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

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*(a + b*x^2
))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*Sqrt[-a - b*x^2]*EllipticF[ArcTa
n[(Sqrt[d]*x)/Sqrt[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])

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Rubi [A]  time = 0.304192, 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*(a + b*x^2
))/(a*(c + d*x^2))]*Sqrt[c + d*x^2]) - (c^(3/2)*Sqrt[-a - b*x^2]*EllipticF[ArcTa
n[(Sqrt[d]*x)/Sqrt[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])

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Rubi in Sympy [A]  time = 41.2471, size = 180, normalized size = 0.84 \[ \frac{\sqrt{a} \sqrt{c + d x^{2}} F\left (\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}\middle | - \frac{a d}{b c} + 1\right )}{\sqrt{b} \sqrt{- \frac{a \left (c + d x^{2}\right )}{c \left (- a - b x^{2}\right )}} \sqrt{- a - b x^{2}}} + \frac{\sqrt{c} \sqrt{d} \sqrt{- a - b x^{2}} E\left (\operatorname{atan}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | 1 - \frac{b c}{a d}\right )}{b \sqrt{\frac{c \left (a + b x^{2}\right )}{a \left (c + d x^{2}\right )}} \sqrt{c + d x^{2}}} - \frac{d x \sqrt{- a - b x^{2}}}{b \sqrt{c + d x^{2}}} \]

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_f(atan(sqrt(b)*x/sqrt(a)), -a*d/(b*c) + 1)/(sq
rt(b)*sqrt(-a*(c + d*x**2)/(c*(-a - b*x**2)))*sqrt(-a - b*x**2)) + sqrt(c)*sqrt(
d)*sqrt(-a - b*x**2)*elliptic_e(atan(sqrt(d)*x/sqrt(c)), 1 - b*c/(a*d))/(b*sqrt(
c*(a + b*x**2)/(a*(c + d*x**2)))*sqrt(c + d*x**2)) - d*x*sqrt(-a - b*x**2)/(b*sq
rt(c + d*x**2))

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Mathematica [A]  time = 0.0767726, 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])

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Maple [A]  time = 0.019, size = 162, normalized size = 0.8 \[{\frac{1}{ \left ( bd{x}^{4}+ad{x}^{2}+c{x}^{2}b+ac \right ) b}\sqrt{d{x}^{2}+c}\sqrt{-b{x}^{2}-a}\sqrt{{\frac{d{x}^{2}+c}{c}}}\sqrt{{\frac{b{x}^{2}+a}{a}}} \left ( ad{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) -c{\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) b-ad{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},\sqrt{{\frac{bc}{ad}}} \right ) \right ){\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]

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
lipticF(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

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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)

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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)

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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)

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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)