3.840 \(\int \frac{1}{x^2 (a-b x^2)^{3/4}} \, dx\)

Optimal. Leaf size=78 \[ \frac{\sqrt{b} \left (1-\frac{b x^2}{a}\right )^{3/4} \text{EllipticF}\left (\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right ),2\right )}{\sqrt{a} \left (a-b x^2\right )^{3/4}}-\frac{\sqrt [4]{a-b x^2}}{a x} \]

[Out]

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

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Rubi [A]  time = 0.0209561, antiderivative size = 78, normalized size of antiderivative = 1., 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 = {325, 233, 232} \[ \frac{\sqrt{b} \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \left (a-b x^2\right )^{3/4}}-\frac{\sqrt [4]{a-b x^2}}{a x} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(a - b*x^2)^(3/4)),x]

[Out]

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

Rule 325

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 233

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Dist[(1 + (b*x^2)/a)^(3/4)/(a + b*x^2)^(3/4), Int[1/(1 + (b*x^2
)/a)^(3/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 232

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2*EllipticF[(1*ArcSin[Rt[-(b/a), 2]*x])/2, 2])/(a^(3/4)*R
t[-(b/a), 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rubi steps

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

Mathematica [C]  time = 0.0094599, size = 50, normalized size = 0.64 \[ -\frac{\left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (-\frac{1}{2},\frac{3}{4};\frac{1}{2};\frac{b x^2}{a}\right )}{x \left (a-b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 0.016, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( -b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{3}{4}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(-b*x^2+a)^(3/4),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (-b x^{2} + a\right )}^{\frac{1}{4}}}{b x^{4} - a x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(-b*x^2+a)^(3/4),x, algorithm="fricas")

[Out]

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

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Sympy [C]  time = 0.88379, size = 29, normalized size = 0.37 \begin{align*} - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{a^{\frac{3}{4}} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**2/(-b*x**2+a)**(3/4),x)

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{3}{4}} x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^2/(-b*x^2+a)^(3/4),x, algorithm="giac")

[Out]

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