\(\int \frac {\sqrt {d-e x} \sqrt {d+e x}}{(a+c x^2)^{3/2}} \, dx\) [28]

Optimal result

Integrand size = 31, antiderivative size = 98 \[ \int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {d-e x} \sqrt {d+e x} E\left (\arctan \left (\frac {\sqrt {c} x}{\sqrt {a}}\right )|1+\frac {a e^2}{c d^2}\right )}{\sqrt {a} \sqrt {c} \sqrt {a+c x^2} \sqrt {\frac {a \left (d^2-e^2 x^2\right )}{d^2 \left (a+c x^2\right )}}} \] Output:

(-e*x+d)^(1/2)*(e*x+d)^(1/2)*EllipticE(c^(1/2)*x/a^(1/2)/(1+c*x^2/a)^(1/2) 
,(1+a*e^2/c/d^2)^(1/2))/a^(1/2)/c^(1/2)/(c*x^2+a)^(1/2)/(a*(-e^2*x^2+d^2)/ 
d^2/(c*x^2+a))^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 21.19 (sec) , antiderivative size = 201, normalized size of antiderivative = 2.05 \[ \int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {\frac {c}{a}} \left (\sqrt {\frac {c}{a}} x \left (d^2-e^2 x^2\right )+i d^2 \sqrt {1+\frac {c x^2}{a}} \sqrt {1-\frac {e^2 x^2}{d^2}} E\left (i \text {arcsinh}\left (\sqrt {\frac {c}{a}} x\right )|-\frac {a e^2}{c d^2}\right )-i d^2 \sqrt {1+\frac {c x^2}{a}} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {\frac {c}{a}} x\right ),-\frac {a e^2}{c d^2}\right )\right )}{c \sqrt {d-e x} \sqrt {d+e x} \sqrt {a+c x^2}} \] Input:

Integrate[(Sqrt[d - e*x]*Sqrt[d + e*x])/(a + c*x^2)^(3/2),x]
 

Output:

(Sqrt[c/a]*(Sqrt[c/a]*x*(d^2 - e^2*x^2) + I*d^2*Sqrt[1 + (c*x^2)/a]*Sqrt[1 
 - (e^2*x^2)/d^2]*EllipticE[I*ArcSinh[Sqrt[c/a]*x], -((a*e^2)/(c*d^2))] - 
I*d^2*Sqrt[1 + (c*x^2)/a]*Sqrt[1 - (e^2*x^2)/d^2]*EllipticF[I*ArcSinh[Sqrt 
[c/a]*x], -((a*e^2)/(c*d^2))]))/(c*Sqrt[d - e*x]*Sqrt[d + e*x]*Sqrt[a + c* 
x^2])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(251\) vs. \(2(98)=196\).

Time = 0.43 (sec) , antiderivative size = 251, normalized size of antiderivative = 2.56, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.355, Rules used = {648, 314, 25, 27, 389, 323, 323, 321, 331, 330, 327}

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.

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

\(\Big \downarrow \) 648

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

\(\Big \downarrow \) 314

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {x \sqrt {d^2-e^2 x^2}}{a \sqrt {a+c x^2}}-\frac {\int -\frac {e^2 x^2}{\sqrt {c x^2+a} \sqrt {d^2-e^2 x^2}}dx}{a}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {\int \frac {e^2 x^2}{\sqrt {c x^2+a} \sqrt {d^2-e^2 x^2}}dx}{a}+\frac {x \sqrt {d^2-e^2 x^2}}{a \sqrt {a+c x^2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {e^2 \int \frac {x^2}{\sqrt {c x^2+a} \sqrt {d^2-e^2 x^2}}dx}{a}+\frac {x \sqrt {d^2-e^2 x^2}}{a \sqrt {a+c x^2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 389

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {e^2 \left (\frac {\int \frac {\sqrt {c x^2+a}}{\sqrt {d^2-e^2 x^2}}dx}{c}-\frac {a d \sqrt {\frac {c x^2}{a}+1} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {e x}{d}\right ),-\frac {c d^2}{a e^2}\right )}{c e \sqrt {a+c x^2} \sqrt {d^2-e^2 x^2}}\right )}{a}+\frac {x \sqrt {d^2-e^2 x^2}}{a \sqrt {a+c x^2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 331

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {e^2 \left (\frac {\sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {\sqrt {c x^2+a}}{\sqrt {1-\frac {e^2 x^2}{d^2}}}dx}{c \sqrt {d^2-e^2 x^2}}-\frac {a d \sqrt {\frac {c x^2}{a}+1} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {e x}{d}\right ),-\frac {c d^2}{a e^2}\right )}{c e \sqrt {a+c x^2} \sqrt {d^2-e^2 x^2}}\right )}{a}+\frac {x \sqrt {d^2-e^2 x^2}}{a \sqrt {a+c x^2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 330

