\(\int \frac {A+B x^2+C x^4}{(a c+(b c+a d) x^2+b d x^4)^{3/2}} \, dx\) [8]

Optimal result

Integrand size = 38, antiderivative size = 325 \[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}} \, dx=-\frac {\left (c^2 C-B c d+A d^2\right ) x}{c d (b c-a d) \sqrt {a c+(b c+a d) x^2+b d x^4}}+\frac {\left (A b^2 c d+a^2 c C d+a b \left (c^2 C-2 B c d+A d^2\right )\right ) \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} E\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|1-\frac {a d}{b c}\right )}{a^{3/2} \sqrt {b} d (b c-a d)^2 \sqrt {a c+(b c+a d) x^2+b d x^4}}+\frac {(b B c-2 a c C-2 A b d+a B d) \left (a+b x^2\right ) \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} (b c-a d)^2 \sqrt {a c+(b c+a d) x^2+b d x^4}} \] Output:

-(A*d^2-B*c*d+C*c^2)*x/c/d/(-a*d+b*c)/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)+(A 
*b^2*c*d+a^2*c*C*d+a*b*(A*d^2-2*B*c*d+C*c^2))*(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^(3/2)/b^(1/2)/d/(-a*d+b*c)^2/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)+(-2* 
A*b*d+B*a*d+B*b*c-2*C*a*c)*(b*x^2+a)*(a*(d*x^2+c)/c/(b*x^2+a))^(1/2)*Inver 
seJacobiAM(arctan(b^(1/2)*x/a^(1/2)),(1-a*d/b/c)^(1/2))/a^(1/2)/b^(1/2)/(- 
a*d+b*c)^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 = 12.20 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} d x \left (A \left (a^2 d^2+a b d^2 x^2+b^2 c \left (c+d x^2\right )\right )+a c \left (b c C x^2-b B \left (c+2 d x^2\right )+a \left (2 c C-B d+C d x^2\right )\right )\right )+i c \left (A b^2 c d+a^2 c C d+a b \left (c^2 C-2 B c d+A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i c (-b c+a d) (a c C+A b d-a B d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{b c d (b c-a d)^2 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \] Input:

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

Output:

(Sqrt[b/a]*(Sqrt[b/a]*d*x*(A*(a^2*d^2 + a*b*d^2*x^2 + b^2*c*(c + d*x^2)) + 
 a*c*(b*c*C*x^2 - b*B*(c + 2*d*x^2) + a*(2*c*C - B*d + C*d*x^2))) + I*c*(A 
*b^2*c*d + a^2*c*C*d + a*b*(c^2*C - 2*B*c*d + 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)*(a*c*C + A*b*d - a*B*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2) 
/c]*EllipticF[I*ArcSinh[Sqrt[b/a]*x], (a*d)/(b*c)]))/(b*c*d*(b*c - a*d)^2* 
Sqrt[(a + b*x^2)*(c + d*x^2)])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(674\) vs. \(2(325)=650\).

Time = 0.86 (sec) , antiderivative size = 674, normalized size of antiderivative = 2.07, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2206, 25, 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.

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

\(\Big \downarrow \) 2206

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1511

\(\displaystyle \frac {\frac {\sqrt {a} \sqrt {c} \left (a^2 c C d+a b \left (A d^2-2 B c d+c^2 C\right )+A b^2 c d\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {a} \sqrt {c} \sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}-\frac {\sqrt {a} \sqrt {c} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {c}\right )^2 \left (-\sqrt {a} \sqrt {b} B \sqrt {c} \sqrt {d}+a c C+A b d\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{a c (b c-a d)^2}-\frac {x \left (-\left (x^2 \left (a^2 c C d+a b \left (A d^2-2 B c d+c^2 C\right )+A b^2 c d\right )\right )-A \left (a^2 d^2+b^2 c^2\right )+a c (a B d-2 a c C+b B c)\right )}{a c (b c-a d)^2 \sqrt {x^2 (a d+b c)+a c+b d x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {\left (a^2 c C d+a b \left (A d^2-2 B c d+c^2 C\right )+A b^2 c d\right ) \int \frac {\sqrt {a} \sqrt {c}-\sqrt {b} \sqrt {d} x^2}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}-\frac {\sqrt {a} \sqrt {c} \left (\sqrt {a} \sqrt {d}+\sqrt {b} \sqrt {c}\right )^2 \left (-\sqrt {a} \sqrt {b} B \sqrt {c} \sqrt {d}+a c C+A b d\right ) \int \frac {1}{\sqrt {b d x^4+(b c+a d) x^2+a c}}dx}{\sqrt {b} \sqrt {d}}}{a c (b c-a d)^2}-\frac {x \left (-\left (x^2 \left (a^2 c C d+a b \left (A d^2-2 B c d+c^2 C\right )+A b^2 c d\right )\right )-A \left (a^2 d^2+b^2 c^2\right )+a c (a B d-2 a c C+b B c)\right )}{a c (b c-a d)^2 \sqrt {x^2 (a d+b c)+a c+b d x^4}}\)

