3.1167 \(\int \frac{1}{x^{14} \left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=153 \[ -\frac{112 b^{7/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{39 a^{9/2} \sqrt [4]{a+b x^4}}+\frac{56 b^3}{39 a^4 x \sqrt [4]{a+b x^4}}-\frac{28 b^2}{117 a^3 x^5 \sqrt [4]{a+b x^4}}+\frac{14 b}{117 a^2 x^9 \sqrt [4]{a+b x^4}}-\frac{1}{13 a x^{13} \sqrt [4]{a+b x^4}} \]

[Out]

-1/(13*a*x^13*(a + b*x^4)^(1/4)) + (14*b)/(117*a^2*x^9*(a + b*x^4)^(1/4)) - (28*
b^2)/(117*a^3*x^5*(a + b*x^4)^(1/4)) + (56*b^3)/(39*a^4*x*(a + b*x^4)^(1/4)) - (
112*b^(7/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2
])/(39*a^(9/2)*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.213358, antiderivative size = 153, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{112 b^{7/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{39 a^{9/2} \sqrt [4]{a+b x^4}}+\frac{56 b^3}{39 a^4 x \sqrt [4]{a+b x^4}}-\frac{28 b^2}{117 a^3 x^5 \sqrt [4]{a+b x^4}}+\frac{14 b}{117 a^2 x^9 \sqrt [4]{a+b x^4}}-\frac{1}{13 a x^{13} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^14*(a + b*x^4)^(5/4)),x]

[Out]

-1/(13*a*x^13*(a + b*x^4)^(1/4)) + (14*b)/(117*a^2*x^9*(a + b*x^4)^(1/4)) - (28*
b^2)/(117*a^3*x^5*(a + b*x^4)^(1/4)) + (56*b^3)/(39*a^4*x*(a + b*x^4)^(1/4)) - (
112*b^(7/2)*(1 + a/(b*x^4))^(1/4)*x*EllipticE[ArcCot[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2
])/(39*a^(9/2)*(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{1}{13 a x^{13} \sqrt [4]{a + b x^{4}}} + \frac{14 b}{117 a^{2} x^{9} \sqrt [4]{a + b x^{4}}} - \frac{28 b^{2}}{117 a^{3} x^{5} \sqrt [4]{a + b x^{4}}} - \frac{56 b^{3} x \sqrt [4]{\frac{a}{b x^{4}} + 1} \int ^{\frac{1}{x^{2}}} \frac{1}{\left (\frac{a x^{2}}{b} + 1\right )^{\frac{5}{4}}}\, dx}{39 a^{4} \sqrt [4]{a + b x^{4}}} + \frac{56 b^{3}}{39 a^{4} x \sqrt [4]{a + b x^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**14/(b*x**4+a)**(5/4),x)

[Out]

-1/(13*a*x**13*(a + b*x**4)**(1/4)) + 14*b/(117*a**2*x**9*(a + b*x**4)**(1/4)) -
28*b**2/(117*a**3*x**5*(a + b*x**4)**(1/4)) - 56*b**3*x*(a/(b*x**4) + 1)**(1/4)
*Integral((a*x**2/b + 1)**(-5/4), (x, x**(-2)))/(39*a**4*(a + b*x**4)**(1/4)) +
56*b**3/(39*a**4*x*(a + b*x**4)**(1/4))

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Mathematica [C]  time = 0.089081, size = 105, normalized size = 0.69 \[ \frac{-9 a^4+14 a^3 b x^4-28 a^2 b^2 x^8-224 b^4 x^{16} \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )+168 a b^3 x^{12}+336 b^4 x^{16}}{117 a^5 x^{13} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^14*(a + b*x^4)^(5/4)),x]

[Out]

(-9*a^4 + 14*a^3*b*x^4 - 28*a^2*b^2*x^8 + 168*a*b^3*x^12 + 336*b^4*x^16 - 224*b^
4*x^16*(1 + (b*x^4)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*x^4)/a)])/(11
7*a^5*x^13*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.096, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{14}} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^14/(b*x^4+a)^(5/4),x)

[Out]

int(1/x^14/(b*x^4+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{14}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^14),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^14), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{18} + a x^{14}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^14),x, algorithm="fricas")

[Out]

integral(1/((b*x^18 + a*x^14)*(b*x^4 + a)^(1/4)), x)

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Sympy [A]  time = 30.8014, size = 44, normalized size = 0.29 \[ \frac{\Gamma \left (- \frac{13}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{13}{4}, \frac{5}{4} \\ - \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{5}{4}} x^{13} \Gamma \left (- \frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**14/(b*x**4+a)**(5/4),x)

[Out]

gamma(-13/4)*hyper((-13/4, 5/4), (-9/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(5/4)*
x**13*gamma(-9/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{14}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^14),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^14), x)