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {e^2 \left (\frac {\sqrt {a+c x^2} \sqrt {1-\frac {e^2 x^2}{d^2}} \int \frac {\sqrt {\frac {c x^2}{a}+1}}{\sqrt {1-\frac {e^2 x^2}{d^2}}}dx}{c \sqrt {\frac {c x^2}{a}+1} \sqrt {d^2-e^2 x^2}}-\frac {a d \sqrt {\frac {c x^2}{a}+1} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {e x}{d}\right ),-\frac {c d^2}{a e^2}\right )}{c e \sqrt {a+c x^2} \sqrt {d^2-e^2 x^2}}\right )}{a}+\frac {x \sqrt {d^2-e^2 x^2}}{a \sqrt {a+c x^2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

\(\Big \downarrow \) 327

\(\displaystyle \frac {\sqrt {d-e x} \sqrt {d+e x} \left (\frac {e^2 \left (\frac {d \sqrt {a+c x^2} \sqrt {1-\frac {e^2 x^2}{d^2}} E\left (\arcsin \left (\frac {e x}{d}\right )|-\frac {c d^2}{a e^2}\right )}{c e \sqrt {\frac {c x^2}{a}+1} \sqrt {d^2-e^2 x^2}}-\frac {a d \sqrt {\frac {c x^2}{a}+1} \sqrt {1-\frac {e^2 x^2}{d^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {e x}{d}\right ),-\frac {c d^2}{a e^2}\right )}{c e \sqrt {a+c x^2} \sqrt {d^2-e^2 x^2}}\right )}{a}+\frac {x \sqrt {d^2-e^2 x^2}}{a \sqrt {a+c x^2}}\right )}{\sqrt {d^2-e^2 x^2}}\)

Input:

Int[(Sqrt[d - e*x]*Sqrt[d + e*x])/(a + c*x^2)^(3/2),x]
 

Output:

(Sqrt[d - e*x]*Sqrt[d + e*x]*((x*Sqrt[d^2 - e^2*x^2])/(a*Sqrt[a + c*x^2]) 
+ (e^2*((d*Sqrt[a + c*x^2]*Sqrt[1 - (e^2*x^2)/d^2]*EllipticE[ArcSin[(e*x)/ 
d], -((c*d^2)/(a*e^2))])/(c*e*Sqrt[1 + (c*x^2)/a]*Sqrt[d^2 - e^2*x^2]) - ( 
a*d*Sqrt[1 + (c*x^2)/a]*Sqrt[1 - (e^2*x^2)/d^2]*EllipticF[ArcSin[(e*x)/d], 
 -((c*d^2)/(a*e^2))])/(c*e*Sqrt[a + c*x^2]*Sqrt[d^2 - e^2*x^2])))/a))/Sqrt 
[d^2 - e^2*x^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 314

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] + Simp[1/(2*a* 
(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*p + 3) + d 
*(2*(p + q + 1) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, 
 x]
 

rule 321

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> S 
imp[(1/(Sqrt[a]*Sqrt[c]*Rt[-d/c, 2]))*EllipticF[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] &&  !(NegQ[b/a] && SimplerSqrtQ[-b/a, -d/c])
 

rule 323

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

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 330

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

rule 331

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

rule 389

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

rule 648

Int[((c_) + (d_.)*(x_))^(m_.)*((e_) + (f_.)*(x_))^(n_.)*((a_.) + (b_.)*(x_) 
^2)^(p_), x_Symbol] :> Simp[(c + d*x)^FracPart[m]*((e + f*x)^FracPart[m]/(c 
*e + d*f*x^2)^FracPart[m])   Int[(c*e + d*f*x^2)^m*(a + b*x^2)^p, x], x] /; 
 FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[m, n] && EqQ[d*e + c*f, 0] && 
  !(EqQ[p, 2] && LtQ[m, -1])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(233\) vs. \(2(95)=190\).

Time = 1.79 (sec) , antiderivative size = 234, normalized size of antiderivative = 2.39

