\(\int \frac {1}{(d+e x^2)^{3/2} (d^2-e^2 x^4)} \, dx\) [30]

Optimal result

Integrand size = 26, antiderivative size = 80 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )} \, dx=\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}+\frac {7 x}{12 d^3 \sqrt {d+e x^2}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{4 \sqrt {2} d^3 \sqrt {e}} \] Output:

1/6*x/d^2/(e*x^2+d)^(3/2)+7/12*x/d^3/(e*x^2+d)^(1/2)+1/8*arctanh(2^(1/2)*e 
^(1/2)*x/(e*x^2+d)^(1/2))*2^(1/2)/d^3/e^(1/2)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )} \, dx=\frac {\frac {2 \left (9 d x+7 e x^3\right )}{\left (d+e x^2\right )^{3/2}}+\frac {3 \sqrt {2} \text {arctanh}\left (\frac {d-e x^2+\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {2} d}\right )}{\sqrt {e}}}{24 d^3} \] Input:

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

Output:

((2*(9*d*x + 7*e*x^3))/(d + e*x^2)^(3/2) + (3*Sqrt[2]*ArcTanh[(d - e*x^2 + 
 Sqrt[e]*x*Sqrt[d + e*x^2])/(Sqrt[2]*d)])/Sqrt[e])/(24*d^3)
 

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1388, 316, 25, 27, 402, 27, 291, 221}

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.

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

\(\Big \downarrow \) 1388

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

\(\Big \downarrow \) 316

\(\displaystyle \frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}-\frac {\int -\frac {e \left (5 d-2 e x^2\right )}{\left (d-e x^2\right ) \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {e \left (5 d-2 e x^2\right )}{\left (d-e x^2\right ) \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {5 d-2 e x^2}{\left (d-e x^2\right ) \left (e x^2+d\right )^{3/2}}dx}{6 d^2}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\frac {7 x}{2 d \sqrt {d+e x^2}}-\frac {\int -\frac {3 d^2 e}{\left (d-e x^2\right ) \sqrt {e x^2+d}}dx}{2 d^2 e}}{6 d^2}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {3}{2} \int \frac {1}{\left (d-e x^2\right ) \sqrt {e x^2+d}}dx+\frac {7 x}{2 d \sqrt {d+e x^2}}}{6 d^2}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\frac {3}{2} \int \frac {1}{d-\frac {2 d e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}+\frac {7 x}{2 d \sqrt {d+e x^2}}}{6 d^2}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {2} d \sqrt {e}}+\frac {7 x}{2 d \sqrt {d+e x^2}}}{6 d^2}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}\)

Input:

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

Output:

x/(6*d^2*(d + e*x^2)^(3/2)) + ((7*x)/(2*d*Sqrt[d + e*x^2]) + (3*ArcTanh[(S 
qrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*Sqrt[2]*d*Sqrt[e]))/(6*d^2)
 

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 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[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 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

Maple [A] (verified)

Time = 0.61 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(\frac {14 e^{\frac {3}{2}} x^{3}+3 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {e \,x^{2}+d}\, \sqrt {2}}{2 x \sqrt {e}}\right ) \left (e \,x^{2}+d \right )^{\frac {3}{2}}+18 \sqrt {e}\, d x}{24 \sqrt {e}\, \left (e \,x^{2}+d \right )^{\frac {3}{2}} d^{3}}\)	\(69\)
default	\(\frac {e \left (\frac {1}{2 d \sqrt {\left (x -\frac {\sqrt {d e}}{e}\right )^{2} e +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )+2 d}}-\frac {\sqrt {d e}\, \left (2 e \left (x -\frac {\sqrt {d e}}{e}\right )+2 \sqrt {d e}\right )}{4 d^{2} e \sqrt {\left (x -\frac {\sqrt {d e}}{e}\right )^{2} e +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )+2 d}}-\frac {\sqrt {2}\, \ln \left (\frac {4 d +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )+2 \sqrt {2}\, \sqrt {d}\, \sqrt {\left (x -\frac {\sqrt {d e}}{e}\right )^{2} e +2 \sqrt {d e}\, \left (x -\frac {\sqrt {d e}}{e}\right )+2 d}}{x -\frac {\sqrt {d e}}{e}}\right )}{4 d^{\frac {3}{2}}}\right )}{2 \left (-\sqrt {d e}+\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \sqrt {d e}}-\frac {e \left (\frac {1}{2 d \sqrt {\left (x +\frac {\sqrt {d e}}{e}\right )^{2} e -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )+2 d}}+\frac {\sqrt {d e}\, \left (2 e \left (x +\frac {\sqrt {d e}}{e}\right )-2 \sqrt {d e}\right )}{4 d^{2} e \sqrt {\left (x +\frac {\sqrt {d e}}{e}\right )^{2} e -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )+2 d}}-\frac {\sqrt {2}\, \ln \left (\frac {4 d -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )+2 \sqrt {2}\, \sqrt {d}\, \sqrt {\left (x +\frac {\sqrt {d e}}{e}\right )^{2} e -2 \sqrt {d e}\, \left (x +\frac {\sqrt {d e}}{e}\right )+2 d}}{x +\frac {\sqrt {d e}}{e}}\right )}{4 d^{\frac {3}{2}}}\right )}{2 \left (-\sqrt {d e}+\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \sqrt {d e}}-\frac {e \left (-\frac {1}{3 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right ) \sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}}-\frac {2 e \left (x -\frac {\sqrt {-d e}}{e}\right )+2 \sqrt {-d e}}{3 \sqrt {-d e}\, d \sqrt {\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )}}\right )}{2 \sqrt {-d e}\, \left (-\sqrt {d e}+\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right )}+\frac {e \left (\frac {1}{3 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right ) \sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}}+\frac {2 e \left (x +\frac {\sqrt {-d e}}{e}\right )-2 \sqrt {-d e}}{3 \sqrt {-d e}\, d \sqrt {\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )}}\right )}{2 \sqrt {-d e}\, \left (-\sqrt {d e}+\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right )}\)	\(865\)

