\(\int \frac {(d^2-e^2 x^4)^{3/2}}{(d+e x^2)^{5/2}} \, dx\) [142]

Optimal result

Integrand size = 28, antiderivative size = 131 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {x \sqrt {d^2-e^2 x^4}}{2 \sqrt {d+e x^2}}-\frac {5 d \arctan \left (\frac {\sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{2 \sqrt {e}}+\frac {2 \sqrt {2} d \arctan \left (\frac {\sqrt {2} \sqrt {e} x \sqrt {d+e x^2}}{\sqrt {d^2-e^2 x^4}}\right )}{\sqrt {e}} \] Output:

-1/2*x*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)-5/2*d*arctan(e^(1/2)*x*(e*x^2+ 
d)^(1/2)/(-e^2*x^4+d^2)^(1/2))/e^(1/2)+2*2^(1/2)*d*arctan(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 = 4.77 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.24 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=-\frac {x \sqrt {d^2-e^2 x^4}}{2 \sqrt {d+e x^2}}+\frac {2 \sqrt {2} d \sqrt {d^2-e^2 x^4} \arctan \left (\frac {\sqrt {2} \sqrt {e} x}{\sqrt {d-e x^2}}\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}-\frac {5 i d \log \left (-2 i \sqrt {e} x+\frac {2 \sqrt {d^2-e^2 x^4}}{\sqrt {d+e x^2}}\right )}{2 \sqrt {e}} \] Input:

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

Output:

-1/2*(x*Sqrt[d^2 - e^2*x^4])/Sqrt[d + e*x^2] + (2*Sqrt[2]*d*Sqrt[d^2 - e^2 
*x^4]*ArcTan[(Sqrt[2]*Sqrt[e]*x)/Sqrt[d - e*x^2]])/(Sqrt[e]*Sqrt[d - e*x^2 
]*Sqrt[d + e*x^2]) - (((5*I)/2)*d*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 = 0.45 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.98, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1396, 318, 27, 398, 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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 318

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {\int \frac {d e \left (3 d-5 e x^2\right )}{\sqrt {d-e x^2} \left (e x^2+d\right )}dx}{2 e}-\frac {1}{2} x \sqrt {d-e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{2} d \int \frac {3 d-5 e x^2}{\sqrt {d-e x^2} \left (e x^2+d\right )}dx-\frac {1}{2} x \sqrt {d-e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 398

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{2} d \left (8 d \int \frac {1}{\sqrt {d-e x^2} \left (e x^2+d\right )}dx-5 \int \frac {1}{\sqrt {d-e x^2}}dx\right )-\frac {1}{2} x \sqrt {d-e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{2} d \left (8 d \int \frac {1}{\sqrt {d-e x^2} \left (e x^2+d\right )}dx-5 \int \frac {1}{\frac {e x^2}{d-e x^2}+1}d\frac {x}{\sqrt {d-e x^2}}\right )-\frac {1}{2} x \sqrt {d-e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{2} d \left (8 d \int \frac {1}{\sqrt {d-e x^2} \left (e x^2+d\right )}dx-\frac {5 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d-e x^2}}\right )}{\sqrt {e}}\right )-\frac {1}{2} x \sqrt {d-e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 291

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {1}{2} d \left (8 d \int \frac {1}{\frac {2 d e x^2}{d-e x^2}+d}d\frac {x}{\sqrt {d-e x^2}}-\frac {5 \arctan \left (\frac {\sqrt {e} x}{\sqrt {d-e x^2}}\right )}{\sqrt {e}}\right )-\frac {1}{2} x \sqrt {d-e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

\(\Big \downarrow \) 218

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 318

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

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

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 [A] (verified)

Time = 0.48 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.56

method	result	size
default	\(\frac {\sqrt {-e^{2} x^{4}+d^{2}}\, \left (2 \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}+\sqrt {-d e}\, x +d \right )}{e x +\sqrt {-d e}}\right ) \sqrt {e}\, d^{\frac {3}{2}} \sqrt {2}-2 \sqrt {e}\, d^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {2 e \left (\sqrt {2}\, \sqrt {d}\, \sqrt {-e \,x^{2}+d}-\sqrt {-d e}\, x +d \right )}{e x -\sqrt {-d e}}\right )-\sqrt {-d e}\, \sqrt {-e \,x^{2}+d}\, \sqrt {e}\, x -5 \sqrt {-d e}\, \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right ) d \right )}{2 \sqrt {e \,x^{2}+d}\, \sqrt {-e \,x^{2}+d}\, \sqrt {-d e}\, \sqrt {e}}\)	\(204\)
risch	\(-\frac {x \sqrt {-e \,x^{2}+d}\, \sqrt {\frac {-e^{2} x^{4}+d^{2}}{e \,x^{2}+d}}\, \sqrt {e \,x^{2}+d}}{2 \sqrt {-e^{2} x^{4}+d^{2}}}+\frac {\left (-\frac {5 d \arctan \left (\frac {\sqrt {e}\, x}{\sqrt {-e \,x^{2}+d}}\right )}{2 \sqrt {e}}-\frac {d^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {4 d -2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )+2 \sqrt {2}\, \sqrt {d}\, \sqrt {-\left (x -\frac {\sqrt {-d e}}{e}\right )^{2} e -2 \sqrt {-d e}\, \left (x -\frac {\sqrt {-d e}}{e}\right )+2 d}}{x -\frac {\sqrt {-d e}}{e}}\right )}{\sqrt {-d e}}+\frac {d^{\frac {3}{2}} \sqrt {2}\, \ln \left (\frac {4 d +2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )+2 \sqrt {2}\, \sqrt {d}\, \sqrt {-\left (x +\frac {\sqrt {-d e}}{e}\right )^{2} e +2 \sqrt {-d e}\, \left (x +\frac {\sqrt {-d e}}{e}\right )+2 d}}{x +\frac {\sqrt {-d e}}{e}}\right )}{\sqrt {-d e}}\right ) \sqrt {\frac {-e^{2} x^{4}+d^{2}}{e \,x^{2}+d}}\, \sqrt {e \,x^{2}+d}}{\sqrt {-e^{2} x^{4}+d^{2}}}\)	\(346\)

