3.843 \(\int \frac {1}{\sqrt {a-b x^4}} \, dx\)

Optimal. Leaf size=53 \[ \frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {224, 221} \[ \frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}} \]

Antiderivative was successfully verified.

[In]

Int[1/Sqrt[a - b*x^4],x]

[Out]

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

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[EllipticF[ArcSin[(Rt[-b, 4]*x)/Rt[a, 4]], -1]/(Rt[a, 4]*Rt[
-b, 4]), x] /; FreeQ[{a, b}, x] && NegQ[b/a] && GtQ[a, 0]

Rule 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + (b*x^4)/a]/Sqrt[a + b*x^4], Int[1/Sqrt[1 + (b*x^4)
/a], x], x] /; FreeQ[{a, b}, x] && NegQ[b/a] &&  !GtQ[a, 0]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {a-b x^4}} \, dx &=\frac {\sqrt {1-\frac {b x^4}{a}} \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{\sqrt {a-b x^4}}\\ &=\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{b} \sqrt {a-b x^4}}\\ \end {align*}

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Mathematica [C]  time = 0.04, size = 72, normalized size = 1.36 \[ -\frac {i \sqrt {1-\frac {b x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}} x\right )\right |-1\right )}{\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}} \sqrt {a-b x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/Sqrt[a - b*x^4],x]

[Out]

((-I)*Sqrt[1 - (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[a])]*x], -1])/(Sqrt[-(Sqrt[b]/Sqrt[a])]*Sqrt
[a - b*x^4])

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fricas [F]  time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-b x^{4} + a}}{b x^{4} - a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-b x^{4} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-b*x^4 + a), x)

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maple [A]  time = 0.01, size = 64, normalized size = 1.21 \[ \frac {\sqrt {-\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \sqrt {\frac {\sqrt {b}\, x^{2}}{\sqrt {a}}+1}\, \EllipticF \left (\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, x , i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-b x^{4} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(1/sqrt(-b*x^4 + a), x)

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mupad [B]  time = 1.09, size = 38, normalized size = 0.72 \[ \frac {x\,\sqrt {1-\frac {b\,x^4}{a}}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {1}{2};\ \frac {5}{4};\ \frac {b\,x^4}{a}\right )}{\sqrt {a-b\,x^4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a - b*x^4)^(1/2),x)

[Out]

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

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sympy [A]  time = 1.00, size = 37, normalized size = 0.70 \[ \frac {x \Gamma \left (\frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {1}{2} \\ \frac {5}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*gamma(1/4)*hyper((1/4, 1/2), (5/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*gamma(5/4))

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