3.1130 \(\int \frac{x^{12}}{\left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=129 \[ \frac{3 a^{5/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{8 b^{5/2} \left (a+b x^4\right )^{3/4}}+\frac{3 a^2 x \sqrt [4]{a+b x^4}}{8 b^3}-\frac{3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac{x^9 \sqrt [4]{a+b x^4}}{10 b} \]

[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[Arc
Cot[(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 = 0.163599, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{3 a^{5/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{8 b^{5/2} \left (a+b x^4\right )^{3/4}}+\frac{3 a^2 x \sqrt [4]{a+b x^4}}{8 b^3}-\frac{3 a x^5 \sqrt [4]{a+b x^4}}{20 b^2}+\frac{x^9 \sqrt [4]{a+b x^4}}{10 b} \]

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[Arc
Cot[(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 = 19.0538, size = 117, normalized size = 0.91 \[ \frac{3 a^{\frac{5}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{8 b^{\frac{5}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{3 a^{2} x \sqrt [4]{a + b x^{4}}}{8 b^{3}} - \frac{3 a x^{5} \sqrt [4]{a + b x^{4}}}{20 b^{2}} + \frac{x^{9} \sqrt [4]{a + b x^{4}}}{10 b} \]

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(atan(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 = 0.0558742, size = 90, normalized size = 0.7 \[ \frac{-15 a^3 x \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )+15 a^3 x+9 a^2 b x^5-2 a b^2 x^9+4 b^3 x^{13}}{40 b^3 \left (a+b x^4\right )^{3/4}} \]

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 = 0.037, 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 = 6.98559, size = 37, normalized size = 0.29 \[ \frac{x^{13} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{17}{4}\right )} \]

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(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)