\(\int \frac {1}{(d+e x^2)^{5/2} \sqrt {d^2-e^2 x^4}} \, dx\) [156]

Optimal result

Integrand size = 28, antiderivative size = 128 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {d^2-e^2 x^4}} \, dx=\frac {x \sqrt {d^2-e^2 x^4}}{8 d^2 \left (d+e x^2\right )^{5/2}}+\frac {9 x \sqrt {d^2-e^2 x^4}}{32 d^3 \left (d+e x^2\right )^{3/2}}+\frac {19 \arctan \left (\frac {\sqrt {2} \sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{32 \sqrt {2} d^3 \sqrt {e}} \] Output:

1/8*x*(-e^2*x^4+d^2)^(1/2)/d^2/(e*x^2+d)^(5/2)+9/32*x*(-e^2*x^4+d^2)^(1/2) 
/d^3/(e*x^2+d)^(3/2)+19/64*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^3/e^(1/2)
 

Mathematica [A] (verified)

Time = 3.05 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {d^2-e^2 x^4}} \, dx=\frac {\sqrt {d^2-e^2 x^4} \left (2 \sqrt {e} x \sqrt {d-e x^2} \left (13 d+9 e x^2\right )+19 \sqrt {2} \left (d+e x^2\right )^2 \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d-e x^2}}\right )\right )}{64 d^3 \sqrt {e} \sqrt {d-e x^2} \left (d+e x^2\right )^{5/2}} \] Input:

Integrate[1/((d + e*x^2)^(5/2)*Sqrt[d^2 - e^2*x^4]),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*(2*Sqrt[e]*x*Sqrt[d - e*x^2]*(13*d + 9*e*x^2) + 19*Sq 
rt[2]*(d + e*x^2)^2*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d - e*x^2]]))/(64*d^3* 
Sqrt[e]*Sqrt[d - e*x^2]*(d + e*x^2)^(5/2))
 

Rubi [A] (verified)

Time = 0.45 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.16, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1396, 316, 25, 27, 402, 27, 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.

\(\displaystyle \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {d^2-e^2 x^4}} \, dx\)

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 316

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {x \sqrt {d-e x^2}}{8 d^2 \left (d+e x^2\right )^2}-\frac {\int -\frac {e \left (7 d-2 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^2}dx}{8 d^2 e}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {e \left (7 d-2 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )^2}dx}{8 d^2 e}+\frac {x \sqrt {d-e x^2}}{8 d^2 \left (d+e x^2\right )^2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {7 d-2 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )^2}dx}{8 d^2}+\frac {x \sqrt {d-e x^2}}{8 d^2 \left (d+e x^2\right )^2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {9 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )}-\frac {\int -\frac {19 d^2 e}{\sqrt {d-e x^2} \left (e x^2+d\right )}dx}{4 d^2 e}}{8 d^2}+\frac {x \sqrt {d-e x^2}}{8 d^2 \left (d+e x^2\right )^2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {19}{4} \int \frac {1}{\sqrt {d-e x^2} \left (e x^2+d\right )}dx+\frac {9 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )}}{8 d^2}+\frac {x \sqrt {d-e x^2}}{8 d^2 \left (d+e x^2\right )^2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {19}{4} \int \frac {1}{\frac {2 d e x^2}{d-e x^2}+d}d\frac {x}{\sqrt {d-e x^2}}+\frac {9 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )}}{8 d^2}+\frac {x \sqrt {d-e x^2}}{8 d^2 \left (d+e x^2\right )^2}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 218

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {19 \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d-e x^2}}\right )}{4 \sqrt {2} d \sqrt {e}}+\frac {9 x \sqrt {d-e x^2}}{4 d \left (d+e x^2\right )}}{8 d^2}+\frac {x \sqrt {d-e x^2}}{8 d^2 \left (d+e x^2\right )^2}\right )}{\sqrt {d^2-e^2 x^4}}\)

Input:

