3.9.49 \(\int \frac {1}{x^2 \sqrt {a-b x^4}} \, dx\) [849]

Optimal. Leaf size=128 \[ -\frac {\sqrt {a-b x^4}}{a x}-\frac {\sqrt [4]{b} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}+\frac {\sqrt [4]{b} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a-b x^4}} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.438, Rules used = {331, 313, 230, 227, 1214, 1213, 435} \begin {gather*} \frac {\sqrt [4]{b} \sqrt {1-\frac {b x^4}{a}} F\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}-\frac {\sqrt [4]{b} \sqrt {1-\frac {b x^4}{a}} E\left (\left .\text {ArcSin}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{\sqrt [4]{a} \sqrt {a-b x^4}}-\frac {\sqrt {a-b x^4}}{a x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*Sqrt[a - b*x^4]),x]

[Out]

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

Rule 227

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 230

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]

Rule 313

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, Dist[-q^(-1), Int[1/Sqrt[a + b*x^4]
, x], x] + Dist[1/q, Int[(1 + q*x^2)/Sqrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]

Rule 331

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c
*(m + 1))), x] - Dist[b*((m + n*(p + 1) + 1)/(a*c^n*(m + 1))), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 1213

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

Rule 1214

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> Dist[Sqrt[1 + c*(x^4/a)]/Sqrt[a + c*x^4], In
t[(d + e*x^2)/Sqrt[1 + c*(x^4/a)], x], x] /; FreeQ[{a, c, d, e}, x] && NegQ[c/a] && EqQ[c*d^2 + a*e^2, 0] &&
!GtQ[a, 0]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 10.01, size = 50, normalized size = 0.39 \begin {gather*} -\frac {\sqrt {1-\frac {b x^4}{a}} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{4};\frac {b x^4}{a}\right )}{x \sqrt {a-b x^4}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*Sqrt[a - b*x^4]),x]

[Out]

-((Sqrt[1 - (b*x^4)/a]*Hypergeometric2F1[-1/4, 1/2, 3/4, (b*x^4)/a])/(x*Sqrt[a - b*x^4]))

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Maple [A]
time = 0.15, size = 106, normalized size = 0.83

method	result	size
default	\(-\frac {\sqrt {-b \,x^{4}+a}}{a x}+\frac {\sqrt {b}\, \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(106\)
risch	\(-\frac {\sqrt {-b \,x^{4}+a}}{a x}+\frac {\sqrt {b}\, \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(106\)
elliptic	\(-\frac {\sqrt {-b \,x^{4}+a}}{a x}+\frac {\sqrt {b}\, \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \left (\EllipticF \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )-\EllipticE \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )\right )}{\sqrt {a}\, \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}\)	\(106\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [A]
time = 0.08, size = 68, normalized size = 0.53 \begin {gather*} -\frac {\sqrt {a} x \left (\frac {b}{a}\right )^{\frac {3}{4}} E(\arcsin \left (x \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - \sqrt {a} x \left (\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) + \sqrt {-b x^{4} + a}}{a x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(sqrt(a)*x*(b/a)^(3/4)*elliptic_e(arcsin(x*(b/a)^(1/4)), -1) - sqrt(a)*x*(b/a)^(3/4)*elliptic_f(arcsin(x*(b/a
)^(1/4)), -1) + sqrt(-b*x^4 + a))/(a*x)

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Sympy [A]
time = 0.42, size = 41, normalized size = 0.32 \begin {gather*} \frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt {a} x \Gamma \left (\frac {3}{4}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

gamma(-1/4)*hyper((-1/4, 1/2), (3/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a)*x*gamma(3/4))

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

[Out]

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

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Mupad [B]
time = 1.28, size = 41, normalized size = 0.32 \begin {gather*} -\frac {\sqrt {1-\frac {a}{b\,x^4}}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {3}{4};\ \frac {7}{4};\ \frac {a}{b\,x^4}\right )}{3\,x\,\sqrt {a-b\,x^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a - b*x^4)^(1/2)),x)

[Out]

-((1 - a/(b*x^4))^(1/2)*hypergeom([1/2, 3/4], 7/4, a/(b*x^4)))/(3*x*(a - b*x^4)^(1/2))

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