3.833 \(\int \frac{1}{x^2 \left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=76 \[ -\frac{\sqrt [4]{a+b x^2}}{a x}-\frac{\sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \left (a+b x^2\right )^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.065673, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{\sqrt [4]{a+b x^2}}{a x}-\frac{\sqrt{b} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \left (a+b x^2\right )^{3/4}} \]

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[ArcTan[(Sq
rt[b]*x)/Sqrt[a]]/2, 2])/(Sqrt[a]*(a + b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 7.93875, size = 63, normalized size = 0.83 \[ - \frac{\sqrt [4]{a + b x^{2}}}{a x} - \frac{\sqrt{b} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{\sqrt{a} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x**2+a)**(3/4),x)

[Out]

-(a + b*x**2)**(1/4)/(a*x) - sqrt(b)*(1 + b*x**2/a)**(3/4)*elliptic_f(atan(sqrt(
b)*x/sqrt(a))/2, 2)/(sqrt(a)*(a + b*x**2)**(3/4))

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Mathematica [C]  time = 0.0446168, size = 70, normalized size = 0.92 \[ \frac{-b x^2 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right )-2 \left (a+b x^2\right )}{2 a x \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

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. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/4)*x^2),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. \[{\rm integral}\left (\frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.12821, size = 27, normalized size = 0.36 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac{3}{4}} x} \]

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(I*pi)/a)/(a**(3/4)*x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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