\(\int \frac {(d^2-e^2 x^4)^{3/2}}{(d+e x^2)^4} \, dx\) [50]

Optimal result

Integrand size = 26, antiderivative size = 93 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^4} \, dx=\frac {2 x \sqrt {d^2-e^2 x^4}}{3 \left (d+e x^2\right )^2}+\frac {\sqrt {d} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{3 \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:

2/3*x*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^2+1/3*d^(1/2)*(1-e^2*x^4/d^2)^(1/2)*E 
llipticF(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 = 10.82 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.98 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^4} \, dx=\frac {\frac {2 x \left (d-e x^2\right )}{d+e x^2}-\frac {i \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {-\frac {e}{d}} x\right ),-1\right )}{\sqrt {-\frac {e}{d}}}}{3 \sqrt {d^2-e^2 x^4}} \] Input:

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

Output:

((2*x*(d - e*x^2))/(d + e*x^2) - (I*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[I*Ar 
cSinh[Sqrt[-(e/d)]*x], -1])/Sqrt[-(e/d)])/(3*Sqrt[d^2 - e^2*x^4])
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.48, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {1396, 315, 27, 289, 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.

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

\(\Big \downarrow \) 1396

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \int \frac {\left (d-e x^2\right )^{3/2}}{\left (e x^2+d\right )^{5/2}}dx}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 315

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 289

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

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

Input:

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

Output:

(Sqrt[d^2 - e^2*x^4]*((2*x*Sqrt[d - e*x^2])/(3*(d + e*x^2)^(3/2)) + (Sqrt[ 
d]*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(3* 
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 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

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 315

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), 
x] - Simp[1/(2*a*b*(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S 
imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) 
*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 
1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
 

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 = 3.18 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.00

method	result	size
default	\(\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{3 e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(93\)
elliptic	\(\frac {2 x \sqrt {-e^{2} x^{4}+d^{2}}}{3 e^{2} \left (x^{2}+\frac {d}{e}\right )^{2}}+\frac {\sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(93\)

Input:

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

Output:

2/3*x/e^2*(-e^2*x^4+d^2)^(1/2)/(x^2+d/e)^2+1/3/(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)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.89 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^4} \, dx=\frac {2 \, \sqrt {-e^{2} x^{4} + d^{2}} e x + {\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {\frac {e}{d}} F(\arcsin \left (x \sqrt {\frac {e}{d}}\right )\,|\,-1)}{3 \, {\left (e^{3} x^{4} + 2 \, d e^{2} x^{2} + d^{2} e\right )}} \] Input:

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

Output:

1/3*(2*sqrt(-e^2*x^4 + d^2)*e*x + (e^2*x^4 + 2*d*e*x^2 + d^2)*sqrt(e/d)*el 
liptic_f(arcsin(x*sqrt(e/d)), -1))/(e^3*x^4 + 2*d*e^2*x^2 + d^2*e)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^4} \, dx=\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x +\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{4} x^{8}-2 d \,e^{3} x^{6}+2 d^{3} e \,x^{2}+d^{4}}d x \right ) d^{4}+2 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{4} x^{8}-2 d \,e^{3} x^{6}+2 d^{3} e \,x^{2}+d^{4}}d x \right ) d^{3} e \,x^{2}+\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}}{-e^{4} x^{8}-2 d \,e^{3} x^{6}+2 d^{3} e \,x^{2}+d^{4}}d x \right ) d^{2} e^{2} x^{4}+\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{4} x^{8}-2 d \,e^{3} x^{6}+2 d^{3} e \,x^{2}+d^{4}}d x \right ) d^{2} e^{2}+2 \left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{4} x^{8}-2 d \,e^{3} x^{6}+2 d^{3} e \,x^{2}+d^{4}}d x \right ) d \,e^{3} x^{2}+\left (\int \frac {\sqrt {-e^{2} x^{4}+d^{2}}\, x^{4}}{-e^{4} x^{8}-2 d \,e^{3} x^{6}+2 d^{3} e \,x^{2}+d^{4}}d x \right ) e^{4} x^{4}}{2 e^{2} x^{4}+4 d e \,x^{2}+2 d^{2}} \] Input:

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

Output:

(sqrt(d**2 - e**2*x**4)*x + int(sqrt(d**2 - e**2*x**4)/(d**4 + 2*d**3*e*x* 
*2 - 2*d*e**3*x**6 - e**4*x**8),x)*d**4 + 2*int(sqrt(d**2 - e**2*x**4)/(d* 
*4 + 2*d**3*e*x**2 - 2*d*e**3*x**6 - e**4*x**8),x)*d**3*e*x**2 + int(sqrt( 
d**2 - e**2*x**4)/(d**4 + 2*d**3*e*x**2 - 2*d*e**3*x**6 - e**4*x**8),x)*d* 
*2*e**2*x**4 + int((sqrt(d**2 - e**2*x**4)*x**4)/(d**4 + 2*d**3*e*x**2 - 2 
*d*e**3*x**6 - e**4*x**8),x)*d**2*e**2 + 2*int((sqrt(d**2 - e**2*x**4)*x** 
4)/(d**4 + 2*d**3*e*x**2 - 2*d*e**3*x**6 - e**4*x**8),x)*d*e**3*x**2 + int 
((sqrt(d**2 - e**2*x**4)*x**4)/(d**4 + 2*d**3*e*x**2 - 2*d*e**3*x**6 - e** 
4*x**8),x)*e**4*x**4)/(2*(d**2 + 2*d*e*x**2 + e**2*x**4))