3.9.26 \(\int \frac {1}{x^2 \sqrt [4]{a-b x^2}} \, dx\) [826]

Optimal. Leaf size=79 \[ -\frac {\left (a-b x^2\right )^{3/4}}{a x}-\frac {\sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt [4]{a-b x^2}} \]

[Out]

-(-b*x^2+a)^(3/4)/a/x-(1-b*x^2/a)^(1/4)*(cos(1/2*arcsin(x*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*arcsin(x*b^(1/2)/
a^(1/2)))*EllipticE(sin(1/2*arcsin(x*b^(1/2)/a^(1/2))),2^(1/2))*b^(1/2)/(-b*x^2+a)^(1/4)/a^(1/2)

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Rubi [A]
time = 0.02, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {331, 235, 234} \begin {gather*} -\frac {\sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \text {ArcSin}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt [4]{a-b x^2}}-\frac {\left (a-b x^2\right )^{3/4}}{a x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a - b*x^2)^(1/4)),x]

[Out]

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

Rule 234

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2/(a^(1/4)*Rt[-b/a, 2]))*EllipticE[(1/2)*ArcSin[Rt[-b/a,
2]*x], 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rule 235

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

Rubi steps

\begin {align*} \int \frac {1}{x^2 \sqrt [4]{a-b x^2}} \, dx &=-\frac {\left (a-b x^2\right )^{3/4}}{a x}-\frac {b \int \frac {1}{\sqrt [4]{a-b x^2}} \, dx}{2 a}\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{a x}-\frac {\left (b \sqrt [4]{1-\frac {b x^2}{a}}\right ) \int \frac {1}{\sqrt [4]{1-\frac {b x^2}{a}}} \, dx}{2 a \sqrt [4]{a-b x^2}}\\ &=-\frac {\left (a-b x^2\right )^{3/4}}{a x}-\frac {\sqrt {b} \sqrt [4]{1-\frac {b x^2}{a}} E\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{\sqrt {a} \sqrt [4]{a-b x^2}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 6.69, size = 50, normalized size = 0.63 \begin {gather*} -\frac {\sqrt [4]{1-\frac {b x^2}{a}} \, _2F_1\left (-\frac {1}{2},\frac {1}{4};\frac {1}{2};\frac {b x^2}{a}\right )}{x \sqrt [4]{a-b x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{x^{2} \left (-b \,x^{2}+a \right )^{\frac {1}{4}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^2+a)^(1/4),x, algorithm="maxima")

[Out]

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

[Out]

integral(-(-b*x^2 + a)^(3/4)/(b*x^4 - a*x^2), x)

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Sympy [C] Result contains complex when optimal does not.
time = 0.44, size = 29, normalized size = 0.37 \begin {gather*} - \frac {{{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, \frac {1}{4} \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{\sqrt [4]{a} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Mupad [B]
time = 5.08, size = 41, normalized size = 0.52 \begin {gather*} -\frac {2\,{\left (1-\frac {a}{b\,x^2}\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{4},\frac {3}{4};\ \frac {7}{4};\ \frac {a}{b\,x^2}\right )}{3\,x\,{\left (a-b\,x^2\right )}^{1/4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2*(a - b*x^2)^(1/4)),x)

[Out]

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

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