3.91 \(\int \frac {\sqrt {2+x^2}}{\sqrt {1+x^2} (a+b x^2)} \, dx\)

Optimal. Leaf size=58 \[ \frac {2 \sqrt {x^2+1} \Pi \left (1-\frac {2 b}{a};\left .\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-1\right )}{a \sqrt {\frac {x^2+1}{x^2+2}} \sqrt {x^2+2}} \]

[Out]

2*(1/(2*x^2+4))^(1/2)*(2*x^2+4)^(1/2)*EllipticPi(x*2^(1/2)/(2*x^2+4)^(1/2),1-2*b/a,I)*(x^2+1)^(1/2)/a/((x^2+1)
/(x^2+2))^(1/2)/(x^2+2)^(1/2)

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Rubi [A]  time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {539} \[ \frac {2 \sqrt {x^2+1} \Pi \left (1-\frac {2 b}{a};\left .\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-1\right )}{a \sqrt {\frac {x^2+1}{x^2+2}} \sqrt {x^2+2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 + x^2]/(Sqrt[1 + x^2]*(a + b*x^2)),x]

[Out]

(2*Sqrt[1 + x^2]*EllipticPi[1 - (2*b)/a, ArcTan[x/Sqrt[2]], -1])/(a*Sqrt[(1 + x^2)/(2 + x^2)]*Sqrt[2 + 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+x^2}}{\sqrt {1+x^2} \left (a+b x^2\right )} \, dx &=\frac {2 \sqrt {1+x^2} \Pi \left (1-\frac {2 b}{a};\left .\tan ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-1\right )}{a \sqrt {\frac {1+x^2}{2+x^2}} \sqrt {2+x^2}}\\ \end {align*}

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Mathematica [C]  time = 0.18, size = 50, normalized size = 0.86 \[ -\frac {i \left (a F\left (i \sinh ^{-1}(x)|\frac {1}{2}\right )-(a-2 b) \Pi \left (\frac {b}{a};i \sinh ^{-1}(x)|\frac {1}{2}\right )\right )}{\sqrt {2} a b} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 + x^2]/(Sqrt[1 + x^2]*(a + b*x^2)),x]

[Out]

((-I)*(a*EllipticF[I*ArcSinh[x], 1/2] - (a - 2*b)*EllipticPi[b/a, I*ArcSinh[x], 1/2]))/(Sqrt[2]*a*b)

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fricas [F]  time = 17.87, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{2} + 2} \sqrt {x^{2} + 1}}{b x^{4} + {\left (a + b\right )} x^{2} + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2)^(1/2)/(x^2+1)^(1/2)/(b*x^2+a),x, algorithm="fricas")

[Out]

integral(sqrt(x^2 + 2)*sqrt(x^2 + 1)/(b*x^4 + (a + b)*x^2 + a), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x^{2} + 2}}{{\left (b x^{2} + a\right )} \sqrt {x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2)^(1/2)/(x^2+1)^(1/2)/(b*x^2+a),x, algorithm="giac")

[Out]

integrate(sqrt(x^2 + 2)/((b*x^2 + a)*sqrt(x^2 + 1)), x)

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maple [A]  time = 0.02, size = 64, normalized size = 1.10 \[ -\frac {i \left (a \EllipticF \left (\frac {i \sqrt {2}\, x}{2}, \sqrt {2}\right )-a \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {2 b}{a}, \sqrt {2}\right )+2 b \EllipticPi \left (\frac {i \sqrt {2}\, x}{2}, \frac {2 b}{a}, \sqrt {2}\right )\right )}{a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2+2)^(1/2)/(x^2+1)^(1/2)/(b*x^2+a),x)

[Out]

-I*(a*EllipticF(1/2*I*2^(1/2)*x,2^(1/2))-a*EllipticPi(1/2*I*2^(1/2)*x,2/a*b,2^(1/2))+2*b*EllipticPi(1/2*I*2^(1
/2)*x,2/a*b,2^(1/2)))/a/b

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x^{2} + 2}}{{\left (b x^{2} + a\right )} \sqrt {x^{2} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2+2)^(1/2)/(x^2+1)^(1/2)/(b*x^2+a),x, algorithm="maxima")

[Out]

integrate(sqrt(x^2 + 2)/((b*x^2 + a)*sqrt(x^2 + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {x^2+2}}{\sqrt {x^2+1}\,\left (b\,x^2+a\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2 + 2)^(1/2)/((x^2 + 1)^(1/2)*(a + b*x^2)),x)

[Out]

int((x^2 + 2)^(1/2)/((x^2 + 1)^(1/2)*(a + b*x^2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x^{2} + 2}}{\left (a + b x^{2}\right ) \sqrt {x^{2} + 1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2+2)**(1/2)/(x**2+1)**(1/2)/(b*x**2+a),x)

[Out]

Integral(sqrt(x**2 + 2)/((a + b*x**2)*sqrt(x**2 + 1)), x)

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