\(\int \frac {(d^2-e^2 x^4)^{3/2}}{d-e x^2} \, dx\) [97]

Optimal result

Integrand size = 27, antiderivative size = 156 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{d-e x^2} \, dx=\frac {1}{15} x \left (5 d+3 e x^2\right ) \sqrt {d^2-e^2 x^4}+\frac {2 d^{7/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{5 \sqrt {e} \sqrt {d^2-e^2 x^4}}+\frac {4 d^{7/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{15 \sqrt {e} \sqrt {d^2-e^2 x^4}} \] Output:

1/15*x*(3*e*x^2+5*d)*(-e^2*x^4+d^2)^(1/2)+2/5*d^(7/2)*(1-e^2*x^4/d^2)^(1/2 
)*EllipticE(e^(1/2)*x/d^(1/2),I)/e^(1/2)/(-e^2*x^4+d^2)^(1/2)+4/15*d^(7/2) 
*(1-e^2*x^4/d^2)^(1/2)*EllipticF(e^(1/2)*x/d^(1/2),I)/e^(1/2)/(-e^2*x^4+d^ 
2)^(1/2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.

Time = 0.02 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.56 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{d-e x^2} \, dx=\frac {\sqrt {d^2-e^2 x^4} \left (3 d x \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {1}{4},\frac {5}{4},\frac {e^2 x^4}{d^2}\right )+e x^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},\frac {3}{4},\frac {7}{4},\frac {e^2 x^4}{d^2}\right )\right )}{3 \sqrt {1-\frac {e^2 x^4}{d^2}}} \] Input:

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

Output:

(Sqrt[d^2 - e^2*x^4]*(3*d*x*Hypergeometric2F1[-1/2, 1/4, 5/4, (e^2*x^4)/d^ 
2] + e*x^3*Hypergeometric2F1[-1/2, 3/4, 7/4, (e^2*x^4)/d^2]))/(3*Sqrt[1 - 
(e^2*x^4)/d^2])
 

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.56, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.407, Rules used = {1396, 318, 27, 403, 27, 399, 289, 329, 327, 765, 762}

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.

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

\(\Big \downarrow \) 1396

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

\(\Big \downarrow \) 318

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 403

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 399

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

\(\Big \downarrow \) 289

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

\(\Big \downarrow \) 329

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

\(\Big \downarrow \) 327

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

\(\Big \downarrow \) 765

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

\(\Big \downarrow \) 762

\(\displaystyle \frac {\sqrt {d^2-e^2 x^4} \left (\frac {2}{5} d \left (\frac {1}{3} d \left (\frac {2 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ),-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}+\frac {3 d^{3/2} \sqrt {1-\frac {e^2 x^4}{d^2}} E\left (\left .\arcsin \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )\right |-1\right )}{\sqrt {e} \sqrt {d-e x^2} \sqrt {d+e x^2}}\right )+\frac {4}{3} x \sqrt {d-e x^2} \sqrt {d+e x^2}\right )-\frac {1}{5} x \left (d-e x^2\right )^{3/2} \sqrt {d+e x^2}\right )}{\sqrt {d-e x^2} \sqrt {d+e x^2}}\)

Input:

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

Output:

(Sqrt[d^2 - e^2*x^4]*(-1/5*(x*(d - e*x^2)^(3/2)*Sqrt[d + e*x^2]) + (2*d*(( 
4*x*Sqrt[d - e*x^2]*Sqrt[d + e*x^2])/3 + (d*((3*d^(3/2)*Sqrt[1 - (e^2*x^4) 
/d^2]*EllipticE[ArcSin[(Sqrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2] 
*Sqrt[d + e*x^2]) + (2*d^(3/2)*Sqrt[1 - (e^2*x^4)/d^2]*EllipticF[ArcSin[(S 
qrt[e]*x)/Sqrt[d]], -1])/(Sqrt[e]*Sqrt[d - e*x^2]*Sqrt[d + e*x^2])))/3))/5 
))/(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 289

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim 
p[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c + b*d*x^4)^FracPart 
[p])   Int[(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b* 
c + a*d, 0] &&  !IntegerQ[p]
 

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 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 329

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
a*(Sqrt[1 - b^2*(x^4/a^2)]/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]))   Int[Sqrt[1 
+ b*(x^2/a)]/Sqrt[1 - b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b 
*c + a*d, 0] &&  !(LtQ[a*c, 0] && GtQ[a*b, 0])
 

rule 399

Int[((e_) + (f_.)*(x_)^2)/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_) 
^2]), x_Symbol] :> Simp[f/b   Int[Sqrt[a + b*x^2]/Sqrt[c + d*x^2], x], x] + 
 Simp[(b*e - a*f)/b   Int[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; Fr 
eeQ[{a, b, c, d, e, f}, x] &&  !((PosQ[b/a] && PosQ[d/c]) || (NegQ[b/a] && 
(PosQ[d/c] || (GtQ[a, 0] && ( !GtQ[c, 0] || SimplerSqrtQ[-b/a, -d/c])))))
 

rule 403

Int[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( 
x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + 
 q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1))   Int[(a + b*x^2)^p*(c 
+ d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + 
f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, 
 d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 765

