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

Optimal. Leaf size=81 \[ \frac{2 x}{3 a \left (a-b x^2\right )^{3/4}}+\frac{2 \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} \sqrt{b} \left (a-b x^2\right )^{3/4}} \]

[Out]

(2*x)/(3*a*(a - b*x^2)^(3/4)) + (2*(1 - (b*x^2)/a)^(3/4)*EllipticF[ArcSin[(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 = 0.0587764, antiderivative size = 81, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 x}{3 a \left (a-b x^2\right )^{3/4}}+\frac{2 \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} \sqrt{b} \left (a-b x^2\right )^{3/4}} \]

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[ArcSin[(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 = 7.79778, size = 68, normalized size = 0.84 \[ \frac{2 x}{3 a \left (a - b x^{2}\right )^{\frac{3}{4}}} + \frac{2 \left (1 - \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{3 \sqrt{a} \sqrt{b} \left (a - b x^{2}\right )^{\frac{3}{4}}} \]

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(asin(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 = 0.0484678, size = 56, normalized size = 0.69 \[ \frac{x \left (\left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^2}{a}\right )+2\right )}{3 a \left (a-b x^2\right )^{3/4}} \]

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 = 0.052, 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. \[{\rm integral}\left (-\frac{1}{{\left (b x^{2} - a\right )}{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(-1/((b*x^2 - a)*(-b*x^2 + a)^(3/4)), x)

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Sympy [A]  time = 4.046, size = 26, normalized size = 0.32 \[ \frac{x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{a^{\frac{7}{4}}} \]

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(2*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)