3.1151 \(\int \frac{1}{x^{11} \left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=128 \[ -\frac{77 b^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 a^{7/2} \sqrt [4]{a+b x^4}}-\frac{77 b^2}{120 a^3 x^2 \sqrt [4]{a+b x^4}}+\frac{11 b}{60 a^2 x^6 \sqrt [4]{a+b x^4}}-\frac{1}{10 a x^{10} \sqrt [4]{a+b x^4}} \]

[Out]

-1/(10*a*x^10*(a + b*x^4)^(1/4)) + (11*b)/(60*a^2*x^6*(a + b*x^4)^(1/4)) - (77*b
^2)/(120*a^3*x^2*(a + b*x^4)^(1/4)) - (77*b^(5/2)*(1 + (b*x^4)/a)^(1/4)*Elliptic
E[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(40*a^(7/2)*(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.194603, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{77 b^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 a^{7/2} \sqrt [4]{a+b x^4}}-\frac{77 b^2}{120 a^3 x^2 \sqrt [4]{a+b x^4}}+\frac{11 b}{60 a^2 x^6 \sqrt [4]{a+b x^4}}-\frac{1}{10 a x^{10} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^11*(a + b*x^4)^(5/4)),x]

[Out]

-1/(10*a*x^10*(a + b*x^4)^(1/4)) + (11*b)/(60*a^2*x^6*(a + b*x^4)^(1/4)) - (77*b
^2)/(120*a^3*x^2*(a + b*x^4)^(1/4)) - (77*b^(5/2)*(1 + (b*x^4)/a)^(1/4)*Elliptic
E[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(40*a^(7/2)*(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{1}{10 a x^{10} \sqrt [4]{a + b x^{4}}} + \frac{11 b}{60 a^{2} x^{6} \sqrt [4]{a + b x^{4}}} - \frac{77 b^{2}}{120 a^{3} x^{2} \sqrt [4]{a + b x^{4}}} - \frac{77 b^{3} x^{2}}{40 a^{4} \sqrt [4]{a + b x^{4}}} + \frac{77 b^{3} \int ^{x^{2}} \frac{1}{\sqrt [4]{a + b x^{2}}}\, dx}{80 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**11/(b*x**4+a)**(5/4),x)

[Out]

-1/(10*a*x**10*(a + b*x**4)**(1/4)) + 11*b/(60*a**2*x**6*(a + b*x**4)**(1/4)) -
77*b**2/(120*a**3*x**2*(a + b*x**4)**(1/4)) - 77*b**3*x**2/(40*a**4*(a + b*x**4)
**(1/4)) + 77*b**3*Integral((a + b*x**2)**(-1/4), (x, x**2))/(80*a**4)

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Mathematica [C]  time = 0.078535, size = 94, normalized size = 0.73 \[ \frac{-24 a^3+44 a^2 b x^4+231 b^3 x^{12} \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )-154 a b^2 x^8-462 b^3 x^{12}}{240 a^4 x^{10} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^11*(a + b*x^4)^(5/4)),x]

[Out]

(-24*a^3 + 44*a^2*b*x^4 - 154*a*b^2*x^8 - 462*b^3*x^12 + 231*b^3*x^12*(1 + (b*x^
4)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^4)/a)])/(240*a^4*x^10*(a + b
*x^4)^(1/4))

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Maple [F]  time = 0.084, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{11}} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^11/(b*x^4+a)^(5/4),x)

[Out]

int(1/x^11/(b*x^4+a)^(5/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^11),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^11), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{15} + a x^{11}\right )}{\left (b x^{4} + a\right )}^{\frac{1}{4}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^11),x, algorithm="fricas")

[Out]

integral(1/((b*x^15 + a*x^11)*(b*x^4 + a)^(1/4)), x)

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Sympy [A]  time = 17.7205, size = 32, normalized size = 0.25 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{5}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{10 a^{\frac{5}{4}} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**11/(b*x**4+a)**(5/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{5}{4}} x^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(5/4)*x^11),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(5/4)*x^11), x)