3.1.95 \(\int \frac {\sqrt {2+d x^2}}{(a+b x^2) \sqrt {3+f x^2}} \, dx\) [95]

Optimal. Leaf size=93 \[ \frac {2 \sqrt {3+f x^2} \Pi \left (1-\frac {2 b}{a d};\tan ^{-1}\left (\frac {\sqrt {d} x}{\sqrt {2}}\right )|1-\frac {2 f}{3 d}\right )}{\sqrt {3} a \sqrt {d} \sqrt {2+d x^2} \sqrt {\frac {3+f x^2}{2+d x^2}}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 93, 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 = {553} \begin {gather*} \frac {2 \sqrt {f x^2+3} \Pi \left (1-\frac {2 b}{a d};\text {ArcTan}\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}}} \end {gather*}

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 553

Int[Sqrt[(c_) + (d_.)*(x_)^2]/(((a_) + (b_.)*(x_)^2)*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[c*(Sqrt[e +
f*x^2]/(a*e*Rt[d/c, 2]*Sqrt[c + d*x^2]*Sqrt[c*((e + f*x^2)/(e*(c + d*x^2)))]))*EllipticPi[1 - b*(c/(a*d)), Ar
cTan[Rt[d/c, 2]*x], 1 - c*(f/(d*e))], 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*}

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Mathematica [C] Result contains complex when optimal does not.
time = 1.63, size = 94, normalized size = 1.01 \begin {gather*} -\frac {i \left (a d F\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}} \end {gather*}

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 = 0.14, size = 133, normalized size = 1.43

method	result	size
default	\(\frac {\sqrt {2}\, \left (\EllipticF \left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {\sqrt {2}\, \sqrt {3}\, \sqrt {\frac {d}{f}}}{2}\right ) a d -\EllipticPi \left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {3 b}{a f}, \frac {\sqrt {2}\, \sqrt {-d}\, \sqrt {3}}{2 \sqrt {-f}}\right ) a d +2 \EllipticPi \left (\frac {x \sqrt {3}\, \sqrt {-f}}{3}, \frac {3 b}{a f}, \frac {\sqrt {2}\, \sqrt {-d}\, \sqrt {3}}{2 \sqrt {-f}}\right ) b \right )}{2 \sqrt {-f}\, a b}\)	\(133\)
elliptic	\(\frac {\sqrt {\left (f \,x^{2}+3\right ) \left (d \,x^{2}+2\right )}\, \left (\frac {\sqrt {3 f \,x^{2}+9}\, \sqrt {2 d \,x^{2}+4}\, \EllipticF \left (\frac {x \sqrt {-3 f}}{3}, \frac {\sqrt {-4+\frac {6 d +4 f}{f}}}{2}\right ) d}{2 b \sqrt {-3 f}\, \sqrt {d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}}-\frac {\sqrt {1+\frac {f \,x^{2}}{3}}\, \sqrt {1+\frac {d \,x^{2}}{2}}\, \EllipticPi \left (\sqrt {-\frac {f}{3}}\, x , \frac {3 b}{a f}, \frac {\sqrt {-\frac {d}{2}}}{\sqrt {-\frac {f}{3}}}\right ) d}{b \sqrt {-\frac {f}{3}}\, \sqrt {d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}}+\frac {2 \sqrt {1+\frac {f \,x^{2}}{3}}\, \sqrt {1+\frac {d \,x^{2}}{2}}\, \EllipticPi \left (\sqrt {-\frac {f}{3}}\, x , \frac {3 b}{a f}, \frac {\sqrt {-\frac {d}{2}}}{\sqrt {-\frac {f}{3}}}\right )}{a \sqrt {-\frac {f}{3}}\, \sqrt {d f \,x^{4}+3 d \,x^{2}+2 f \,x^{2}+6}}\right )}{\sqrt {f \,x^{2}+3}\, \sqrt {d \,x^{2}+2}}\)	\(279\)

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,method=_RETURNVERBOSE)

[Out]

1/2*2^(1/2)*(EllipticF(1/3*x*3^(1/2)*(-f)^(1/2),1/2*2^(1/2)*3^(1/2)*(d/f)^(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 = 0.00, 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((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]
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((d*x^2+2)^(1/2)/(b*x^2+a)/(f*x^2+3)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(d*x^2 + 2)*sqrt(f*x^2 + 3)/(b*f*x^4 + (a*f + 3*b)*x^2 + 3*a), x)

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

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 = 0.00, 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((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)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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