt[b]*Sqrt[c]*Sqrt[d]))/4])/(2*b^(3/4)*d^(3/4)*Sqr 
t[a*c + (b*c + a*d)*x^2 + b*d*x^4]))/(15*b*d))/(7*b*d)
 

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 1490

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(2*b*e*p + c*d*(4*p + 3) + c*e*(4*p + 1)*x^2)*((a + b*x^2 + c 
*x^4)^p/(c*(4*p + 1)*(4*p + 3))), x] + Simp[2*(p/(c*(4*p + 1)*(4*p + 3))) 
 Int[Simp[2*a*c*d*(4*p + 3) - a*b*e + (2*a*c*e*(4*p + 1) + b*c*d*(4*p + 3) 
- b^2*e*(2*p + 1))*x^2, x]*(a + b*x^2 + c*x^4)^(p - 1), x], x] /; FreeQ[{a, 
 b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && 
 GtQ[p, 0] && FractionQ[p] && IntegerQ[2*p]
 

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 = 9.64 (sec) , antiderivative size = 594, normalized size of antiderivative = 1.06

method	result	size
elliptic	\(\frac {C \,x^{5} \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{7}+\frac {\left (B b d +a C d +C c b -\frac {C \left (6 a d +6 b c \right )}{7}\right ) x^{3} \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{5 b d}+\frac {\left (A b d +B a d +B b c +\frac {2 C a c}{7}-\frac {\left (B b d +a C d +C c b -\frac {C \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}{3 b d}+\frac {\left (A a c -\frac {\left (A b d +B a d +B b c +\frac {2 C a c}{7}-\frac {\left (B b d +a C d +C c b -\frac {C \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (A a d +A b c +B a c -\frac {3 \left (B b d +a C d +C c b -\frac {C \left (6 a d +6 b c \right )}{7}\right ) a c}{5 b d}-\frac {\left (A b d +B a d +B b c +\frac {2 C a c}{7}-\frac {\left (B b d +a C d +C c b -\frac {C \left (6 a d +6 b c \right )}{7}\right ) \left (4 a d +4 b c \right )}{5 b d}\right ) \left (2 a d +2 b c \right )}{3 b d}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}\)	\(594\)
risch	\(\frac {x \left (15 C \,b^{2} d^{2} x^{4}+21 B \,b^{2} d^{2} x^{2}+3 C a b \,d^{2} x^{2}+3 C \,b^{2} c d \,x^{2}+35 A \,b^{2} d^{2}+7 B b \,d^{2} a +7 B \,b^{2} c d -4 a^{2} C \,d^{2}+2 C a b c d -4 C \,b^{2} c^{2}\right ) \left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}{105 b^{2} d^{2} \sqrt {\left (b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}}+\frac {-\frac {\left (35 A a \,b^{2} d^{3}+35 A \,b^{3} c \,d^{2}-14 B \,a^{2} b \,d^{3}+14 B a \,b^{2} c \,d^{2}-14 B \,b^{3} c^{2} d +8 C \,a^{3} d^{3}-5 C \,a^{2} b c \,d^{2}-5 C a \,b^{2} c^{2} d +8 C \,b^{3} c^{3}\right ) c \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}+\frac {4 C a \,b^{2} c^{3} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}+\frac {4 a^{3} c C \,d^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}+\frac {70 A a \,b^{2} c \,d^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {7 B a \,b^{2} c^{2} d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {7 B \,a^{2} b c \,d^{2} \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}-\frac {2 C \,a^{2} b \,c^{2} d \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}}{105 b^{2} d^{2}}\)	\(937\)
default	\(\text {Expression too large to display}\)	\(1042\)

Input:

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

Output:

