3.3.90 \(\int \frac {1}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx\) [290]

Optimal. Leaf size=78 \[ \frac {\sqrt {2+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{\sqrt {2} \sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}} \]

[Out]

1/2*(1/(3*d*x^2+9))^(1/2)*(3*d*x^2+9)^(1/2)*EllipticF(x*d^(1/2)*3^(1/2)/(3*d*x^2+9)^(1/2),1/2*(4-6*b/d)^(1/2))
*2^(1/2)*(b*x^2+2)^(1/2)/d^(1/2)/((b*x^2+2)/(d*x^2+3))^(1/2)/(d*x^2+3)^(1/2)

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Rubi [A]
time = 0.01, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {429} \begin {gather*} \frac {\sqrt {b x^2+2} F\left (\text {ArcTan}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{\sqrt {2} \sqrt {d} \sqrt {d x^2+3} \sqrt {\frac {b x^2+2}{d x^2+3}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2]),x]

[Out]

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

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*
Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {2+b x^2} \sqrt {3+d x^2}} \, dx &=\frac {\sqrt {2+b x^2} F\left (\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {3}}\right )|1-\frac {3 b}{2 d}\right )}{\sqrt {2} \sqrt {d} \sqrt {\frac {2+b x^2}{3+d x^2}} \sqrt {3+d x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.57, size = 37, normalized size = 0.47 \begin {gather*} \frac {F\left (\sin ^{-1}\left (\frac {\sqrt {-b} x}{\sqrt {2}}\right )|\frac {2 d}{3 b}\right )}{\sqrt {3} \sqrt {-b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[2 + b*x^2]*Sqrt[3 + d*x^2]),x]

[Out]

EllipticF[ArcSin[(Sqrt[-b]*x)/Sqrt[2]], (2*d)/(3*b)]/(Sqrt[3]*Sqrt[-b])

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Maple [A]
time = 0.08, size = 38, normalized size = 0.49

method	result	size
default	\(\frac {\sqrt {2}\, \EllipticF \left (\frac {x \sqrt {3}\, \sqrt {-d}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {b}{d}}}{2}\right )}{2 \sqrt {-d}}\)	\(38\)
elliptic	\(\frac {\sqrt {\left (b \,x^{2}+2\right ) \left (d \,x^{2}+3\right )}\, \sqrt {3 d \,x^{2}+9}\, \sqrt {2 b \,x^{2}+4}\, \EllipticF \left (\frac {x \sqrt {-3 d}}{3}, \frac {\sqrt {-4+\frac {6 b +4 d}{d}}}{2}\right )}{2 \sqrt {b \,x^{2}+2}\, \sqrt {d \,x^{2}+3}\, \sqrt {-3 d}\, \sqrt {b d \,x^{4}+3 b \,x^{2}+2 d \,x^{2}+6}}\)	\(112\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+2)^(1/2)/(d*x^2+3)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/2*2^(1/2)*EllipticF(1/3*x*3^(1/2)*(-d)^(1/2),1/2*2^(1/2)*3^(1/2)*(b/d)^(1/2))/(-d)^(1/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+2)^(1/2)/(d*x^2+3)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(b*x^2 + 2)*sqrt(d*x^2 + 3)), x)

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Fricas [A]
time = 0.45, size = 34, normalized size = 0.44 \begin {gather*} -\frac {\sqrt {6} \sqrt {2} \sqrt {-b} {\rm ellipticF}\left (\frac {1}{2} \, \sqrt {2} \sqrt {-b} x, \frac {2 \, d}{3 \, b}\right )}{6 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+2)^(1/2)/(d*x^2+3)^(1/2),x, algorithm="fricas")

[Out]

-1/6*sqrt(6)*sqrt(2)*sqrt(-b)*ellipticF(1/2*sqrt(2)*sqrt(-b)*x, 2/3*d/b)/b

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {b x^{2} + 2} \sqrt {d x^{2} + 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**2+2)**(1/2)/(d*x**2+3)**(1/2),x)

[Out]

Integral(1/(sqrt(b*x**2 + 2)*sqrt(d*x**2 + 3)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+2)^(1/2)/(d*x^2+3)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(b*x^2 + 2)*sqrt(d*x^2 + 3)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\sqrt {b\,x^2+2}\,\sqrt {d\,x^2+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x^2 + 2)^(1/2)*(d*x^2 + 3)^(1/2)),x)

[Out]

int(1/((b*x^2 + 2)^(1/2)*(d*x^2 + 3)^(1/2)), x)

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