\(\int \frac {(c d-b e-c e x^2)^2}{(\frac {-c d^2+b d e}{e^2}+b x^2+c x^4)^{3/2}} \, dx\) [17]

Optimal result

Integrand size = 48, antiderivative size = 124 \[ \int \frac {\left (c d-b e-c e x^2\right )^2}{\left (\frac {-c d^2+b d e}{e^2}+b x^2+c x^4\right )^{3/2}} \, dx=-\frac {e^{3/2} (c d-b e) \left (d+e x^2\right ) \sqrt {\frac {d \left (1-\frac {c e x^2}{c d-b e}\right )}{d+e x^2}} E\left (\arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )|\frac {2 c d-b e}{c d-b e}\right )}{d^{3/2} \sqrt {-\frac {d (c d-b e)}{e^2}+b x^2+c x^4}} \] Output:

-e^(3/2)*(-b*e+c*d)*(e*x^2+d)*(d*(1-c*e*x^2/(-b*e+c*d))/(e*x^2+d))^(1/2)*E 
llipticE(e^(1/2)*x/d^(1/2)/(1+e*x^2/d)^(1/2),((-b*e+2*c*d)/(-b*e+c*d))^(1/ 
2))/d^(3/2)/(-d*(-b*e+c*d)/e^2+b*x^2+c*x^4)^(1/2)
                                                                                    
                                                                                    
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.33 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.90 \[ \int \frac {\left (c d-b e-c e x^2\right )^2}{\left (\frac {-c d^2+b d e}{e^2}+b x^2+c x^4\right )^{3/2}} \, dx=\frac {e \sqrt {\frac {e}{d}} \left (\sqrt {\frac {e}{d}} x \left (-c d+b e+c e x^2\right )-i (c d-b e) \sqrt {\frac {-c d+b e+c e x^2}{-c d+b e}} \sqrt {1+\frac {e x^2}{d}} E\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right )|\frac {c d}{-c d+b e}\right )+i (c d-b e) \sqrt {\frac {-c d+b e+c e x^2}{-c d+b e}} \sqrt {1+\frac {e x^2}{d}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {e}{d}} x\right ),\frac {c d}{-c d+b e}\right )\right )}{\sqrt {\frac {\left (d+e x^2\right ) \left (-c d+b e+c e x^2\right )}{e^2}}} \] Input:

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

Output:

(e*Sqrt[e/d]*(Sqrt[e/d]*x*(-(c*d) + b*e + c*e*x^2) - I*(c*d - b*e)*Sqrt[(- 
(c*d) + b*e + c*e*x^2)/(-(c*d) + b*e)]*Sqrt[1 + (e*x^2)/d]*EllipticE[I*Arc 
Sinh[Sqrt[e/d]*x], (c*d)/(-(c*d) + b*e)] + I*(c*d - b*e)*Sqrt[(-(c*d) + b* 
e + c*e*x^2)/(-(c*d) + b*e)]*Sqrt[1 + (e*x^2)/d]*EllipticF[I*ArcSinh[Sqrt[ 
e/d]*x], (c*d)/(-(c*d) + b*e)]))/Sqrt[((d + e*x^2)*(-(c*d) + b*e + c*e*x^2 
))/e^2]
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(399\) vs. \(2(124)=248\).

Time = 0.69 (sec) , antiderivative size = 399, normalized size of antiderivative = 3.22, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.229, Rules used = {1395, 314, 25, 27, 389, 323, 323, 321, 331, 330, 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.

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

\(\Big \downarrow \) 1395

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

\(\Big \downarrow \) 314

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 389

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(Sqrt[-(d/e^2) - x^2/e]*Sqrt[c*d - b*e - c*e*x^2]*(-((e^2*x*Sqrt[c*d - b*e 
 - c*e*x^2])/(d*Sqrt[-(d/e^2) - x^2/e])) - (c*e^3*(-((Sqrt[e]*Sqrt[c*d - b 
*e]*Sqrt[-(d/e^2) - x^2/e]*Sqrt[1 - (c*e*x^2)/(c*d - b*e)]*EllipticE[ArcSi 
n[(Sqrt[c]*Sqrt[e]*x)/Sqrt[c*d - b*e]], -1 + (b*e)/(c*d)])/(Sqrt[c]*Sqrt[c 
*d - b*e - c*e*x^2]*Sqrt[1 + (e*x^2)/d])) - (d*Sqrt[c*d - b*e]*Sqrt[1 + (e 
*x^2)/d]*Sqrt[1 - (c*e*x^2)/(c*d - b*e)]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[e] 
*x)/Sqrt[c*d - b*e]], -1 + (b*e)/(c*d)])/(Sqrt[c]*e^(3/2)*Sqrt[-(d/e^2) - 
x^2/e]*Sqrt[c*d - b*e - c*e*x^2])))/d))/Sqrt[-((d*(c*d - b*e))/e^2) + b*x^ 
2 + c*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 314

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] + Simp[1/(2*a* 
(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*p + 3) + d 
*(2*(p + q + 1) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, 
 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 323

