3.1074 \(\int \frac{\left (a+b x^4\right )^{5/4}}{x^8} \, dx\)

Optimal. Leaf size=101 \[ -\frac{5 b^{5/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 \sqrt{a} \left (a+b x^4\right )^{3/4}}-\frac{\left (a+b x^4\right )^{5/4}}{7 x^7}-\frac{5 b \sqrt [4]{a+b x^4}}{21 x^3} \]

[Out]

(-5*b*(a + b*x^4)^(1/4))/(21*x^3) - (a + b*x^4)^(5/4)/(7*x^7) - (5*b^(5/2)*(1 +
a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(21*Sqrt[a]*
(a + b*x^4)^(3/4))

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Rubi [A]  time = 0.123628, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{5 b^{5/2} x^3 \left (\frac{a}{b x^4}+1\right )^{3/4} F\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{21 \sqrt{a} \left (a+b x^4\right )^{3/4}}-\frac{\left (a+b x^4\right )^{5/4}}{7 x^7}-\frac{5 b \sqrt [4]{a+b x^4}}{21 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(5/4)/x^8,x]

[Out]

(-5*b*(a + b*x^4)^(1/4))/(21*x^3) - (a + b*x^4)^(5/4)/(7*x^7) - (5*b^(5/2)*(1 +
a/(b*x^4))^(3/4)*x^3*EllipticF[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(21*Sqrt[a]*
(a + b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 14.4431, size = 92, normalized size = 0.91 \[ - \frac{5 b \sqrt [4]{a + b x^{4}}}{21 x^{3}} - \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{7 x^{7}} - \frac{5 b^{\frac{5}{2}} x^{3} \left (\frac{a}{b x^{4}} + 1\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{a}}{\sqrt{b} x^{2}} \right )}}{2}\middle | 2\right )}{21 \sqrt{a} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5*b*(a + b*x**4)**(1/4)/(21*x**3) - (a + b*x**4)**(5/4)/(7*x**7) - 5*b**(5/2)*x
**3*(a/(b*x**4) + 1)**(3/4)*elliptic_f(atan(sqrt(a)/(sqrt(b)*x**2))/2, 2)/(21*sq
rt(a)*(a + b*x**4)**(3/4))

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Mathematica [C]  time = 0.0497967, size = 80, normalized size = 0.79 \[ \frac{-3 a^2+5 b^2 x^8 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{5}{4};-\frac{b x^4}{a}\right )-11 a b x^4-8 b^2 x^8}{21 x^7 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*a^2 - 11*a*b*x^4 - 8*b^2*x^8 + 5*b^2*x^8*(1 + (b*x^4)/a)^(3/4)*Hypergeometri
c2F1[1/4, 3/4, 5/4, -((b*x^4)/a)])/(21*x^7*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{8}} \left ( b{x}^{4}+a \right ) ^{{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(5/4)/x^8,x)

[Out]

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

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

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)

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

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^4 + a)^(5/4)/x^8, x)

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Sympy [A]  time = 10.9264, size = 31, normalized size = 0.31 \[ - \frac{b^{\frac{5}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b**(5/4)*hyper((-5/4, 1/2), (3/2,), a*exp_polar(I*pi)/(b*x**4))/(2*x**2)

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

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)