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

Optimal. Leaf size=214 \[ -\frac{c^{3/2} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{-c-d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{d x \sqrt{a+b x^2}}{b \sqrt{-c-d x^2}}+\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{b \sqrt{-c-d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

[Out]

-((d*x*Sqrt[a + b*x^2])/(b*Sqrt[-c - d*x^2])) + (Sqrt[c]*Sqrt[d]*Sqrt[a + b*x^2]
*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[-c - d*x^2]*Sq
rt[(c*(a + b*x^2))/(a*(c + d*x^2))]) - (c^(3/2)*Sqrt[a + b*x^2]*EllipticF[ArcTan
[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[-c - d*x^2]*Sqrt[(c*(a
+ b*x^2))/(a*(c + d*x^2))])

_______________________________________________________________________________________

Rubi [A]  time = 0.298888, antiderivative size = 214, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{c^{3/2} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{-c-d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{d x \sqrt{a+b x^2}}{b \sqrt{-c-d x^2}}+\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{b \sqrt{-c-d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

-((d*x*Sqrt[a + b*x^2])/(b*Sqrt[-c - d*x^2])) + (Sqrt[c]*Sqrt[d]*Sqrt[a + b*x^2]
*EllipticE[ArcTan[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(b*Sqrt[-c - d*x^2]*Sq
rt[(c*(a + b*x^2))/(a*(c + d*x^2))]) - (c^(3/2)*Sqrt[a + b*x^2]*EllipticF[ArcTan
[(Sqrt[d]*x)/Sqrt[c]], 1 - (b*c)/(a*d)])/(a*Sqrt[d]*Sqrt[-c - d*x^2]*Sqrt[(c*(a
+ b*x^2))/(a*(c + d*x^2))])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 40.4029, size = 177, normalized size = 0.83 \[ - \frac{\sqrt{a} \sqrt{- c - d x^{2}} E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{\sqrt{b} \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} + \frac{x \sqrt{- c - d x^{2}}}{\sqrt{a + b x^{2}}} - \frac{c^{\frac{3}{2}} \sqrt{a + b x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{a \sqrt{d} \sqrt{- \frac{c \left (a + b x^{2}\right )}{a \left (- c - d x^{2}\right )}} \sqrt{- c - d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-d*x**2-c)**(1/2)/(b*x**2+a)**(1/2),x)

[Out]

-sqrt(a)*sqrt(-c - d*x**2)*elliptic_e(atan(sqrt(b)*x/sqrt(a)), -a*d/(b*c) + 1)/(
sqrt(b)*sqrt(a*(c + d*x**2)/(c*(a + b*x**2)))*sqrt(a + b*x**2)) + x*sqrt(-c - d*
x**2)/sqrt(a + b*x**2) - c**(3/2)*sqrt(a + b*x**2)*elliptic_f(atan(sqrt(d)*x/sqr
t(c)), 1 - b*c/(a*d))/(a*sqrt(d)*sqrt(-c*(a + b*x**2)/(a*(-c - d*x**2)))*sqrt(-c
- d*x**2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0682434, size = 89, normalized size = 0.42 \[ \frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{-c-d x^2} E\left (\sin ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|\frac{a d}{b c}\right )}{\sqrt{-\frac{b}{a}} \sqrt{a+b x^2} \sqrt{\frac{c+d x^2}{c}}} \]

Antiderivative was successfully verified.

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

[Out]

(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])

_______________________________________________________________________________________

Maple [A]  time = 0.024, size = 161, normalized size = 0.8 \[{\frac{1}{ \left ( bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac \right ) b}\sqrt{-d{x}^{2}-c}\sqrt{b{x}^{2}+a}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{b{x}^{2}+a}{a}}} \left ( ad{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) -ad{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) +c{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) b \right ){\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-d x^{2} - c}}{\sqrt{b x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-d*x^2 - c)/sqrt(b*x^2 + a),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{-d x^{2} - c}}{\sqrt{b x^{2} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-d*x^2 - c)/sqrt(b*x^2 + a),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- c - d x^{2}}}{\sqrt{a + b x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-d x^{2} - c}}{\sqrt{b x^{2} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-d*x^2 - c)/sqrt(b*x^2 + a),x, algorithm="giac")

[Out]

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