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

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 330

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

rule 331

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

rule 389

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] 
 :> Simp[1/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] - Simp[a/b   Int 
[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && N 
eQ[b*c - a*d, 0] &&  !SimplerSqrtQ[-b/a, -d/c]
 

rule 1395

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e_.)*( 
x_)^(n_))^(q_.), x_Symbol] :> Simp[(a + b*x^n + c*x^(2*n))^FracPart[p]/((d 
+ e*x^n)^FracPart[p]*(a/d + c*(x^n/e))^FracPart[p])   Int[u*(d + e*x^n)^(p 
+ q)*(a/d + (c/e)*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && E 
qQ[n2, 2*n] && EqQ[c*d^2 - b*d*e + a*e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 
1] && EqQ[n, 2])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(382\) vs. \(2(125)=250\).

Time = 12.12 (sec) , antiderivative size = 383, normalized size of antiderivative = 3.09

method	result	size
elliptic	\(\frac {\left (c e \,x^{2}+b e -c d \right ) e^{2} x}{d \sqrt {\frac {\left (x^{2}+\frac {d}{e}\right ) \left (c e \,x^{2}+b e -c d \right )}{e}}}+\frac {\left (c \,e^{2}+\frac {\left (b e -2 c d \right ) e^{2}}{d}-\frac {\left (b e -c d \right ) e^{2}}{d}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )}{\sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}}+\frac {2 e^{3} c \left (\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}\right ) \sqrt {1+\frac {c e \,x^{2}}{b e -c d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {c e}{b e -c d}}, \sqrt {-1+\frac {b e}{c d}}\right )\right )}{d \sqrt {-\frac {c e}{b e -c d}}\, \sqrt {c \,x^{4}+b \,x^{2}+\frac {b d}{e}-\frac {c \,d^{2}}{e^{2}}}\, \left (b +\frac {b e -2 c d}{e}\right )}\)	\(383\)
default	\(\text {Expression too large to display}\)	\(1497\)

Input:

int((-c*e*x^2-b*e+c*d)^2/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(3/2),x,method=_R 
ETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.95 \[ \int \frac {\left (c d-b e-c e x^2\right )^2}{\left (\frac {-c d^2+b d e}{e^2}+b x^2+c x^4\right )^{3/2}} \, dx=\frac {{\left (c e^{4} x^{2} + c d e^{3}\right )} \sqrt {-c d^{2} + b d e} \sqrt {\frac {c e}{c d - b e}} E(\arcsin \left (\sqrt {\frac {c e}{c d - b e}} x\right )\,|\,-\frac {c d - b e}{c d}) - {\left (c e^{4} x^{2} + c d e^{3}\right )} \sqrt {-c d^{2} + b d e} \sqrt {\frac {c e}{c d - b e}} F(\arcsin \left (\sqrt {\frac {c e}{c d - b e}} x\right )\,|\,-\frac {c d - b e}{c d}) + {\left (c d e^{4} - b e^{5}\right )} x \sqrt {\frac {c e^{2} x^{4} + b e^{2} x^{2} - c d^{2} + b d e}{e^{2}}}}{c d^{3} - b d^{2} e + {\left (c d^{2} e - b d e^{2}\right )} x^{2}} \] Input:

integrate((-c*e*x^2-b*e+c*d)^2/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(3/2),x, al 
gorithm="fricas")
 

Output:

((c*e^4*x^2 + c*d*e^3)*sqrt(-c*d^2 + b*d*e)*sqrt(c*e/(c*d - b*e))*elliptic 
_e(arcsin(sqrt(c*e/(c*d - b*e))*x), -(c*d - b*e)/(c*d)) - (c*e^4*x^2 + c*d 
*e^3)*sqrt(-c*d^2 + b*d*e)*sqrt(c*e/(c*d - b*e))*elliptic_f(arcsin(sqrt(c* 
e/(c*d - b*e))*x), -(c*d - b*e)/(c*d)) + (c*d*e^4 - b*e^5)*x*sqrt((c*e^2*x 
^4 + b*e^2*x^2 - c*d^2 + b*d*e)/e^2))/(c*d^3 - b*d^2*e + (c*d^2*e - b*d*e^ 
2)*x^2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((-c*e*x^2-b*e+c*d)^2/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(3/2),x, al 
gorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((-c*e*x^2-b*e+c*d)^2/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(3/2),x, al 
gorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

int((b*e - c*d + c*e*x^2)^2/(b*x^2 - (c*d^2 - b*d*e)/e^2 + c*x^4)^(3/2),x)
 

Output:

int((b*e - c*d + c*e*x^2)^2/(b*x^2 - (c*d^2 - b*d*e)/e^2 + c*x^4)^(3/2), x 
)
 

Reduce [F]

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

int((-c*e*x^2-b*e+c*d)^2/((b*d*e-c*d^2)/e^2+b*x^2+c*x^4)^(3/2),x)
 

Output:

int((sqrt(d + e*x**2)*sqrt(b*e - c*d + c*e*x**2))/(d**2 + 2*d*e*x**2 + e** 
2*x**4),x)*e**3