\(\int \frac {1}{(a c+(b c-a d) x^2-b d x^4)^{5/2}} \, dx\) [24]

Optimal result

Integrand size = 27, antiderivative size = 471 \[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}} \, dx=\frac {x \left (b^2 c^2+a^2 d^2-b d (b c-a d) x^2\right )}{3 a c (b c+a d)^2 \left (a c+(b c-a d) x^2-b d x^4\right )^{3/2}}+\frac {x \left (2 b^4 c^4+9 a b^3 c^3 d-2 a^2 b^2 c^2 d^2+9 a^3 b c d^3+2 a^4 d^4-2 b d (b c-a d) \left (8 a b c d+(b c-a d)^2\right ) x^2\right )}{3 a^2 c^2 (b c+a d)^4 \sqrt {a c+(b c-a d) x^2-b d x^4}}+\frac {2 \sqrt {d} (b c-a d) \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{3 a c^{3/2} (b c+a d)^4 \sqrt {a c+(b c-a d) x^2-b d x^4}}-\frac {\sqrt {d} \left (b^2 c^2-9 a b c d-2 a^2 d^2\right ) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{3 a c^{3/2} (b c+a d)^3 \sqrt {a c+(b c-a d) x^2-b d x^4}} \] Output:

1/3*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)^(3/2)+1/3*x*(2*b^4*c^4+9*a*b^3*c^3*d-2*a^2*b^2*c^2*d^2+9*a^3 
*b*c*d^3+2*a^4*d^4-2*b*d*(-a*d+b*c)*(8*a*b*c*d+(-a*d+b*c)^2)*x^2)/a^2/c^2/ 
(a*d+b*c)^4/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(1/2)+2/3*d^(1/2)*(-a*d+b*c)*(a^2 
*d^2+6*a*b*c*d+b^2*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))/a/c^(3/2)/(a*d+b*c)^4/(a*c+(-a*d+b*c)*x^2- 
b*d*x^4)^(1/2)-1/3*d^(1/2)*(-2*a^2*d^2-9*a*b*c*d+b^2*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))/a/c^(3/2 
)/(a*d+b*c)^3/(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 = 7.66 (sec) , antiderivative size = 358, normalized size of antiderivative = 0.76 \[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}} \, dx=\frac {\sqrt {\frac {b}{a}} x \left (a^2 c d^3 (b c+a d) \left (a+b x^2\right )^2+2 a^2 d^3 (5 b c+a d) \left (a+b x^2\right )^2 \left (c-d x^2\right )+a b^3 c^2 (b c+a d) \left (c-d x^2\right )^2+2 b^3 c^2 (b c+5 a d) \left (a+b x^2\right ) \left (c-d x^2\right )^2\right )+i b c \left (a+b x^2\right ) \sqrt {1+\frac {b x^2}{a}} \left (-c+d x^2\right ) \sqrt {1-\frac {d x^2}{c}} \left (-2 \left (b^3 c^3+5 a b^2 c^2 d-5 a^2 b c d^2-a^3 d^3\right ) E\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )+\left (2 b^3 c^3+11 a b^2 c^2 d+8 a^2 b c d^2-a^3 d^3\right ) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {b}{a}} x\right ),-\frac {a d}{b c}\right )\right )}{3 a^2 \sqrt {\frac {b}{a}} c^2 (b c+a d)^4 \left (\left (a+b x^2\right ) \left (c-d x^2\right )\right )^{3/2}} \] Input:

Integrate[(a*c + (b*c - a*d)*x^2 - b*d*x^4)^(-5/2),x]
 

Output:

