\(\int \frac {\sqrt {d^2-e^2 x^4}}{d+e x^2} \, dx\) [18]

Optimal result

Integrand size = 26, antiderivative size = 121 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{d+e x^2} \, dx=-\frac {d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {2 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 10.26 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{d+e x^2} \, dx=\frac {i d \sqrt {1-\frac {e^2 x^4}{d^2}} \left (E\left (\left .i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right )\right |-1\right )-2 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )\right )}{\sqrt {-\frac {e}{d}} \sqrt {d^2-e^2 x^4}} \] Input:

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

Output:

(I*d*Sqrt[1 - (e^2*x^4)/d^2]*(EllipticE[I*ArcSinh[Sqrt[-(e/d)]*x], -1] - 2 
*EllipticF[I*ArcSinh[Sqrt[-(e/d)]*x], -1]))/(Sqrt[-(e/d)]*Sqrt[d^2 - e^2*x 
^4])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.45, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1396, 326, 289, 329, 327, 765, 762}

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.

\(\displaystyle \int \frac {\sqrt {d^2-e^2 x^4}}{d+e x^2} \, dx\)

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 326

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

\(\Big \downarrow \) 289

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

\(\Big \downarrow \) 329

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

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2 d \sqrt {d^2-e^2 x^4} \int \frac {1}{\sqrt {d^2-e^2 x^4}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 765

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2 d \sqrt {1-\frac {e^2 x^4}{d^2}} \int \frac {1}{\sqrt {1-\frac {e^2 x^4}{d^2}}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 762

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

Input:

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

Output:

(Sqrt[d^2 - e^2*x^4]*(-((d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticE[ArcSin[ 
(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2])) + (2 
*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1 
])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2])))/(Sqrt[d - e*x^2]*Sqrt[d + e 
*x^2])
 

Defintions of rubi rules used

rule 289

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim 
p[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c + b*d*x^4)^FracPart 
[p])   Int[(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b* 
c + a*d, 0] &&  !IntegerQ[p]
 

rule 326

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

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 329

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
a*(Sqrt[1 - b^2*(x^4/a^2)]/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]))   Int[Sqrt[1 
+ b*(x^2/a)]/Sqrt[1 - b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b 
*c + a*d, 0] &&  !(LtQ[a*c, 0] && GtQ[a*b, 0])
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 0]
 

rule 1396

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_)*((d_) + (e_.)*(x_)^(n_))^(q_.), x 
_Symbol] :> Simp[(a + 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, c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a* 
e^2, 0] &&  !IntegerQ[p] &&  !(EqQ[q, 1] && EqQ[n, 2])
 

Maple [A] (verified)

Time = 1.21 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.13

method	result	size
default	\(\frac {d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(137\)
elliptic	\(\frac {d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(137\)

Input:

int((-e^2*x^4+d^2)^(1/2)/(e*x^2+d),x,method=_RETURNVERBOSE)
 

Output:

d/(e/d)^(1/2)*(1-e*x^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/(-e^2*x^4+d^2)^(1/2)*Ell 
ipticF(x*(e/d)^(1/2),I)+d/(e/d)^(1/2)*(1-e*x^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/ 
(-e^2*x^4+d^2)^(1/2)*(EllipticF(x*(e/d)^(1/2),I)-EllipticE(x*(e/d)^(1/2),I 
))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.75 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{d+e x^2} \, dx=\frac {\sqrt {-e^{2}} d x \sqrt {\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - \sqrt {-e^{2}} {\left (d - e\right )} x \sqrt {\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) + \sqrt {-e^{2} x^{4} + d^{2}} e}{e^{2} x} \] Input:

integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d),x, algorithm="fricas")
 

Output:

(sqrt(-e^2)*d*x*sqrt(d/e)*elliptic_e(arcsin(sqrt(d/e)/x), -1) - sqrt(-e^2) 
*(d - e)*x*sqrt(d/e)*elliptic_f(arcsin(sqrt(d/e)/x), -1) + sqrt(-e^2*x^4 + 
 d^2)*e)/(e^2*x)
 

Sympy [F]

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

integrate((-e**2*x**4+d**2)**(1/2)/(e*x**2+d),x)
 

Output:

Integral(sqrt(-(-d + e*x**2)*(d + e*x**2))/(d + e*x**2), x)
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {d^2-e^2 x^4}}{d+e x^2} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

\[ \int \frac {\sqrt {d^2-e^2 x^4}}{d+e x^2} \, dx=\int { \frac {\sqrt {-e^{2} x^{4} + d^{2}}}{e x^{2} + d} \,d x } \] Input:

integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d),x, algorithm="giac")
 

Output:

integrate(sqrt(-e^2*x^4 + d^2)/(e*x^2 + d), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {d^2-e^2 x^4}}{d+e x^2} \, dx=\int \frac {\sqrt {d^2-e^2\,x^4}}{e\,x^2+d} \,d x \] Input:

int((d^2 - e^2*x^4)^(1/2)/(d + e*x^2),x)
 

Output:

int((d^2 - e^2*x^4)^(1/2)/(d + e*x^2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {d^2-e^2 x^4}}{d+e x^2} \, dx=\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{e \,x^{2}+d}d x \] Input:

int((-e^2*x^4+d^2)^(1/2)/(e*x^2+d),x)
 

Output:

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