3.1131 \(\int \frac{x^8}{\left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=105 \[ -\frac{5 a^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 b^{3/2} \left (a+b x^4\right )^{3/4}}-\frac{5 a x \sqrt [4]{a+b x^4}}{12 b^2}+\frac{x^5 \sqrt [4]{a+b x^4}}{6 b} \]

[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[ArcCot[(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 = 0.133324, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{5 a^{3/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{12 b^{3/2} \left (a+b x^4\right )^{3/4}}-\frac{5 a x \sqrt [4]{a+b x^4}}{12 b^2}+\frac{x^5 \sqrt [4]{a+b x^4}}{6 b} \]

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[ArcCot[(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 = 15.15, size = 94, normalized size = 0.9 \[ - \frac{5 a^{\frac{3}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{12 b^{\frac{3}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} - \frac{5 a x \sqrt [4]{a + b x^{4}}}{12 b^{2}} + \frac{x^{5} \sqrt [4]{a + b x^{4}}}{6 b} \]

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(atan(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 = 0.0498434, size = 79, normalized size = 0.75 \[ \frac{5 a^2 x \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )-5 a^2 x-3 a b x^5+2 b^2 x^9}{12 b^2 \left (a+b x^4\right )^{3/4}} \]

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 = 0.037, 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 = 3.55493, size = 37, normalized size = 0.35 \[ \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{13}{4}\right )} \]

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(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)