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

Optimal. Leaf size=191 \[ \frac{a \sqrt{d x^2+2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{\sqrt{2} \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{b x \sqrt{d x^2+2}}{d \sqrt{f x^2+3}}-\frac{\sqrt{2} b \sqrt{d x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{d \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}} \]

[Out]

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

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Rubi [A]  time = 0.343267, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{a \sqrt{d x^2+2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{\sqrt{2} \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{b x \sqrt{d x^2+2}}{d \sqrt{f x^2+3}}-\frac{\sqrt{2} b \sqrt{d x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{d \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 37.8688, size = 180, normalized size = 0.94 \[ \frac{\sqrt{3} a \sqrt{d x^{2} + 2} F\left (\operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{f} x}{3} \right )}\middle | - \frac{3 d}{2 f} + 1\right )}{2 \sqrt{f} \sqrt{\frac{3 d x^{2} + 6}{2 f x^{2} + 6}} \sqrt{f x^{2} + 3}} + \frac{b x \sqrt{f x^{2} + 3}}{f \sqrt{d x^{2} + 2}} - \frac{\sqrt{2} b \sqrt{f x^{2} + 3} E\left (\operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{d} x}{2} \right )}\middle | 1 - \frac{2 f}{3 d}\right )}{\sqrt{d} f \sqrt{\frac{2 f x^{2} + 6}{3 d x^{2} + 6}} \sqrt{d x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.138845, size = 81, normalized size = 0.42 \[ -\frac{i \left ((a f-3 b) F\left (i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )+3 b E\left (i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )\right )}{\sqrt{3} \sqrt{d} f} \]

Antiderivative was successfully verified.

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

[Out]

((-I)*(3*b*EllipticE[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)] + (-3*b + a*f)
*EllipticF[I*ArcSinh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)]))/(Sqrt[3]*Sqrt[d]*f)

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Maple [A]  time = 0.051, size = 105, normalized size = 0.6 \[{\frac{\sqrt{2}}{2\,d} \left ({\it EllipticF} \left ({\frac{x\sqrt{3}}{3}\sqrt{-f}},{\frac{\sqrt{3}\sqrt{2}}{2}\sqrt{{\frac{d}{f}}}} \right ) ad-2\,{\it EllipticF} \left ( 1/3\,x\sqrt{3}\sqrt{-f},1/2\,\sqrt{3}\sqrt{2}\sqrt{{\frac{d}{f}}} \right ) b+2\,{\it EllipticE} \left ( 1/3\,x\sqrt{3}\sqrt{-f},1/2\,\sqrt{3}\sqrt{2}\sqrt{{\frac{d}{f}}} \right ) b \right ){\frac{1}{\sqrt{-f}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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