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

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

Input:

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

Output:

-((x*(a*c*(b*B*c - 2*a*c*C + a*B*d) - A*(b^2*c^2 + a^2*d^2) - (A*b^2*c*d + 
 a^2*c*C*d + a*b*(c^2*C - 2*B*c*d + A*d^2))*x^2))/(a*c*(b*c - a*d)^2*Sqrt[ 
a*c + (b*c + a*d)*x^2 + b*d*x^4])) + (((A*b^2*c*d + a^2*c*C*d + a*b*(c^2*C 
 - 2*B*c*d + A*d^2))*(-((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] + Sqr 
t[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] + Sqrt[a]*Sqrt[d])^2*(a*c*C - Sqrt[a]*Sqrt[ 
b]*B*Sqrt[c]*Sqrt[d] + A*b*d)*(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)*Sqrt[a*c + 
 (b*c + a*d)*x^2 + b*d*x^4]))/(a*c*(b*c - a*d)^2)
 

Defintions of rubi rules used

rule 25

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

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 1416

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

rule 1509

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

rule 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]
 

Maple [A] (verified)

Time = 3.88 (sec) , antiderivative size = 572, normalized size of antiderivative = 1.76

method	result	size
elliptic	\(-\frac {2 b d \left (-\frac {\left (A a b \,d^{2}+A \,b^{2} c d -2 B a c d b +C \,a^{2} c d +a b \,c^{2} C \right ) x^{3}}{2 b d a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (A \,a^{2} d^{2}+A \,b^{2} c^{2}-B \,a^{2} c d -B a b \,c^{2}+2 C \,a^{2} c^{2}\right ) x}{2 b d a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\right )}{\sqrt {\left (x^{4}+\frac {\left (a d +b c \right ) x^{2}}{b d}+\frac {a c}{b d}\right ) b d}}+\frac {\left (\frac {C}{b d}+\frac {A b d -C a c}{b d a c}-\frac {A \,a^{2} d^{2}+A \,b^{2} c^{2}-B \,a^{2} c d -B a b \,c^{2}+2 C \,a^{2} c^{2}}{a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}+\frac {\left (A a b \,d^{2}+A \,b^{2} c d -2 B a c d b +C \,a^{2} c d +a b \,c^{2} C \right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}\, d}\)	\(572\)
default	\(\text {Expression too large to display}\)	\(1182\)

Input:

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

Output:

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

Leaf count of result is larger than twice the leaf count of optimal. 807 vs. \(2 (317) = 634\).

Time = 0.09 (sec) , antiderivative size = 807, normalized size of antiderivative = 2.48 \[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

