\(\int \frac {1}{(a c+(b c+a d) x^2+b d x^4)^{3/2}} \, dx\) [16]

Optimal result

Integrand size = 25, antiderivative size = 267 \[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}} \, dx=\frac {b x}{a (b c-a d) \sqrt {a c+(b c+a d) x^2+b d x^4}}+\frac {\sqrt {d} (b c+a d) \sqrt {a c+(b c+a d) x^2+b d x^4} E\left (\arctan \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|1-\frac {b c}{a d}\right )}{a \sqrt {c} (b c-a d)^2 \sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \left (c+d x^2\right )}-\frac {2 \sqrt {a} \sqrt {b} d \left (c+d x^2\right ) \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),1-\frac {a d}{b c}\right )}{c (b c-a d)^2 \sqrt {\frac {a \left (c+d x^2\right )}{c \left (a+b x^2\right )}} \sqrt {a c+(b c+a d) x^2+b d x^4}} \] Output:

b*x/a/(-a*d+b*c)/(a*c+(a*d+b*c)*x^2+b*d*x^4)^(1/2)+d^(1/2)*(a*d+b*c)*(a*c+ 
(a*d+b*c)*x^2+b*d*x^4)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2)/(1+d*x^2/c)^(1/2) 
,(1-b*c/a/d)^(1/2))/a/c^(1/2)/(-a*d+b*c)^2/(c*(b*x^2+a)/a/(d*x^2+c))^(1/2) 
/(d*x^2+c)-2*a^(1/2)*b^(1/2)*d*(d*x^2+c)*InverseJacobiAM(arctan(b^(1/2)*x/ 
a^(1/2)),(1-a*d/b/c)^(1/2))/c/(-a*d+b*c)^2/(a*(d*x^2+c)/c/(b*x^2+a))^(1/2) 
/(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 = 6.22 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} x \left (a^2 d^2+a b d^2 x^2+b^2 c \left (c+d x^2\right )\right )+i b c (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|\frac {a d}{b c}\right )+i b c (-b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),\frac {a d}{b c}\right )\right )}{b c (b c-a d)^2 \sqrt {\left (a+b x^2\right ) \left (c+d x^2\right )}} \] Input:

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

Output:

(Sqrt[b/a]*(Sqrt[b/a]*x*(a^2*d^2 + a*b*d^2*x^2 + b^2*c*(c + d*x^2)) + I*b* 
c*(b*c + a*d)*Sqrt[1 + (b*x^2)/a]*Sqrt[1 + (d*x^2)/c]*EllipticE[I*ArcSinh[ 
Sqrt[b/a]*x], (a*d)/(b*c)] + I*b*c*(-(b*c) + a*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*(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. \(571\) vs. \(2(267)=534\).

Time = 0.64 (sec) , antiderivative size = 571, normalized size of antiderivative = 2.14, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1405, 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.

\(\displaystyle \int \frac {1}{\left (x^2 (a d+b c)+a c+b d x^4\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 1405

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1511

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1416

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

\(\Big \downarrow \) 1509

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

Input:

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

Output:

(x*(b^2*c^2 + a^2*d^2 + b*d*(b*c + a*d)*x^2))/(a*c*(b*c - a*d)^2*Sqrt[a*c 
+ (b*c + a*d)*x^2 + b*d*x^4]) - (b*d*(-(((b*c + a*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*ArcT 
an[(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*(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*Arc 
Tan[(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 27

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

rule 1405

Int[((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(-x)*(b^2 
- 2*a*c + b*c*x^2)*((a + b*x^2 + c*x^4)^(p + 1)/(2*a*(p + 1)*(b^2 - 4*a*c)) 
), x] + Simp[1/(2*a*(p + 1)*(b^2 - 4*a*c))   Int[(b^2 - 2*a*c + 2*(p + 1)*( 
b^2 - 4*a*c) + b*c*(4*p + 7)*x^2)*(a + b*x^2 + c*x^4)^(p + 1), x], x] /; Fr 
eeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && IntegerQ[2*p]
 

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]
 

Maple [A] (verified)

