3.1254 \(\int \frac{x^{12}}{\left (a-b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=134 \[ -\frac{3 a^{5/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{8 b^{5/2} \left (a-b x^4\right )^{3/4}}-\frac{3 a^2 x \sqrt [4]{a-b x^4}}{8 b^3}-\frac{3 a x^5 \sqrt [4]{a-b x^4}}{20 b^2}-\frac{x^9 \sqrt [4]{a-b x^4}}{10 b} \]

[Out]

(-3*a^2*x*(a - b*x^4)^(1/4))/(8*b^3) - (3*a*x^5*(a - b*x^4)^(1/4))/(20*b^2) - (x
^9*(a - b*x^4)^(1/4))/(10*b) - (3*a^(5/2)*(1 - a/(b*x^4))^(3/4)*x^3*EllipticF[Ar
cCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(8*b^(5/2)*(a - b*x^4)^(3/4))

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Rubi [A]  time = 0.172759, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312 \[ -\frac{3 a^{5/2} x^3 \left (1-\frac{a}{b x^4}\right )^{3/4} F\left (\left .\frac{1}{2} \csc ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{8 b^{5/2} \left (a-b x^4\right )^{3/4}}-\frac{3 a^2 x \sqrt [4]{a-b x^4}}{8 b^3}-\frac{3 a x^5 \sqrt [4]{a-b x^4}}{20 b^2}-\frac{x^9 \sqrt [4]{a-b x^4}}{10 b} \]

Antiderivative was successfully verified.

[In]  Int[x^12/(a - b*x^4)^(3/4),x]

[Out]

(-3*a^2*x*(a - b*x^4)^(1/4))/(8*b^3) - (3*a*x^5*(a - b*x^4)^(1/4))/(20*b^2) - (x
^9*(a - b*x^4)^(1/4))/(10*b) - (3*a^(5/2)*(1 - a/(b*x^4))^(3/4)*x^3*EllipticF[Ar
cCsc[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(8*b^(5/2)*(a - b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 23.5756, size = 119, normalized size = 0.89 \[ - \frac{3 a^{\frac{5}{2}} x^{3} \left (- \frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{8 b^{\frac{5}{2}} \left (a - b x^{4}\right )^{\frac{3}{4}}} - \frac{3 a^{2} x \sqrt [4]{a - b x^{4}}}{8 b^{3}} - \frac{3 a x^{5} \sqrt [4]{a - b x^{4}}}{20 b^{2}} - \frac{x^{9} \sqrt [4]{a - b x^{4}}}{10 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**12/(-b*x**4+a)**(3/4),x)

[Out]

-3*a**(5/2)*x**3*(-a/(b*x**4) + 1)**(3/4)*elliptic_f(asin(sqrt(a)/(sqrt(b)*x**2)
)/2, 2)/(8*b**(5/2)*(a - b*x**4)**(3/4)) - 3*a**2*x*(a - b*x**4)**(1/4)/(8*b**3)
- 3*a*x**5*(a - b*x**4)**(1/4)/(20*b**2) - x**9*(a - b*x**4)**(1/4)/(10*b)

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Mathematica [C]  time = 0.0633525, size = 91, normalized size = 0.68 \[ \frac{15 a^3 x \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};\frac{b x^4}{a}\right )-15 a^3 x+9 a^2 b x^5+2 a b^2 x^9+4 b^3 x^{13}}{40 b^3 \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^12/(a - b*x^4)^(3/4),x]

[Out]

(-15*a^3*x + 9*a^2*b*x^5 + 2*a*b^2*x^9 + 4*b^3*x^13 + 15*a^3*x*(1 - (b*x^4)/a)^(
3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, (b*x^4)/a])/(40*b^3*(a - b*x^4)^(3/4))

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \[ \int{{x}^{12} \left ( -b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^12/(-b*x^4+a)^(3/4),x)

[Out]

int(x^12/(-b*x^4+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{12}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^12/(-b*x^4 + a)^(3/4),x, algorithm="maxima")

[Out]

integrate(x^12/(-b*x^4 + a)^(3/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{12}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^12/(-b*x^4 + a)^(3/4),x, algorithm="fricas")

[Out]

integral(x^12/(-b*x^4 + a)^(3/4), x)

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Sympy [A]  time = 7.22891, size = 39, normalized size = 0.29 \[ \frac{x^{13} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{17}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**12/(-b*x**4+a)**(3/4),x)

[Out]

x**13*gamma(13/4)*hyper((3/4, 13/4), (17/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a**
(3/4)*gamma(17/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{12}}{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^12/(-b*x^4 + a)^(3/4),x, algorithm="giac")

[Out]

integrate(x^12/(-b*x^4 + a)^(3/4), x)