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

Optimal. Leaf size=56 \[ \frac{2 \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{a+b x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0350618, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{\sqrt{a} \sqrt{b} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(-5/4),x]

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.035376, size = 55, normalized size = 0.98 \[ \frac{2 x-x \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )}{a \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(-5/4),x]

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)^(5/4),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(-5/4), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^2 + a)^(-5/4), x)

_______________________________________________________________________________________

Sympy [A]  time = 2.77653, size = 24, normalized size = 0.43 \[ \frac{x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{5}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x*hyper((1/2, 5/4), (3/2,), b*x**2*exp_polar(I*pi)/a)/a**(5/4)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(-5/4), x)