\(\int \frac {d-e x^2}{\sqrt {d^2-e^2 x^4}} \, dx\) [17]

Optimal result

Integrand size = 25, antiderivative size = 121 \[ \int \frac {d-e x^2}{\sqrt {d^2-e^2 x^4}} \, dx=-\frac {d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {2 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d^2-e^2 x^4}} \] 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 higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 10.05 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.73 \[ \int \frac {d-e x^2}{\sqrt {d^2-e^2 x^4}} \, dx=\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} \left (3 d x \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {e^2 x^4}{d^2}\right )-e x^3 \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {e^2 x^4}{d^2}\right )\right )}{3 \sqrt {d^2-e^2 x^4}} \] Input:

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

Output:

(Sqrt[1 - (e^2*x^4)/d^2]*(3*d*x*Hypergeometric2F1[1/4, 1/2, 5/4, (e^2*x^4) 
/d^2] - e*x^3*Hypergeometric2F1[1/2, 3/4, 7/4, (e^2*x^4)/d^2]))/(3*Sqrt[d^ 
2 - e^2*x^4])
 

Rubi [A] (verified)

Time = 0.42 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.74, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {1390, 1389, 326, 284, 327, 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.

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

\(\Big \downarrow \) 1390

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

\(\Big \downarrow \) 1389

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

\(\Big \downarrow \) 326

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

\(\Big \downarrow \) 284

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 762

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 284

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

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 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 1389

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

rule 1390

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

Maple [A] (verified)

Time = 2.18 (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*x^2+d)/(-e^2*x^4+d^2)^(1/2),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 = 0.08 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.75 \[ \int \frac {d-e x^2}{\sqrt {d^2-e^2 x^4}} \, dx=\frac {\sqrt {-e^{2}} d x \sqrt {\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - \sqrt {-e^{2}} {\left (d - e\right )} x \sqrt {\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) + \sqrt {-e^{2} x^{4} + d^{2}} e}{e^{2} x} \] Input:

integrate((-e*x^2+d)/(-e^2*x^4+d^2)^(1/2),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 [A] (verification not implemented)

Time = 0.85 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.64 \[ \int \frac {d-e x^2}{\sqrt {d^2-e^2 x^4}} \, dx=\frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} - \frac {e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 d \Gamma \left (\frac {7}{4}\right )} \] Input:

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

Output:

x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), e**2*x**4*exp_polar(2*I*pi)/d**2)/( 
4*gamma(5/4)) - e*x**3*gamma(3/4)*hyper((1/2, 3/4), (7/4,), e**2*x**4*exp_ 
polar(2*I*pi)/d**2)/(4*d*gamma(7/4))
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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