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

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 = 0.0361127, 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, Rules used = {539} \[ \frac{2 \sqrt{f x^2+3} \Pi \left (1-\frac{2 b}{a d};\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|1-\frac{2 f}{3 d}\right )}{\sqrt{3} a \sqrt{d} \sqrt{d x^2+2} \sqrt{\frac{f x^2+3}{d x^2+2}}} \]

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)])

Rule 539

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(c*Sqrt[e +
f*x^2]*EllipticPi[1 - (b*c)/(a*d), ArcTan[Rt[d/c, 2]*x], 1 - (c*f)/(d*e)])/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sq
rt[(c*(e + f*x^2))/(e*(c + d*x^2))]), x] /; FreeQ[{a, b, c, d, e, f}, x] && PosQ[d/c]

Rubi steps

\begin{align*} \int \frac{\sqrt{2+d x^2}}{\left (a+b x^2\right ) \sqrt{3+f x^2}} \, dx &=\frac{2 \sqrt{3+f x^2} \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{2+d x^2} \sqrt{\frac{3+f x^2}{2+d x^2}}}\\ \end{align*}

Mathematica [C]  time = 0.198352, size = 94, normalized size = 1.01 \[ -\frac{i \left (a d \text{EllipticF}\left (i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right ),\frac{2 f}{3 d}\right )+(2 b-a d) \Pi \left (\frac{2 b}{a d};i \sinh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{2}}\right )|\frac{2 f}{3 d}\right )\right )}{\sqrt{3} a b \sqrt{d}} \]

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*ArcS
inh[(Sqrt[d]*x)/Sqrt[2]], (2*f)/(3*d)]))/(Sqrt[3]*a*b*Sqrt[d])

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Maple [A]  time = 0.02, size = 133, normalized size = 1.4 \begin{align*}{\frac{\sqrt{2}}{2\,ab} \left ({\it EllipticF} \left ({\frac{x\sqrt{3}}{3}\sqrt{-f}},{\frac{\sqrt{2}\sqrt{3}}{2}\sqrt{{\frac{d}{f}}}} \right ) ad-{\it EllipticPi} \left ({\frac{x\sqrt{3}}{3}\sqrt{-f}},3\,{\frac{b}{af}},{\frac{\sqrt{2}\sqrt{3}}{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}}}} \end{align*}

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*2^(1/2)*3^(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. \begin{align*} \int \frac{\sqrt{d x^{2} + 2}}{{\left (b x^{2} + a\right )} \sqrt{f x^{2} + 3}}\,{d x} \end{align*}

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, 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. \begin{align*} \text{Timed out} \end{align*}

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, algorithm="fricas")

[Out]

Timed out

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

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

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, algorithm="giac")

[Out]

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