(Sqrt[b/a]*x*(a^2*c*d^3*(b*c + a*d)*(a + b*x^2)^2 + 2*a^2*d^3*(5*b*c + a*d 
)*(a + b*x^2)^2*(c - d*x^2) + a*b^3*c^2*(b*c + a*d)*(c - d*x^2)^2 + 2*b^3* 
c^2*(b*c + 5*a*d)*(a + b*x^2)*(c - d*x^2)^2) + I*b*c*(a + b*x^2)*Sqrt[1 + 
(b*x^2)/a]*(-c + d*x^2)*Sqrt[1 - (d*x^2)/c]*(-2*(b^3*c^3 + 5*a*b^2*c^2*d - 
 5*a^2*b*c*d^2 - a^3*d^3)*EllipticE[I*ArcSinh[Sqrt[b/a]*x], -((a*d)/(b*c)) 
] + (2*b^3*c^3 + 11*a*b^2*c^2*d + 8*a^2*b*c*d^2 - a^3*d^3)*EllipticF[I*Arc 
Sinh[Sqrt[b/a]*x], -((a*d)/(b*c))]))/(3*a^2*Sqrt[b/a]*c^2*(b*c + a*d)^4*(( 
a + b*x^2)*(c - d*x^2))^(3/2))
 

Rubi [A] (verified)

Time = 1.15 (sec) , antiderivative size = 442, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1405, 25, 1492, 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.

\(\displaystyle \int \frac {1}{\left (x^2 (b c-a d)+a c-b d x^4\right )^{5/2}} \, dx\)

\(\Big \downarrow \) 1405

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 1492

\(\displaystyle \frac {\frac {x \left (2 a^4 d^4+9 a^3 b c d^3-2 b d x^2 (b c-a d) \left (a^2 d^2+6 a b c d+b^2 c^2\right )-2 a^2 b^2 c^2 d^2+9 a b^3 c^3 d+2 b^4 c^4\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}-\frac {\int -\frac {b d \left (2 (b c-a d) \left (b^2 c^2+6 a b d c+a^2 d^2\right ) x^2+a c \left (b^2 c^2+18 a b d c+a^2 d^2\right )\right )}{\sqrt {-b d x^4+(b c-a d) x^2+a c}}dx}{a c (a d+b c)^2}}{3 a c (a d+b c)^2}+\frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {b d \left (2 (b c-a d) \left (b^2 c^2+6 a b d c+a^2 d^2\right ) x^2+a c \left (b^2 c^2+18 a b d c+a^2 d^2\right )\right )}{\sqrt {-b d x^4+(b c-a d) x^2+a c}}dx}{a c (a d+b c)^2}+\frac {x \left (2 a^4 d^4+9 a^3 b c d^3-2 b d x^2 (b c-a d) \left (a^2 d^2+6 a b c d+b^2 c^2\right )-2 a^2 b^2 c^2 d^2+9 a b^3 c^3 d+2 b^4 c^4\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}+\frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {b d \int \frac {2 (b c-a d) \left (b^2 c^2+6 a b d c+a^2 d^2\right ) x^2+a c \left (b^2 c^2+18 a b d c+a^2 d^2\right )}{\sqrt {-b d x^4+(b c-a d) x^2+a c}}dx}{a c (a d+b c)^2}+\frac {x \left (2 a^4 d^4+9 a^3 b c d^3-2 b d x^2 (b c-a d) \left (a^2 d^2+6 a b c d+b^2 c^2\right )-2 a^2 b^2 c^2 d^2+9 a b^3 c^3 d+2 b^4 c^4\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}+\frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}\)

\(\Big \downarrow \) 1514

\(\displaystyle \frac {\frac {b d \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \int \frac {2 (b c-a d) \left (b^2 c^2+6 a b d c+a^2 d^2\right ) x^2+a c \left (b^2 c^2+18 a b d c+a^2 d^2\right )}{\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}}}dx}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \left (2 a^4 d^4+9 a^3 b c d^3-2 b d x^2 (b c-a d) \left (a^2 d^2+6 a b c d+b^2 c^2\right )-2 a^2 b^2 c^2 d^2+9 a b^3 c^3 d+2 b^4 c^4\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}+\frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}\)

\(\Big \downarrow \) 399

\(\displaystyle \frac {\frac {b d \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (\frac {2 a (b c-a d) \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {d x^2}{c}}}dx}{b}-\frac {a (a d+b c) \left (-2 a^2 d^2-9 a b c d+b^2 c^2\right ) \int \frac {1}{\sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}}}dx}{b}\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \left (2 a^4 d^4+9 a^3 b c d^3-2 b d x^2 (b c-a d) \left (a^2 d^2+6 a b c d+b^2 c^2\right )-2 a^2 b^2 c^2 d^2+9 a b^3 c^3 d+2 b^4 c^4\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}+\frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}\)

