\(\int (d-e x^2) \sqrt {d^2-e^2 x^4} \, dx\) [86]

Optimal result

Integrand size = 25, antiderivative size = 156 \[ \int \left (d-e x^2\right ) \sqrt {d^2-e^2 x^4} \, dx=\frac {1}{15} x \left (5 d-3 e x^2\right ) \sqrt {d^2-e^2 x^4}-\frac {2 d^{7/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{5 \sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {16 d^{7/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{15 \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:

1/15*x*(-3*e*x^2+5*d)*(-e^2*x^4+d^2)^(1/2)-2/5*d^(7/2)*(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)+16/15*d^(7/ 
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 = 0.01 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.56 \[ \int \left (d-e x^2\right ) \sqrt {d^2-e^2 x^4} \, dx=\frac {\sqrt {d^2-e^2 x^4} \left (3 d x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {e^2 x^4}{d^2}\right )-e x^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {e^2 x^4}{d^2}\right )\right )}{3 \sqrt {1-\frac {e^2 x^4}{d^2}}} \] Input:

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

Output:

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

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.

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

\(\Big \downarrow \) 1571

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

Input:

Int[(d - e*x^2)*Sqrt[d^2 - e^2*x^4],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 = 3.97 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.10

method	result	size
risch	\(\frac {x \left (-3 e \,x^{2}+5 d \right ) \sqrt {-e^{2} x^{4}+d^{2}}}{15}+\frac {2 d^{2} \left (\frac {5 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {3 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )}{15}\)	\(172\)
elliptic	\(-\frac {e \,x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{5}+\frac {d x \sqrt {-e^{2} x^{4}+d^{2}}}{3}+\frac {2 d^{3} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {2 d^{3} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{5 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(181\)
default	\(d \left (\frac {x \sqrt {-e^{2} x^{4}+d^{2}}}{3}+\frac {2 d^{2} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )-e \left (\frac {x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{5}-\frac {2 d^{3} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{5 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}\, e}\right )\)	\(189\)

Input:

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

Output:

1/15*x*(-3*e*x^2+5*d)*(-e^2*x^4+d^2)^(1/2)+2/15*d^2*(5*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)+3*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 = 126, normalized size of antiderivative = 0.81 \[ \int \left (d-e x^2\right ) \sqrt {d^2-e^2 x^4} \, dx=\frac {6 \, \sqrt {-e^{2}} d^{3} x \sqrt {\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - 2 \, {\left (3 \, d^{3} - 5 \, d^{2} e\right )} \sqrt {-e^{2}} x \sqrt {\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - {\left (3 \, e^{3} x^{4} - 5 \, d e^{2} x^{2} - 6 \, d^{2} e\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{15 \, e^{2} x} \] Input:

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

Output:

1/15*(6*sqrt(-e^2)*d^3*x*sqrt(d/e)*elliptic_e(arcsin(sqrt(d/e)/x), -1) - 2 
*(3*d^3 - 5*d^2*e)*sqrt(-e^2)*x*sqrt(d/e)*elliptic_f(arcsin(sqrt(d/e)/x), 
-1) - (3*e^3*x^4 - 5*d*e^2*x^2 - 6*d^2*e)*sqrt(-e^2*x^4 + d^2))/(e^2*x)
 

Sympy [A] (verification not implemented)

Time = 1.03 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.54 \[ \int \left (d-e x^2\right ) \sqrt {d^2-e^2 x^4} \, dx=\frac {d^{2} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} - \frac {d e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \] Input:

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \left (d-e x^2\right ) \sqrt {d^2-e^2 x^4} \, dx=\int \sqrt {d^2-e^2\,x^4}\,\left (d-e\,x^2\right ) \,d x \] 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]

\[ \int \left (d-e x^2\right ) \sqrt {d^2-e^2 x^4} \, dx=\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, d x}{3}-\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e \,x^{3}}{5}+\frac {2 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{3}}{3}-\frac {2 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{2}}{-e^{2} x^{4}+d^{2}}d x \right ) d^{2} e}{5} \] Input:

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

Output:

(5*sqrt(d**2 - e**2*x**4)*d*x - 3*sqrt(d**2 - e**2*x**4)*e*x**3 + 10*int(s 
qrt(d**2 - e**2*x**4)/(d**2 - e**2*x**4),x)*d**3 - 6*int((sqrt(d**2 - e**2 
*x**4)*x**2)/(d**2 - e**2*x**4),x)*d**2*e)/15