Int[1/((d + e*x^2)^(5/2)*Sqrt[d^2 - e^2*x^4]),x]
 

Output:

(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*((x*Sqrt[d - e*x^2])/(8*d^2*(d + e*x^2)^2 
) + ((9*x*Sqrt[d - e*x^2])/(4*d*(d + e*x^2)) + (19*ArcTan[(Sqrt[2]*Sqrt[e] 
*x)/Sqrt[d - e*x^2]])/(4*Sqrt[2]*d*Sqrt[e]))/(8*d^2)))/Sqrt[d^2 - e^2*x^4]
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 316

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

rule 402

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*(x 
_)^2), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ 
(q + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) 
 Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) 
*(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b 
, c, d, e, f, q}, x] && LtQ[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. \(710\) vs. \(2(102)=204\).

Time = 0.84 (sec) , antiderivative size = 711, normalized size of antiderivative = 5.55

method	result	size
default	\(-\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e^{\frac {9}{2}} \left (19 \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^{\frac {5}{2}} x^{4} \sqrt {d}-19 \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^{\frac {5}{2}} x^{4} \sqrt {d}+38 \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}} e^{\frac {3}{2}} x^{2}-38 \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}} e^{\frac {3}{2}} x^{2}+16 \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) e^{2} x^{4} \sqrt {-d e}-16 \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {\frac {\left (e x +\sqrt {d e}\right ) \left (-e x +\sqrt {d e}\right )}{e}}}\right ) e^{2} x^{4} \sqrt {-d e}-36 \sqrt {-e \,x^{2}+d}\, e^{\frac {3}{2}} \sqrt {-d e}\, x^{3}+19 \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 {5}{2}} \sqrt {e}-19 \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 {5}{2}} \sqrt {e}+32 \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) d e \,x^{2} \sqrt {-d e}-32 \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {\frac {\left (e x +\sqrt {d e}\right ) \left (-e x +\sqrt {d e}\right )}{e}}}\right ) d e \,x^{2} \sqrt {-d e}-52 \sqrt {-e \,x^{2}+d}\, d \sqrt {e}\, \sqrt {-d e}\, x +16 \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) d^{2} \sqrt {-d e}-16 \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {\frac {\left (e x +\sqrt {d e}\right ) \left (-e x +\sqrt {d e}\right )}{e}}}\right ) d^{2} \sqrt {-d e}\right )}{16 \sqrt {e \,x^{2}+d}\, \sqrt {-e \,x^{2}+d}\, \left (\sqrt {d e}-\sqrt {-d e}\right )^{3} \left (\sqrt {d e}+\sqrt {-d e}\right )^{3} \left (e x -\sqrt {-d e}\right )^{2} \left (e x +\sqrt {-d e}\right )^{2} \sqrt {-d e}}\)	\(711\)

Input:

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

Output:

