\(\int \frac {(d-e x^2)^{3/2}}{(d^2-e^2 x^4)^{5/2}} \, dx\) [169]

Optimal result

Integrand size = 29, antiderivative size = 131 \[ \int \frac {\left (d-e x^2\right )^{3/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {x \left (d-e x^2\right )^{3/2}}{6 d^2 \left (d^2-e^2 x^4\right )^{3/2}}+\frac {7 x \sqrt {d-e x^2}}{12 d^3 \sqrt {d^2-e^2 x^4}}+\frac {\text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x \sqrt {d-e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{4 \sqrt {2} d^3 \sqrt {e}} \] Output:

1/6*x*(-e*x^2+d)^(3/2)/d^2/(-e^2*x^4+d^2)^(3/2)+7/12*x*(-e*x^2+d)^(1/2)/d^ 
3/(-e^2*x^4+d^2)^(1/2)+1/8*arctanh(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 = 5.17 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92 \[ \int \frac {\left (d-e x^2\right )^{3/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\frac {\sqrt {d^2-e^2 x^4} \left (2 \sqrt {e} x \sqrt {d+e x^2} \left (9 d+7 e x^2\right )+3 \sqrt {2} \left (d+e x^2\right )^2 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )\right )}{24 d^3 \sqrt {e} \sqrt {d-e x^2} \left (d+e x^2\right )^{5/2}} \] Input:

Integrate[(d - e*x^2)^(3/2)/(d^2 - e^2*x^4)^(5/2),x]
 

Output:

(Sqrt[d^2 - e^2*x^4]*(2*Sqrt[e]*x*Sqrt[d + e*x^2]*(9*d + 7*e*x^2) + 3*Sqrt 
[2]*(d + e*x^2)^2*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]]))/(24*d^3*S 
qrt[e]*Sqrt[d - e*x^2]*(d + e*x^2)^(5/2))
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.276, Rules used = {1396, 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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 316

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}-\frac {\int -\frac {e \left (5 d-2 e x^2\right )}{\left (d-e x^2\right ) \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {e \left (5 d-2 e x^2\right )}{\left (d-e x^2\right ) \left (e x^2+d\right )^{3/2}}dx}{6 d^2 e}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\int \frac {5 d-2 e x^2}{\left (d-e x^2\right ) \left (e x^2+d\right )^{3/2}}dx}{6 d^2}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 402

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {7 x}{2 d \sqrt {d+e x^2}}-\frac {\int -\frac {3 d^2 e}{\left (d-e x^2\right ) \sqrt {e x^2+d}}dx}{2 d^2 e}}{6 d^2}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3}{2} \int \frac {1}{\left (d-e x^2\right ) \sqrt {e x^2+d}}dx+\frac {7 x}{2 d \sqrt {d+e x^2}}}{6 d^2}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3}{2} \int \frac {1}{d-\frac {2 d e x^2}{e x^2+d}}d\frac {x}{\sqrt {e x^2+d}}+\frac {7 x}{2 d \sqrt {d+e x^2}}}{6 d^2}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\sqrt {d-e x^2} \sqrt {d+e x^2} \left (\frac {\frac {3 \text {arctanh}\left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d+e x^2}}\right )}{2 \sqrt {2} d \sqrt {e}}+\frac {7 x}{2 d \sqrt {d+e x^2}}}{6 d^2}+\frac {x}{6 d^2 \left (d+e x^2\right )^{3/2}}\right )}{\sqrt {d^2-e^2 x^4}}\)

Input:

Int[(d - e*x^2)^(3/2)/(d^2 - e^2*x^4)^(5/2),x]
 

Output:

(Sqrt[d - e*x^2]*Sqrt[d + e*x^2]*(x/(6*d^2*(d + e*x^2)^(3/2)) + ((7*x)/(2* 
d*Sqrt[d + e*x^2]) + (3*ArcTanh[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d + e*x^2]])/(2*S 
qrt[2]*d*Sqrt[e]))/(6*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 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 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. \(605\) vs. \(2(105)=210\).

