3.1075 \(\int \frac{\left (a+b x^4\right )^{5/4}}{x^{12}} \, dx\)

Optimal. Leaf size=125 \[ \frac{10 b^{7/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{231 a^{3/2} \left (a+b x^4\right )^{3/4}}-\frac{5 b^2 \sqrt [4]{a+b x^4}}{231 a x^3}-\frac{\left (a+b x^4\right )^{5/4}}{11 x^{11}}-\frac{5 b \sqrt [4]{a+b x^4}}{77 x^7} \]

[Out]

(-5*b*(a + b*x^4)^(1/4))/(77*x^7) - (5*b^2*(a + b*x^4)^(1/4))/(231*a*x^3) - (a +
b*x^4)^(5/4)/(11*x^11) + (10*b^(7/2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot
[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(231*a^(3/2)*(a + b*x^4)^(3/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.159633, antiderivative size = 125, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{10 b^{7/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{231 a^{3/2} \left (a+b x^4\right )^{3/4}}-\frac{5 b^2 \sqrt [4]{a+b x^4}}{231 a x^3}-\frac{\left (a+b x^4\right )^{5/4}}{11 x^{11}}-\frac{5 b \sqrt [4]{a+b x^4}}{77 x^7} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(5/4)/x^12,x]

[Out]

(-5*b*(a + b*x^4)^(1/4))/(77*x^7) - (5*b^2*(a + b*x^4)^(1/4))/(231*a*x^3) - (a +
b*x^4)^(5/4)/(11*x^11) + (10*b^(7/2)*(1 + a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot
[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(231*a^(3/2)*(a + b*x^4)^(3/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 18.4706, size = 112, normalized size = 0.9 \[ - \frac{5 b \sqrt [4]{a + b x^{4}}}{77 x^{7}} - \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{11 x^{11}} - \frac{5 b^{2} \sqrt [4]{a + b x^{4}}}{231 a x^{3}} + \frac{10 b^{\frac{7}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{231 a^{\frac{3}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*b*(a + b*x**4)**(1/4)/(77*x**7) - (a + b*x**4)**(5/4)/(11*x**11) - 5*b**2*(a
+ b*x**4)**(1/4)/(231*a*x**3) + 10*b**(7/2)*x**3*(a/(b*x**4) + 1)**(3/4)*ellipti
c_f(atan(sqrt(a)/(sqrt(b)*x**2))/2, 2)/(231*a**(3/2)*(a + b*x**4)**(3/4))

_______________________________________________________________________________________

Mathematica [C]  time = 0.0576552, size = 94, normalized size = 0.75 \[ \frac{-21 a^3-57 a^2 b x^4-10 b^3 x^{12} \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )-41 a b^2 x^8-5 b^3 x^{12}}{231 a x^{11} \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-21*a^3 - 57*a^2*b*x^4 - 41*a*b^2*x^8 - 5*b^3*x^12 - 10*b^3*x^12*(1 + (b*x^4)/a
)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^4)/a)])/(231*a*x^11*(a + b*x^4)^
(3/4))

_______________________________________________________________________________________

Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{12}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(5/4)/x^12,x)

[Out]

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

_______________________________________________________________________________________

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

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. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{12}}, x\right ) \]

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^4 + a)^(5/4)/x^12, x)

_______________________________________________________________________________________

Sympy [A]  time = 22.0856, size = 31, normalized size = 0.25 \[ - \frac{b^{\frac{5}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**(5/4)*hyper((-5/4, 3/2), (5/2,), a*exp_polar(I*pi)/(b*x**4))/(6*x**6)

_______________________________________________________________________________________

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

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)