3.3.72 \(\int \frac {\sqrt {a-b x^2}}{\sqrt {-c-d x^2}} \, dx\) [272]

3.3.72.1 Optimal result

Integrand size = 27, antiderivative size = 194 \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c-d x^2}} \, dx=\frac {\sqrt {a} \sqrt {b} \sqrt {1-\frac {b x^2}{a}} \sqrt {-c-d x^2} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{d \sqrt {a-b x^2} \sqrt {1+\frac {d x^2}{c}}}+\frac {\sqrt {a} (b c+a d) \sqrt {1-\frac {b x^2}{a}} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {b} d \sqrt {a-b x^2} \sqrt {-c-d x^2}} \]

output

EllipticE(x*b^(1/2)/a^(1/2),(-a*d/b/c)^(1/2))*a^(1/2)*b^(1/2)*(1-b*x^2/a)^ 
(1/2)*(-d*x^2-c)^(1/2)/d/(-b*x^2+a)^(1/2)/(1+d*x^2/c)^(1/2)+(a*d+b*c)*Elli 
pticF(x*b^(1/2)/a^(1/2),(-a*d/b/c)^(1/2))*a^(1/2)*(1-b*x^2/a)^(1/2)*(1+d*x 
^2/c)^(1/2)/d/b^(1/2)/(-b*x^2+a)^(1/2)/(-d*x^2-c)^(1/2)
 

3.3.72.2 Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c-d x^2}} \, dx=\frac {\sqrt {a-b x^2} \sqrt {\frac {c+d x^2}{c}} E\left (\arcsin \left (\sqrt {-\frac {d}{c}} x\right )|-\frac {b c}{a d}\right )}{\sqrt {-\frac {d}{c}} \sqrt {\frac {a-b x^2}{a}} \sqrt {-c-d x^2}} \]

input

Integrate[Sqrt[a - b*x^2]/Sqrt[-c - d*x^2],x]
 

output

(Sqrt[a - b*x^2]*Sqrt[(c + d*x^2)/c]*EllipticE[ArcSin[Sqrt[-(d/c)]*x], -(( 
b*c)/(a*d))])/(Sqrt[-(d/c)]*Sqrt[(a - b*x^2)/a]*Sqrt[-c - d*x^2])
 

3.3.72.3 Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {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.

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

\(\Big \downarrow \) 326

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

input

Int[Sqrt[a - b*x^2]/Sqrt[-c - d*x^2],x]
 

output

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

3.3.72.3.1 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]
 

3.3.72.4 Maple [A] (verified)

Time = 2.57 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.56

method	result	size
default	\(\frac {\sqrt {-b \,x^{2}+a}\, \sqrt {-d \,x^{2}-c}\, a \sqrt {\frac {d \,x^{2}+c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right )}{\left (b d \,x^{4}-a d \,x^{2}+c b \,x^{2}-a c \right ) \sqrt {-\frac {d}{c}}}\)	\(109\)
elliptic	\(\frac {\sqrt {-\left (-b \,x^{2}+a \right ) \left (d \,x^{2}+c \right )}\, \left (\frac {a \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {b d \,x^{4}-a d \,x^{2}+c b \,x^{2}-a c}}-\frac {a \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \left (F\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-E\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {b d \,x^{4}-a d \,x^{2}+c b \,x^{2}-a c}}\right )}{\sqrt {-b \,x^{2}+a}\, \sqrt {-d \,x^{2}-c}}\)	\(263\)

input

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

output

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

3.3.72.5 Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c-d x^2}} \, dx=-\frac {\sqrt {b d} a x \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) - \sqrt {b d} {\left (a - b\right )} x \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) + \sqrt {-b x^{2} + a} \sqrt {-d x^{2} - c} b}{b d x} \]

input

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

output

-(sqrt(b*d)*a*x*sqrt(a/b)*elliptic_e(arcsin(sqrt(a/b)/x), -b*c/(a*d)) - sq 
rt(b*d)*(a - b)*x*sqrt(a/b)*elliptic_f(arcsin(sqrt(a/b)/x), -b*c/(a*d)) + 
sqrt(-b*x^2 + a)*sqrt(-d*x^2 - c)*b)/(b*d*x)
 

3.3.72.6 Sympy [F]

\[ \int \frac {\sqrt {a-b x^2}}{\sqrt {-c-d x^2}} \, dx=\int \frac {\sqrt {a - b x^{2}}}{\sqrt {- c - d x^{2}}}\, dx \]

input

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

output

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

3.3.72.7 Maxima [F]

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

input

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

output

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

3.3.72.8 Giac [F]

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

input

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

output

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

3.3.72.9 Mupad [F(-1)]

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

input

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

output

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