\(\int \frac {1}{(a c+(b c-a d) x^2-b d x^4)^{3/2}} \, dx\) [23]

Optimal result

Integrand size = 27, antiderivative size = 283 \[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{3/2}} \, dx=\frac {x \left (b^2 c^2+a^2 d^2-b d (b c-a d) x^2\right )}{a c (b c+a d)^2 \sqrt {a c+(b c-a d) x^2-b d x^4}}+\frac {\sqrt {d} (b c-a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{\sqrt {c} (b c+a d)^2 \sqrt {a c+(b c-a d) x^2-b d x^4}}+\frac {\sqrt {d} \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{\sqrt {c} (b c+a d) \sqrt {a c+(b c-a d) x^2-b d x^4}} \] Output:

x*(b^2*c^2+a^2*d^2-b*d*(-a*d+b*c)*x^2)/a/c/(a*d+b*c)^2/(a*c+(-a*d+b*c)*x^2 
-b*d*x^4)^(1/2)+d^(1/2)*(-a*d+b*c)*(1+b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)*Ell 
ipticE(d^(1/2)*x/c^(1/2),(-b*c/a/d)^(1/2))/c^(1/2)/(a*d+b*c)^2/(a*c+(-a*d+ 
b*c)*x^2-b*d*x^4)^(1/2)+d^(1/2)*(1+b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)*Ellipt 
icF(d^(1/2)*x/c^(1/2),(-b*c/a/d)^(1/2))/c^(1/2)/(a*d+b*c)/(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 = 6.41 (sec) , antiderivative size = 226, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} x \left (a^2 d^2+a b d^2 x^2+b^2 c \left (c-d x^2\right )\right )-i b c (-b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )-i b c (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )\right )}{b c (b c+a d)^2 \sqrt {\left (a+b x^2\right ) \left (c-d x^2\right )}} \] 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*ArcSi 
nh[Sqrt[b/a]*x], -((a*d)/(b*c))] - I*b*c*(b*c + a*d)*Sqrt[1 + (b*x^2)/a]*S 
qrt[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 [A] (verified)

Time = 0.68 (sec) , antiderivative size = 243, normalized size of antiderivative = 0.86, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1405, 25, 27, 1514, 399, 321, 327}

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.

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

\(\Big \downarrow \) 1405

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1514

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

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*Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/ 
c]*((a*Sqrt[c]*(b*c - a*d)*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/ 
(a*d))])/(b*Sqrt[d]) + (a*Sqrt[c]*(b*c + a*d)*EllipticF[ArcSin[(Sqrt[d]*x) 
/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[d])))/(a*c*(b*c + a*d)^2*Sqrt[a*c + (b 
*c - a*d)*x^2 - b*d*x^4])
 

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 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[ArcSin[Rt[-d/c, 2]*x], b*(c 
/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 
0] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

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 1514

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

Maple [A] (verified)

Time = 0.63 (sec) , antiderivative size = 440, normalized size of antiderivative = 1.55

method	result	size
default	\(\frac {2 b d \left (\frac {\left (a d -b c \right ) x^{3}}{2 a c \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )}+\frac {\left (a^{2} d^{2}+b^{2} c^{2}\right ) x}{2 a c \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) b d}\right )}{\sqrt {-\left (x^{4}+\frac {\left (a d -b c \right ) x^{2}}{b d}-\frac {a c}{b d}\right ) b d}}+\frac {\left (\frac {1}{a c}-\frac {a^{2} d^{2}+b^{2} c^{2}}{a c \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )}\right ) \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}+\frac {\left (a d -b c \right ) d \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{c \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}\)	\(440\)
elliptic	\(\frac {2 b d \left (\frac {\left (a d -b c \right ) x^{3}}{2 a c \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )}+\frac {\left (a^{2} d^{2}+b^{2} c^{2}\right ) x}{2 a c \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) b d}\right )}{\sqrt {-\left (x^{4}+\frac {\left (a d -b c \right ) x^{2}}{b d}-\frac {a c}{b d}\right ) b d}}+\frac {\left (\frac {1}{a c}-\frac {a^{2} d^{2}+b^{2} c^{2}}{a c \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )}\right ) \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}+\frac {\left (a d -b c \right ) d \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{c \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}\)	\(440\)

