3.1076 \(\int \frac{\left (a+b x^4\right )^{5/4}}{x^{16}} \, dx\)

Optimal. Leaf size=149 \[ -\frac{4 b^{9/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{231 a^{5/2} \left (a+b x^4\right )^{3/4}}+\frac{2 b^3 \sqrt [4]{a+b x^4}}{231 a^2 x^3}-\frac{b^2 \sqrt [4]{a+b x^4}}{231 a x^7}-\frac{\left (a+b x^4\right )^{5/4}}{15 x^{15}}-\frac{b \sqrt [4]{a+b x^4}}{33 x^{11}} \]

[Out]

-(b*(a + b*x^4)^(1/4))/(33*x^11) - (b^2*(a + b*x^4)^(1/4))/(231*a*x^7) + (2*b^3*
(a + b*x^4)^(1/4))/(231*a^2*x^3) - (a + b*x^4)^(5/4)/(15*x^15) - (4*b^(9/2)*(1 +
a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(231*a^(5/2
)*(a + b*x^4)^(3/4))

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Rubi [A]  time = 0.195359, antiderivative size = 149, 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 \[ -\frac{4 b^{9/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{231 a^{5/2} \left (a+b x^4\right )^{3/4}}+\frac{2 b^3 \sqrt [4]{a+b x^4}}{231 a^2 x^3}-\frac{b^2 \sqrt [4]{a+b x^4}}{231 a x^7}-\frac{\left (a+b x^4\right )^{5/4}}{15 x^{15}}-\frac{b \sqrt [4]{a+b x^4}}{33 x^{11}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(5/4)/x^16,x]

[Out]

-(b*(a + b*x^4)^(1/4))/(33*x^11) - (b^2*(a + b*x^4)^(1/4))/(231*a*x^7) + (2*b^3*
(a + b*x^4)^(1/4))/(231*a^2*x^3) - (a + b*x^4)^(5/4)/(15*x^15) - (4*b^(9/2)*(1 +
a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(231*a^(5/2
)*(a + b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 22.8774, size = 133, normalized size = 0.89 \[ - \frac{b \sqrt [4]{a + b x^{4}}}{33 x^{11}} - \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{15 x^{15}} - \frac{b^{2} \sqrt [4]{a + b x^{4}}}{231 a x^{7}} + \frac{2 b^{3} \sqrt [4]{a + b x^{4}}}{231 a^{2} x^{3}} - \frac{4 b^{\frac{9}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{231 a^{\frac{5}{2}} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(5/4)/x**16,x)

[Out]

-b*(a + b*x**4)**(1/4)/(33*x**11) - (a + b*x**4)**(5/4)/(15*x**15) - b**2*(a + b
*x**4)**(1/4)/(231*a*x**7) + 2*b**3*(a + b*x**4)**(1/4)/(231*a**2*x**3) - 4*b**(
9/2)*x**3*(a/(b*x**4) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x**2))/2, 2)/
(231*a**(5/2)*(a + b*x**4)**(3/4))

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Mathematica [C]  time = 0.0634472, size = 105, normalized size = 0.7 \[ \frac{-77 a^4-189 a^3 b x^4-117 a^2 b^2 x^8+20 b^4 x^{16} \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )+5 a b^3 x^{12}+10 b^4 x^{16}}{1155 a^2 x^{15} \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(5/4)/x^16,x]

[Out]

(-77*a^4 - 189*a^3*b*x^4 - 117*a^2*b^2*x^8 + 5*a*b^3*x^12 + 10*b^4*x^16 + 20*b^4
*x^16*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/4, 3/4, 5/4, -((b*x^4)/a)])/(115
5*a^2*x^15*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{16}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(5/4)/x^16,x)

[Out]

int((b*x^4+a)^(5/4)/x^16,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{16}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^16,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(5/4)/x^16, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{16}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^16,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(5/4)/x^16, x)

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Sympy [A]  time = 36.7347, size = 46, normalized size = 0.31 \[ \frac{a^{\frac{5}{4}} \Gamma \left (- \frac{15}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{15}{4}, - \frac{5}{4} \\ - \frac{11}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{15} \Gamma \left (- \frac{11}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(5/4)/x**16,x)

[Out]

a**(5/4)*gamma(-15/4)*hyper((-15/4, -5/4), (-11/4,), b*x**4*exp_polar(I*pi)/a)/(
4*x**15*gamma(-11/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{16}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(5/4)/x^16,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(5/4)/x^16, x)