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

Optimal. Leaf size=191 \[ \frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{\frac{b x^2}{a}+1} \sqrt{d x^2-c}}-\frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{a+b x^2} \sqrt{d x^2-c}} \]

[Out]

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

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Rubi [A]  time = 0.37521, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.24 \[ \frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{\frac{b x^2}{a}+1} \sqrt{d x^2-c}}-\frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{a+b x^2} \sqrt{d x^2-c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 75.2671, size = 162, normalized size = 0.85 \[ \frac{\sqrt{c} \sqrt{d} \sqrt{1 - \frac{d x^{2}}{c}} \sqrt{a + b x^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{b c}{a d}\right )}{b \sqrt{1 + \frac{b x^{2}}{a}} \sqrt{- c + d x^{2}}} - \frac{\sqrt{c} \sqrt{1 + \frac{b x^{2}}{a}} \sqrt{1 - \frac{d x^{2}}{c}} \left (a d + b c\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{a + b x^{2}} \sqrt{- c + d x^{2}}} \]

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

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Mathematica [A]  time = 0.0669913, size = 90, normalized size = 0.47 \[ \frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{d x^2-c} E\left (\sin ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )}{\sqrt{-\frac{b}{a}} \sqrt{a+b x^2} \sqrt{\frac{c-d x^2}{c}}} \]

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 = 0.022, size = 109, normalized size = 0.6 \[{\frac{c}{-bd{x}^{4}-ad{x}^{2}+c{x}^{2}b+ac}\sqrt{d{x}^{2}-c}\sqrt{b{x}^{2}+a}\sqrt{{\frac{b{x}^{2}+a}{a}}}\sqrt{-{\frac{d{x}^{2}-c}{c}}}{\it EllipticE} \left ( x\sqrt{-{\frac{b}{a}}},\sqrt{-{\frac{ad}{bc}}} \right ){\frac{1}{\sqrt{-{\frac{b}{a}}}}}} \]

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]

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

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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)