3.839 \(\int \frac{x^8}{\sqrt{a-b x^4}} \, dx\)

Optimal. Leaf size=100 \[ \frac{5 a^{9/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 b^{9/4} \sqrt{a-b x^4}}-\frac{5 a x \sqrt{a-b x^4}}{21 b^2}-\frac{x^5 \sqrt{a-b x^4}}{7 b} \]

[Out]

(-5*a*x*Sqrt[a - b*x^4])/(21*b^2) - (x^5*Sqrt[a - b*x^4])/(7*b) + (5*a^(9/4)*Sqr
t[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(21*b^(9/4)*Sqrt[a
- b*x^4])

_______________________________________________________________________________________

Rubi [A]  time = 0.100349, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188 \[ \frac{5 a^{9/4} \sqrt{1-\frac{b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{21 b^{9/4} \sqrt{a-b x^4}}-\frac{5 a x \sqrt{a-b x^4}}{21 b^2}-\frac{x^5 \sqrt{a-b x^4}}{7 b} \]

Antiderivative was successfully verified.

[In]  Int[x^8/Sqrt[a - b*x^4],x]

[Out]

(-5*a*x*Sqrt[a - b*x^4])/(21*b^2) - (x^5*Sqrt[a - b*x^4])/(7*b) + (5*a^(9/4)*Sqr
t[1 - (b*x^4)/a]*EllipticF[ArcSin[(b^(1/4)*x)/a^(1/4)], -1])/(21*b^(9/4)*Sqrt[a
- b*x^4])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.8651, size = 88, normalized size = 0.88 \[ \frac{5 a^{\frac{9}{4}} \sqrt{1 - \frac{b x^{4}}{a}} F\left (\operatorname{asin}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | -1\right )}{21 b^{\frac{9}{4}} \sqrt{a - b x^{4}}} - \frac{5 a x \sqrt{a - b x^{4}}}{21 b^{2}} - \frac{x^{5} \sqrt{a - b x^{4}}}{7 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(-b*x**4+a)**(1/2),x)

[Out]

5*a**(9/4)*sqrt(1 - b*x**4/a)*elliptic_f(asin(b**(1/4)*x/a**(1/4)), -1)/(21*b**(
9/4)*sqrt(a - b*x**4)) - 5*a*x*sqrt(a - b*x**4)/(21*b**2) - x**5*sqrt(a - b*x**4
)/(7*b)

_______________________________________________________________________________________

Mathematica [C]  time = 0.160883, size = 122, normalized size = 1.22 \[ \frac{x \sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} \left (-5 a^2+2 a b x^4+3 b^2 x^8\right )-5 i a^2 \sqrt{1-\frac{b x^4}{a}} F\left (\left .i \sinh ^{-1}\left (\sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )}{21 b^2 \sqrt{-\frac{\sqrt{b}}{\sqrt{a}}} \sqrt{a-b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/Sqrt[a - b*x^4],x]

[Out]

(Sqrt[-(Sqrt[b]/Sqrt[a])]*x*(-5*a^2 + 2*a*b*x^4 + 3*b^2*x^8) - (5*I)*a^2*Sqrt[1
- (b*x^4)/a]*EllipticF[I*ArcSinh[Sqrt[-(Sqrt[b]/Sqrt[a])]*x], -1])/(21*Sqrt[-(Sq
rt[b]/Sqrt[a])]*b^2*Sqrt[a - b*x^4])

_______________________________________________________________________________________

Maple [A]  time = 0.027, size = 107, normalized size = 1.1 \[ -{\frac{{x}^{5}}{7\,b}\sqrt{-b{x}^{4}+a}}-{\frac{5\,ax}{21\,{b}^{2}}\sqrt{-b{x}^{4}+a}}+{\frac{5\,{a}^{2}}{21\,{b}^{2}}\sqrt{1-{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}{\it EllipticF} \left ( x\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ){\frac{1}{\sqrt{{1\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{-b{x}^{4}+a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(-b*x^4+a)^(1/2),x)

[Out]

-1/7*x^5*(-b*x^4+a)^(1/2)/b-5/21*a*x*(-b*x^4+a)^(1/2)/b^2+5/21*a^2/b^2/(1/a^(1/2
)*b^(1/2))^(1/2)*(1-b^(1/2)*x^2/a^(1/2))^(1/2)*(1+b^(1/2)*x^2/a^(1/2))^(1/2)/(-b
*x^4+a)^(1/2)*EllipticF(x*(1/a^(1/2)*b^(1/2))^(1/2),I)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\sqrt{-b x^{4} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(-b*x^4 + a),x, algorithm="maxima")

[Out]

integrate(x^8/sqrt(-b*x^4 + a), x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{8}}{\sqrt{-b x^{4} + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(-b*x^4 + a),x, algorithm="fricas")

[Out]

integral(x^8/sqrt(-b*x^4 + a), x)

_______________________________________________________________________________________

Sympy [A]  time = 3.27009, size = 39, normalized size = 0.39 \[ \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt{a} \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(-b*x**4+a)**(1/2),x)

[Out]

x**9*gamma(9/4)*hyper((1/2, 9/4), (13/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*sqrt(a
)*gamma(13/4))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{\sqrt{-b x^{4} + a}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/sqrt(-b*x^4 + a),x, algorithm="giac")

[Out]

integrate(x^8/sqrt(-b*x^4 + a), x)