Input:

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

Output:

1/24*(14*e^(3/2)*x^3+3*2^(1/2)*arctanh(1/2*(e*x^2+d)^(1/2)/x*2^(1/2)/e^(1/ 
2))*(e*x^2+d)^(3/2)+18*e^(1/2)*d*x)/e^(1/2)/(e*x^2+d)^(3/2)/d^3
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 127 vs. \(2 (60) = 120\).

Time = 0.09 (sec) , antiderivative size = 279, normalized size of antiderivative = 3.49 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )} \, dx=\left [\frac {3 \, \sqrt {2} {\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {e} \log \left (\frac {17 \, e^{2} x^{4} + 14 \, d e x^{2} + 4 \, \sqrt {2} {\left (3 \, e x^{3} + d x\right )} \sqrt {e x^{2} + d} \sqrt {e} + d^{2}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right ) + 8 \, {\left (7 \, e^{2} x^{3} + 9 \, d e x\right )} \sqrt {e x^{2} + d}}{96 \, {\left (d^{3} e^{3} x^{4} + 2 \, d^{4} e^{2} x^{2} + d^{5} e\right )}}, -\frac {3 \, \sqrt {2} {\left (e^{2} x^{4} + 2 \, d e x^{2} + d^{2}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {2} {\left (3 \, e x^{2} + d\right )} \sqrt {e x^{2} + d} \sqrt {-e}}{4 \, {\left (e^{2} x^{3} + d e x\right )}}\right ) - 4 \, {\left (7 \, e^{2} x^{3} + 9 \, d e x\right )} \sqrt {e x^{2} + d}}{48 \, {\left (d^{3} e^{3} x^{4} + 2 \, d^{4} e^{2} x^{2} + d^{5} e\right )}}\right ] \] Input:

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

Output:

[1/96*(3*sqrt(2)*(e^2*x^4 + 2*d*e*x^2 + d^2)*sqrt(e)*log((17*e^2*x^4 + 14* 
d*e*x^2 + 4*sqrt(2)*(3*e*x^3 + d*x)*sqrt(e*x^2 + d)*sqrt(e) + d^2)/(e^2*x^ 
4 - 2*d*e*x^2 + d^2)) + 8*(7*e^2*x^3 + 9*d*e*x)*sqrt(e*x^2 + d))/(d^3*e^3* 
x^4 + 2*d^4*e^2*x^2 + d^5*e), -1/48*(3*sqrt(2)*(e^2*x^4 + 2*d*e*x^2 + d^2) 
*sqrt(-e)*arctan(1/4*sqrt(2)*(3*e*x^2 + d)*sqrt(e*x^2 + d)*sqrt(-e)/(e^2*x 
^3 + d*e*x)) - 4*(7*e^2*x^3 + 9*d*e*x)*sqrt(e*x^2 + d))/(d^3*e^3*x^4 + 2*d 
^4*e^2*x^2 + d^5*e)]
 

Sympy [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )} \, dx=- \int \frac {1}{- d^{3} \sqrt {d + e x^{2}} - d^{2} e x^{2} \sqrt {d + e x^{2}} + d e^{2} x^{4} \sqrt {d + e x^{2}} + e^{3} x^{6} \sqrt {d + e x^{2}}}\, dx \] Input:

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

Output:

