\(\int \frac {\sqrt {-a+b x^2}}{\sqrt {c+d x^2}} \, dx\) [184]

Optimal result

Integrand size = 25, antiderivative size = 192 \[ \int \frac {\sqrt {-a+b x^2}}{\sqrt {c+d x^2}} \, dx=-\frac {\sqrt {b} c \sqrt {-a+b x^2} \sqrt {1+\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )|-\frac {a d}{b c}\right )}{\sqrt {a} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2}}+\frac {(b c+a d) \sqrt {-a+b x^2} \sqrt {1+\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ),-\frac {a d}{b c}\right )}{\sqrt {a} \sqrt {b} d \sqrt {1-\frac {b x^2}{a}} \sqrt {c+d x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.72 (sec) , antiderivative size = 90, 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])
 

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.99, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, 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.

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

\(\Big \downarrow \) 326

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

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[(Sqr 
t[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])
 

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 = 0.97 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.56

method	result	size
default	\(\frac {\sqrt {b \,x^{2}-a}\, \sqrt {x^{2} d +c}\, a \sqrt {\frac {x^{2} d +c}{c}}\, \sqrt {\frac {-b \,x^{2}+a}{a}}\, \operatorname {EllipticE}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right )}{\left (-b d \,x^{4}+a d \,x^{2}-x^{2} b c +a c \right ) \sqrt {-\frac {d}{c}}}\)	\(107\)
elliptic	\(\frac {\sqrt {-\left (-b \,x^{2}+a \right ) \left (x^{2} d +c \right )}\, \left (-\frac {a \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {b d \,x^{4}-a d \,x^{2}+x^{2} b c -a c}}+\frac {a \sqrt {1+\frac {d \,x^{2}}{c}}\, \sqrt {1-\frac {b \,x^{2}}{a}}\, \left (\operatorname {EllipticF}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-\operatorname {EllipticE}\left (x \sqrt {-\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {-\frac {d}{c}}\, \sqrt {b d \,x^{4}-a d \,x^{2}+x^{2} b c -a c}}\right )}{\sqrt {b \,x^{2}-a}\, \sqrt {x^{2} d +c}}\)	\(261\)

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

Fricas [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {-a+b x^2}}{\sqrt {c+d x^2}} \, dx=\frac {\sqrt {b d} a x \sqrt {\frac {a}{b}} E(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) - \sqrt {b d} {\left (a - b\right )} x \sqrt {\frac {a}{b}} F(\arcsin \left (\frac {\sqrt {\frac {a}{b}}}{x}\right )\,|\,-\frac {b c}{a d}) + \sqrt {b x^{2} - a} \sqrt {d x^{2} + c} b}{b d x} \] 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)) - sqr 
t(b*d)*(a - b)*x*sqrt(a/b)*elliptic_f(arcsin(sqrt(a/b)/x), -b*c/(a*d)) + s 
qrt(b*x^2 - a)*sqrt(d*x^2 + c)*b)/(b*d*x)
 

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)
 

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)
 

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)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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