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

Optimal. Leaf size=298 \[ \frac{3 \sqrt{d x^2+2} (2 b-a d) \Pi \left (1-\frac{3 b}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{\sqrt{2} a b \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{f x \sqrt{d x^2+2}}{b \sqrt{f x^2+3}}+\frac{3 d \sqrt{d x^2+2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{\sqrt{2} b \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}-\frac{\sqrt{2} \sqrt{f} \sqrt{d x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{b \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}} \]

[Out]

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

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Rubi [A]  time = 0.587406, antiderivative size = 298, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{3 \sqrt{d x^2+2} (2 b-a d) \Pi \left (1-\frac{3 b}{a f};\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{\sqrt{2} a b \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}+\frac{f x \sqrt{d x^2+2}}{b \sqrt{f x^2+3}}+\frac{3 d \sqrt{d x^2+2} F\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{\sqrt{2} b \sqrt{f} \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}}-\frac{\sqrt{2} \sqrt{f} \sqrt{d x^2+2} E\left (\tan ^{-1}\left (\frac{\sqrt{f} x}{\sqrt{3}}\right )|1-\frac{3 d}{2 f}\right )}{b \sqrt{f x^2+3} \sqrt{\frac{d x^2+2}{f x^2+3}}} \]

Antiderivative was successfully verified.

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

[Out]

(f*x*Sqrt[2 + d*x^2])/(b*Sqrt[3 + f*x^2]) - (Sqrt[2]*Sqrt[f]*Sqrt[2 + d*x^2]*Ell
ipticE[ArcTan[(Sqrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(b*Sqrt[(2 + d*x^2)/(3 + f
*x^2)]*Sqrt[3 + f*x^2]) + (3*d*Sqrt[2 + d*x^2]*EllipticF[ArcTan[(Sqrt[f]*x)/Sqrt
[3]], 1 - (3*d)/(2*f)])/(Sqrt[2]*b*Sqrt[f]*Sqrt[(2 + d*x^2)/(3 + f*x^2)]*Sqrt[3
+ f*x^2]) + (3*(2*b - a*d)*Sqrt[2 + d*x^2]*EllipticPi[1 - (3*b)/(a*f), ArcTan[(S
qrt[f]*x)/Sqrt[3]], 1 - (3*d)/(2*f)])/(Sqrt[2]*a*b*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 = 66.3375, size = 277, normalized size = 0.93 \[ - \frac{\sqrt{2} \sqrt{d} \sqrt{f x^{2} + 3} E\left (\operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{d} x}{2} \right )}\middle | 1 - \frac{2 f}{3 d}\right )}{b \sqrt{\frac{2 f x^{2} + 6}{3 d x^{2} + 6}} \sqrt{d x^{2} + 2}} + \frac{d x \sqrt{f x^{2} + 3}}{b \sqrt{d x^{2} + 2}} + \frac{3 \sqrt{3} d \sqrt{d x^{2} + 2} F\left (\operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{f} x}{3} \right )}\middle | - \frac{3 d}{2 f} + 1\right )}{2 b \sqrt{f} \sqrt{\frac{3 d x^{2} + 6}{2 f x^{2} + 6}} \sqrt{f x^{2} + 3}} + \frac{3 \sqrt{3} \left (- a d + 2 b\right ) \sqrt{d x^{2} + 2} \Pi \left (1 - \frac{3 b}{a f}; \operatorname{atan}{\left (\frac{\sqrt{3} \sqrt{f} x}{3} \right )}\middle | - \frac{3 d}{2 f} + 1\right )}{2 a b \sqrt{f} \sqrt{\frac{3 d x^{2} + 6}{2 f x^{2} + 6}} \sqrt{f x^{2} + 3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.284124, size = 134, normalized size = 0.45 \[ \frac{i \left ((a d-2 b) \left ((3 b-a f) \Pi \left (\frac{2 b}{a d};i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )+a f F\left (i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )\right )-3 a b d E\left (i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )\right )}{\sqrt{3} a b^2 \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.043, size = 293, normalized size = 1. \[ -{\frac{\sqrt{2}}{2\,a{b}^{2}} \left ({\it EllipticF} \left ({\frac{x\sqrt{3}}{3}\sqrt{-f}},{\frac{\sqrt{3}\sqrt{2}}{2}\sqrt{{\frac{d}{f}}}} \right ){a}^{2}df-{a}^{2}{\it EllipticPi} \left ({\frac{x\sqrt{3}}{3}\sqrt{-f}},3\,{\frac{b}{af}},{\frac{\sqrt{3}\sqrt{2}}{2}\sqrt{-d}{\frac{1}{\sqrt{-f}}}} \right ) df-3\,{\it EllipticF} \left ( 1/3\,x\sqrt{3}\sqrt{-f},1/2\,\sqrt{3}\sqrt{2}\sqrt{{\frac{d}{f}}} \right ) dba-2\,f{\it EllipticE} \left ( 1/3\,x\sqrt{3}\sqrt{-f},1/2\,\sqrt{3}\sqrt{2}\sqrt{{\frac{d}{f}}} \right ) ba+3\,{\it EllipticPi} \left ( 1/3\,x\sqrt{3}\sqrt{-f},3\,{\frac{b}{af}},1/2\,{\frac{\sqrt{2}\sqrt{-d}\sqrt{3}}{\sqrt{-f}}} \right ) dba+2\,{\it EllipticPi} \left ( 1/3\,x\sqrt{3}\sqrt{-f},3\,{\frac{b}{af}},1/2\,{\frac{\sqrt{2}\sqrt{-d}\sqrt{3}}{\sqrt{-f}}} \right ) fba-6\,{\it EllipticPi} \left ( 1/3\,x\sqrt{3}\sqrt{-f},3\,{\frac{b}{af}},1/2\,{\frac{\sqrt{2}\sqrt{-d}\sqrt{3}}{\sqrt{-f}}} \right ){b}^{2} \right ){\frac{1}{\sqrt{-f}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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^2*
d*f-a^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))*d*f-3*EllipticF(1/3*x*3^(1/2)*(-f)^(1/2),1/2*3^(1/2)*2^(1/2)*(1/f
*d)^(1/2))*d*b*a-2*f*EllipticE(1/3*x*3^(1/2)*(-f)^(1/2),1/2*3^(1/2)*2^(1/2)*(1/f
*d)^(1/2))*b*a+3*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))*d*b*a+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))*f*b*a-6*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^2)*2^(1/2)/a/(-f)^(1/2)/
b^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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