3.1.96 \(\int \frac {1}{(a+b x^2) \sqrt {2+d x^2} \sqrt {3+f x^2}} \, dx\) [96]

Optimal. Leaf size=49 \[ \frac {\Pi \left (\frac {2 b}{a d};\sin ^{-1}\left (\frac {\sqrt {-d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )}{\sqrt {3} a \sqrt {-d}} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 49, normalized size of antiderivative = 1.00, 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 = {551} \begin {gather*} \frac {\Pi \left (\frac {2 b}{a d};\text {ArcSin}\left (\frac {\sqrt {-d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )}{\sqrt {3} a \sqrt {-d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

EllipticPi[(2*b)/(a*d), ArcSin[(Sqrt[-d]*x)/Sqrt[2]], (2*f)/(3*d)]/(Sqrt[3]*a*Sqrt[-d])

Rule 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b*(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c,
d, e, f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && SimplerSqrtQ[-f/e, -d/c])

Rubi steps

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

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Mathematica [A]
time = 1.86, size = 49, normalized size = 1.00 \begin {gather*} \frac {\Pi \left (\frac {2 b}{a d};\sin ^{-1}\left (\frac {\sqrt {-d} x}{\sqrt {2}}\right )|\frac {2 f}{3 d}\right )}{\sqrt {3} a \sqrt {-d}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

EllipticPi[(2*b)/(a*d), ArcSin[(Sqrt[-d]*x)/Sqrt[2]], (2*f)/(3*d)]/(Sqrt[3]*a*Sqrt[-d])

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Maple [A]
time = 0.14, size = 53, normalized size = 1.08

method	result	size
default	\(\frac {\sqrt {2}\, \EllipticPi \left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {3 b}{a f}, \frac {\sqrt {2}\, \sqrt {-d}\, \sqrt {3}}{2 \sqrt {-f}}\right )}{2 \sqrt {-f}\, a}\)	\(53\)
elliptic	\(\frac {\sqrt {\left (f \,x^{2}+3\right ) \left (d \,x^{2}+2\right )}\, \sqrt {1+\frac {f \,x^{2}}{3}}\, \sqrt {1+\frac {d \,x^{2}}{2}}\, \EllipticPi \left (\sqrt {-\frac {f}{3}}\, x , \frac {3 b}{a f}, \frac {\sqrt {-\frac {d}{2}}}{\sqrt {-\frac {f}{3}}}\right )}{\sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}\, a \sqrt {-\frac {f}{3}}\, \sqrt {d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}}\)	\(115\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*2^(1/2)*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)^(1/2)/
a

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

[Out]

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

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((a + b*x**2)*sqrt(d*x**2 + 2)*sqrt(f*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+a)/(d*x^2+2)^(1/2)/(f*x^2+3)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{\left (b\,x^2+a\right )\,\sqrt {d\,x^2+2}\,\sqrt {f\,x^2+3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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