3.847 \(\int \frac{1}{x^2 \left (a+b x^2\right )^{5/4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0676975, 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 \[ -\frac{3 \sqrt{b} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{a^{3/2} \sqrt [4]{a+b x^2}}-\frac{1}{a x \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^2*(a + b*x^2)^(5/4)),x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{3 b \int \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{2 a} - \frac{1}{a x \sqrt [4]{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**2/(b*x**2+a)**(5/4),x)

[Out]

-3*b*Integral((a + b*x**2)**(-5/4), x)/(2*a) - 1/(a*x*(a + b*x**2)**(1/4))

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Mathematica [C]  time = 0.0490377, size = 71, normalized size = 0.93 \[ \frac{3 b x^2 \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )-2 \left (a+3 b x^2\right )}{2 a^2 x \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^2*(a + b*x^2)^(5/4)),x]

[Out]

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

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(b*x^2+a)^(5/4),x)

[Out]

int(1/x^2/(b*x^2+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/4)*x^2),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(5/4)*x^2), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/4)*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.08966, size = 27, normalized size = 0.36 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{5}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac{5}{4}} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**2/(b*x**2+a)**(5/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(5/4)*x^2),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(5/4)*x^2), x)