3.992 \(\int \frac{x^2}{\sqrt{2+b x^2} \sqrt{3+d x^2}} \, dx\)

Optimal. Leaf size=110 \[ \frac{x \sqrt{b x^2+2}}{b \sqrt{d x^2+3}}-\frac{\sqrt{2} \sqrt{b x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{b \sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}} \]

[Out]

(x*Sqrt[2 + b*x^2])/(b*Sqrt[3 + d*x^2]) - (Sqrt[2]*Sqrt[2 + b*x^2]*EllipticE[Arc
Tan[(Sqrt[d]*x)/Sqrt[3]], 1 - (3*b)/(2*d)])/(b*Sqrt[d]*Sqrt[(2 + b*x^2)/(3 + d*x
^2)]*Sqrt[3 + d*x^2])

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Rubi [A]  time = 0.156639, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{x \sqrt{b x^2+2}}{b \sqrt{d x^2+3}}-\frac{\sqrt{2} \sqrt{b x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{b \sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2]),x]

[Out]

(x*Sqrt[2 + b*x^2])/(b*Sqrt[3 + d*x^2]) - (Sqrt[2]*Sqrt[2 + b*x^2]*EllipticE[Arc
Tan[(Sqrt[d]*x)/Sqrt[3]], 1 - (3*b)/(2*d)])/(b*Sqrt[d]*Sqrt[(2 + b*x^2)/(3 + d*x
^2)]*Sqrt[3 + d*x^2])

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Rubi in Sympy [A]  time = 19.5007, size = 99, normalized size = 0.9 \[ \frac{x \sqrt{d x^{2} + 3}}{d \sqrt{b x^{2} + 2}} - \frac{\sqrt{2} \sqrt{d x^{2} + 3} E\left (\operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{b} x}{2} \right )}\middle | 1 - \frac{2 d}{3 b}\right )}{\sqrt{b} d \sqrt{\frac{2 d x^{2} + 6}{3 b x^{2} + 6}} \sqrt{b x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x**2+2)**(1/2)/(d*x**2+3)**(1/2),x)

[Out]

x*sqrt(d*x**2 + 3)/(d*sqrt(b*x**2 + 2)) - sqrt(2)*sqrt(d*x**2 + 3)*elliptic_e(at
an(sqrt(2)*sqrt(b)*x/2), 1 - 2*d/(3*b))/(sqrt(b)*d*sqrt((2*d*x**2 + 6)/(3*b*x**2
+ 6))*sqrt(b*x**2 + 2))

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Mathematica [C]  time = 0.0716663, size = 72, normalized size = 0.65 \[ -\frac{i \sqrt{3} \left (E\left (i \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2}}\right )|\frac{2 d}{3 b}\right )-F\left (i \sinh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2}}\right )|\frac{2 d}{3 b}\right )\right )}{\sqrt{b} d} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2]),x]

[Out]

((-I)*Sqrt[3]*(EllipticE[I*ArcSinh[(Sqrt[b]*x)/Sqrt[2]], (2*d)/(3*b)] - Elliptic
F[I*ArcSinh[(Sqrt[b]*x)/Sqrt[2]], (2*d)/(3*b)]))/(Sqrt[b]*d)

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Maple [A]  time = 0.028, size = 70, normalized size = 0.6 \[{\frac{\sqrt{2}}{b} \left ( -{\it EllipticF} \left ({\frac{x\sqrt{3}}{3}\sqrt{-d}},{\frac{\sqrt{3}\sqrt{2}}{2}\sqrt{{\frac{b}{d}}}} \right ) +{\it EllipticE} \left ({\frac{x\sqrt{3}}{3}\sqrt{-d}},{\frac{\sqrt{3}\sqrt{2}}{2}\sqrt{{\frac{b}{d}}}} \right ) \right ){\frac{1}{\sqrt{-d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^2+2)^(1/2)/(d*x^2+3)^(1/2),x)

[Out]

(-EllipticF(1/3*x*3^(1/2)*(-d)^(1/2),1/2*3^(1/2)*2^(1/2)*(b/d)^(1/2))+EllipticE(
1/3*x*3^(1/2)*(-d)^(1/2),1/2*3^(1/2)*2^(1/2)*(b/d)^(1/2)))*2^(1/2)/(-d)^(1/2)/b

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{b x^{2} + 2} \sqrt{d x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x^2 + 2)*sqrt(d*x^2 + 3)),x, algorithm="maxima")

[Out]

integrate(x^2/(sqrt(b*x^2 + 2)*sqrt(d*x^2 + 3)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{2}}{\sqrt{b x^{2} + 2} \sqrt{d x^{2} + 3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x^2 + 2)*sqrt(d*x^2 + 3)),x, algorithm="fricas")

[Out]

integral(x^2/(sqrt(b*x^2 + 2)*sqrt(d*x^2 + 3)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{b x^{2} + 2} \sqrt{d x^{2} + 3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x**2+2)**(1/2)/(d*x**2+3)**(1/2),x)

[Out]

Integral(x**2/(sqrt(b*x**2 + 2)*sqrt(d*x**2 + 3)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\sqrt{b x^{2} + 2} \sqrt{d x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(sqrt(b*x^2 + 2)*sqrt(d*x^2 + 3)),x, algorithm="giac")

[Out]

integrate(x^2/(sqrt(b*x^2 + 2)*sqrt(d*x^2 + 3)), x)