-Integral(1/(-d**3*sqrt(d + e*x**2) - d**2*e*x**2*sqrt(d + e*x**2) + d*e** 
2*x**4*sqrt(d + e*x**2) + e**3*x**6*sqrt(d + e*x**2)), x)
 

Maxima [F]

\[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )} \, dx=\int { -\frac {1}{{\left (e^{2} x^{4} - d^{2}\right )} {\left (e x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.41 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )} \, dx=\frac {x {\left (\frac {7 \, e x^{2}}{d^{3}} + \frac {9}{d^{2}}\right )}}{12 \, {\left (e x^{2} + d\right )}^{\frac {3}{2}}} + \frac {\sqrt {2} \log \left (\frac {{\left | 2 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} - 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}{{\left | 2 \, {\left (\sqrt {e} x - \sqrt {e x^{2} + d}\right )}^{2} + 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}\right )}{16 \, d^{2} \sqrt {e} {\left | d \right |}} \] Input:

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

Output:

1/12*x*(7*e*x^2/d^3 + 9/d^2)/(e*x^2 + d)^(3/2) + 1/16*sqrt(2)*log(abs(2*(s 
qrt(e)*x - sqrt(e*x^2 + d))^2 - 4*sqrt(2)*abs(d) - 6*d)/abs(2*(sqrt(e)*x - 
 sqrt(e*x^2 + d))^2 + 4*sqrt(2)*abs(d) - 6*d))/(d^2*sqrt(e)*abs(d))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )} \, dx=\int \frac {1}{\left (d^2-e^2\,x^4\right )\,{\left (e\,x^2+d\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 542, normalized size of antiderivative = 6.78 \[ \int \frac {1}{\left (d+e x^2\right )^{3/2} \left (d^2-e^2 x^4\right )} \, dx=\frac {36 \sqrt {e \,x^{2}+d}\, d e x +28 \sqrt {e \,x^{2}+d}\, e^{2} x^{3}-3 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{2}-6 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d e \,x^{2}-3 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e^{2} x^{4}+3 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{2}+6 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d e \,x^{2}+3 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}-\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e^{2} x^{4}+3 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{2}+6 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d e \,x^{2}+3 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}-\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e^{2} x^{4}-3 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d^{2}-6 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) d e \,x^{2}-3 \sqrt {e}\, \sqrt {2}\, \mathrm {log}\left (\frac {\sqrt {e \,x^{2}+d}+\sqrt {d}\, \sqrt {2}+\sqrt {d}+\sqrt {e}\, x}{\sqrt {d}}\right ) e^{2} x^{4}-20 \sqrt {e}\, d^{2}-40 \sqrt {e}\, d e \,x^{2}-20 \sqrt {e}\, e^{2} x^{4}}{48 d^{3} e \left (e^{2} x^{4}+2 d e \,x^{2}+d^{2}\right )} \] Input:

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

Output:

(36*sqrt(d + e*x**2)*d*e*x + 28*sqrt(d + e*x**2)*e**2*x**3 - 3*sqrt(e)*sqr 
t(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d 
))*d**2 - 6*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) - sqrt 
(d) + sqrt(e)*x)/sqrt(d))*d*e*x**2 - 3*sqrt(e)*sqrt(2)*log((sqrt(d + e*x** 
2) - sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*e**2*x**4 + 3*sqrt(e) 
*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sq 
rt(d))*d**2 + 6*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) - sqrt(d)*sqrt(2) + 
sqrt(d) + sqrt(e)*x)/sqrt(d))*d*e*x**2 + 3*sqrt(e)*sqrt(2)*log((sqrt(d + e 
*x**2) - sqrt(d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*e**2*x**4 + 3*sqr 
t(e)*sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x 
)/sqrt(d))*d**2 + 6*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2 
) - sqrt(d) + sqrt(e)*x)/sqrt(d))*d*e*x**2 + 3*sqrt(e)*sqrt(2)*log((sqrt(d 
 + e*x**2) + sqrt(d)*sqrt(2) - sqrt(d) + sqrt(e)*x)/sqrt(d))*e**2*x**4 - 3 
*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sqrt(2) + sqrt(d) + sqrt( 
e)*x)/sqrt(d))*d**2 - 6*sqrt(e)*sqrt(2)*log((sqrt(d + e*x**2) + sqrt(d)*sq 
rt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*d*e*x**2 - 3*sqrt(e)*sqrt(2)*log((sq 
rt(d + e*x**2) + sqrt(d)*sqrt(2) + sqrt(d) + sqrt(e)*x)/sqrt(d))*e**2*x**4 
 - 20*sqrt(e)*d**2 - 40*sqrt(e)*d*e*x**2 - 20*sqrt(e)*e**2*x**4)/(48*d**3* 
e*(d**2 + 2*d*e*x**2 + e**2*x**4))