1/7*C*x^5*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/5*(B*b*d+a*C*d+C*c*b-1/7*C 
*(6*a*d+6*b*c))/b/d*x^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+1/3*(A*b*d+B*a 
*d+B*b*c+2/7*C*a*c-1/5*(B*b*d+a*C*d+C*c*b-1/7*C*(6*a*d+6*b*c))/b/d*(4*a*d+ 
4*b*c))/b/d*x*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)+(A*a*c-1/3*(A*b*d+B*a*d+ 
B*b*c+2/7*C*a*c-1/5*(B*b*d+a*C*d+C*c*b-1/7*C*(6*a*d+6*b*c))/b/d*(4*a*d+4*b 
*c))/b/d*a*c)/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*(1+d*x^2/c)^(1/2)/(b*d*x^4+a* 
d*x^2+b*c*x^2+a*c)^(1/2)*EllipticF(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2) 
)-(A*a*d+A*b*c+B*a*c-3/5*(B*b*d+a*C*d+C*c*b-1/7*C*(6*a*d+6*b*c))/b/d*a*c-1 
/3*(A*b*d+B*a*d+B*b*c+2/7*C*a*c-1/5*(B*b*d+a*C*d+C*c*b-1/7*C*(6*a*d+6*b*c) 
)/b/d*(4*a*d+4*b*c))/b/d*(2*a*d+2*b*c))*c/(-b/a)^(1/2)*(1+b*x^2/a)^(1/2)*( 
1+d*x^2/c)^(1/2)/(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)/d*(EllipticF(x*(-b/a) 
^(1/2),(-1+(a*d+b*c)/c/b)^(1/2))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/ 
b)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 517, normalized size of antiderivative = 0.92 \[ \int \left (A+B x^2+C x^4\right ) \sqrt {a c+(b c+a d) x^2+b d x^4} \, dx=-\frac {{\left (8 \, C b^{3} c^{4} - {\left (5 \, C a b^{2} + 14 \, B b^{3}\right )} c^{3} d - {\left (5 \, C a^{2} b - 14 \, B a b^{2} - 35 \, A b^{3}\right )} c^{2} d^{2} + {\left (8 \, C a^{3} - 14 \, B a^{2} b + 35 \, A a b^{2}\right )} c d^{3}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (8 \, C b^{3} c^{4} - {\left (5 \, C a b^{2} + 14 \, B b^{3}\right )} c^{3} d - {\left (5 \, C a^{2} b - 2 \, {\left (7 \, B + 2 \, C\right )} a b^{2} - 35 \, A b^{3}\right )} c^{2} d^{2} + {\left (8 \, C a^{3} - 2 \, {\left (7 \, B + C\right )} a^{2} b + 7 \, {\left (5 \, A - B\right )} a b^{2}\right )} c d^{3} + {\left (4 \, C a^{3} - 7 \, B a^{2} b + 70 \, A a b^{2}\right )} d^{4}\right )} \sqrt {b d} x \sqrt {-\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {-\frac {c}{d}}}{x}\right )\,|\,\frac {a d}{b c}) - {\left (15 \, C b^{3} d^{4} x^{6} + 8 \, C b^{3} c^{3} d - {\left (5 \, C a b^{2} + 14 \, B b^{3}\right )} c^{2} d^{2} - {\left (5 \, C a^{2} b - 14 \, B a b^{2} - 35 \, A b^{3}\right )} c d^{3} + {\left (8 \, C a^{3} - 14 \, B a^{2} b + 35 \, A a b^{2}\right )} d^{4} + 3 \, {\left (C b^{3} c d^{3} + {\left (C a b^{2} + 7 \, B b^{3}\right )} d^{4}\right )} x^{4} - {\left (4 \, C b^{3} c^{2} d^{2} - {\left (2 \, C a b^{2} + 7 \, B b^{3}\right )} c d^{3} + {\left (4 \, C a^{2} b - 7 \, B a b^{2} - 35 \, A b^{3}\right )} d^{4}\right )} x^{2}\right )} \sqrt {b d x^{4} + {\left (b c + a d\right )} x^{2} + a c}}{105 \, b^{3} d^{4} x} \] Input:

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

Output:

-1/105*((8*C*b^3*c^4 - (5*C*a*b^2 + 14*B*b^3)*c^3*d - (5*C*a^2*b - 14*B*a* 
b^2 - 35*A*b^3)*c^2*d^2 + (8*C*a^3 - 14*B*a^2*b + 35*A*a*b^2)*c*d^3)*sqrt( 
b*d)*x*sqrt(-c/d)*elliptic_e(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (8*C*b^3*c 
^4 - (5*C*a*b^2 + 14*B*b^3)*c^3*d - (5*C*a^2*b - 2*(7*B + 2*C)*a*b^2 - 35* 
A*b^3)*c^2*d^2 + (8*C*a^3 - 2*(7*B + C)*a^2*b + 7*(5*A - B)*a*b^2)*c*d^3 + 
 (4*C*a^3 - 7*B*a^2*b + 70*A*a*b^2)*d^4)*sqrt(b*d)*x*sqrt(-c/d)*elliptic_f 
(arcsin(sqrt(-c/d)/x), a*d/(b*c)) - (15*C*b^3*d^4*x^6 + 8*C*b^3*c^3*d - (5 
*C*a*b^2 + 14*B*b^3)*c^2*d^2 - (5*C*a^2*b - 14*B*a*b^2 - 35*A*b^3)*c*d^3 + 
 (8*C*a^3 - 14*B*a^2*b + 35*A*a*b^2)*d^4 + 3*(C*b^3*c*d^3 + (C*a*b^2 + 7*B 
*b^3)*d^4)*x^4 - (4*C*b^3*c^2*d^2 - (2*C*a*b^2 + 7*B*b^3)*c*d^3 + (4*C*a^2 
*b - 7*B*a*b^2 - 35*A*b^3)*d^4)*x^2)*sqrt(b*d*x^4 + (b*c + a*d)*x^2 + a*c) 
)/(b^3*d^4*x)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a**2*c*d**2*x + 42*sqrt(c + d*x**2 
)*sqrt(a + b*x**2)*a*b**2*d**2*x + 2*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b 
*c**2*d*x + 3*sqrt(c + d*x**2)*sqrt(a + b*x**2)*a*b*c*d**2*x**3 + 7*sqrt(c 
 + d*x**2)*sqrt(a + b*x**2)*b**3*c*d*x + 21*sqrt(c + d*x**2)*sqrt(a + b*x* 
*2)*b**3*d**2*x**3 - 4*sqrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**3*x + 3*s 
qrt(c + d*x**2)*sqrt(a + b*x**2)*b**2*c**2*d*x**3 + 15*sqrt(c + d*x**2)*sq 
rt(a + b*x**2)*b**2*c*d**2*x**5 + 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2) 
*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**3*c*d**3 + 21*int((sqr 
t(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4 
),x)*a**2*b**2*d**3 - 5*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c 
+ a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**2*b*c**2*d**2 + 49*int((sqrt(c + d 
*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a* 
b**3*c*d**2 - 5*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x* 
*2 + b*c*x**2 + b*d*x**4),x)*a*b**2*c**3*d - 14*int((sqrt(c + d*x**2)*sqrt 
(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*b**4*c**2*d + 
 8*int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a*c + a*d*x**2 + b*c*x**2 
 + b*d*x**4),x)*b**3*c**4 + 4*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c 
 + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**3*c**2*d**2 + 63*int((sqrt(c + d* 
x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + b*c*x**2 + b*d*x**4),x)*a**2*b** 
2*c*d**2 - 2*int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a*c + a*d*x**2 + ...