\(\int \frac {\sqrt {d^2-e^2 x^4}}{(d+e x^2)^{3/2}} \, dx\) [134]

Optimal result

Integrand size = 28, antiderivative size = 94 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=-\frac {\arctan \left (\frac {\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{\sqrt {e}}+\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{\sqrt {e}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 2.51 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.30 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\frac {\frac {\sqrt {2} \sqrt {d^2-e^2 x^4} \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d-e x^2}}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}-i \log \left (-2 i \sqrt {e} x+\frac {2 \sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}}\right )}{\sqrt {e}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.40 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {1396, 301, 224, 216, 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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 301

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 216

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 216

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A 
rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a 
, 0] || GtQ[b, 0])
 

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 224

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

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 301

Int[((a_) + (b_.)*(x_)^2)^(p_.)/((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[b/ 
d   Int[(a + b*x^2)^(p - 1), x], x] - Simp[(b*c - a*d)/d   Int[(a + b*x^2)^ 
(p - 1)/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] 
&& GtQ[p, 0] && (EqQ[p, 1/2] || EqQ[Denominator[p], 4] || (EqQ[p, 2/3] && E 
qQ[b*c + 3*a*d, 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(179\) vs. \(2(74)=148\).

Time = 0.43 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.91

method	result	size
default	\(-\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, \left (\sqrt {d}\, \sqrt {2}\, \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right ) \sqrt {e}-\sqrt {d}\, \sqrt {2}\, \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) \sqrt {e}+2 \sqrt {-d e}\, \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right )\right )}{2 \sqrt {e \,x^{2}+d}\, \sqrt {-e \,x^{2}+d}\, \sqrt {-d e}\, \sqrt {e}}\)	\(180\)

Input:

int((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 279, normalized size of antiderivative = 2.97 \[ \int \frac {\sqrt {d^2-e^2 x^4}}{\left (d+e x^2\right )^{3/2}} \, dx=\left [\frac {\sqrt {2} e \sqrt {-\frac {1}{e}} \log \left (-\frac {3 \, e^{2} x^{4} + 2 \, d e x^{2} + 2 \, \sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} e x \sqrt {-\frac {1}{e}} - d^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right ) - \sqrt {-e} \log \left (-\frac {2 \, e^{2} x^{4} + d e x^{2} + 2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e x^{2} + d}\right )}{2 \, e}, -\frac {\sqrt {2} \sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right ) - \sqrt {e} \arctan \left (\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right )}{e}\right ] \] Input:

integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(3/2),x, algorithm="fricas")
 

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(3/2),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate((-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(3/2),x, algorithm="giac")
 

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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