Time = 0.57 (sec) , antiderivative size = 606, normalized size of antiderivative = 4.63

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, e^{2} \left (3 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {e \,x^{2}+d}+\sqrt {d e}\, x +d \right )}{e x -\sqrt {d e}}\right ) \sqrt {2}\, d e \,x^{2} \sqrt {e \,x^{2}+d}\, \sqrt {-\frac {\left (e x +\sqrt {-d e}\right ) \left (-e x +\sqrt {-d e}\right )}{e}}-3 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {e \,x^{2}+d}-\sqrt {d e}\, x +d \right )}{e x +\sqrt {d e}}\right ) \sqrt {2}\, d e \,x^{2} \sqrt {e \,x^{2}+d}\, \sqrt {-\frac {\left (e x +\sqrt {-d e}\right ) \left (-e x +\sqrt {-d e}\right )}{e}}+16 e \,x^{3} \sqrt {d e}\, \sqrt {e \,x^{2}+d}\, \sqrt {d}+12 e \,x^{3} \sqrt {d e}\, \sqrt {d}\, \sqrt {-\frac {\left (e x +\sqrt {-d e}\right ) \left (-e x +\sqrt {-d e}\right )}{e}}+3 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {e \,x^{2}+d}+\sqrt {d e}\, x +d \right )}{e x -\sqrt {d e}}\right ) \sqrt {2}\, d^{2} \sqrt {e \,x^{2}+d}\, \sqrt {-\frac {\left (e x +\sqrt {-d e}\right ) \left (-e x +\sqrt {-d e}\right )}{e}}-3 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {e \,x^{2}+d}-\sqrt {d e}\, x +d \right )}{e x +\sqrt {d e}}\right ) \sqrt {2}\, d^{2} \sqrt {e \,x^{2}+d}\, \sqrt {-\frac {\left (e x +\sqrt {-d e}\right ) \left (-e x +\sqrt {-d e}\right )}{e}}+24 d^{\frac {3}{2}} x \sqrt {d e}\, \sqrt {e \,x^{2}+d}+12 d^{\frac {3}{2}} x \sqrt {d e}\, \sqrt {-\frac {\left (e x +\sqrt {-d e}\right ) \left (-e x +\sqrt {-d e}\right )}{e}}\right )}{24 d^{\frac {5}{2}} \sqrt {-e \,x^{2}+d}\, \left (e \,x^{2}+d \right ) \left (\sqrt {d e}-\sqrt {-d e}\right ) \left (\sqrt {d e}+\sqrt {-d e}\right ) \sqrt {d e}\, \left (e x -\sqrt {-d e}\right ) \sqrt {-\frac {\left (e x +\sqrt {-d e}\right ) \left (-e x +\sqrt {-d e}\right )}{e}}\, \left (e x +\sqrt {-d e}\right )}\)	\(606\)

Input:

int((-e*x^2+d)^(3/2)/(-e^2*x^4+d^2)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

