\(\int \frac {A+B x^2+C x^4}{(a c+(b c-a d) x^2-b d x^4)^{3/2}} \, dx\) [13]

Optimal result

Integrand size = 40, antiderivative size = 378 \[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c-a d) x^2-b d x^4\right )^{3/2}} \, dx=\frac {x \left (A \left (b^2 c^2+a^2 d^2\right )-a c (b B c-a (2 c C+B d))-\left (A b^2 c d+a^2 c C d-a b \left (c^2 C+2 B c d+A d^2\right )\right ) x^2\right )}{a c (b c+a d)^2 \sqrt {a c+(b c-a d) x^2-b d x^4}}+\frac {\left (A b^2 c d+a^2 c C d-a b \left (c^2 C+2 B c d+A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {c} \sqrt {d} (b c+a d)^2 \sqrt {a c+(b c-a d) x^2-b d x^4}}+\frac {(b B c-a c C+A b d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {c} \sqrt {d} (b c+a d) \sqrt {a c+(b c-a d) x^2-b d x^4}} \] Output:

x*(A*(a^2*d^2+b^2*c^2)-a*c*(B*b*c-a*(B*d+2*C*c))-(A*b^2*c*d+a^2*c*C*d-a*b* 
(A*d^2+2*B*c*d+C*c^2))*x^2)/a/c/(a*d+b*c)^2/(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))*(1+b*x^2/a)^(1/2)*(1- 
d*x^2/c)^(1/2)*EllipticE(d^(1/2)*x/c^(1/2),(-b*c/a/d)^(1/2))/b/c^(1/2)/d^( 
1/2)/(a*d+b*c)^2/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2)+(A*b*d+B*b*c-C*a*c)*(1 
+b*x^2/a)^(1/2)*(1-d*x^2/c)^(1/2)*EllipticF(d^(1/2)*x/c^(1/2),(-b*c/a/d)^( 
1/2))/b/c^(1/2)/d^(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 = 12.36 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.83 \[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c-a d) x^2-b d x^4\right )^{3/2}} \, dx=\frac {\sqrt {\frac {b}{a}} \left (\sqrt {\frac {b}{a}} d x \left (A \left (a^2 d^2+a b d^2 x^2+b^2 c \left (c-d x^2\right )\right )+a c \left (2 a c C+a d \left (B-C x^2\right )+b \left (-B c+c C x^2+2 B d x^2\right )\right )\right )+i c \left (A b^2 c d+a^2 c C d-a b \left (c^2 C+2 B c d+A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i c (b c+a d) (a c C-A b d+a B d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )\right )}{b c d (b c+a d)^2 \sqrt {\left (a+b x^2\right ) \left (c-d x^2\right )}} \] 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*(2*a*c*C + a*d*(B - C*x^2) + b*(-(B*c) + c*C*x^2 + 2*B*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 [A] (verified)

Time = 0.64 (sec) , antiderivative size = 330, normalized size of antiderivative = 0.87, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2206, 25, 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.

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

\(\Big \downarrow \) 2206

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1514

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

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*(b^2*c^2 + a^2*d^2) - a*c*(b*B*c - a*(2*c*C + B*d)) - (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]) + (Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c] 
*((a*Sqrt[c]*(A*b^2*c*d + a^2*c*C*d - a*b*(c^2*C + 2*B*c*d + A*d^2))*Ellip 
ticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[d]) + (a*Sqrt[c 
]*(b*c + a*d)*(b*B*c - a*c*C + A*b*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 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 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]
 

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 = 3.70 (sec) , antiderivative size = 584, normalized size of antiderivative = 1.54

