\(\int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx\) [27]

Optimal result

Integrand size = 31, antiderivative size = 189 \[ \int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx=-\frac {a e \sqrt {d-e x} \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} E\left (\arcsin \left (\frac {e x}{d}\right )|-\frac {c d^2}{a e^2}\right )}{c d \sqrt {a+c x^2} \sqrt {1-\frac {e^2 x^2}{d^2}}}+\frac {\left (\frac {d}{e}+\frac {a e}{c d}\right ) \sqrt {d-e x} \sqrt {d+e x} \sqrt {1+\frac {c x^2}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {e x}{d}\right ),-\frac {c d^2}{a e^2}\right )}{\sqrt {a+c x^2} \sqrt {1-\frac {e^2 x^2}{d^2}}} \] Output:

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

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 13.66 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.56 \[ \int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx=-\frac {i d^2 \sqrt {1+\frac {c x^2}{a}} \sqrt {1-\frac {e^2 x^2}{d^2}} E\left (i \text {arcsinh}\left (\sqrt {\frac {c}{a}} x\right )|-\frac {a e^2}{c d^2}\right )}{\sqrt {\frac {c}{a}} \sqrt {d-e x} \sqrt {d+e x} \sqrt {a+c x^2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {648, 326, 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.

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

\(\Big \downarrow \) 648

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

\(\Big \downarrow \) 326

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

(Sqrt[d - e*x]*Sqrt[d + e*x]*(-((d*e*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*Sqrt[1 + (c*x^2)/a]* 
Sqrt[d^2 - e^2*x^2])) + (d*(c*d^2 + a*e^2)*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])))/Sqrt[d^2 - e^2*x^2]
 

Defintions of rubi rules used

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 326

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

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 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 [A] (verified)

Time = 1.44 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.05

method	result	size
default	\(\frac {\sqrt {-e x +d}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}\, \sqrt {\frac {-e^{2} x^{2}+d^{2}}{d^{2}}}\, \sqrt {\frac {c \,x^{2}+a}{a}}\, \left (a \,e^{2} \operatorname {EllipticF}\left (x \sqrt {\frac {e^{2}}{d^{2}}}, \sqrt {-\frac {c \,d^{2}}{a \,e^{2}}}\right )+d^{2} \operatorname {EllipticF}\left (x \sqrt {\frac {e^{2}}{d^{2}}}, \sqrt {-\frac {c \,d^{2}}{a \,e^{2}}}\right ) c -a \,e^{2} \operatorname {EllipticE}\left (x \sqrt {\frac {e^{2}}{d^{2}}}, \sqrt {-\frac {c \,d^{2}}{a \,e^{2}}}\right )\right )}{\left (-c \,e^{2} x^{4}-a \,e^{2} x^{2}+c \,d^{2} x^{2}+a \,d^{2}\right ) \sqrt {\frac {e^{2}}{d^{2}}}\, c}\)	\(198\)
elliptic	\(\frac {\sqrt {\left (-e^{2} x^{2}+d^{2}\right ) \left (c \,x^{2}+a \right )}\, \left (\frac {d^{2} \sqrt {1-\frac {e^{2} x^{2}}{d^{2}}}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {e^{2}}{d^{2}}}, \sqrt {-1-\frac {-a \,e^{2}+c \,d^{2}}{a \,e^{2}}}\right )}{\sqrt {\frac {e^{2}}{d^{2}}}\, \sqrt {-c \,e^{2} x^{4}-a \,e^{2} x^{2}+c \,d^{2} x^{2}+a \,d^{2}}}+\frac {e^{2} a \sqrt {1-\frac {e^{2} x^{2}}{d^{2}}}\, \sqrt {1+\frac {c \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {e^{2}}{d^{2}}}, \sqrt {-1-\frac {-a \,e^{2}+c \,d^{2}}{a \,e^{2}}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {e^{2}}{d^{2}}}, \sqrt {-1-\frac {-a \,e^{2}+c \,d^{2}}{a \,e^{2}}}\right )\right )}{\sqrt {\frac {e^{2}}{d^{2}}}\, \sqrt {-c \,e^{2} x^{4}-a \,e^{2} x^{2}+c \,d^{2} x^{2}+a \,d^{2}}\, c}\right )}{\sqrt {-e x +d}\, \sqrt {e x +d}\, \sqrt {c \,x^{2}+a}}\)	\(312\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx=\frac {\sqrt {c x^{2} + a} \sqrt {e x + d} \sqrt {-e x + d} e^{3} + {\left (d^{3} x E(\arcsin \left (\frac {d}{e x}\right )\,|\,-\frac {a e^{2}}{c d^{2}}) - {\left (d^{3} - d e^{2}\right )} x F(\arcsin \left (\frac {d}{e x}\right )\,|\,-\frac {a e^{2}}{c d^{2}})\right )} \sqrt {-c e^{2}}}{c e^{3} x} \] Input:

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

Output:

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

Sympy [F]

\[ \int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {d - e x} \sqrt {d + e x}}{\sqrt {a + c x^{2}}}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx=\int { \frac {\sqrt {e x + d} \sqrt {-e x + d}}{\sqrt {c x^{2} + a}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx=\int { \frac {\sqrt {e x + d} \sqrt {-e x + d}}{\sqrt {c x^{2} + a}} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {d-e x} \sqrt {d+e x}}{\sqrt {a+c x^2}} \, dx=\int \frac {\sqrt {e x +d}\, \sqrt {-e x +d}\, \sqrt {c \,x^{2}+a}}{c \,x^{2}+a}d x \] Input:

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

Output:

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