\(\int \frac {\sqrt {d^2-e^2 x^4}}{(d+e x^2)^{5/2}} \, dx\) [135]

Optimal result

Integrand size = 28, antiderivative size = 93 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {x \sqrt {d^2-e^2 x^4}}{2 d \left (d+e x^2\right )^{3/2}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{2 \sqrt {2} d \sqrt {e}} \] Output:

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

Mathematica [A] (verified)

Time = 2.84 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.18 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2 x^4} \left (2 \sqrt {e} x \sqrt {d-e x^2}+\sqrt {2} \left (d+e x^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d-e x^2}}\right )\right )}{4 d \sqrt {e} \sqrt {d-e x^2} \left (d+e x^2\right )^{3/2}} \] Input:

Integrate[Sqrt[d^2 - e^2*x^4]/(d + e*x^2)^(5/2),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*(2*Sqrt[e]*x*Sqrt[d - e*x^2] + Sqrt[2]*(d + e*x^2)*Ar 
cTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d - e*x^2]]))/(4*d*Sqrt[e]*Sqrt[d - e*x^2]*( 
d + e*x^2)^(3/2))
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.20, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {1396, 292, 291, 218}

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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 292

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

Int[Sqrt[d^2 - e^2*x^4]/(d + e*x^2)^(5/2),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*((x*Sqrt[d - e*x^2])/(2*d*(d + e*x^2)) + ArcTan[(Sqrt 
[2]*Sqrt[e]*x)/Sqrt[d - e*x^2]]/(2*Sqrt[2]*d*Sqrt[e])))/(Sqrt[d - e*x^2]*S 
qrt[d + e*x^2])
 

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 292

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Si 
mp[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] - Simp[c*(q/( 
a*(p + 1)))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1), x], x] /; FreeQ[ 
{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q + 1) + 1, 0] && Gt 
Q[q, 0] && NeQ[p, -1]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(307\) vs. \(2(73)=146\).

Time = 0.46 (sec) , antiderivative size = 308, normalized size of antiderivative = 3.31

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e \left (-\sqrt {2}\, \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right ) e \,x^{2} \sqrt {d}+\sqrt {2}\, \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) e \,x^{2} \sqrt {d}-\sqrt {2}\, \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right ) d^{\frac {3}{2}}+\sqrt {2}\, \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) d^{\frac {3}{2}}+4 \sqrt {-d e}\, \sqrt {-e \,x^{2}+d}\, x \right )}{8 d \sqrt {e \,x^{2}+d}\, \sqrt {-e \,x^{2}+d}\, \left (e x -\sqrt {-d e}\right ) \left (e x +\sqrt {-d e}\right ) \sqrt {-d e}}\)	\(308\)

Input:

int((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/8*(-e^2*x^4+d^2)^(1/2)*e*(-2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(-e*x^2+d)^(1 
/2)-(-d*e)^(1/2)*x+d)/(e*x-(-d*e)^(1/2)))*e*x^2*d^(1/2)+2^(1/2)*ln(2*e*(2^ 
(1/2)*d^(1/2)*(-e*x^2+d)^(1/2)+(-d*e)^(1/2)*x+d)/(e*x+(-d*e)^(1/2)))*e*x^2 
*d^(1/2)-2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(-e*x^2+d)^(1/2)-(-d*e)^(1/2)*x+d 
)/(e*x-(-d*e)^(1/2)))*d^(3/2)+2^(1/2)*ln(2*e*(2^(1/2)*d^(1/2)*(-e*x^2+d)^( 
1/2)+(-d*e)^(1/2)*x+d)/(e*x+(-d*e)^(1/2)))*d^(3/2)+4*(-d*e)^(1/2)*(-e*x^2+ 
d)^(1/2)*x)/d/(e*x^2+d)^(1/2)/(-e*x^2+d)^(1/2)/(e*x-(-d*e)^(1/2))/(e*x+(-d 
*e)^(1/2))/(-d*e)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 294, normalized size of antiderivative = 3.16 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{5/2}} \, dx=\left [\frac {4 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} e x - \sqrt {2} {\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {-e} \log \left (-\frac {3 \, e^{2} x^{4} + 2 \, d e x^{2} - 2 \, \sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right )}{8 \, {\left (d e^{3} x^{4} + 2 \, d^{2} e^{2} x^{2} + d^{3} e\right )}}, \frac {2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} e x - \sqrt {2} {\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right )}{4 \, {\left (d e^{3} x^{4} + 2 \, d^{2} e^{2} x^{2} + d^{3} e\right )}}\right ] \] Input:

integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(5/2),x, algorithm="fricas")
 

Output:

[1/8*(4*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*e*x - sqrt(2)*(e^2*x^4 + 2*d* 
e*x^2 + d^2)*sqrt(-e)*log(-(3*e^2*x^4 + 2*d*e*x^2 - 2*sqrt(2)*sqrt(-e^2*x^ 
4 + d^2)*sqrt(e*x^2 + d)*sqrt(-e)*x - d^2)/(e^2*x^4 + 2*d*e*x^2 + d^2)))/( 
d*e^3*x^4 + 2*d^2*e^2*x^2 + d^3*e), 1/4*(2*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 
 + d)*e*x - sqrt(2)*(e^2*x^4 + 2*d*e*x^2 + d^2)*sqrt(e)*arctan(sqrt(2)*sqr 
t(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*sqrt(e)*x/(e^2*x^4 - d^2)))/(d*e^3*x^4 + 
 2*d^2*e^2*x^2 + d^3*e)]
 

Sympy [F]

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

integrate((-e**2*x**4+d**2)**(1/2)/(e*x**2+d)**(5/2),x)
 

Output:

Integral(sqrt(-(-d + e*x**2)*(d + e*x**2))/(d + e*x**2)**(5/2), x)
 

Maxima [F]

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

integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(5/2),x, algorithm="maxima")
 

Output:

integrate(sqrt(-e^2*x^4 + d^2)/(e*x^2 + d)^(5/2), x)
 

Giac [F]

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

integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(5/2),x, algorithm="giac")
 

Output:

integrate(sqrt(-e^2*x^4 + d^2)/(e*x^2 + d)^(5/2), x)
 

Mupad [F(-1)]

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

int((d^2 - e^2*x^4)^(1/2)/(d + e*x^2)^(5/2),x)
 

Output:

int((d^2 - e^2*x^4)^(1/2)/(d + e*x^2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}\, x +2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}}{-e^{4} x^{8}-2 d \,e^{3} x^{6}+2 d^{3} e \,x^{2}+d^{4}}d x \right ) d^{4}+4 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}}{-e^{4} x^{8}-2 d \,e^{3} x^{6}+2 d^{3} e \,x^{2}+d^{4}}d x \right ) d^{3} e \,x^{2}+2 \left (\int \frac {\sqrt {e \,x^{2}+d}\, \sqrt {-e^{2} x^{4}+d^{2}}}{-e^{4} x^{8}-2 d \,e^{3} x^{6}+2 d^{3} e \,x^{2}+d^{4}}d x \right ) d^{2} e^{2} x^{4}}{3 d \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

int((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(5/2),x)
 

Output:

(sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x + 2*int((sqrt(d + e*x**2)*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 + 4*int((sqrt(d + e*x**2)*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 + 2*int((sqrt(d + e*x**2)*sq 
rt(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)/(3*d*(d**2 + 2*d*e*x**2 + e**2*x**4))