1/24*(-e^2*x^4+d^2)^(1/2)*e^2/d^(5/2)*(3*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)))*2^(1/2)*d*e*x^2*(e*x^2+d)^(1/2) 
*(-(e*x+(-d*e)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2)-3*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)))*2^(1/2)*d*e*x^2*(e* 
x^2+d)^(1/2)*(-(e*x+(-d*e)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2)+16*e*x^3*(d 
*e)^(1/2)*(e*x^2+d)^(1/2)*d^(1/2)+12*e*x^3*(d*e)^(1/2)*d^(1/2)*(-(e*x+(-d* 
e)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2)+3*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)))*2^(1/2)*d^2*(e*x^2+d)^(1/2)*(-( 
e*x+(-d*e)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2)-3*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)))*2^(1/2)*d^2*(e*x^2+d)^( 
1/2)*(-(e*x+(-d*e)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2)+24*d^(3/2)*x*(d*e)^ 
(1/2)*(e*x^2+d)^(1/2)+12*d^(3/2)*x*(d*e)^(1/2)*(-(e*x+(-d*e)^(1/2))/e*(-e* 
x+(-d*e)^(1/2)))^(1/2))/(-e*x^2+d)^(1/2)/(e*x^2+d)/((d*e)^(1/2)-(-d*e)^(1/ 
2))/((d*e)^(1/2)+(-d*e)^(1/2))/(d*e)^(1/2)/(e*x-(-d*e)^(1/2))/(-(e*x+(-d*e 
)^(1/2))/e*(-e*x+(-d*e)^(1/2)))^(1/2)/(e*x+(-d*e)^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 372, normalized size of antiderivative = 2.84 \[ \int \frac {\left (d-e x^2\right )^{3/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\left [\frac {3 \, \sqrt {2} {\left (e^{3} x^{6} + d e^{2} x^{4} - d^{2} e x^{2} - d^{3}\right )} \sqrt {e} \log \left (-\frac {3 \, e^{2} x^{4} - 2 \, d e x^{2} - 2 \, \sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {e} x - d^{2}}{e^{2} x^{4} - 2 \, d e x^{2} + d^{2}}\right ) - 4 \, \sqrt {-e^{2} x^{4} + d^{2}} {\left (7 \, e^{2} x^{3} + 9 \, d e x\right )} \sqrt {-e x^{2} + d}}{48 \, {\left (d^{3} e^{4} x^{6} + d^{4} e^{3} x^{4} - d^{5} e^{2} x^{2} - d^{6} e\right )}}, \frac {3 \, \sqrt {2} {\left (e^{3} x^{6} + d e^{2} x^{4} - d^{2} e x^{2} - d^{3}\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {-e x^{2} + d} \sqrt {-e} x}{e^{2} x^{4} - d^{2}}\right ) - 2 \, \sqrt {-e^{2} x^{4} + d^{2}} {\left (7 \, e^{2} x^{3} + 9 \, d e x\right )} \sqrt {-e x^{2} + d}}{24 \, {\left (d^{3} e^{4} x^{6} + d^{4} e^{3} x^{4} - d^{5} e^{2} x^{2} - d^{6} e\right )}}\right ] \] Input:

integrate((-e*x^2+d)^(3/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="fricas")
 

Output:

[1/48*(3*sqrt(2)*(e^3*x^6 + d*e^2*x^4 - d^2*e*x^2 - d^3)*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)) - 4*sqrt(-e^2*x^4 + d^2)*(7*e^2*x 
^3 + 9*d*e*x)*sqrt(-e*x^2 + d))/(d^3*e^4*x^6 + d^4*e^3*x^4 - d^5*e^2*x^2 - 
 d^6*e), 1/24*(3*sqrt(2)*(e^3*x^6 + d*e^2*x^4 - 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)*(7*e^2*x^3 + 9*d*e*x)*sqrt(-e*x^2 + d))/(d 
^3*e^4*x^6 + d^4*e^3*x^4 - d^5*e^2*x^2 - d^6*e)]
 

Sympy [F]

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

integrate((-e*x**2+d)**(3/2)/(-e**2*x**4+d**2)**(5/2),x)
 

Output:

Integral((d - e*x**2)**(3/2)/(-(-d + e*x**2)*(d + e*x**2))**(5/2), x)
 

Maxima [F]

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

integrate((-e*x^2+d)^(3/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="maxima")
 

Output:

integrate((-e*x^2 + d)^(3/2)/(-e^2*x^4 + d^2)^(5/2), x)
 

Giac [F]

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

integrate((-e*x^2+d)^(3/2)/(-e^2*x^4+d^2)^(5/2),x, algorithm="giac")
 

Output:

integrate((-e*x^2 + d)^(3/2)/(-e^2*x^4 + d^2)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (d-e x^2\right )^{3/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, dx=\int \frac {{\left (d-e\,x^2\right )}^{3/2}}{{\left (d^2-e^2\,x^4\right )}^{5/2}} \,d x \] Input:

int((d - e*x^2)^(3/2)/(d^2 - e^2*x^4)^(5/2),x)
 

Output:

int((d - e*x^2)^(3/2)/(d^2 - e^2*x^4)^(5/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 542, normalized size of antiderivative = 4.14 \[ \int \frac {\left (d-e x^2\right )^{3/2}}{\left (d^2-e^2 x^4\right )^{5/2}} \, 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((-e*x^2+d)^(3/2)/(-e^2*x^4+d^2)^(5/2),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))