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

Optimal. Leaf size=189 \[ \frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}}-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2}} \]

[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 = 0.386785, antiderivative size = 189, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{\sqrt{c} \sqrt{\frac{b x^2}{a}+1} \sqrt{1-\frac{d x^2}{c}} (a d+b c) F\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{a+b x^2} \sqrt{c-d x^2}}-\frac{\sqrt{c} \sqrt{d} \sqrt{a+b x^2} \sqrt{1-\frac{d x^2}{c}} E\left (\sin ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )|-\frac{b c}{a d}\right )}{b \sqrt{\frac{b x^2}{a}+1} \sqrt{c-d x^2}} \]

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 = 74.5856, size = 162, normalized size = 0.86 \[ - \frac{\sqrt{c} \sqrt{d} \sqrt{1 - \frac{d x^{2}}{c}} \sqrt{a + b x^{2}} E\left (\operatorname{asin}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{b c}{a d}\right )}{b \sqrt{1 + \frac{b x^{2}}{a}} \sqrt{c - d x^{2}}} + \frac{\sqrt{c} \sqrt{1 + \frac{b x^{2}}{a}} \sqrt{1 - \frac{d x^{2}}{c}} \left (a d + b c\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt{d} x}{\sqrt{c}} \right )}\middle | - \frac{b c}{a d}\right )}{b \sqrt{d} \sqrt{a + b x^{2}} \sqrt{c - d x^{2}}} \]

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/s
qrt(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 = 0.0799097, size = 89, normalized size = 0.47 \[ \frac{\sqrt{\frac{a+b x^2}{a}} \sqrt{c-d x^2} E\left (\sin ^{-1}\left (\sqrt{-\frac{b}{a}} x\right )|-\frac{a d}{b c}\right )}{\sqrt{-\frac{b}{a}} \sqrt{a+b x^2} \sqrt{\frac{c-d x^2}{c}}} \]

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 = 0.017, size = 164, normalized size = 0.9 \[{\frac{1}{ \left ( bd{x}^{4}+ad{x}^{2}-c{x}^{2}b-ac \right ) b} \left ( -ad{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) -c{\it EllipticF} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) b+ad{\it EllipticE} \left ( x\sqrt{{\frac{d}{c}}},\sqrt{-{\frac{bc}{ad}}} \right ) \right ) \sqrt{-d{x}^{2}+c}\sqrt{b{x}^{2}+a}\sqrt{-{\frac{d{x}^{2}-c}{c}}}\sqrt{{\frac{b{x}^{2}+a}{a}}}{\frac{1}{\sqrt{{\frac{d}{c}}}}}} \]

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]

(-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)))*(-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)/(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)