3.1092 \(\int \frac{1}{x^{11} \sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=152 \[ -\frac{7 b^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 a^{5/2} \sqrt [4]{a+b x^4}}+\frac{7 b^3 x^2}{40 a^3 \sqrt [4]{a+b x^4}}-\frac{7 b^2 \left (a+b x^4\right )^{3/4}}{40 a^3 x^2}+\frac{7 b \left (a+b x^4\right )^{3/4}}{60 a^2 x^6}-\frac{\left (a+b x^4\right )^{3/4}}{10 a x^{10}} \]

[Out]

(7*b^3*x^2)/(40*a^3*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(10*a*x^10) + (7*b*(a
+ b*x^4)^(3/4))/(60*a^2*x^6) - (7*b^2*(a + b*x^4)^(3/4))/(40*a^3*x^2) - (7*b^(5
/2)*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(40*a^(
5/2)*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.21462, antiderivative size = 152, 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 \[ -\frac{7 b^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 a^{5/2} \sqrt [4]{a+b x^4}}+\frac{7 b^3 x^2}{40 a^3 \sqrt [4]{a+b x^4}}-\frac{7 b^2 \left (a+b x^4\right )^{3/4}}{40 a^3 x^2}+\frac{7 b \left (a+b x^4\right )^{3/4}}{60 a^2 x^6}-\frac{\left (a+b x^4\right )^{3/4}}{10 a x^{10}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^11*(a + b*x^4)^(1/4)),x]

[Out]

(7*b^3*x^2)/(40*a^3*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(10*a*x^10) + (7*b*(a
+ b*x^4)^(3/4))/(60*a^2*x^6) - (7*b^2*(a + b*x^4)^(3/4))/(40*a^3*x^2) - (7*b^(5
/2)*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(40*a^(
5/2)*(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{10 a x^{10}} - \frac{7 b^{3} \int ^{x^{2}} \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{80 a^{2}} + \frac{7 b \left (a + b x^{4}\right )^{\frac{3}{4}}}{60 a^{2} x^{6}} + \frac{7 b^{3} x^{2}}{40 a^{3} \sqrt [4]{a + b x^{4}}} - \frac{7 b^{2} \left (a + b x^{4}\right )^{\frac{3}{4}}}{40 a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**11/(b*x**4+a)**(1/4),x)

[Out]

-(a + b*x**4)**(3/4)/(10*a*x**10) - 7*b**3*Integral((a + b*x**2)**(-5/4), (x, x*
*2))/(80*a**2) + 7*b*(a + b*x**4)**(3/4)/(60*a**2*x**6) + 7*b**3*x**2/(40*a**3*(
a + b*x**4)**(1/4)) - 7*b**2*(a + b*x**4)**(3/4)/(40*a**3*x**2)

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Mathematica [C]  time = 0.0664678, size = 94, normalized size = 0.62 \[ \frac{-24 a^3+4 a^2 b x^4+21 b^3 x^{12} \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )-14 a b^2 x^8-42 b^3 x^{12}}{240 a^3 x^{10} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^11*(a + b*x^4)^(1/4)),x]

[Out]

(-24*a^3 + 4*a^2*b*x^4 - 14*a*b^2*x^8 - 42*b^3*x^12 + 21*b^3*x^12*(1 + (b*x^4)/a
)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^4)/a)])/(240*a^3*x^10*(a + b*x^4
)^(1/4))

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Maple [F]  time = 0.048, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{11}}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^11/(b*x^4+a)^(1/4),x)

[Out]

int(1/x^11/(b*x^4+a)^(1/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^11),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(1/4)*x^11), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{11}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^11),x, algorithm="fricas")

[Out]

integral(1/((b*x^4 + a)^(1/4)*x^11), x)

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Sympy [A]  time = 8.88039, size = 32, normalized size = 0.21 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{1}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{10 \sqrt [4]{a} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**11/(b*x**4+a)**(1/4),x)

[Out]

-hyper((-5/2, 1/4), (-3/2,), b*x**4*exp_polar(I*pi)/a)/(10*a**(1/4)*x**10)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} x^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(1/4)*x^11),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(1/4)*x^11), x)