3.858 \(\int \frac{1}{\left (a+b x^2\right )^{9/4}} \, dx\)

Optimal. Leaf size=78 \[ \frac{6 \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt{b} \sqrt [4]{a+b x^2}}+\frac{2 x}{5 a \left (a+b x^2\right )^{5/4}} \]

[Out]

(2*x)/(5*a*(a + b*x^2)^(5/4)) + (6*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[
b]*x)/Sqrt[a]]/2, 2])/(5*a^(3/2)*Sqrt[b]*(a + b*x^2)^(1/4))

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Rubi [A]  time = 0.0557094, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{6 \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 a^{3/2} \sqrt{b} \sqrt [4]{a+b x^2}}+\frac{2 x}{5 a \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(-9/4),x]

[Out]

(2*x)/(5*a*(a + b*x^2)^(5/4)) + (6*(1 + (b*x^2)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[
b]*x)/Sqrt[a]]/2, 2])/(5*a^(3/2)*Sqrt[b]*(a + b*x^2)^(1/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)**(9/4),x)

[Out]

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

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Mathematica [C]  time = 0.089354, size = 72, normalized size = 0.92 \[ \frac{-3 x \left (a+b x^2\right ) \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )+8 a x+6 b x^3}{5 a^2 \left (a+b x^2\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(-9/4),x]

[Out]

(8*a*x + 6*b*x^3 - 3*x*(a + b*x^2)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4,
1/2, 3/2, -((b*x^2)/a)])/(5*a^2*(a + b*x^2)^(5/4))

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \[ \int \left ( b{x}^{2}+a \right ) ^{-{\frac{9}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)^(9/4),x)

[Out]

int(1/(b*x^2+a)^(9/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(-9/4),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(-9/4), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(-9/4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.24556, size = 24, normalized size = 0.31 \[ \frac{x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{9}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**2+a)**(9/4),x)

[Out]

x*hyper((1/2, 9/4), (3/2,), b*x**2*exp_polar(I*pi)/a)/a**(9/4)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{9}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(-9/4),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(-9/4), x)