3.1166 \(\int \frac{1}{x^{10} \left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=129 \[ \frac{8 b^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 a^{7/2} \sqrt [4]{a+b x^4}}-\frac{4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}+\frac{2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac{1}{9 a x^9 \sqrt [4]{a+b x^4}} \]

[Out]

-1/(9*a*x^9*(a + b*x^4)^(1/4)) + (2*b)/(9*a^2*x^5*(a + b*x^4)^(1/4)) - (4*b^2)/(
3*a^3*x*(a + b*x^4)^(1/4)) + (8*b^(5/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcCot
[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(3*a^(7/2)*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.178053, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{8 b^{5/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{3 a^{7/2} \sqrt [4]{a+b x^4}}-\frac{4 b^2}{3 a^3 x \sqrt [4]{a+b x^4}}+\frac{2 b}{9 a^2 x^5 \sqrt [4]{a+b x^4}}-\frac{1}{9 a x^9 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^10*(a + b*x^4)^(5/4)),x]

[Out]

-1/(9*a*x^9*(a + b*x^4)^(1/4)) + (2*b)/(9*a^2*x^5*(a + b*x^4)^(1/4)) - (4*b^2)/(
3*a^3*x*(a + b*x^4)^(1/4)) + (8*b^(5/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcCot
[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(3*a^(7/2)*(a + b*x^4)^(1/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**10/(b*x**4+a)**(5/4),x)

[Out]

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

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Mathematica [C]  time = 0.0756725, size = 94, normalized size = 0.73 \[ \frac{-a^3+2 a^2 b x^4+16 b^3 x^{12} \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-12 a b^2 x^8-24 b^3 x^{12}}{9 a^4 x^9 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^10*(a + b*x^4)^(5/4)),x]

[Out]

(-a^3 + 2*a^2*b*x^4 - 12*a*b^2*x^8 - 24*b^3*x^12 + 16*b^3*x^12*(1 + (b*x^4)/a)^(
1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^4)/a)])/(9*a^4*x^9*(a + b*x^4)^(1/4
))

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Maple [F]  time = 0.087, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{10}} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^10/(b*x^4+a)^(5/4),x)

[Out]

int(1/x^10/(b*x^4+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^10),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^10), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{14} + a x^{10}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^10),x, algorithm="fricas")

[Out]

integral(1/((b*x^14 + a*x^10)*(b*x^4 + a)^(1/4)), x)

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Sympy [A]  time = 14.8666, size = 44, normalized size = 0.34 \[ \frac{\Gamma \left (- \frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{9}{4}, \frac{5}{4} \\ - \frac{5}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{5}{4}} x^{9} \Gamma \left (- \frac{5}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**10/(b*x**4+a)**(5/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{10}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^10),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^10), x)