\(\Big \downarrow \) 321

\(\displaystyle \frac {\frac {b d \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (\frac {2 a (b c-a d) \left (a^2 d^2+6 a b c d+b^2 c^2\right ) \int \frac {\sqrt {\frac {b x^2}{a}+1}}{\sqrt {1-\frac {d x^2}{c}}}dx}{b}-\frac {a \sqrt {c} (a d+b c) \left (-2 a^2 d^2-9 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {d}}\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \left (2 a^4 d^4+9 a^3 b c d^3-2 b d x^2 (b c-a d) \left (a^2 d^2+6 a b c d+b^2 c^2\right )-2 a^2 b^2 c^2 d^2+9 a b^3 c^3 d+2 b^4 c^4\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}+\frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {x \left (a^2 d^2-b d x^2 (b c-a d)+b^2 c^2\right )}{3 a c (a d+b c)^2 \left (x^2 (b c-a d)+a c-b d x^4\right )^{3/2}}+\frac {\frac {b d \sqrt {\frac {b x^2}{a}+1} \sqrt {1-\frac {d x^2}{c}} \left (\frac {2 a \sqrt {c} (b c-a d) \left (a^2 d^2+6 a b c d+b^2 c^2\right ) E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {d}}-\frac {a \sqrt {c} (a d+b c) \left (-2 a^2 d^2-9 a b c d+b^2 c^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {d}}\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}+\frac {x \left (2 a^4 d^4+9 a^3 b c d^3-2 b d x^2 (b c-a d) \left (a^2 d^2+6 a b c d+b^2 c^2\right )-2 a^2 b^2 c^2 d^2+9 a b^3 c^3 d+2 b^4 c^4\right )}{a c (a d+b c)^2 \sqrt {x^2 (b c-a d)+a c-b d x^4}}}{3 a c (a d+b c)^2}\)

Input:

Int[(a*c + (b*c - a*d)*x^2 - b*d*x^4)^(-5/2),x]
 

Output:

(x*(b^2*c^2 + a^2*d^2 - b*d*(b*c - a*d)*x^2))/(3*a*c*(b*c + a*d)^2*(a*c + 
(b*c - a*d)*x^2 - b*d*x^4)^(3/2)) + ((x*(2*b^4*c^4 + 9*a*b^3*c^3*d - 2*a^2 
*b^2*c^2*d^2 + 9*a^3*b*c*d^3 + 2*a^4*d^4 - 2*b*d*(b*c - a*d)*(b^2*c^2 + 6* 
a*b*c*d + a^2*d^2)*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]*((2*a*Sqrt[c]*(b*c 
 - a*d)*(b^2*c^2 + 6*a*b*c*d + a^2*d^2)*EllipticE[ArcSin[(Sqrt[d]*x)/Sqrt[ 
c]], -((b*c)/(a*d))])/(b*Sqrt[d]) - (a*Sqrt[c]*(b*c + a*d)*(b^2*c^2 - 9*a* 
b*c*d - 2*a^2*d^2)*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]))/ 
(3*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 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 1492

Int[((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symb 
ol] :> Simp[x*(a*b*e - d*(b^2 - 2*a*c) - c*(b*d - 2*a*e)*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[Simp[(2*p + 3)*d*b^2 - a*b*e - 2*a*c*d*(4*p + 5) + (4*p + 
7)*(d*b - 2*a*e)*c*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] && 
 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 = 0.64 (sec) , antiderivative size = 843, normalized size of antiderivative = 1.79

