\(\int (A+B x^2+C x^4) \sqrt {a c+(b c-a d) x^2-b d x^4} \, dx\) [11]

Optimal result

Integrand size = 40, antiderivative size = 463 \[ \int \left (A+B x^2+C x^4\right ) \sqrt {a c+(b c-a d) x^2-b d x^4} \, dx=\frac {x \left (5 b d (a c C+7 A b d)-(b c-a d) (4 b c C+7 b B d-4 a C d)+3 b d (4 b c C+7 b B d-4 a C d) x^2\right ) \sqrt {a c+(b c-a d) x^2-b d x^4}}{105 b^2 d^2}-\frac {C x \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2}}{7 b d}+\frac {a \sqrt {c} \left (5 b d (b c-a d) (a c C+7 A b d)+6 a b c d (4 b c C+7 b B d-4 a C d)+2 (b c-a d)^2 (4 b c C+7 b B d-4 a C d)\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{105 b^3 d^{5/2} \sqrt {a c+(b c-a d) x^2-b d x^4}}+\frac {a \sqrt {c} (b c+a d) \left (8 a^2 C d^2+a b d (c C-14 B d)-b^2 \left (4 c^2 C+7 B c d-35 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{105 b^3 d^{5/2} \sqrt {a c+(b c-a d) x^2-b d x^4}} \] Output:

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*a*c^(1/2)*(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))*(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^3/d^(5/2)/(a*c+(-a*d+b*c)*x^2-b*d*x^4 
)^(1/2)+1/105*a*c^(1/2)*(a*d+b*c)*(8*a^2*C*d^2+a*b*d*(-14*B*d+C*c)-b^2*(-3 
5*A*d^2+7*B*c*d+4*C*c^2))*(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^3/d^(5/2)/(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 = 4.86 (sec) , antiderivative size = 393, normalized size of antiderivative = 0.85 \[ \int \left (A+B x^2+C x^4\right ) \sqrt {a c+(b c-a d) x^2-b d x^4} \, dx=\frac {\sqrt {\frac {b}{a}} d x \left (a+b x^2\right ) \left (-c+d x^2\right ) \left (4 a^2 C d^2+a b d \left (2 c C-7 B d-3 C d x^2\right )+b^2 \left (4 c^2 C+c d \left (7 B+3 C x^2\right )-d^2 \left (35 A+21 B x^2+15 C x^4\right )\right )\right )-i c \left (8 a^3 C d^3+a^2 b d^2 (5 c C-14 B d)+a b^2 d \left (-5 c^2 C-14 B c d+35 A d^2\right )-b^3 c \left (8 c^2 C+14 B c d+35 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+i c (b c+a d) \left (4 a^2 C d^2-a b d (c C+7 B d)-b^2 \left (8 c^2 C+14 B c d+35 A d^2\right )\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )}{105 a^2 \left (\frac {b}{a}\right )^{5/2} d^3 \sqrt {\left (a+b x^2\right ) \left (c-d x^2\right )}} \] 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) + a*b^2*d*( 
-5*c^2*C - 14*B*c*d + 35*A*d^2) - b^3*c*(8*c^2*C + 14*B*c*d + 35*A*d^2))*S 
qrt[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]*E 
llipticF[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 = 0.81 (sec) , antiderivative size = 423, normalized size of antiderivative = 0.91, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2207, 25, 1490, 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.

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

\(\Big \downarrow \) 2207

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1490

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1514

\(\displaystyle \frac {\frac {\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \int \frac {\left (2 (4 b c C-4 a d C+7 b B d) (b c-a d)^2+5 b d (a c C+7 A b d) (b c-a d)+6 a b c d (4 b c C-4 a d C+7 b B d)\right ) x^2+a c (10 b d (a c C+7 A b d)+(b c-a d) (4 b c C-4 a d C+7 b B d))}{\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}}}dx}{15 b d \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \sqrt {x^2 (b c-a d)+a c-b d x^4} \left (5 b d (a c C+7 A b d)+3 b d x^2 (-4 a C d+7 b B d+4 b c C)-(b c-a d) (-4 a C d+7 b B d+4 b c C)\right )}{15 b d}}{7 b d}-\frac {C x \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}{7 b d}\)

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

-1/7*(C*x*(a*c + (b*c - a*d)*x^2 - b*d*x^4)^(3/2))/(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) 
 + (Sqrt[1 + (b*x^2)/a]*Sqrt[1 - (d*x^2)/c]*((a*Sqrt[c]*(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))*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[c]], 
 -((b*c)/(a*d))])/(b*Sqrt[d]) + (a*Sqrt[c]*(b*c + a*d)*(8*a^2*C*d^2 + a*b* 
d*(c*C - 14*B*d) - b^2*(4*c^2*C + 7*B*c*d - 35*A*d^2))*EllipticF[ArcSin[(S 
qrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[d])))/(15*b*d*Sqrt[a*c + (b*c 
 - a*d)*x^2 - b*d*x^4]))/(7*b*d)
 