-1/16*(-e^2*x^4+d^2)^(1/2)*e^(9/2)*(19*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^(5/2)*x^4*d^(1/2)-19 
*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^(5/2)*x^4*d^(1/2)+38*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)*e^(3/2)*x^2-38*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)*e^(3/2)*x^2+16*arctan(e^(1/2)*x/(-e*x^2+d)^(1/2))*e^2* 
x^4*(-d*e)^(1/2)-16*arctan(e^(1/2)*x/((e*x+(d*e)^(1/2))/e*(-e*x+(d*e)^(1/2 
)))^(1/2))*e^2*x^4*(-d*e)^(1/2)-36*(-e*x^2+d)^(1/2)*e^(3/2)*(-d*e)^(1/2)*x 
^3+19*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^(5/2)*e^(1/2)-19*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^(5/2)*e^(1/2)+32*arct 
an(e^(1/2)*x/(-e*x^2+d)^(1/2))*d*e*x^2*(-d*e)^(1/2)-32*arctan(e^(1/2)*x/(( 
e*x+(d*e)^(1/2))/e*(-e*x+(d*e)^(1/2)))^(1/2))*d*e*x^2*(-d*e)^(1/2)-52*(-e* 
x^2+d)^(1/2)*d*e^(1/2)*(-d*e)^(1/2)*x+16*arctan(e^(1/2)*x/(-e*x^2+d)^(1/2) 
)*d^2*(-d*e)^(1/2)-16*arctan(e^(1/2)*x/((e*x+(d*e)^(1/2))/e*(-e*x+(d*e)^(1 
/2)))^(1/2))*d^2*(-d*e)^(1/2))/(e*x^2+d)^(1/2)/(-e*x^2+d)^(1/2)/((d*e)^(1/ 
2)-(-d*e)^(1/2))^3/((d*e)^(1/2)+(-d*e)^(1/2))^3/(e*x-(-d*e)^(1/2))^2/(e*x+ 
(-d*e)^(1/2))^2/(-d*e)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.86 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {d^2-e^2 x^4}} \, dx=\left [-\frac {19 \, \sqrt {2} {\left (e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}\right )} \sqrt {-e} \log \left (-\frac {3 \, e^{2} x^{4} + 2 \, d e x^{2} - 2 \, \sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right ) - 4 \, \sqrt {-e^{2} x^{4} + d^{2}} {\left (9 \, e^{2} x^{3} + 13 \, d e x\right )} \sqrt {e x^{2} + d}}{128 \, {\left (d^{3} e^{4} x^{6} + 3 \, d^{4} e^{3} x^{4} + 3 \, d^{5} e^{2} x^{2} + d^{6} e\right )}}, -\frac {19 \, \sqrt {2} {\left (e^{3} x^{6} + 3 \, d e^{2} x^{4} + 3 \, d^{2} e x^{2} + d^{3}\right )} \sqrt {e} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right ) - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} {\left (9 \, e^{2} x^{3} + 13 \, d e x\right )} \sqrt {e x^{2} + d}}{64 \, {\left (d^{3} e^{4} x^{6} + 3 \, d^{4} e^{3} x^{4} + 3 \, d^{5} e^{2} x^{2} + d^{6} e\right )}}\right ] \] Input:

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

Output:

[-1/128*(19*sqrt(2)*(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^2 + d^3)*sqrt(-e)*l 
og(-(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)) - 4*sqrt(-e^2*x^4 + d^2)* 
(9*e^2*x^3 + 13*d*e*x)*sqrt(e*x^2 + d))/(d^3*e^4*x^6 + 3*d^4*e^3*x^4 + 3*d 
^5*e^2*x^2 + d^6*e), -1/64*(19*sqrt(2)*(e^3*x^6 + 3*d*e^2*x^4 + 3*d^2*e*x^ 
2 + d^3)*sqrt(e)*arctan(sqrt(2)*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*sqrt( 
e)*x/(e^2*x^4 - d^2)) - 2*sqrt(-e^2*x^4 + d^2)*(9*e^2*x^3 + 13*d*e*x)*sqrt 
(e*x^2 + d))/(d^3*e^4*x^6 + 3*d^4*e^3*x^4 + 3*d^5*e^2*x^2 + d^6*e)]
 

Sympy [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {d^2-e^2 x^4}} \, dx=\int \frac {1}{\sqrt {- \left (- d + e x^{2}\right ) \left (d + e x^{2}\right )} \left (d + e x^{2}\right )^{\frac {5}{2}}}\, dx \] Input:

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

Output:

