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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]
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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]