3.857 \(\int \frac{1}{\left (a+b x^2\right )^{7/4}} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0533191, 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 \[ \frac{2 x}{3 a \left (a+b x^2\right )^{3/4}}+\frac{2 \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} \sqrt{b} \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(-7/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.97665, size = 68, normalized size = 0.87 \[ \frac{2 x}{3 a \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{2 \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{3 \sqrt{a} \sqrt{b} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+a)**(7/4),x)

[Out]

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

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Mathematica [C]  time = 0.0478071, size = 55, normalized size = 0.71 \[ \frac{x \left (\left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right )+2\right )}{3 a \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(-7/4),x]

[Out]

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

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Maple [F]  time = 0.044, size = 0, normalized size = 0. \[ \int \left ( b{x}^{2}+a \right ) ^{-{\frac{7}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2+a)^(7/4),x)

[Out]

int(1/(b*x^2+a)^(7/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(-7/4),x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(-7/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(-7/4),x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(-7/4), x)

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Sympy [A]  time = 4.04126, size = 24, normalized size = 0.31 \[ \frac{x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**2+a)**(7/4),x)

[Out]

x*hyper((1/2, 7/4), (3/2,), b*x**2*exp_polar(I*pi)/a)/a**(7/4)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(-7/4),x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(-7/4), x)