method	result	size
default	\(\frac {\left (\frac {\left (a d -b c \right ) x^{3}}{3 a c \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) b d}+\frac {\left (a^{2} d^{2}+b^{2} c^{2}\right ) x}{3 a c \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) b^{2} d^{2}}\right ) \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}{\left (x^{4}+\frac {\left (a d -b c \right ) x^{2}}{b d}-\frac {a c}{b d}\right )^{2}}+\frac {2 b d \left (\frac {\left (a d -b c \right ) \left (a^{2} d^{2}+6 a b c d +b^{2} c^{2}\right ) x^{3}}{3 a^{2} c^{2} \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )^{2}}+\frac {\left (2 a^{4} d^{4}+9 a^{3} b c \,d^{3}-2 a^{2} b^{2} c^{2} d^{2}+9 a \,b^{3} c^{3} d +2 b^{4} c^{4}\right ) x}{6 a^{2} c^{2} \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )^{2} 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 {\frac {2}{3} a^{2} d^{2}+2 a b c d +\frac {2}{3} b^{2} c^{2}}{\left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) a^{2} c^{2}}-\frac {2 a^{4} d^{4}+9 a^{3} b c \,d^{3}-2 a^{2} b^{2} c^{2} d^{2}+9 a \,b^{3} c^{3} d +2 b^{4} c^{4}}{3 a^{2} c^{2} \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )^{2}}\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 (\frac {4 b d \left (a^{3} d^{3}+5 a^{2} b c \,d^{2}-5 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{3 \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )^{2} a^{2} c^{2}}-\frac {2 b d \left (a d -b c \right ) \left (a^{2} d^{2}+6 a b c d +b^{2} c^{2}\right )}{a^{2} c^{2} \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )^{2}}\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}\)	\(843\)
elliptic	\(\frac {\left (\frac {\left (a d -b c \right ) x^{3}}{3 a c \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) b d}+\frac {\left (a^{2} d^{2}+b^{2} c^{2}\right ) x}{3 a c \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) b^{2} d^{2}}\right ) \sqrt {-b d \,x^{4}-x^{2} d a +b c \,x^{2}+a c}}{\left (x^{4}+\frac {\left (a d -b c \right ) x^{2}}{b d}-\frac {a c}{b d}\right )^{2}}+\frac {2 b d \left (\frac {\left (a d -b c \right ) \left (a^{2} d^{2}+6 a b c d +b^{2} c^{2}\right ) x^{3}}{3 a^{2} c^{2} \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )^{2}}+\frac {\left (2 a^{4} d^{4}+9 a^{3} b c \,d^{3}-2 a^{2} b^{2} c^{2} d^{2}+9 a \,b^{3} c^{3} d +2 b^{4} c^{4}\right ) x}{6 a^{2} c^{2} \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )^{2} 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 {\frac {2}{3} a^{2} d^{2}+2 a b c d +\frac {2}{3} b^{2} c^{2}}{\left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right ) a^{2} c^{2}}-\frac {2 a^{4} d^{4}+9 a^{3} b c \,d^{3}-2 a^{2} b^{2} c^{2} d^{2}+9 a \,b^{3} c^{3} d +2 b^{4} c^{4}}{3 a^{2} c^{2} \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )^{2}}\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 (\frac {4 b d \left (a^{3} d^{3}+5 a^{2} b c \,d^{2}-5 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}{3 \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )^{2} a^{2} c^{2}}-\frac {2 b d \left (a d -b c \right ) \left (a^{2} d^{2}+6 a b c d +b^{2} c^{2}\right )}{a^{2} c^{2} \left (a^{2} d^{2}+2 a b c d +b^{2} c^{2}\right )^{2}}\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}\)	\(843\)

Input:

