3.1043 \(\int \frac{\left (a+b x^4\right )^{3/4}}{x^{10}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{2 b^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{15 a^{3/2} \sqrt [4]{a+b x^4}}+\frac{2 b^2}{15 a x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{9 x^9}-\frac{b \left (a+b x^4\right )^{3/4}}{15 a x^5} \]

[Out]

(2*b^2)/(15*a*x*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(9*x^9) - (b*(a + b*x^4)^
(3/4))/(15*a*x^5) - (2*b^(5/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcCot[(Sqrt[b]
*x^2)/Sqrt[a]]/2, 2])/(15*a^(3/2)*(a + b*x^4)^(1/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.174998, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.467 \[ -\frac{2 b^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{15 a^{3/2} \sqrt [4]{a+b x^4}}+\frac{2 b^2}{15 a x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{9 x^9}-\frac{b \left (a+b x^4\right )^{3/4}}{15 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(3/4)/x^10,x]

[Out]

(2*b^2)/(15*a*x*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(9*x^9) - (b*(a + b*x^4)^
(3/4))/(15*a*x^5) - (2*b^(5/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcCot[(Sqrt[b]
*x^2)/Sqrt[a]]/2, 2])/(15*a^(3/2)*(a + b*x^4)^(1/4))

_______________________________________________________________________________________

Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{9 x^{9}} + \frac{b^{2} x \sqrt [4]{\frac{a}{b x^{4}} + 1} \int ^{\frac{1}{x^{2}}} \frac{1}{\sqrt [4]{\frac{a x^{2}}{b} + 1}}\, dx}{15 a \sqrt [4]{a + b x^{4}}} - \frac{b \left (a + b x^{4}\right )^{\frac{3}{4}}}{15 a x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(3/4)/x**10,x)

[Out]

-(a + b*x**4)**(3/4)/(9*x**9) + b**2*x*(a/(b*x**4) + 1)**(1/4)*Integral((a*x**2/
b + 1)**(-1/4), (x, x**(-2)))/(15*a*(a + b*x**4)**(1/4)) - b*(a + b*x**4)**(3/4)
/(15*a*x**5)

_______________________________________________________________________________________

Mathematica [C]  time = 0.0546576, size = 94, normalized size = 0.75 \[ \frac{-5 a^3-8 a^2 b x^4-4 b^3 x^{12} \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+3 a b^2 x^8+6 b^3 x^{12}}{45 a^2 x^9 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(3/4)/x^10,x]

[Out]

(-5*a^3 - 8*a^2*b*x^4 + 3*a*b^2*x^8 + 6*b^3*x^12 - 4*b^3*x^12*(1 + (b*x^4)/a)^(1
/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^4)/a)])/(45*a^2*x^9*(a + b*x^4)^(1/4
))

_______________________________________________________________________________________

Maple [F]  time = 0.053, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{10}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(3/4)/x^10,x)

[Out]

int((b*x^4+a)^(3/4)/x^10,x)

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^10,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/4)/x^10, x)

_______________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{10}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^10,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(3/4)/x^10, x)

_______________________________________________________________________________________

Sympy [A]  time = 10.5797, size = 31, normalized size = 0.25 \[ - \frac{b^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{6 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(3/4)/x**10,x)

[Out]

-b**(3/4)*hyper((-3/4, 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. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^10,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/x^10, x)