3.993 \(\int \frac{x^2}{\sqrt{4-x^2} \sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=87 \[ \frac{\sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{d \sqrt{\frac{d x^2}{c}+1}}-\frac{c \sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{d \sqrt{c+d x^2}} \]

[Out]

(Sqrt[c + d*x^2]*EllipticE[ArcSin[x/2], (-4*d)/c])/(d*Sqrt[1 + (d*x^2)/c]) - (c*
Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[x/2], (-4*d)/c])/(d*Sqrt[c + d*x^2])

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Rubi [A]  time = 0.233913, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\sqrt{c+d x^2} E\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{d \sqrt{\frac{d x^2}{c}+1}}-\frac{c \sqrt{\frac{d x^2}{c}+1} F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )}{d \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]

[Out]

(Sqrt[c + d*x^2]*EllipticE[ArcSin[x/2], (-4*d)/c])/(d*Sqrt[1 + (d*x^2)/c]) - (c*
Sqrt[1 + (d*x^2)/c]*EllipticF[ArcSin[x/2], (-4*d)/c])/(d*Sqrt[c + d*x^2])

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Rubi in Sympy [A]  time = 39.3365, size = 71, normalized size = 0.82 \[ - \frac{c \sqrt{1 + \frac{d x^{2}}{c}} F\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | - \frac{4 d}{c}\right )}{d \sqrt{c + d x^{2}}} + \frac{\sqrt{c + d x^{2}} E\left (\operatorname{asin}{\left (\frac{x}{2} \right )}\middle | - \frac{4 d}{c}\right )}{d \sqrt{1 + \frac{d x^{2}}{c}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(-x**2+4)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

-c*sqrt(1 + d*x**2/c)*elliptic_f(asin(x/2), -4*d/c)/(d*sqrt(c + d*x**2)) + sqrt(
c + d*x**2)*elliptic_e(asin(x/2), -4*d/c)/(d*sqrt(1 + d*x**2/c))

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Mathematica [A]  time = 0.082787, size = 59, normalized size = 0.68 \[ \frac{c \sqrt{\frac{d x^2}{c}+1} \left (E\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )-F\left (\sin ^{-1}\left (\frac{x}{2}\right )|-\frac{4 d}{c}\right )\right )}{d \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(Sqrt[4 - x^2]*Sqrt[c + d*x^2]),x]

[Out]

(c*Sqrt[1 + (d*x^2)/c]*(EllipticE[ArcSin[x/2], (-4*d)/c] - EllipticF[ArcSin[x/2]
, (-4*d)/c]))/(d*Sqrt[c + d*x^2])

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Maple [A]  time = 0.024, size = 59, normalized size = 0.7 \[{\frac{c}{d}\sqrt{{\frac{d{x}^{2}+c}{c}}} \left ( -{\it EllipticF} \left ({\frac{x}{2}},2\,\sqrt{-{\frac{d}{c}}} \right ) +{\it EllipticE} \left ({\frac{x}{2}},2\,\sqrt{-{\frac{d}{c}}} \right ) \right ){\frac{1}{\sqrt{d{x}^{2}+c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-x^2+4)^(1/2)/(d*x^2+c)^(1/2),x)

[Out]

1/(d*x^2+c)^(1/2)*c*((d*x^2+c)/c)^(1/2)*(-EllipticF(1/2*x,2*(-d/c)^(1/2))+Ellipt
icE(1/2*x,2*(-d/c)^(1/2)))/d

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{d x^{2} + c} \sqrt{-x^{2} + 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)),x, algorithm="maxima")

[Out]

integrate(x^2/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{d x^{2} + c} \sqrt{-x^{2} + 4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)),x, algorithm="fricas")

[Out]

integral(x^2/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(-x**2+4)**(1/2)/(d*x**2+c)**(1/2),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{d x^{2} + c} \sqrt{-x^{2} + 4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)),x, algorithm="giac")

[Out]

integrate(x^2/(sqrt(d*x^2 + c)*sqrt(-x^2 + 4)), x)