Input:

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

Output:

1/2*(-e^2*x^4+d^2)^(1/2)/(e*x^2+d)^(1/2)*(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^(3/2)*2^(1/2)-2 
*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*e)^(1/2)*(-e*x^2+d)^(1/2)*e^(1/2)*x-5*(- 
d*e)^(1/2)*arctan(e^(1/2)*x/(-e*x^2+d)^(1/2))*d)/(-e*x^2+d)^(1/2)/(-d*e)^( 
1/2)/e^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 401, normalized size of antiderivative = 3.06 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\left [-\frac {2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} e x - 4 \, \sqrt {2} {\left (d e^{2} x^{2} + d^{2} e\right )} \sqrt {-\frac {1}{e}} \log \left (-\frac {3 \, e^{2} x^{4} + 2 \, d e x^{2} + 2 \, \sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} e x \sqrt {-\frac {1}{e}} - d^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}\right ) + 5 \, {\left (d e x^{2} + d^{2}\right )} \sqrt {-e} \log \left (-\frac {2 \, e^{2} x^{4} + d e x^{2} + 2 \, \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {-e} x - d^{2}}{e x^{2} + d}\right )}{4 \, {\left (e^{2} x^{2} + d e\right )}}, -\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} e x - 5 \, {\left (d e x^{2} + d^{2}\right )} \sqrt {e} \arctan \left (\frac {\sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right ) + \frac {4 \, \sqrt {2} {\left (d e^{2} x^{2} + d^{2} e\right )} \arctan \left (\frac {\sqrt {2} \sqrt {-e^{2} x^{4} + d^{2}} \sqrt {e x^{2} + d} \sqrt {e} x}{e^{2} x^{4} - d^{2}}\right )}{\sqrt {e}}}{2 \, {\left (e^{2} x^{2} + d e\right )}}\right ] \] Input:

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

Output:

[-1/4*(2*sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*e*x - 4*sqrt(2)*(d*e^2*x^2 + 
 d^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)) + 
 5*(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^2*x^2 + d*e), -1/2 
*(sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*e*x - 5*(d*e*x^2 + d^2)*sqrt(e)*arc 
tan(sqrt(-e^2*x^4 + d^2)*sqrt(e*x^2 + d)*sqrt(e)*x/(e^2*x^4 - d^2)) + 4*sq 
rt(2)*(d*e^2*x^2 + d^2*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))/sqrt(e))/(e^2*x^2 + d*e)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.26 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{\left (d+e x^2\right )^{5/2}} \, dx=\frac {\sqrt {2} d^{2} \log \left (\frac {{\left | 2 \, {\left (\sqrt {-e} x - \sqrt {-e x^{2} + d}\right )}^{2} - 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}{{\left | 2 \, {\left (\sqrt {-e} x - \sqrt {-e x^{2} + d}\right )}^{2} + 4 \, \sqrt {2} {\left | d \right |} - 6 \, d \right |}}\right )}{\sqrt {-e} {\left | d \right |}} - \frac {1}{2} \, \sqrt {-e x^{2} + d} x + \frac {d \log \left ({\left (\sqrt {-e} x - \sqrt {-e x^{2} + d}\right )}^{2}\right )}{\sqrt {-e}} + \frac {d \log \left ({\left | -\sqrt {-e} x + \sqrt {-e x^{2} + d} \right |}\right )}{2 \, \sqrt {-e}} \] Input:

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

Output:

sqrt(2)*d^2*log(abs(2*(sqrt(-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)) 
/(sqrt(-e)*abs(d)) - 1/2*sqrt(-e*x^2 + d)*x + d*log((sqrt(-e)*x - sqrt(-e* 
x^2 + d))^2)/sqrt(-e) + 1/2*d*log(abs(-sqrt(-e)*x + sqrt(-e*x^2 + d)))/sqr 
t(-e)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(2*sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x + 5*int((sqrt(d + e*x**2)*sqr 
t(d**2 - e**2*x**4)*x**4)/(d**3 + d**2*e*x**2 - d*e**2*x**4 - e**3*x**6),x 
)*d*e**2 + 5*int((sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4)*x**4)/(d**3 + d* 
*2*e*x**2 - d*e**2*x**4 - e**3*x**6),x)*e**3*x**2 - int((sqrt(d + e*x**2)* 
sqrt(d**2 - e**2*x**4))/(d**3 + d**2*e*x**2 - d*e**2*x**4 - e**3*x**6),x)* 
d**3 - int((sqrt(d + e*x**2)*sqrt(d**2 - e**2*x**4))/(d**3 + d**2*e*x**2 - 
 d*e**2*x**4 - e**3*x**6),x)*d**2*e*x**2)/(d + e*x**2)