3.862 \(\int \frac{1}{\left (a-b x^2\right )^{11/4}} \, dx\)

Optimal. Leaf size=101 \[ \frac{10 \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{21 a^{3/2} \sqrt{b} \left (a-b x^2\right )^{3/4}}+\frac{10 x}{21 a^2 \left (a-b x^2\right )^{3/4}}+\frac{2 x}{7 a \left (a-b x^2\right )^{7/4}} \]

[Out]

(2*x)/(7*a*(a - b*x^2)^(7/4)) + (10*x)/(21*a^2*(a - b*x^2)^(3/4)) + (10*(1 - (b*
x^2)/a)^(3/4)*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(21*a^(3/2)*Sqrt[b]*(
a - b*x^2)^(3/4))

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Rubi [A]  time = 0.0770378, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{10 \left (1-\frac{b x^2}{a}\right )^{3/4} F\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{21 a^{3/2} \sqrt{b} \left (a-b x^2\right )^{3/4}}+\frac{10 x}{21 a^2 \left (a-b x^2\right )^{3/4}}+\frac{2 x}{7 a \left (a-b x^2\right )^{7/4}} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^2)^(-11/4),x]

[Out]

(2*x)/(7*a*(a - b*x^2)^(7/4)) + (10*x)/(21*a^2*(a - b*x^2)^(3/4)) + (10*(1 - (b*
x^2)/a)^(3/4)*EllipticF[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(21*a^(3/2)*Sqrt[b]*(
a - b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 9.91093, size = 87, normalized size = 0.86 \[ \frac{2 x}{7 a \left (a - b x^{2}\right )^{\frac{7}{4}}} + \frac{10 x}{21 a^{2} \left (a - b x^{2}\right )^{\frac{3}{4}}} + \frac{10 \left (1 - \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{21 a^{\frac{3}{2}} \sqrt{b} \left (a - b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(-b*x**2+a)**(11/4),x)

[Out]

2*x/(7*a*(a - b*x**2)**(7/4)) + 10*x/(21*a**2*(a - b*x**2)**(3/4)) + 10*(1 - b*x
**2/a)**(3/4)*elliptic_f(asin(sqrt(b)*x/sqrt(a))/2, 2)/(21*a**(3/2)*sqrt(b)*(a -
b*x**2)**(3/4))

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Mathematica [C]  time = 0.0948414, size = 77, normalized size = 0.76 \[ \frac{5 x \left (a-b x^2\right ) \left (1-\frac{b x^2}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};\frac{b x^2}{a}\right )+2 x \left (8 a-5 b x^2\right )}{21 a^2 \left (a-b x^2\right )^{7/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^2)^(-11/4),x]

[Out]

(2*x*(8*a - 5*b*x^2) + 5*x*(a - b*x^2)*(1 - (b*x^2)/a)^(3/4)*Hypergeometric2F1[1
/2, 3/4, 3/2, (b*x^2)/a])/(21*a^2*(a - b*x^2)^(7/4))

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Maple [F]  time = 0.051, size = 0, normalized size = 0. \[ \int \left ( -b{x}^{2}+a \right ) ^{-{\frac{11}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(-b*x^2+a)^(11/4),x)

[Out]

int(1/(-b*x^2+a)^(11/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{11}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(-11/4),x, algorithm="maxima")

[Out]

integrate((-b*x^2 + a)^(-11/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b^{2} x^{4} - 2 \, a b x^{2} + a^{2}\right )}{\left (-b x^{2} + a\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(-11/4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 15.5896, size = 26, normalized size = 0.26 \[ \frac{x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{11}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{a^{\frac{11}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(-b*x**2+a)**(11/4),x)

[Out]

x*hyper((1/2, 11/4), (3/2,), b*x**2*exp_polar(2*I*pi)/a)/a**(11/4)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (-b x^{2} + a\right )}^{\frac{11}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^2 + a)^(-11/4),x, algorithm="giac")

[Out]

integrate((-b*x^2 + a)^(-11/4), x)