3.1069 \(\int x^{12} \left (a+b x^4\right )^{5/4} \, dx\)

Optimal. Leaf size=171 \[ \frac{5 a^{9/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{672 b^{5/2} \left (a+b x^4\right )^{3/4}}+\frac{5 a^4 x \sqrt [4]{a+b x^4}}{672 b^3}-\frac{a^3 x^5 \sqrt [4]{a+b x^4}}{336 b^2}+\frac{a^2 x^9 \sqrt [4]{a+b x^4}}{504 b}+\frac{1}{18} x^{13} \left (a+b x^4\right )^{5/4}+\frac{5}{252} a x^{13} \sqrt [4]{a+b x^4} \]

[Out]

(5*a^4*x*(a + b*x^4)^(1/4))/(672*b^3) - (a^3*x^5*(a + b*x^4)^(1/4))/(336*b^2) +
(a^2*x^9*(a + b*x^4)^(1/4))/(504*b) + (5*a*x^13*(a + b*x^4)^(1/4))/252 + (x^13*(
a + b*x^4)^(5/4))/18 + (5*a^(9/2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sq
rt[b]*x^2)/Sqrt[a]]/2, 2])/(672*b^(5/2)*(a + b*x^4)^(3/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.244322, antiderivative size = 171, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{5 a^{9/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{672 b^{5/2} \left (a+b x^4\right )^{3/4}}+\frac{5 a^4 x \sqrt [4]{a+b x^4}}{672 b^3}-\frac{a^3 x^5 \sqrt [4]{a+b x^4}}{336 b^2}+\frac{a^2 x^9 \sqrt [4]{a+b x^4}}{504 b}+\frac{1}{18} x^{13} \left (a+b x^4\right )^{5/4}+\frac{5}{252} a x^{13} \sqrt [4]{a+b x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^12*(a + b*x^4)^(5/4),x]

[Out]

(5*a^4*x*(a + b*x^4)^(1/4))/(672*b^3) - (a^3*x^5*(a + b*x^4)^(1/4))/(336*b^2) +
(a^2*x^9*(a + b*x^4)^(1/4))/(504*b) + (5*a*x^13*(a + b*x^4)^(1/4))/252 + (x^13*(
a + b*x^4)^(5/4))/18 + (5*a^(9/2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sq
rt[b]*x^2)/Sqrt[a]]/2, 2])/(672*b^(5/2)*(a + b*x^4)^(3/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 27.9464, size = 155, normalized size = 0.91 \[ \frac{5 a^{\frac{9}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{672 b^{\frac{5}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} + \frac{5 a^{4} x \sqrt [4]{a + b x^{4}}}{672 b^{3}} - \frac{a^{3} x^{5} \sqrt [4]{a + b x^{4}}}{336 b^{2}} + \frac{a^{2} x^{9} \sqrt [4]{a + b x^{4}}}{504 b} + \frac{5 a x^{13} \sqrt [4]{a + b x^{4}}}{252} + \frac{x^{13} \left (a + b x^{4}\right )^{\frac{5}{4}}}{18} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**12*(b*x**4+a)**(5/4),x)

[Out]

5*a**(9/2)*x**3*(a/(b*x**4) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x**2))/
2, 2)/(672*b**(5/2)*(a + b*x**4)**(3/4)) + 5*a**4*x*(a + b*x**4)**(1/4)/(672*b**
3) - a**3*x**5*(a + b*x**4)**(1/4)/(336*b**2) + a**2*x**9*(a + b*x**4)**(1/4)/(5
04*b) + 5*a*x**13*(a + b*x**4)**(1/4)/252 + x**13*(a + b*x**4)**(5/4)/18

_______________________________________________________________________________________

Mathematica [C]  time = 0.0620271, size = 112, normalized size = 0.65 \[ \frac{-15 a^5 x \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )+15 a^5 x+9 a^4 b x^5-2 a^3 b^2 x^9+156 a^2 b^3 x^{13}+264 a b^4 x^{17}+112 b^5 x^{21}}{2016 b^3 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^12*(a + b*x^4)^(5/4),x]

[Out]

(15*a^5*x + 9*a^4*b*x^5 - 2*a^3*b^2*x^9 + 156*a^2*b^3*x^13 + 264*a*b^4*x^17 + 11
2*b^5*x^21 - 15*a^5*x*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((
b*x^4)/a)])/(2016*b^3*(a + b*x^4)^(3/4))

_______________________________________________________________________________________

Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int{x}^{12} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^12*(b*x^4+a)^(5/4),x)

[Out]

int(x^12*(b*x^4+a)^(5/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{12}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^12,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(5/4)*x^12, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{16} + a x^{12}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^12,x, algorithm="fricas")

[Out]

integral((b*x^16 + a*x^12)*(b*x^4 + a)^(1/4), x)

_______________________________________________________________________________________

Sympy [A]  time = 34.2498, size = 39, normalized size = 0.23 \[ \frac{a^{\frac{5}{4}} x^{13} \Gamma \left (\frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{13}{4} \\ \frac{17}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{17}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**12*(b*x**4+a)**(5/4),x)

[Out]

a**(5/4)*x**13*gamma(13/4)*hyper((-5/4, 13/4), (17/4,), b*x**4*exp_polar(I*pi)/a
)/(4*gamma(17/4))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{12}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^12,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(5/4)*x^12, x)