method	result	size
elliptic	\(\frac {2 b d \left (\frac {\left (A a b \,d^{2}-A \,b^{2} c d +2 B a c d b -C \,a^{2} c d +a b \,c^{2} C \right ) x^{3}}{2 b d a c \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )}+\frac {\left (A \,a^{2} d^{2}+A \,b^{2} c^{2}+B \,a^{2} c d -B a b \,c^{2}+2 C \,a^{2} c^{2}\right ) x}{2 b d a c \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )}\right )}{\sqrt {-\left (x^{4}+\frac {\left (a d -b c \right ) x^{2}}{b d}-\frac {a c}{b d}\right ) b d}}+\frac {\left (-\frac {C}{b d}+\frac {A b d +C a c}{b d a c}-\frac {A \,a^{2} d^{2}+A \,b^{2} c^{2}+B \,a^{2} c d -B a b \,c^{2}+2 C \,a^{2} c^{2}}{a c \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )}\right ) \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}+\frac {\left (A a b \,d^{2}-A \,b^{2} c d +2 B a c d b -C \,a^{2} c d +a b \,c^{2} C \right ) \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{c \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) \sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}\, b}\)	\(584\)
default	\(\text {Expression too large to display}\)	\(1219\)

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=_RETURNVER 
BOSE)
 

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))/(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))+(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)/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)/b*(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 811 vs. \(2 (352) = 704\).

Time = 0.11 (sec) , antiderivative size = 811, normalized size of antiderivative = 2.15 \[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c-a d) x^2-b d x^4\right )^{3/2}} \, dx =\text {Too large to display} \] 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*c^3*d + A*a^2*b*c*d^3 - (C*a^3 - 2*B*a^2*b + A*a*b^2)*c^2*d^2 - 
 (C*a*b^2*c^2*d^2 + A*a*b^2*d^4 - (C*a^2*b - 2*B*a*b^2 + A*b^3)*c*d^3)*x^4 
 + (C*a*b^2*c^3*d - A*a^2*b*d^4 - (2*C*a^2*b - 2*B*a*b^2 + A*b^3)*c^2*d^2 
+ (C*a^3 - 2*B*a^2*b + 2*A*a*b^2)*c*d^3)*x^2)*sqrt(a*c)*sqrt(d/c)*elliptic 
_e(arcsin(x*sqrt(d/c)), -b*c/(a*d)) - (A*a^2*b*c*d^3 - (2*C*a^2*b - B*a*b^ 
2)*c^4 - ((B - C)*a^2*b - 2*A*a*b^2)*c^3*d - (C*a^3 - 2*B*a^2*b + A*a*b^2) 
*c^2*d^2 - (A*a*b^2*d^4 - (2*C*a*b^2 - B*b^3)*c^3*d - ((B - C)*a*b^2 - 2*A 
*b^3)*c^2*d^2 - (C*a^2*b - 2*B*a*b^2 + A*b^3)*c*d^3)*x^4 - (A*a^2*b*d^4 + 
(2*C*a*b^2 - B*b^3)*c^4 - (2*C*a^2*b - (2*B - C)*a*b^2 + 2*A*b^3)*c^3*d - 
((B - 2*C)*a^2*b - 2*(A - B)*a*b^2 - A*b^3)*c^2*d^2 - (C*a^3 - 2*B*a^2*b + 
 2*A*a*b^2)*c*d^3)*x^2)*sqrt(a*c)*sqrt(d/c)*elliptic_f(arcsin(x*sqrt(d/c)) 
, -b*c/(a*d)) - sqrt(-b*d*x^4 + (b*c - a*d)*x^2 + a*c)*((C*a*b^2*c^3*d + A 
*a*b^2*c*d^3 - (C*a^2*b - 2*B*a*b^2 + A*b^3)*c^2*d^2)*x^3 + (B*a^2*b*c^2*d 
^2 + A*a^2*b*c*d^3 + (2*C*a^2*b - B*a*b^2 + A*b^3)*c^3*d)*x))/(a^2*b^3*c^5 
*d + 2*a^3*b^2*c^4*d^2 + a^4*b*c^3*d^3 - (a*b^4*c^4*d^2 + 2*a^2*b^3*c^3*d^ 
3 + a^3*b^2*c^2*d^4)*x^4 + (a*b^4*c^5*d + a^2*b^3*c^4*d^2 - a^3*b^2*c^3*d^ 
3 - a^4*b*c^2*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. \[ \int \frac {A+B x^2+C x^4}{\left (a c+(b c-a d) x^2-b d x^4\right )^{3/2}} \, dx=\int \frac {C\,x^4+B\,x^2+A}{{\left (-b\,d\,x^4+\left (b\,c-a\,d\right )\,x^2+a\,c\right )}^{3/2}} \,d x \] 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 - 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**...