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 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 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 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 = 11.38 (sec) , antiderivative size = 624, normalized size of antiderivative = 1.35

method	result	size
elliptic	\(\frac {C \,x^{5} \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}{7}-\frac {\left (-B b d -a C d +C c b -\frac {C \left (-6 a d +6 b c \right )}{7}\right ) x^{3} \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}{5 b d}-\frac {\left (-A b d -B a d +B b c +\frac {2 C a c}{7}+\frac {\left (-B b d -a C d +C c b -\frac {C \left (-6 a d +6 b c \right )}{7}\right ) \left (-4 a d +4 b c \right )}{5 b d}\right ) x \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}{3 b d}+\frac {\left (A a c +\frac {\left (-A b d -B a d +B b c +\frac {2 C a c}{7}+\frac {\left (-B b d -a C d +C c b -\frac {C \left (-6 a d +6 b c \right )}{7}\right ) \left (-4 a d +4 b c \right )}{5 b d}\right ) a c}{3 b d}\right ) \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}-\frac {\left (-A a d +A b c +B a c +\frac {3 \left (-B b d -a C d +C c b -\frac {C \left (-6 a d +6 b c \right )}{7}\right ) a c}{5 b d}+\frac {\left (-A b d -B a d +B b c +\frac {2 C a c}{7}+\frac {\left (-B b d -a C d +C c b -\frac {C \left (-6 a d +6 b c \right )}{7}\right ) \left (-4 a d +4 b c \right )}{5 b d}\right ) \left (-2 a d +2 b c \right )}{3 b d}\right ) a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}\, b}\)	\(624\)
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 ) a \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 )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}\, b}+\frac {4 C a \,b^{2} c^{3} \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 {4 a^{3} c C \,d^{2} \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 {70 A a \,b^{2} c \,d^{2} \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 {7 B a \,b^{2} c^{2} d \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 {7 B \,a^{2} b c \,d^{2} \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 {2 C \,a^{2} b \,c^{2} d \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}}}{105 b^{2} d^{2}}\)	\(962\)
default	\(\text {Expression too large to display}\)	\(1075\)

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=_RETURNVER 
BOSE)
 

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)/(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*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))*a/(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 [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 518, normalized size of antiderivative = 1.12 \[ \int \left (A+B x^2+C x^4\right ) \sqrt {a c+(b c-a d) x^2-b d x^4} \, dx=-\frac {{\left (8 \, C b^{3} c^{4} + {\left (5 \, C a b^{2} + 14 \, B b^{3}\right )} c^{3} d - {\left (5 \, C a^{2} b - 14 \, B a b^{2} - 35 \, A b^{3}\right )} c^{2} d^{2} - {\left (8 \, C a^{3} - 14 \, B a^{2} b + 35 \, A a b^{2}\right )} c d^{3}\right )} \sqrt {-b d} x \sqrt {\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c}) - {\left (8 \, C b^{3} c^{4} + {\left (5 \, C a b^{2} + 14 \, B b^{3}\right )} c^{3} d - {\left (5 \, C a^{2} b - 2 \, {\left (7 \, B + 2 \, C\right )} a b^{2} - 35 \, A b^{3}\right )} c^{2} d^{2} - {\left (8 \, C a^{3} - 2 \, {\left (7 \, B + C\right )} a^{2} b + 7 \, {\left (5 \, A - B\right )} a b^{2}\right )} c d^{3} + {\left (4 \, C a^{3} - 7 \, B a^{2} b + 70 \, A a b^{2}\right )} d^{4}\right )} \sqrt {-b d} x \sqrt {\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c}) - {\left (15 \, C b^{3} d^{4} x^{6} - 8 \, C b^{3} c^{3} d - {\left (5 \, C a b^{2} + 14 \, B b^{3}\right )} c^{2} d^{2} + {\left (5 \, C a^{2} b - 14 \, B a b^{2} - 35 \, A b^{3}\right )} c d^{3} + {\left (8 \, C a^{3} - 14 \, B a^{2} b + 35 \, A a b^{2}\right )} d^{4} - 3 \, {\left (C b^{3} c d^{3} - {\left (C a b^{2} + 7 \, B b^{3}\right )} d^{4}\right )} x^{4} - {\left (4 \, C b^{3} c^{2} d^{2} + {\left (2 \, C a b^{2} + 7 \, B b^{3}\right )} c d^{3} + {\left (4 \, C a^{2} b - 7 \, B a b^{2} - 35 \, A b^{3}\right )} d^{4}\right )} x^{2}\right )} \sqrt {-b d x^{4} + {\left (b c - a d\right )} x^{2} + a c}}{105 \, b^{3} d^{4} x} \] 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. \[ \int \left (A+B x^2+C x^4\right ) \sqrt {a c+(b c-a d) x^2-b d x^4} \, dx=\int \sqrt {-b\,d\,x^4+\left (b\,c-a\,d\right )\,x^2+a\,c}\,\left (C\,x^4+B\,x^2+A\right ) \,d x \] 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 + ...