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[Sqrt[1 + b*(x^4/a)]/Sqrt 
[a + b*x^4]   Int[1/Sqrt[1 + b*(x^4/a)], x], x] /; FreeQ[{a, b}, x] && NegQ 
[b/a] &&  !GtQ[a, 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 [A] (verified)

Time = 2.63 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.10

method	result	size
risch	\(\frac {x \left (3 e \,x^{2}+5 d \right ) \sqrt {-e^{2} x^{4}+d^{2}}}{15}+\frac {2 d^{2} \left (\frac {5 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {3 d \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{\sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\right )}{15}\)	\(172\)
default	\(\frac {e \,x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{5}+\frac {d x \sqrt {-e^{2} x^{4}+d^{2}}}{3}+\frac {2 d^{3} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {2 d^{3} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{5 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(181\)
elliptic	\(\frac {e \,x^{3} \sqrt {-e^{2} x^{4}+d^{2}}}{5}+\frac {d x \sqrt {-e^{2} x^{4}+d^{2}}}{3}+\frac {2 d^{3} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )}{3 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}-\frac {2 d^{3} \sqrt {1-\frac {e \,x^{2}}{d}}\, \sqrt {1+\frac {e \,x^{2}}{d}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e}{d}}, i\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e}{d}}, i\right )\right )}{5 \sqrt {\frac {e}{d}}\, \sqrt {-e^{2} x^{4}+d^{2}}}\)	\(181\)

Input:

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

Output:

1/15*x*(3*e*x^2+5*d)*(-e^2*x^4+d^2)^(1/2)+2/15*d^2*(5*d/(e/d)^(1/2)*(1-e*x 
^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/(-e^2*x^4+d^2)^(1/2)*EllipticF(x*(e/d)^(1/2) 
,I)-3*d/(e/d)^(1/2)*(1-e*x^2/d)^(1/2)*(1+e*x^2/d)^(1/2)/(-e^2*x^4+d^2)^(1/ 
2)*(EllipticF(x*(e/d)^(1/2),I)-EllipticE(x*(e/d)^(1/2),I)))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.81 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{d-e x^2} \, dx=-\frac {6 \, \sqrt {-e^{2}} d^{3} x \sqrt {\frac {d}{e}} E(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - 2 \, {\left (3 \, d^{3} + 5 \, d^{2} e\right )} \sqrt {-e^{2}} x \sqrt {\frac {d}{e}} F(\arcsin \left (\frac {\sqrt {\frac {d}{e}}}{x}\right )\,|\,-1) - {\left (3 \, e^{3} x^{4} + 5 \, d e^{2} x^{2} - 6 \, d^{2} e\right )} \sqrt {-e^{2} x^{4} + d^{2}}}{15 \, e^{2} x} \] Input:

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

Output:

-1/15*(6*sqrt(-e^2)*d^3*x*sqrt(d/e)*elliptic_e(arcsin(sqrt(d/e)/x), -1) - 
2*(3*d^3 + 5*d^2*e)*sqrt(-e^2)*x*sqrt(d/e)*elliptic_f(arcsin(sqrt(d/e)/x), 
 -1) - (3*e^3*x^4 + 5*d*e^2*x^2 - 6*d^2*e)*sqrt(-e^2*x^4 + d^2))/(e^2*x)
 

Sympy [A] (verification not implemented)

Time = 2.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.54 \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{d-e x^2} \, dx=\frac {d^{2} x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {5}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {5}{4}\right )} + \frac {d e x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {3}{4} \\ \frac {7}{4} \end {matrix}\middle | {\frac {e^{2} x^{4} e^{2 i \pi }}{d^{2}}} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \] Input:

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

Output:

d**2*x*gamma(1/4)*hyper((-1/2, 1/4), (5/4,), e**2*x**4*exp_polar(2*I*pi)/d 
**2)/(4*gamma(5/4)) + d*e*x**3*gamma(3/4)*hyper((-1/2, 3/4), (7/4,), e**2* 
x**4*exp_polar(2*I*pi)/d**2)/(4*gamma(7/4))
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\left (d^2-e^2 x^4\right )^{3/2}}{d-e x^2} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

(5*sqrt(d**2 - e**2*x**4)*d*x + 3*sqrt(d**2 - e**2*x**4)*e*x**3 + 10*int(s 
qrt(d**2 - e**2*x**4)/(d**2 - e**2*x**4),x)*d**3 + 6*int((sqrt(d**2 - e**2 
*x**4)*x**2)/(d**2 - e**2*x**4),x)*d**2*e)/15