3.1070 \(\int x^8 \left (a+b x^4\right )^{5/4} \, dx\)

Optimal. Leaf size=147 \[ -\frac{5 a^{7/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{336 b^{3/2} \left (a+b x^4\right )^{3/4}}-\frac{5 a^3 x \sqrt [4]{a+b x^4}}{336 b^2}+\frac{a^2 x^5 \sqrt [4]{a+b x^4}}{168 b}+\frac{1}{14} x^9 \left (a+b x^4\right )^{5/4}+\frac{1}{28} a x^9 \sqrt [4]{a+b x^4} \]

[Out]

(-5*a^3*x*(a + b*x^4)^(1/4))/(336*b^2) + (a^2*x^5*(a + b*x^4)^(1/4))/(168*b) + (
a*x^9*(a + b*x^4)^(1/4))/28 + (x^9*(a + b*x^4)^(5/4))/14 - (5*a^(7/2)*(1 + a/(b*
x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(336*b^(3/2)*(a +
b*x^4)^(3/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.199449, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{5 a^{7/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{336 b^{3/2} \left (a+b x^4\right )^{3/4}}-\frac{5 a^3 x \sqrt [4]{a+b x^4}}{336 b^2}+\frac{a^2 x^5 \sqrt [4]{a+b x^4}}{168 b}+\frac{1}{14} x^9 \left (a+b x^4\right )^{5/4}+\frac{1}{28} a x^9 \sqrt [4]{a+b x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^8*(a + b*x^4)^(5/4),x]

[Out]

(-5*a^3*x*(a + b*x^4)^(1/4))/(336*b^2) + (a^2*x^5*(a + b*x^4)^(1/4))/(168*b) + (
a*x^9*(a + b*x^4)^(1/4))/28 + (x^9*(a + b*x^4)^(5/4))/14 - (5*a^(7/2)*(1 + a/(b*
x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(336*b^(3/2)*(a +
b*x^4)^(3/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 22.7772, size = 131, normalized size = 0.89 \[ - \frac{5 a^{\frac{7}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{336 b^{\frac{3}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} - \frac{5 a^{3} x \sqrt [4]{a + b x^{4}}}{336 b^{2}} + \frac{a^{2} x^{5} \sqrt [4]{a + b x^{4}}}{168 b} + \frac{a x^{9} \sqrt [4]{a + b x^{4}}}{28} + \frac{x^{9} \left (a + b x^{4}\right )^{\frac{5}{4}}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8*(b*x**4+a)**(5/4),x)

[Out]

-5*a**(7/2)*x**3*(a/(b*x**4) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x**2))
/2, 2)/(336*b**(3/2)*(a + b*x**4)**(3/4)) - 5*a**3*x*(a + b*x**4)**(1/4)/(336*b*
*2) + a**2*x**5*(a + b*x**4)**(1/4)/(168*b) + a*x**9*(a + b*x**4)**(1/4)/28 + x*
*9*(a + b*x**4)**(5/4)/14

_______________________________________________________________________________________

Mathematica [C]  time = 0.0554723, size = 101, normalized size = 0.69 \[ \frac{5 a^4 x \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )-5 a^4 x-3 a^3 b x^5+38 a^2 b^2 x^9+60 a b^3 x^{13}+24 b^4 x^{17}}{336 b^2 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8*(a + b*x^4)^(5/4),x]

[Out]

(-5*a^4*x - 3*a^3*b*x^5 + 38*a^2*b^2*x^9 + 60*a*b^3*x^13 + 24*b^4*x^17 + 5*a^4*x
*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^4)/a)])/(336*b^2*
(a + b*x^4)^(3/4))

_______________________________________________________________________________________

Maple [F]  time = 0.039, size = 0, normalized size = 0. \[ \int{x}^{8} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8*(b*x^4+a)^(5/4),x)

[Out]

int(x^8*(b*x^4+a)^(5/4),x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{8}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^8,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(5/4)*x^8, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{12} + a x^{8}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^8,x, algorithm="fricas")

[Out]

integral((b*x^12 + a*x^8)*(b*x^4 + a)^(1/4), x)

_______________________________________________________________________________________

Sympy [A]  time = 20.5637, size = 39, normalized size = 0.27 \[ \frac{a^{\frac{5}{4}} x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8*(b*x**4+a)**(5/4),x)

[Out]

a**(5/4)*x**9*gamma(9/4)*hyper((-5/4, 9/4), (13/4,), b*x**4*exp_polar(I*pi)/a)/(
4*gamma(13/4))

_______________________________________________________________________________________

GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{8}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)*x^8,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(5/4)*x^8, x)