\(\int \frac {d-e x^2}{(d^2-e^2 x^4)^{3/2}} \, dx\) [114]

Optimal result

Integrand size = 25, antiderivative size = 95 \[ \int \frac {d-e x^2}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {x \left (d-e x^2\right )}{2 d^2 \sqrt {d^2-e^2 x^4}}+\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{2 \sqrt {d} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:

1/2*x*(-e*x^2+d)/d^2/(-e^2*x^4+d^2)^(1/2)+1/2*(1-e^2*x^4/d^2)^(1/2)*Ellipt 
icE(e^(1/2)*x/d^(1/2),I)/d^(1/2)/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 = 0.02 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.18 \[ \int \frac {d-e x^2}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {3 d x+3 d x \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {e^2 x^4}{d^2}\right )-2 e x^3 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {e^2 x^4}{d^2}\right )}{6 d^2 \sqrt {d^2-e^2 x^4}} \] Input:

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

Output:

(3*d*x + 3*d*x*Sqrt[1 - (e^2*x^4)/d^2]*Hypergeometric2F1[1/4, 1/2, 5/4, (e 
^2*x^4)/d^2] - 2*e*x^3*Sqrt[1 - (e^2*x^4)/d^2]*Hypergeometric2F1[3/4, 3/2, 
 7/4, (e^2*x^4)/d^2])/(6*d^2*Sqrt[d^2 - e^2*x^4])
 

Rubi [F]

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.

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

\(\Big \downarrow \) 1571

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

Input:

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

Output:

$Aborted
 

Defintions of rubi rules used

rule 1571

Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> U 
nintegrable[(d + e*x^2)^q*(a + c*x^4)^p, x] /; FreeQ[{a, c, d, e, p, q}, x]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (77 ) = 154\).

Time = 2.14 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.98

method	result	size
elliptic	\(\frac {\left (-e^{2} x^{2}+d e \right ) x}{2 d^{2} e \sqrt {\left (x^{2}+\frac {d}{e}\right ) \left (-e^{2} x^{2}+d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(188\)
default	\(d \left (\frac {x}{2 d^{2} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{2 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )-e \left (\frac {x^{3}}{2 d^{2} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}\, e}\right )\)	\(205\)

Input:

int((-e*x^2+d)/(-e^2*x^4+d^2)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

1/2*(-e^2*x^2+d*e)/d^2*x/e/((x^2+d/e)*(-e^2*x^2+d*e))^(1/2)+1/2/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)-1/2/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 = 110, normalized size of antiderivative = 1.16 \[ \int \frac {d-e x^2}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {\sqrt {-e^{2} x^{4} + d^{2}} e x + {\left (e^{2} x^{2} + d e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) + {\left ({\left (d e - e^{2}\right )} x^{2} + d^{2} - d e\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1)}{2 \, {\left (d^{2} e^{2} x^{2} + d^{3} e\right )}} \] Input:

integrate((-e*x^2+d)/(-e^2*x^4+d^2)^(3/2),x, algorithm="fricas")
 

Output:

1/2*(sqrt(-e^2*x^4 + d^2)*e*x + (e^2*x^2 + d*e)*sqrt(e/d)*elliptic_e(arcsi 
n(x*sqrt(e/d)), -1) + ((d*e - e^2)*x^2 + d^2 - d*e)*sqrt(e/d)*elliptic_f(a 
rcsin(x*sqrt(e/d)), -1))/(d^2*e^2*x^2 + d^3*e)
 

Sympy [A] (verification not implemented)

Time = 2.71 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.87 \[ \int \frac {d-e x^2}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 d^{2} \Gamma \left (\frac {5}{4}\right )} - \frac {e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 d^{3} \Gamma \left (\frac {7}{4}\right )} \] Input:

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

Output:

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

Maxima [F]

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

integrate((-e*x^2+d)/(-e^2*x^4+d^2)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((-e*x^2+d)/(-e^2*x^4+d^2)^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {d-e x^2}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{3} x^{6}-d \,e^{2} x^{4}+d^{2} e \,x^{2}+d^{3}}d x \] Input:

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

Output:

int(sqrt(d**2 - e**2*x**4)/(d**3 + d**2*e*x**2 - d*e**2*x**4 - e**3*x**6), 
x)