Time = 0.60 (sec) , antiderivative size = 428, normalized size of antiderivative = 1.60

method	result	size
default	\(-\frac {2 b d \left (-\frac {\left (a d +b c \right ) x^{3}}{2 a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (a^{2} d^{2}+b^{2} c^{2}\right ) x}{2 a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b d}\right )}{\sqrt {\left (x^{4}+\frac {\left (a d +b c \right ) x^{2}}{b d}+\frac {a c}{b d}\right ) b d}}+\frac {\left (\frac {1}{a c}-\frac {a^{2} d^{2}+b^{2} c^{2}}{a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}+\frac {\left (a d +b c \right ) b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}\)	\(428\)
elliptic	\(-\frac {2 b d \left (-\frac {\left (a d +b c \right ) x^{3}}{2 a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}-\frac {\left (a^{2} d^{2}+b^{2} c^{2}\right ) x}{2 a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b d}\right )}{\sqrt {\left (x^{4}+\frac {\left (a d +b c \right ) x^{2}}{b d}+\frac {a c}{b d}\right ) b d}}+\frac {\left (\frac {1}{a c}-\frac {a^{2} d^{2}+b^{2} c^{2}}{a c \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}\right ) \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )}{\sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}+\frac {\left (a d +b c \right ) b \sqrt {1+\frac {b \,x^{2}}{a}}\, \sqrt {1+\frac {d \,x^{2}}{c}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {b}{a}}, \sqrt {-1+\frac {a d +b c}{c b}}\right )\right )}{a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \sqrt {-\frac {b}{a}}\, \sqrt {b d \,x^{4}+x^{2} d a +b c \,x^{2}+a c}}\)	\(428\)

Input:

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

Output:

-2*b*d*(-1/2/a/c*(a*d+b*c)/(a^2*d^2-2*a*b*c*d+b^2*c^2)*x^3-1/2*(a^2*d^2+b^ 
2*c^2)/a/c/(a^2*d^2-2*a*b*c*d+b^2*c^2)/b/d*x)/((x^4+(a*d+b*c)/b/d*x^2+a*c/ 
b/d)*b*d)^(1/2)+(1/a/c-(a^2*d^2+b^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))+1/a*(a*d+b*c 
)/(a^2*d^2-2*a*b*c*d+b^2*c^2)*b/(-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))-EllipticE(x*(-b/a)^(1/2),(-1+(a*d+b*c)/c/b)^(1/2)))
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.54 \[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}} \, dx=-\frac {{\left (a b^{2} c^{2} + a^{2} b c d + {\left (b^{3} c d + a b^{2} d^{2}\right )} x^{4} + {\left (b^{3} c^{2} + 2 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{2}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} E(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - {\left (a b^{2} c^{2} + {\left (b^{3} c d + {\left (2 \, a^{2} b + a b^{2}\right )} d^{2}\right )} x^{4} + {\left (2 \, a^{3} + a^{2} b\right )} c d + {\left (b^{3} c^{2} + 2 \, {\left (a^{2} b + a b^{2}\right )} c d + {\left (2 \, a^{3} + a^{2} b\right )} d^{2}\right )} x^{2}\right )} \sqrt {a c} \sqrt {-\frac {b}{a}} F(\arcsin \left (x \sqrt {-\frac {b}{a}}\right )\,|\,\frac {a d}{b c}) - \sqrt {b d x^{4} + {\left (b c + a d\right )} x^{2} + a c} {\left ({\left (a b^{2} c d + a^{2} b d^{2}\right )} x^{3} + {\left (a b^{2} c^{2} + a^{3} d^{2}\right )} x\right )}}{a^{3} b^{2} c^{4} - 2 \, a^{4} b c^{3} d + a^{5} c^{2} d^{2} + {\left (a^{2} b^{3} c^{3} d - 2 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3}\right )} x^{4} + {\left (a^{2} b^{3} c^{4} - a^{3} b^{2} c^{3} d - a^{4} b c^{2} d^{2} + a^{5} c d^{3}\right )} x^{2}} \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}} \, dx=\int \frac {1}{\left (a c + b d x^{4} + x^{2} \left (a d + b c\right )\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

Integral((a*c + b*d*x**4 + x**2*(a*d + b*c))**(-3/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b d x^{4} + {\left (b c + a d\right )} x^{2} + a c\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}} \, dx=\int { \frac {1}{{\left (b d x^{4} + {\left (b c + a d\right )} x^{2} + a c\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,d\,x^4+\left (a\,d+b\,c\right )\,x^2+a\,c\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {1}{\left (a c+(b c+a d) x^2+b d x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{b^{2} d^{2} x^{8}+2 a b \,d^{2} x^{6}+2 b^{2} c d \,x^{6}+a^{2} d^{2} x^{4}+4 a b c d \,x^{4}+b^{2} c^{2} x^{4}+2 a^{2} c d \,x^{2}+2 a b \,c^{2} x^{2}+a^{2} c^{2}}d x \] Input:

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

Output:

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)