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

Optimal. Leaf size=203 \[ \frac{x \sqrt{a+b x^2}}{\sqrt{-c-d x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{-c-d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{c} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{-c-d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.295679, antiderivative size = 203, 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 \[ \frac{x \sqrt{a+b x^2}}{\sqrt{-c-d x^2}}+\frac{\sqrt{c} \sqrt{a+b x^2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{-c-d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}}-\frac{\sqrt{c} \sqrt{a+b x^2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|1-\frac{b c}{a d}\right )}{\sqrt{d} \sqrt{-c-d x^2} \sqrt{\frac{c \left (a+b x^2\right )}{a \left (c+d x^2\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 42.7143, size = 180, normalized size = 0.89 \[ \frac{\sqrt{a} \sqrt{b} \sqrt{- c - d x^{2}} E\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{d \sqrt{\frac{a \left (c + d x^{2}\right )}{c \left (a + b x^{2}\right )}} \sqrt{a + b x^{2}}} - \frac{b x \sqrt{- c - d x^{2}}}{d \sqrt{a + b x^{2}}} + \frac{\sqrt{c} \sqrt{a + b x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{\sqrt{d} \sqrt{- \frac{c \left (a + b x^{2}\right )}{a \left (- c - d x^{2}\right )}} \sqrt{- c - d x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0804456, size = 89, normalized size = 0.44 \[ \frac{\sqrt{a+b x^2} \sqrt{\frac{c+d x^2}{c}} E\left (\sin ^{-1}\left (\sqrt{-\frac{d}{c}} x\right )|\frac{b c}{a d}\right )}{\sqrt{-\frac{d}{c}} \sqrt{\frac{a+b x^2}{a}} \sqrt{-c-d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(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])

_______________________________________________________________________________________

Maple [A]  time = 0.019, size = 108, normalized size = 0.5 \[{\frac{a}{-bd{x}^{4}-ad{x}^{2}-c{x}^{2}b-ac}\sqrt{b{x}^{2}+a}\sqrt{-d{x}^{2}-c}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{b{x}^{2}+a}{a}}}{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ){\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(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)*Ellip
ticE(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)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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