int(1/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

(1/3/a/c*(a*d-b*c)/(a^2*d^2+2*a*b*c*d+b^2*c^2)/b/d*x^3+1/3*(a^2*d^2+b^2*c^ 
2)/a/c/(a^2*d^2+2*a*b*c*d+b^2*c^2)/b^2/d^2*x)*(-b*d*x^4-a*d*x^2+b*c*x^2+a* 
c)^(1/2)/(x^4+(a*d-b*c)/b/d*x^2-a*c/b/d)^2+2*b*d*(1/3*(a*d-b*c)*(a^2*d^2+6 
*a*b*c*d+b^2*c^2)/a^2/c^2/(a^2*d^2+2*a*b*c*d+b^2*c^2)^2*x^3+1/6*(2*a^4*d^4 
+9*a^3*b*c*d^3-2*a^2*b^2*c^2*d^2+9*a*b^3*c^3*d+2*b^4*c^4)/a^2/c^2/(a^2*d^2 
+2*a*b*c*d+b^2*c^2)^2/b/d*x)/(-(x^4+(a*d-b*c)/b/d*x^2-a*c/b/d)*b*d)^(1/2)+ 
(2/3*(a^2*d^2+3*a*b*c*d+b^2*c^2)/(a^2*d^2+2*a*b*c*d+b^2*c^2)/a^2/c^2-1/3*( 
2*a^4*d^4+9*a^3*b*c*d^3-2*a^2*b^2*c^2*d^2+9*a*b^3*c^3*d+2*b^4*c^4)/a^2/c^2 
/(a^2*d^2+2*a*b*c*d+b^2*c^2)^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))-(4/3*b*d*(a^3*d^3+5*a^2*b*c*d^2-5*a*b^2*c^2*d-b^3*c^3) 
/(a^2*d^2+2*a*b*c*d+b^2*c^2)^2/a^2/c^2-2*b*d*(a*d-b*c)*(a^2*d^2+6*a*b*c*d+ 
b^2*c^2)/a^2/c^2/(a^2*d^2+2*a*b*c*d+b^2*c^2)^2)*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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1472 vs. \(2 (434) = 868\).

Time = 0.17 (sec) , antiderivative size = 1472, normalized size of antiderivative = 3.13 \[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

integrate(1/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(5/2),x, algorithm="fricas")
 

Output:

1/3*(2*(a^2*b^3*c^5*d + 5*a^3*b^2*c^4*d^2 - 5*a^4*b*c^3*d^3 - a^5*c^2*d^4 
+ (b^5*c^3*d^3 + 5*a*b^4*c^2*d^4 - 5*a^2*b^3*c*d^5 - a^3*b^2*d^6)*x^8 - 2* 
(b^5*c^4*d^2 + 4*a*b^4*c^3*d^3 - 10*a^2*b^3*c^2*d^4 + 4*a^3*b^2*c*d^5 + a^ 
4*b*d^6)*x^6 + (b^5*c^5*d + a*b^4*c^4*d^2 - 24*a^2*b^3*c^3*d^3 + 24*a^3*b^ 
2*c^2*d^4 - a^4*b*c*d^5 - a^5*d^6)*x^4 + 2*(a*b^4*c^5*d + 4*a^2*b^3*c^4*d^ 
2 - 10*a^3*b^2*c^3*d^3 + 4*a^4*b*c^2*d^4 + a^5*c*d^5)*x^2)*sqrt(a*c)*sqrt( 
d/c)*elliptic_e(arcsin(x*sqrt(d/c)), -b*c/(a*d)) + (a^2*b^3*c^6 + 10*a^4*b 
*c^3*d^3 + 2*a^5*c^2*d^4 + (b^5*c^4*d^2 + 10*a^2*b^3*c*d^5 + 2*a^3*b^2*d^6 
 + 2*(9*a*b^4 - b^5)*c^3*d^3 + (a^2*b^3 - 10*a*b^4)*c^2*d^4)*x^8 + 2*(9*a^ 
3*b^2 - a^2*b^3)*c^5*d + (a^4*b - 10*a^3*b^2)*c^4*d^2 - 2*(b^5*c^5*d - 8*a 
^3*b^2*c*d^5 - 2*a^4*b*d^6 + (17*a*b^4 - 2*b^5)*c^4*d^2 - (17*a^2*b^3 + 8* 
a*b^4)*c^3*d^3 - (a^3*b^2 - 20*a^2*b^3)*c^2*d^4)*x^6 + (b^5*c^6 + 2*a^4*b* 
c*d^5 + 2*a^5*d^6 + 2*(7*a*b^4 - b^5)*c^5*d - 2*(35*a^2*b^3 + a*b^4)*c^4*d 
^2 + 2*(7*a^3*b^2 + 24*a^2*b^3)*c^3*d^3 + (a^4*b - 48*a^3*b^2)*c^2*d^4)*x^ 
4 + 2*(a*b^4*c^6 - 8*a^4*b*c^2*d^4 - 2*a^5*c*d^5 + (17*a^2*b^3 - 2*a*b^4)* 
c^5*d - (17*a^3*b^2 + 8*a^2*b^3)*c^4*d^2 - (a^4*b - 20*a^3*b^2)*c^3*d^3)*x 
^2)*sqrt(a*c)*sqrt(d/c)*elliptic_f(arcsin(x*sqrt(d/c)), -b*c/(a*d)) + (2*( 
b^5*c^4*d^2 + 5*a*b^4*c^3*d^3 - 5*a^2*b^3*c^2*d^4 - a^3*b^2*c*d^5)*x^7 - ( 
4*b^5*c^5*d + 17*a*b^4*c^4*d^2 - 22*a^2*b^3*c^3*d^3 + 17*a^3*b^2*c^2*d^4 + 
 4*a^4*b*c*d^5)*x^5 + 2*(b^5*c^6 + 2*a*b^4*c^5*d - 11*a^2*b^3*c^4*d^2 +...
 

