3.1255 \(\int \frac{x^8}{\left (a-b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=109 \[ -\frac{5 a^{3/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 b^{3/2} \left (a-b x^4\right )^{3/4}}-\frac{5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac{x^5 \sqrt [4]{a-b x^4}}{6 b} \]

[Out]

(-5*a*x*(a - b*x^4)^(1/4))/(12*b^2) - (x^5*(a - b*x^4)^(1/4))/(6*b) - (5*a^(3/2)
*(1 - a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(12*b^
(3/2)*(a - b*x^4)^(3/4))

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Rubi [A]  time = 0.139454, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{5 a^{3/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 b^{3/2} \left (a-b x^4\right )^{3/4}}-\frac{5 a x \sqrt [4]{a-b x^4}}{12 b^2}-\frac{x^5 \sqrt [4]{a-b x^4}}{6 b} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a - b*x^4)^(3/4),x]

[Out]

(-5*a*x*(a - b*x^4)^(1/4))/(12*b^2) - (x^5*(a - b*x^4)^(1/4))/(6*b) - (5*a^(3/2)
*(1 - a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(12*b^
(3/2)*(a - b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 19.2726, size = 95, normalized size = 0.87 \[ - \frac{5 a^{\frac{3}{2}} x^{3} \left (- \frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{12 b^{\frac{3}{2}} \left (a - b x^{4}\right )^{\frac{3}{4}}} - \frac{5 a x \sqrt [4]{a - b x^{4}}}{12 b^{2}} - \frac{x^{5} \sqrt [4]{a - b x^{4}}}{6 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(-b*x**4+a)**(3/4),x)

[Out]

-5*a**(3/2)*x**3*(-a/(b*x**4) + 1)**(3/4)*elliptic_f(asin(sqrt(a)/(sqrt(b)*x**2)
)/2, 2)/(12*b**(3/2)*(a - b*x**4)**(3/4)) - 5*a*x*(a - b*x**4)**(1/4)/(12*b**2)
- x**5*(a - b*x**4)**(1/4)/(6*b)

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Mathematica [C]  time = 0.0557986, size = 80, normalized size = 0.73 \[ \frac{5 a^2 x \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^4}{a}\right )-5 a^2 x+3 a b x^5+2 b^2 x^9}{12 b^2 \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a - b*x^4)^(3/4),x]

[Out]

(-5*a^2*x + 3*a*b*x^5 + 2*b^2*x^9 + 5*a^2*x*(1 - (b*x^4)/a)^(3/4)*Hypergeometric
2F1[1/4, 3/4, 5/4, (b*x^4)/a])/(12*b^2*(a - b*x^4)^(3/4))

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{{x}^{8} \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(-b*x^4+a)^(3/4),x)

[Out]

int(x^8/(-b*x^4+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(-b*x^4 + a)^(3/4),x, algorithm="maxima")

[Out]

integrate(x^8/(-b*x^4 + a)^(3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{8}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(-b*x^4 + a)^(3/4),x, algorithm="fricas")

[Out]

integral(x^8/(-b*x^4 + a)^(3/4), x)

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Sympy [A]  time = 3.70027, size = 39, normalized size = 0.36 \[ \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(-b*x**4+a)**(3/4),x)

[Out]

x**9*gamma(9/4)*hyper((3/4, 9/4), (13/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a**(3/
4)*gamma(13/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(-b*x^4 + a)^(3/4),x, algorithm="giac")

[Out]

integrate(x^8/(-b*x^4 + a)^(3/4), x)