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.49 \[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{3/2}} \, dx=\frac {{\left (a b c^{2} d - a^{2} c d^{2} - {\left (b^{2} c d^{2} - a b d^{3}\right )} x^{4} + {\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{2}\right )} \sqrt {a c} \sqrt {\frac {d}{c}} E(\arcsin \left (x \sqrt {\frac {d}{c}}\right )\,|\,-\frac {b c}{a d}) + {\left (2 \, a b c^{3} - a b c^{2} d + a^{2} c d^{2} - {\left (2 \, b^{2} c^{2} d - b^{2} c d^{2} + a b d^{3}\right )} x^{4} + {\left (2 \, b^{2} c^{3} + 2 \, a b c d^{2} - a^{2} d^{3} - {\left (2 \, a b + b^{2}\right )} c^{2} d\right )} x^{2}\right )} \sqrt {a c} \sqrt {\frac {d}{c}} F(\arcsin \left (x \sqrt {\frac {d}{c}}\right )\,|\,-\frac {b c}{a d}) - \sqrt {-b d x^{4} + {\left (b c - a d\right )} x^{2} + a c} {\left ({\left (b^{2} c^{2} d - a b c d^{2}\right )} x^{3} - {\left (b^{2} c^{3} + a^{2} c d^{2}\right )} x\right )}}{a^{2} b^{2} c^{5} + 2 \, a^{3} b c^{4} d + a^{4} c^{3} d^{2} - {\left (a b^{3} c^{4} d + 2 \, a^{2} b^{2} c^{3} d^{2} + a^{3} b c^{2} d^{3}\right )} x^{4} + {\left (a b^{3} c^{5} + a^{2} b^{2} c^{4} d - a^{3} b c^{3} d^{2} - a^{4} c^{2} d^{3}\right )} x^{2}} \] Input:

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

Output:

((a*b*c^2*d - a^2*c*d^2 - (b^2*c*d^2 - a*b*d^3)*x^4 + (b^2*c^2*d - 2*a*b*c 
*d^2 + a^2*d^3)*x^2)*sqrt(a*c)*sqrt(d/c)*elliptic_e(arcsin(x*sqrt(d/c)), - 
b*c/(a*d)) + (2*a*b*c^3 - a*b*c^2*d + a^2*c*d^2 - (2*b^2*c^2*d - b^2*c*d^2 
 + a*b*d^3)*x^4 + (2*b^2*c^3 + 2*a*b*c*d^2 - a^2*d^3 - (2*a*b + b^2)*c^2*d 
)*x^2)*sqrt(a*c)*sqrt(d/c)*elliptic_f(arcsin(x*sqrt(d/c)), -b*c/(a*d)) - s 
qrt(-b*d*x^4 + (b*c - a*d)*x^2 + a*c)*((b^2*c^2*d - a*b*c*d^2)*x^3 - (b^2* 
c^3 + a^2*c*d^2)*x))/(a^2*b^2*c^5 + 2*a^3*b*c^4*d + a^4*c^3*d^2 - (a*b^3*c 
^4*d + 2*a^2*b^2*c^3*d^2 + a^3*b*c^2*d^3)*x^4 + (a*b^3*c^5 + a^2*b^2*c^4*d 
 - a^3*b*c^3*d^2 - a^4*c^2*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. \[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{3/2}} \, dx=\int \frac {1}{{\left (-b\,d\,x^4+\left (b\,c-a\,d\right )\,x^2+a\,c\right )}^{3/2}} \,d x \] 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)