Sympy [F]

\[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}} \, dx=\int \frac {1}{\left (a c - b d x^{4} + x^{2} \left (- a d + b c\right )\right )^{\frac {5}{2}}}\, dx \] Input:

integrate(1/(a*c+(-a*d+b*c)*x**2-b*d*x**4)**(5/2),x)
 

Output:

Integral((a*c - b*d*x**4 + x**2*(-a*d + b*c))**(-5/2), x)
 

Maxima [F]

\[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-b d x^{4} + {\left (b c - a d\right )} x^{2} + a c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(5/2),x, algorithm="maxima")
 

Output:

integrate((-b*d*x^4 + (b*c - a*d)*x^2 + a*c)^(-5/2), x)
 

Giac [F]

\[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}} \, dx=\int { \frac {1}{{\left (-b d x^{4} + {\left (b c - a d\right )} x^{2} + a c\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate(1/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(5/2),x, algorithm="giac")
 

Output:

integrate((-b*d*x^4 + (b*c - a*d)*x^2 + a*c)^(-5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}} \, dx=\int \frac {1}{{\left (-b\,d\,x^4+\left (b\,c-a\,d\right )\,x^2+a\,c\right )}^{5/2}} \,d x \] Input:

int(1/(a*c - x^2*(a*d - b*c) - b*d*x^4)^(5/2),x)
 

Output:

int(1/(a*c - x^2*(a*d - b*c) - b*d*x^4)^(5/2), x)
 

Reduce [F]

\[ \int \frac {1}{\left (a c+(b c-a d) x^2-b d x^4\right )^{5/2}} \, dx=\int \frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}}{-b^{3} d^{3} x^{12}-3 a \,b^{2} d^{3} x^{10}+3 b^{3} c \,d^{2} x^{10}-3 a^{2} b \,d^{3} x^{8}+9 a \,b^{2} c \,d^{2} x^{8}-3 b^{3} c^{2} d \,x^{8}-a^{3} d^{3} x^{6}+9 a^{2} b c \,d^{2} x^{6}-9 a \,b^{2} c^{2} d \,x^{6}+b^{3} c^{3} x^{6}+3 a^{3} c \,d^{2} x^{4}-9 a^{2} b \,c^{2} d \,x^{4}+3 a \,b^{2} c^{3} x^{4}-3 a^{3} c^{2} d \,x^{2}+3 a^{2} b \,c^{3} x^{2}+a^{3} c^{3}}d x \] Input:

int(1/(a*c+(-a*d+b*c)*x^2-b*d*x^4)^(5/2),x)
 

Output:

int((sqrt(c - d*x**2)*sqrt(a + b*x**2))/(a**3*c**3 - 3*a**3*c**2*d*x**2 + 
3*a**3*c*d**2*x**4 - a**3*d**3*x**6 + 3*a**2*b*c**3*x**2 - 9*a**2*b*c**2*d 
*x**4 + 9*a**2*b*c*d**2*x**6 - 3*a**2*b*d**3*x**8 + 3*a*b**2*c**3*x**4 - 9 
*a*b**2*c**2*d*x**6 + 9*a*b**2*c*d**2*x**8 - 3*a*b**2*d**3*x**10 + b**3*c* 
*3*x**6 - 3*b**3*c**2*d*x**8 + 3*b**3*c*d**2*x**10 - b**3*d**3*x**12),x)