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

Optimal. Leaf size=74 \[ \frac{2 x}{b \sqrt [4]{a+b x^2}}-\frac{4 \sqrt{a} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{b^{3/2} \sqrt [4]{a+b x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0671062, antiderivative size = 74, 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 \[ \frac{2 x}{b \sqrt [4]{a+b x^2}}-\frac{4 \sqrt{a} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{b^{3/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x^2)^(5/4),x]

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [C]  time = 0.0481904, size = 53, normalized size = 0.72 \[ \frac{2 x \left (\sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )-1\right )}{b \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{{x}^{2} \left ( b{x}^{2}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^2+a)^(5/4),x)

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(b*x^2 + a)^(5/4), x)

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^2/(b*x^2 + a)^(5/4), x)

_______________________________________________________________________________________

Sympy [A]  time = 2.82698, size = 27, normalized size = 0.36 \[ \frac{x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3 a^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^2/(b*x^2 + a)^(5/4), x)