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

Optimal. Leaf size=182 \[ \frac{x \sqrt{b x^2+2}}{\sqrt{d x^2+3}}+\frac{\sqrt{2} \sqrt{b x^2+2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{\sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}-\frac{\sqrt{2} \sqrt{b x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{\sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}} \]

[Out]

(x*Sqrt[2 + b*x^2])/Sqrt[3 + d*x^2] - (Sqrt[2]*Sqrt[2 + b*x^2]*EllipticE[ArcTan[
(Sqrt[d]*x)/Sqrt[3]], 1 - (3*b)/(2*d)])/(Sqrt[d]*Sqrt[(2 + b*x^2)/(3 + d*x^2)]*S
qrt[3 + d*x^2]) + (Sqrt[2]*Sqrt[2 + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[3]]
, 1 - (3*b)/(2*d)])/(Sqrt[d]*Sqrt[(2 + b*x^2)/(3 + d*x^2)]*Sqrt[3 + d*x^2])

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Rubi [A]  time = 0.246113, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ \frac{x \sqrt{b x^2+2}}{\sqrt{d x^2+3}}+\frac{\sqrt{2} \sqrt{b x^2+2} F\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{\sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}}-\frac{\sqrt{2} \sqrt{b x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{3}}\right )|1-\frac{3 b}{2 d}\right )}{\sqrt{d} \sqrt{d x^2+3} \sqrt{\frac{b x^2+2}{d x^2+3}}} \]

Antiderivative was successfully verified.

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

[Out]

(x*Sqrt[2 + b*x^2])/Sqrt[3 + d*x^2] - (Sqrt[2]*Sqrt[2 + b*x^2]*EllipticE[ArcTan[
(Sqrt[d]*x)/Sqrt[3]], 1 - (3*b)/(2*d)])/(Sqrt[d]*Sqrt[(2 + b*x^2)/(3 + d*x^2)]*S
qrt[3 + d*x^2]) + (Sqrt[2]*Sqrt[2 + b*x^2]*EllipticF[ArcTan[(Sqrt[d]*x)/Sqrt[3]]
, 1 - (3*b)/(2*d)])/(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 = 35.2931, size = 175, normalized size = 0.96 \[ - \frac{\sqrt{2} \sqrt{b} \sqrt{d x^{2} + 3} E\left (\operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{b} x}{2} \right )}\middle | 1 - \frac{2 d}{3 b}\right )}{d \sqrt{\frac{2 d x^{2} + 6}{3 b x^{2} + 6}} \sqrt{b x^{2} + 2}} + \frac{b x \sqrt{d x^{2} + 3}}{d \sqrt{b x^{2} + 2}} + \frac{\sqrt{3} \sqrt{b x^{2} + 2} F\left (\operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{d} x}{3} \right )}\middle | - \frac{3 b}{2 d} + 1\right )}{\sqrt{d} \sqrt{\frac{3 b x^{2} + 6}{2 d x^{2} + 6}} \sqrt{d x^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*sqrt(b)*sqrt(d*x**2 + 3)*elliptic_e(atan(sqrt(2)*sqrt(b)*x/2), 1 - 2*d/
(3*b))/(d*sqrt((2*d*x**2 + 6)/(3*b*x**2 + 6))*sqrt(b*x**2 + 2)) + b*x*sqrt(d*x**
2 + 3)/(d*sqrt(b*x**2 + 2)) + sqrt(3)*sqrt(b*x**2 + 2)*elliptic_f(atan(sqrt(3)*s
qrt(d)*x/3), -3*b/(2*d) + 1)/(sqrt(d)*sqrt((3*b*x**2 + 6)/(2*d*x**2 + 6))*sqrt(d
*x**2 + 3))

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Mathematica [A]  time = 0.0391646, size = 37, normalized size = 0.2 \[ \frac{\sqrt{2} E\left (\sin ^{-1}\left (\frac{\sqrt{-d} x}{\sqrt{3}}\right )|\frac{3 b}{2 d}\right )}{\sqrt{-d}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2]*EllipticE[ArcSin[(Sqrt[-d]*x)/Sqrt[3]], (3*b)/(2*d)])/Sqrt[-d]

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Maple [A]  time = 0.023, size = 37, normalized size = 0.2 \[{\sqrt{2}{\it EllipticE} \left ({\frac{x\sqrt{3}}{3}\sqrt{-d}},{\frac{\sqrt{3}\sqrt{2}}{2}\sqrt{{\frac{b}{d}}}} \right ){\frac{1}{\sqrt{-d}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(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. \[ \int \frac{\sqrt{b x^{2} + 2}}{\sqrt{d x^{2} + 3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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