\(\int \frac {d+e x^2}{(d^2-e^2 x^4)^{3/2}} \, dx\) [63]

Optimal result

Integrand size = 24, antiderivative size = 153 \[ \int \frac {d+e x^2}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {x \left (d+e x^2\right )}{2 d^2 \sqrt {d^2-e^2 x^4}}-\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{2 \sqrt {d} \sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {\sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {d} \sqrt {e} \sqrt {d^2-e^2 x^4}} \] 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)*Ellipti 
cE(e^(1/2)*x/d^(1/2),I)/d^(1/2)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)+(1-e^2*x^4/d^ 
2)^(1/2)*EllipticF(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 = 0.73 \[ \int \frac {d+e x^2}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {3 d x+3 d x \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},\frac {e^2 x^4}{d^2}\right )+2 e x^3 \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},\frac {e^2 x^4}{d^2}\right )}{6 d^2 \sqrt {d^2-e^2 x^4}} \] 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 [A] (verified)

Time = 2.30 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.25

method	result	size
elliptic	\(-\frac {\left (-e^{2} x^{2}-d e \right ) x}{2 d^{2} e \sqrt {\left (x^{2}-\frac {d}{e}\right ) \left (-e^{2} x^{2}-d e \right )}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(191\)
default	\(d \left (\frac {x}{2 d^{2} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{2 d^{2} \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )+e \left (\frac {x^{3}}{2 d^{2} \sqrt {-\left (x^{4}-\frac {d^{2}}{e^{2}}\right ) e^{2}}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{2 d \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}\, e}\right )\)	\(204\)

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 = 113, normalized size of antiderivative = 0.74 \[ \int \frac {d+e x^2}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=-\frac {\sqrt {-e^{2} x^{4} + d^{2}} e x + {\left (e^{2} x^{2} - d e\right )} \sqrt {\frac {e}{d}} E(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1) - {\left ({\left (d e + e^{2}\right )} x^{2} - d^{2} - d e\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1)}{2 \, {\left (d^{2} e^{2} x^{2} - d^{3} e\right )}} \] 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(arcs 
in(x*sqrt(e/d)), -1) - ((d*e + e^2)*x^2 - d^2 - d*e)*sqrt(e/d)*elliptic_f( 
arcsin(x*sqrt(e/d)), -1))/(d^2*e^2*x^2 - d^3*e)
 

Sympy [A] (verification not implemented)

Time = 2.33 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.54 \[ \int \frac {d+e x^2}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 d^{2} \Gamma \left (\frac {5}{4}\right )} + \frac {e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 d^{3} \Gamma \left (\frac {7}{4}\right )} \] 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. \[ \int \frac {d+e x^2}{\left (d^2-e^2 x^4\right )^{3/2}} \, dx=\int \frac {e\,x^2+d}{{\left (d^2-e^2\,x^4\right )}^{3/2}} \,d x \] 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)