Integral(1/(sqrt(-(-d + e*x**2)*(d + e*x**2))*(d + e*x**2)**(5/2)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {d^2-e^2 x^4}} \, dx=\int { \frac {1}{\sqrt {-e^{2} x^{4} + d^{2}} {\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {d^2-e^2 x^4}} \, dx=\int { \frac {1}{\sqrt {-e^{2} x^{4} + d^{2}} {\left (e x^{2} + d\right )}^{\frac {5}{2}}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {d^2-e^2 x^4}} \, dx=\int \frac {1}{\sqrt {d^2-e^2\,x^4}\,{\left (e\,x^2+d\right )}^{5/2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 345, normalized size of antiderivative = 2.70 \[ \int \frac {1}{\left (d+e x^2\right )^{5/2} \sqrt {d^2-e^2 x^4}} \, dx=\frac {38 \sqrt {e}\, \sqrt {2}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {e}\, x}{\sqrt {d}}\right )}{2}\right )}{\sqrt {2}+1}\right ) d^{2}+76 \sqrt {e}\, \sqrt {2}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {e}\, x}{\sqrt {d}}\right )}{2}\right )}{\sqrt {2}+1}\right ) d e \,x^{2}+38 \sqrt {e}\, \sqrt {2}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {e}\, x}{\sqrt {d}}\right )}{2}\right )}{\sqrt {2}+1}\right ) e^{2} x^{4}+52 \sqrt {-e \,x^{2}+d}\, d e x +36 \sqrt {-e \,x^{2}+d}\, e^{2} x^{3}-19 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {2}\, i +\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {e}\, x}{\sqrt {d}}\right )}{2}\right )+i \right ) d^{2} i -38 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {2}\, i +\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {e}\, x}{\sqrt {d}}\right )}{2}\right )+i \right ) d e i \,x^{2}-19 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (-\sqrt {2}\, i +\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {e}\, x}{\sqrt {d}}\right )}{2}\right )+i \right ) e^{2} i \,x^{4}+19 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {2}\, i +\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {e}\, x}{\sqrt {d}}\right )}{2}\right )-i \right ) d^{2} i +38 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {2}\, i +\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {e}\, x}{\sqrt {d}}\right )}{2}\right )-i \right ) d e i \,x^{2}+19 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\sqrt {2}\, i +\tan \left (\frac {\mathit {asin} \left (\frac {\sqrt {e}\, x}{\sqrt {d}}\right )}{2}\right )-i \right ) e^{2} i \,x^{4}}{128 d^{3} e \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

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

Output:

(38*sqrt(e)*sqrt(2)*atan(tan(asin((sqrt(e)*x)/sqrt(d))/2)/(sqrt(2) + 1))*d 
**2 + 76*sqrt(e)*sqrt(2)*atan(tan(asin((sqrt(e)*x)/sqrt(d))/2)/(sqrt(2) + 
1))*d*e*x**2 + 38*sqrt(e)*sqrt(2)*atan(tan(asin((sqrt(e)*x)/sqrt(d))/2)/(s 
qrt(2) + 1))*e**2*x**4 + 52*sqrt(d - e*x**2)*d*e*x + 36*sqrt(d - e*x**2)*e 
**2*x**3 - 19*sqrt(e)*sqrt(2)*log( - sqrt(2)*i + tan(asin((sqrt(e)*x)/sqrt 
(d))/2) + i)*d**2*i - 38*sqrt(e)*sqrt(2)*log( - sqrt(2)*i + tan(asin((sqrt 
(e)*x)/sqrt(d))/2) + i)*d*e*i*x**2 - 19*sqrt(e)*sqrt(2)*log( - sqrt(2)*i + 
 tan(asin((sqrt(e)*x)/sqrt(d))/2) + i)*e**2*i*x**4 + 19*sqrt(e)*sqrt(2)*lo 
g(sqrt(2)*i + tan(asin((sqrt(e)*x)/sqrt(d))/2) - i)*d**2*i + 38*sqrt(e)*sq 
rt(2)*log(sqrt(2)*i + tan(asin((sqrt(e)*x)/sqrt(d))/2) - i)*d*e*i*x**2 + 1 
9*sqrt(e)*sqrt(2)*log(sqrt(2)*i + tan(asin((sqrt(e)*x)/sqrt(d))/2) - i)*e* 
*2*i*x**4)/(128*d**3*e*(d**2 + 2*d*e*x**2 + e**2*x**4))