\(\int \frac {1}{(a+b x^2) \sqrt {4-5 x^4}} \, dx\) [170]

Optimal result

Integrand size = 21, antiderivative size = 40 \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {4-5 x^4}} \, dx=\frac {\operatorname {EllipticPi}\left (-\frac {2 b}{\sqrt {5} a},\arcsin \left (\frac {\sqrt [4]{5} x}{\sqrt {2}}\right ),-1\right )}{\sqrt {2} \sqrt [4]{5} a} \]

[Out]

1/10*EllipticPi(1/2*5^(1/4)*x*2^(1/2),-2/5*b/a*5^(1/2),I)*5^(3/4)/a*2^(1/2)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {1227, 551} \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {4-5 x^4}} \, dx=\frac {\operatorname {EllipticPi}\left (-\frac {2 b}{\sqrt {5} a},\arcsin \left (\frac {\sqrt [4]{5} x}{\sqrt {2}}\right ),-1\right )}{\sqrt {2} \sqrt [4]{5} a} \]

[In]

Int[1/((a + b*x^2)*Sqrt[4 - 5*x^4]),x]

[Out]

EllipticPi[(-2*b)/(Sqrt[5]*a), ArcSin[(5^(1/4)*x)/Sqrt[2]], -1]/(Sqrt[2]*5^(1/4)*a)

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])

Rule 1227

Int[1/(((d_) + (e_.)*(x_)^2)*Sqrt[(a_) + (c_.)*(x_)^4]), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[Sqrt[-c],
Int[1/((d + e*x^2)*Sqrt[q + c*x^2]*Sqrt[q - c*x^2]), x], x]] /; FreeQ[{a, c, d, e}, x] && GtQ[a, 0] && LtQ[c,
0]

Rubi steps \begin{align*} \text {integral}& = \sqrt {5} \int \frac {1}{\sqrt {2 \sqrt {5}-5 x^2} \sqrt {2 \sqrt {5}+5 x^2} \left (a+b x^2\right )} \, dx \\ & = \frac {\Pi \left (-\frac {2 b}{\sqrt {5} a};\left .\sin ^{-1}\left (\frac {\sqrt [4]{5} x}{\sqrt {2}}\right )\right |-1\right )}{\sqrt {2} \sqrt [4]{5} a} \\ \end{align*}

Mathematica [A] (verified)

Time = 10.21 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {4-5 x^4}} \, dx=\frac {\operatorname {EllipticPi}\left (-\frac {2 b}{\sqrt {5} a},\arcsin \left (\frac {\sqrt [4]{5} x}{\sqrt {2}}\right ),-1\right )}{\sqrt {2} \sqrt [4]{5} a} \]

[In]

Integrate[1/((a + b*x^2)*Sqrt[4 - 5*x^4]),x]

[Out]

EllipticPi[(-2*b)/(Sqrt[5]*a), ArcSin[(5^(1/4)*x)/Sqrt[2]], -1]/(Sqrt[2]*5^(1/4)*a)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(78\) vs. \(2(32)=64\).

Time = 1.08 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.98

method	result	size
default	\(\frac {\sqrt {2}\, 5^{\frac {3}{4}} \sqrt {1-\frac {x^{2} \sqrt {5}}{2}}\, \sqrt {1+\frac {x^{2} \sqrt {5}}{2}}\, \Pi \left (\frac {5^{\frac {1}{4}} x \sqrt {2}}{2}, -\frac {2 b \sqrt {5}}{5 a}, \frac {\sqrt {-\frac {\sqrt {5}}{2}}\, \sqrt {2}\, 5^{\frac {3}{4}}}{5}\right )}{5 a \sqrt {-5 x^{4}+4}}\)	\(79\)
elliptic	\(\frac {\sqrt {2}\, 5^{\frac {3}{4}} \sqrt {1-\frac {x^{2} \sqrt {5}}{2}}\, \sqrt {1+\frac {x^{2} \sqrt {5}}{2}}\, \Pi \left (\frac {5^{\frac {1}{4}} x \sqrt {2}}{2}, -\frac {2 b \sqrt {5}}{5 a}, \frac {\sqrt {-\frac {\sqrt {5}}{2}}\, \sqrt {2}\, 5^{\frac {3}{4}}}{5}\right )}{5 a \sqrt {-5 x^{4}+4}}\)	\(79\)

[In]

int(1/(b*x^2+a)/(-5*x^4+4)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/5/a*2^(1/2)*5^(3/4)*(1-1/2*x^2*5^(1/2))^(1/2)*(1+1/2*x^2*5^(1/2))^(1/2)/(-5*x^4+4)^(1/2)*EllipticPi(1/2*5^(1
/4)*x*2^(1/2),-2/5*b/a*5^(1/2),1/5*(-1/2*5^(1/2))^(1/2)*2^(1/2)*5^(3/4))

Fricas [F]

\[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {4-5 x^4}} \, dx=\int { \frac {1}{\sqrt {-5 \, x^{4} + 4} {\left (b x^{2} + a\right )}} \,d x } \]

[In]

integrate(1/(b*x^2+a)/(-5*x^4+4)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-5*x^4 + 4)/(5*b*x^6 + 5*a*x^4 - 4*b*x^2 - 4*a), x)

Sympy [F]

\[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {4-5 x^4}} \, dx=\int \frac {1}{\sqrt {4 - 5 x^{4}} \left (a + b x^{2}\right )}\, dx \]

[In]

integrate(1/(b*x**2+a)/(-5*x**4+4)**(1/2),x)

[Out]

Integral(1/(sqrt(4 - 5*x**4)*(a + b*x**2)), x)

Maxima [F]

\[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {4-5 x^4}} \, dx=\int { \frac {1}{\sqrt {-5 \, x^{4} + 4} {\left (b x^{2} + a\right )}} \,d x } \]

[In]

integrate(1/(b*x^2+a)/(-5*x^4+4)^(1/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-5*x^4 + 4)*(b*x^2 + a)), x)

Giac [F]

\[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {4-5 x^4}} \, dx=\int { \frac {1}{\sqrt {-5 \, x^{4} + 4} {\left (b x^{2} + a\right )}} \,d x } \]

[In]

integrate(1/(b*x^2+a)/(-5*x^4+4)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-5*x^4 + 4)*(b*x^2 + a)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^2\right ) \sqrt {4-5 x^4}} \, dx=\int \frac {1}{\left (b\,x^2+a\right )\,\sqrt {4-5\,x^4}} \,d x \]

[In]

int(1/((a + b*x^2)*(4 - 5*x^4)^(1/2)),x)

[Out]

int(1/((a + b*x^2)*(4 - 5*x^4)^(1/2)), x)