\(\int \frac {\sqrt {-c+d x^2}}{\sqrt {-a-b x^2}} \, dx\) [202]

Optimal result

Integrand size = 28, antiderivative size = 198 \[ \int \frac {\sqrt {-c+d x^2}}{\sqrt {-a-b x^2}} \, dx=-\frac {a \sqrt {d} \sqrt {1+\frac {b x^2}{a}} \sqrt {-c+d x^2} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {c} \sqrt {-a-b x^2} \sqrt {1-\frac {d x^2}{c}}}+\frac {(b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {-c+d x^2} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {c} \sqrt {d} \sqrt {-a-b x^2} \sqrt {1-\frac {d x^2}{c}}} \] Output:

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

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.47 \[ \int \frac {\sqrt {-c+d x^2}}{\sqrt {-a-b x^2}} \, dx=\frac {\sqrt {\frac {a+b x^2}{a}} \sqrt {-c+d x^2} E\left (\arcsin \left (\sqrt {-\frac {b}{a}} x\right )|-\frac {a d}{b c}\right )}{\sqrt {-\frac {b}{a}} \sqrt {-a-b x^2} \sqrt {\frac {c-d x^2}{c}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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.

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

\(\Big \downarrow \) 326

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

Input:

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

Output:

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

Maple [A] (verified)

Time = 1.32 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.84

method	result	size
default	\(\frac {\left (-a d \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right )-c \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) b +a d \operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right )\right ) \sqrt {x^{2} d -c}\, \sqrt {-b \,x^{2}-a}\, \sqrt {\frac {-x^{2} d +c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}}{\left (-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c \right ) \sqrt {\frac {d}{c}}\, b}\)	\(166\)
elliptic	\(\frac {\sqrt {\left (-x^{2} d +c \right ) \left (b \,x^{2}+a \right )}\, \left (-\frac {c \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}}-\frac {d a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+x^{2} b c +a c}\, b}\right )}{\sqrt {-b \,x^{2}-a}\, \sqrt {x^{2} d -c}}\)	\(263\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {-c+d x^2}}{\sqrt {-a-b x^2}} \, dx=-\frac {\sqrt {-b d} c x \sqrt {\frac {c}{d}} E(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c}) - \sqrt {-b d} {\left (c - d\right )} x \sqrt {\frac {c}{d}} F(\arcsin \left (\frac {\sqrt {\frac {c}{d}}}{x}\right )\,|\,-\frac {a d}{b c}) + \sqrt {-b x^{2} - a} \sqrt {d x^{2} - c} d}{b d x} \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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