method	result	size
default	\(\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}\, \left (-\sqrt {\frac {e^{2}}{d^{2}}}\, c \,e^{2} x^{3}-\sqrt {\frac {-e^{2} x^{2}+d^{2}}{d^{2}}}\, \sqrt {\frac {c \,x^{2}+a}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e^{2}}{d^{2}}}, \sqrt {-\frac {c \,d^{2}}{a \,e^{2}}}\right ) a \,e^{2}+\sqrt {\frac {-e^{2} x^{2}+d^{2}}{d^{2}}}\, \sqrt {\frac {c \,x^{2}+a}{a}}\, \operatorname {EllipticE}\left (x \sqrt {\frac {e^{2}}{d^{2}}}, \sqrt {-\frac {c \,d^{2}}{a \,e^{2}}}\right ) a \,e^{2}+\sqrt {\frac {e^{2}}{d^{2}}}\, c \,d^{2} x \right )}{\left (-c \,e^{2} x^{4}-a \,e^{2} x^{2}+c \,d^{2} x^{2}+a \,d^{2}\right ) c a \sqrt {\frac {e^{2}}{d^{2}}}}\)	\(234\)
elliptic	\(\frac {\sqrt {\left (-e^{2} x^{2}+d^{2}\right ) \left (c \,x^{2}+a \right )}\, \left (\frac {\left (-x^{2} c \,e^{2}+c \,d^{2}\right ) x}{c a \sqrt {\left (x^{2}+\frac {a}{c}\right ) \left (-x^{2} c \,e^{2}+c \,d^{2}\right )}}+\frac {\left (-\frac {e^{2}}{c}+\frac {a \,e^{2}+c \,d^{2}}{a c}-\frac {d^{2}}{a}\right ) \sqrt {1-\frac {e^{2} x^{2}}{d^{2}}}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e^{2}}{d^{2}}}, \sqrt {-1-\frac {-a \,e^{2}+c \,d^{2}}{a \,e^{2}}}\right )}{\sqrt {\frac {e^{2}}{d^{2}}}\, \sqrt {-c \,e^{2} x^{4}-a \,e^{2} x^{2}+c \,d^{2} x^{2}+a \,d^{2}}}-\frac {e^{2} \sqrt {1-\frac {e^{2} x^{2}}{d^{2}}}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e^{2}}{d^{2}}}, \sqrt {-1-\frac {-a \,e^{2}+c \,d^{2}}{a \,e^{2}}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e^{2}}{d^{2}}}, \sqrt {-1-\frac {-a \,e^{2}+c \,d^{2}}{a \,e^{2}}}\right )\right )}{\sqrt {\frac {e^{2}}{d^{2}}}\, \sqrt {-c \,e^{2} x^{4}-a \,e^{2} x^{2}+c \,d^{2} x^{2}+a \,d^{2}}\, c}\right )}{\sqrt {-e x +d}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\)	\(394\)

Input:

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

Output:

(-e*x+d)^(1/2)*(e*x+d)^(1/2)*(c*x^2+a)^(1/2)*(-(e^2/d^2)^(1/2)*c*e^2*x^3-( 
(-e^2*x^2+d^2)/d^2)^(1/2)*((c*x^2+a)/a)^(1/2)*EllipticF(x*(e^2/d^2)^(1/2), 
(-c*d^2/a/e^2)^(1/2))*a*e^2+((-e^2*x^2+d^2)/d^2)^(1/2)*((c*x^2+a)/a)^(1/2) 
*EllipticE(x*(e^2/d^2)^(1/2),(-c*d^2/a/e^2)^(1/2))*a*e^2+(e^2/d^2)^(1/2)*c 
*d^2*x)/(-c*e^2*x^4-a*e^2*x^2+c*d^2*x^2+a*d^2)/c/a/(e^2/d^2)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.37 \[ \int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\left (a+c x^2\right )^{3/2}} \, dx=\frac {\sqrt {c x^{2} + a} \sqrt {e x + d} \sqrt {-e x + d} c d^{3} x + \sqrt {a d^{2}} {\left ({\left (c e^{3} x^{2} + a e^{3}\right )} E(\arcsin \left (\frac {e x}{d}\right )\,|\,-\frac {c d^{2}}{a e^{2}}) - {\left (c e^{3} x^{2} + a e^{3}\right )} F(\arcsin \left (\frac {e x}{d}\right )\,|\,-\frac {c d^{2}}{a e^{2}})\right )}}{a c^{2} d^{3} x^{2} + a^{2} c d^{3}} \] Input:

integrate((-e*x+d)^(1/2)*(e*x+d)^(1/2)/(c*x^2+a)^(3/2),x, algorithm="frica 
s")
 

Output:

(sqrt(c*x^2 + a)*sqrt(e*x + d)*sqrt(-e*x + d)*c*d^3*x + sqrt(a*d^2)*((c*e^ 
3*x^2 + a*e^3)*elliptic_e(arcsin(e*x/d), -c*d^2/(a*e^2)) - (c*e^3*x^2 + a* 
e^3)*elliptic_f(arcsin(e*x/d), -c*d^2/(a*e^2))))/(a*c^2*d^3*x^2 + a^2*c*d^ 
3)
 

Sympy [F]

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

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

Output:

Integral(sqrt(d - e*x)*sqrt(d + e*x)/(a + c*x**2)**(3/2), x)
 

Maxima [F]

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

integrate((-e*x+d)^(1/2)*(e*x+d)^(1/2)/(c*x^2+a)^(3/2),x, algorithm="maxim 
a")
 

Output:

integrate(sqrt(e*x + d)*sqrt(-e*x + d)/(c*x^2 + a)^(3/2), x)
 

Giac [F(-1)]

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

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

Output:

Timed out
                                                                                    
                                                                                    
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

int((sqrt(d + e*x)*sqrt(d - e*x)*sqrt(a + c*x**2))/(a**2 + 2*a*c*x**2 + c* 
*2*x**4),x)