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

Optimal. Leaf size=93 \[ \frac{2 \sqrt{f x^2+3} \Pi \left (1-\frac{2 b}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|1-\frac{2 f}{3 d}\right )}{\sqrt{3} a \sqrt{d} \sqrt{d x^2+2} \sqrt{\frac{f x^2+3}{d x^2+2}}} \]

[Out]

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

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Rubi [A]  time = 0.13467, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.031 \[ \frac{2 \sqrt{f x^2+3} \Pi \left (1-\frac{2 b}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|1-\frac{2 f}{3 d}\right )}{\sqrt{3} a \sqrt{d} \sqrt{d x^2+2} \sqrt{\frac{f x^2+3}{d x^2+2}}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.2507, size = 85, normalized size = 0.91 \[ \frac{2 \sqrt{2} \sqrt{f x^{2} + 3} \Pi \left (1 - \frac{2 b}{a d}; \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{d} x}{2} \right )}\middle | 1 - \frac{2 f}{3 d}\right )}{3 a \sqrt{d} \sqrt{\frac{2 f x^{2} + 6}{3 d x^{2} + 6}} \sqrt{d x^{2} + 2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(2)*sqrt(f*x**2 + 3)*elliptic_pi(1 - 2*b/(a*d), atan(sqrt(2)*sqrt(d)*x/2),
1 - 2*f/(3*d))/(3*a*sqrt(d)*sqrt((2*f*x**2 + 6)/(3*d*x**2 + 6))*sqrt(d*x**2 + 2
))

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Mathematica [C]  time = 0.110367, size = 94, normalized size = 1.01 \[ -\frac{i \left ((2 b-a d) \Pi \left (\frac{2 b}{a d};i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )+a d F\left (i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )\right )}{\sqrt{3} a b \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.032, size = 133, normalized size = 1.4 \[{\frac{\sqrt{2}}{2\,ab} \left ({\it EllipticF} \left ({\frac{x\sqrt{3}}{3}\sqrt{-f}},{\frac{\sqrt{3}\sqrt{2}}{2}\sqrt{{\frac{d}{f}}}} \right ) ad-{\it EllipticPi} \left ({\frac{x\sqrt{3}}{3}\sqrt{-f}},3\,{\frac{b}{af}},{\frac{\sqrt{3}\sqrt{2}}{2}\sqrt{-d}{\frac{1}{\sqrt{-f}}}} \right ) ad+2\,{\it EllipticPi} \left ( 1/3\,x\sqrt{3}\sqrt{-f},3\,{\frac{b}{af}},1/2\,{\frac{\sqrt{2}\sqrt{-d}\sqrt{3}}{\sqrt{-f}}} \right ) b \right ){\frac{1}{\sqrt{-f}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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