3.3.86 \(\int \frac {\sqrt {c-d x^2}}{\sqrt {a+b x^2}} \, dx\) [286]

3.3.86.1 Optimal result

Integrand size = 24, antiderivative size = 189 \[ \int \frac {\sqrt {c-d x^2}}{\sqrt {a+b x^2}} \, dx=-\frac {\sqrt {c} \sqrt {d} \sqrt {a+b x^2} \sqrt {1-\frac {d x^2}{c}} E\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right )|-\frac {b c}{a d}\right )}{b \sqrt {1+\frac {b x^2}{a}} \sqrt {c-d x^2}}+\frac {\sqrt {c} (b c+a d) \sqrt {1+\frac {b x^2}{a}} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {d} x}{\sqrt {c}}\right ),-\frac {b c}{a d}\right )}{b \sqrt {d} \sqrt {a+b x^2} \sqrt {c-d x^2}} \]

output

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

3.3.86.2 Mathematica [A] (verified)

Time = 0.85 (sec) , antiderivative size = 89, 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])
 

3.3.86.3 Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, 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.

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

\(\Big \downarrow \) 326

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 323

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

\(\Big \downarrow \) 321

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

\(\Big \downarrow \) 331

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

\(\Big \downarrow \) 330

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

\(\Big \downarrow \) 327

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

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[(S 
qrt[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]*Ellipti 
cF[ArcSin[(Sqrt[d]*x)/Sqrt[c]], -((b*c)/(a*d))])/(b*Sqrt[d]*Sqrt[a + b*x^2 
]*Sqrt[c - d*x^2])
 

3.3.86.3.1 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]
 

3.3.86.4 Maple [A] (verified)

Time = 2.43 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.85

method	result	size
default	\(\frac {\sqrt {-d \,x^{2}+c}\, \sqrt {b \,x^{2}+a}\, \sqrt {\frac {-d \,x^{2}+c}{c}}\, \sqrt {\frac {b \,x^{2}+a}{a}}\, \left (a d F\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right )+c F\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right ) b -a d E\left (x \sqrt {\frac {d}{c}}, \sqrt {-\frac {b c}{a d}}\right )\right )}{\left (-b d \,x^{4}-a d \,x^{2}+c b \,x^{2}+a c \right ) \sqrt {\frac {d}{c}}\, b}\)	\(161\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+a \right ) \left (-d \,x^{2}+c \right )}\, \left (\frac {c \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, F\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+c b \,x^{2}+a c}}+\frac {d a \sqrt {1-\frac {d \,x^{2}}{c}}\, \sqrt {1+\frac {b \,x^{2}}{a}}\, \left (F\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )-E\left (x \sqrt {\frac {d}{c}}, \sqrt {-1-\frac {-a d +b c}{a d}}\right )\right )}{\sqrt {\frac {d}{c}}\, \sqrt {-b d \,x^{4}-a d \,x^{2}+c b \,x^{2}+a c}\, b}\right )}{\sqrt {b \,x^{2}+a}\, \sqrt {-d \,x^{2}+c}}\)	\(257\)

input

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

output

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

3.3.86.5 Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 115, 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)) - sq 
rt(-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)
 

3.3.86.6 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)
 

3.3.86.7 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)
 

3.3.86.8 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)
 

3.3.86.9 Mupad [F(-1)]

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

input

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

output

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