-((C*a^2*b^2*c^3 + A*a^2*b^2*c*d^2 + (C*a*b^3*c^2*d + A*a*b^3*d^3 + (C*a^2 
*b^2 - 2*B*a*b^3 + A*b^4)*c*d^2)*x^4 + (C*a^3*b - 2*B*a^2*b^2 + A*a*b^3)*c 
^2*d + (C*a*b^3*c^3 + A*a^2*b^2*d^3 + (2*C*a^2*b^2 - 2*B*a*b^3 + A*b^4)*c^ 
2*d + (C*a^3*b - 2*B*a^2*b^2 + 2*A*a*b^3)*c*d^2)*x^2)*sqrt(a*c)*sqrt(-b/a) 
*elliptic_e(arcsin(x*sqrt(-b/a)), a*d/(b*c)) - (C*a^2*b^2*c^3 + (C*a*b^3*c 
^2*d + (2*C*a^3*b - (B - C)*a^2*b^2 - 2*B*a*b^3 + A*b^4)*c*d^2 - (B*a^3*b 
- 2*A*a^2*b^2 - A*a*b^3)*d^3)*x^4 + (2*C*a^4 - (B - C)*a^3*b - 2*B*a^2*b^2 
 + A*a*b^3)*c^2*d - (B*a^4 - 2*A*a^3*b - A*a^2*b^2)*c*d^2 + (C*a*b^3*c^3 + 
 (2*C*a^3*b - (B - 2*C)*a^2*b^2 - 2*B*a*b^3 + A*b^4)*c^2*d + (2*C*a^4 - (2 
*B - C)*a^3*b + 2*(A - B)*a^2*b^2 + 2*A*a*b^3)*c*d^2 - (B*a^4 - 2*A*a^3*b 
- A*a^2*b^2)*d^3)*x^2)*sqrt(a*c)*sqrt(-b/a)*elliptic_f(arcsin(x*sqrt(-b/a) 
), a*d/(b*c)) - sqrt(b*d*x^4 + (b*c + a*d)*x^2 + a*c)*((C*a^2*b^2*c^2*d + 
A*a^2*b^2*d^3 + (C*a^3*b - 2*B*a^2*b^2 + A*a*b^3)*c*d^2)*x^3 - (B*a^3*b*c* 
d^2 - A*a^3*b*d^3 - (2*C*a^3*b - B*a^2*b^2 + A*a*b^3)*c^2*d)*x))/(a^3*b^3* 
c^4*d - 2*a^4*b^2*c^3*d^2 + a^5*b*c^2*d^3 + (a^2*b^4*c^3*d^2 - 2*a^3*b^3*c 
^2*d^3 + a^4*b^2*c*d^4)*x^4 + (a^2*b^4*c^4*d - a^3*b^3*c^3*d^2 - a^4*b^2*c 
^2*d^3 + a^5*b*c*d^4)*x^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

( - sqrt(c + d*x**2)*sqrt(a + b*x**2)*c*x + int((sqrt(c + d*x**2)*sqrt(a + 
 b*x**2)*x**2)/(a**2*c**2 + 2*a**2*c*d*x**2 + a**2*d**2*x**4 + 2*a*b*c**2* 
x**2 + 4*a*b*c*d*x**4 + 2*a*b*d**2*x**6 + b**2*c**2*x**4 + 2*b**2*c*d*x**6 
 + b**2*d**2*x**8),x)*a*b**2*c*d + int((sqrt(c + d*x**2)*sqrt(a + b*x**2)* 
x**2)/(a**2*c**2 + 2*a**2*c*d*x**2 + a**2*d**2*x**4 + 2*a*b*c**2*x**2 + 4* 
a*b*c*d*x**4 + 2*a*b*d**2*x**6 + b**2*c**2*x**4 + 2*b**2*c*d*x**6 + b**2*d 
**2*x**8),x)*a*b**2*d**2*x**2 + int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x** 
2)/(a**2*c**2 + 2*a**2*c*d*x**2 + a**2*d**2*x**4 + 2*a*b*c**2*x**2 + 4*a*b 
*c*d*x**4 + 2*a*b*d**2*x**6 + b**2*c**2*x**4 + 2*b**2*c*d*x**6 + b**2*d**2 
*x**8),x)*b**3*c*d*x**2 + int((sqrt(c + d*x**2)*sqrt(a + b*x**2)*x**2)/(a* 
*2*c**2 + 2*a**2*c*d*x**2 + a**2*d**2*x**4 + 2*a*b*c**2*x**2 + 4*a*b*c*d*x 
**4 + 2*a*b*d**2*x**6 + b**2*c**2*x**4 + 2*b**2*c*d*x**6 + b**2*d**2*x**8) 
,x)*b**3*d**2*x**4 + int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c**2 + 
2*a**2*c*d*x**2 + a**2*d**2*x**4 + 2*a*b*c**2*x**2 + 4*a*b*c*d*x**4 + 2*a* 
b*d**2*x**6 + b**2*c**2*x**4 + 2*b**2*c*d*x**6 + b**2*d**2*x**8),x)*a**2*b 
*c*d + int((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c**2 + 2*a**2*c*d*x** 
2 + a**2*d**2*x**4 + 2*a*b*c**2*x**2 + 4*a*b*c*d*x**4 + 2*a*b*d**2*x**6 + 
b**2*c**2*x**4 + 2*b**2*c*d*x**6 + b**2*d**2*x**8),x)*a**2*b*d**2*x**2 + i 
nt((sqrt(c + d*x**2)*sqrt(a + b*x**2))/(a**2*c**2 + 2*a**2*c*d*x**2 + a**2 
*d**2*x**4 + 2*a*b*c**2*x**2 + 4*a*b*c*d*x**4 + 2*a*b*d**2*x**6 + b**2*...