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

Optimal. Leaf size=88 \[ \frac{x \sqrt{c+d x^2}}{d \sqrt{x^2+4}}-\frac{\sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{d \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}} \]

[Out]

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

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Rubi [A]  time = 0.126099, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{x \sqrt{c+d x^2}}{d \sqrt{x^2+4}}-\frac{\sqrt{c+d x^2} E\left (\tan ^{-1}\left (\frac{x}{2}\right )|1-\frac{4 d}{c}\right )}{d \sqrt{x^2+4} \sqrt{\frac{c+d x^2}{c \left (x^2+4\right )}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.8391, size = 75, normalized size = 0.85 \[ \frac{x \sqrt{c + d x^{2}}}{d \sqrt{x^{2} + 4}} - \frac{2 \sqrt{c + d x^{2}} E\left (\operatorname{atan}{\left (\frac{x}{2} \right )}\middle | 1 - \frac{4 d}{c}\right )}{d \sqrt{\frac{4 c + 4 d x^{2}}{c \left (x^{2} + 4\right )}} \sqrt{x^{2} + 4}} \]

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]

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

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Mathematica [C]  time = 0.0722538, size = 70, normalized size = 0.8 \[ -\frac{i c \sqrt{\frac{d x^2}{c}+1} \left (E\left (i \sinh ^{-1}\left (\frac{x}{2}\right )|\frac{4 d}{c}\right )-F\left (i \sinh ^{-1}\left (\frac{x}{2}\right )|\frac{4 d}{c}\right )\right )}{d \sqrt{c+d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*c*Sqrt[1 + (d*x^2)/c]*(EllipticE[I*ArcSinh[x/2], (4*d)/c] - EllipticF[I*Ar
cSinh[x/2], (4*d)/c]))/(d*Sqrt[c + d*x^2])

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Maple [A]  time = 0.023, size = 76, normalized size = 0.9 \[ -2\,{\frac{1}{\sqrt{d{x}^{2}+c}}\sqrt{{\frac{d{x}^{2}+c}{c}}} \left ({\it EllipticF} \left ( x\sqrt{-{\frac{d}{c}}},1/2\,\sqrt{{\frac{c}{d}}} \right ) -{\it EllipticE} \left ( x\sqrt{-{\frac{d}{c}}},1/2\,\sqrt{{\frac{c}{d}}} \right ) \right ){\frac{1}{\sqrt{-{\frac{d}{c}}}}}} \]

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]

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

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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. \[ \int \frac{x^{2}}{\sqrt{c + d x^{2}} \sqrt{x^{2} + 4}}\, dx \]

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(c + d*x**2)*sqrt(x**2 + 4)), 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)