3.1028 \(\int \frac{\left (a+b x^4\right )^{3/4}}{x^{11}} \, dx\)

Optimal. Leaf size=149 \[ \frac{3 b^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 a^{3/2} \sqrt [4]{a+b x^4}}-\frac{3 b^3 x^2}{40 a^2 \sqrt [4]{a+b x^4}}+\frac{3 b^2 \left (a+b x^4\right )^{3/4}}{40 a^2 x^2}-\frac{\left (a+b x^4\right )^{3/4}}{10 x^{10}}-\frac{b \left (a+b x^4\right )^{3/4}}{20 a x^6} \]

[Out]

(-3*b^3*x^2)/(40*a^2*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(10*x^10) - (b*(a +
b*x^4)^(3/4))/(20*a*x^6) + (3*b^2*(a + b*x^4)^(3/4))/(40*a^2*x^2) + (3*b^(5/2)*(
1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(40*a^(3/2)*
(a + b*x^4)^(1/4))

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Rubi [A]  time = 0.216134, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{3 b^{5/2} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{40 a^{3/2} \sqrt [4]{a+b x^4}}-\frac{3 b^3 x^2}{40 a^2 \sqrt [4]{a+b x^4}}+\frac{3 b^2 \left (a+b x^4\right )^{3/4}}{40 a^2 x^2}-\frac{\left (a+b x^4\right )^{3/4}}{10 x^{10}}-\frac{b \left (a+b x^4\right )^{3/4}}{20 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(3/4)/x^11,x]

[Out]

(-3*b^3*x^2)/(40*a^2*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(10*x^10) - (b*(a +
b*x^4)^(3/4))/(20*a*x^6) + (3*b^2*(a + b*x^4)^(3/4))/(40*a^2*x^2) + (3*b^(5/2)*(
1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(Sqrt[b]*x^2)/Sqrt[a]]/2, 2])/(40*a^(3/2)*
(a + b*x^4)^(1/4))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{10 x^{10}} + \frac{3 b^{3} \int ^{x^{2}} \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{80 a} - \frac{b \left (a + b x^{4}\right )^{\frac{3}{4}}}{20 a x^{6}} - \frac{3 b^{3} x^{2}}{40 a^{2} \sqrt [4]{a + b x^{4}}} + \frac{3 b^{2} \left (a + b x^{4}\right )^{\frac{3}{4}}}{40 a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(3/4)/x**11,x)

[Out]

-(a + b*x**4)**(3/4)/(10*x**10) + 3*b**3*Integral((a + b*x**2)**(-5/4), (x, x**2
))/(80*a) - b*(a + b*x**4)**(3/4)/(20*a*x**6) - 3*b**3*x**2/(40*a**2*(a + b*x**4
)**(1/4)) + 3*b**2*(a + b*x**4)**(3/4)/(40*a**2*x**2)

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Mathematica [C]  time = 0.0559941, size = 94, normalized size = 0.63 \[ \frac{-8 a^3-12 a^2 b x^4-3 b^3 x^{12} \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )+2 a b^2 x^8+6 b^3 x^{12}}{80 a^2 x^{10} \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(3/4)/x^11,x]

[Out]

(-8*a^3 - 12*a^2*b*x^4 + 2*a*b^2*x^8 + 6*b^3*x^12 - 3*b^3*x^12*(1 + (b*x^4)/a)^(
1/4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^4)/a)])/(80*a^2*x^10*(a + b*x^4)^(1
/4))

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Maple [F]  time = 0.05, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{11}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(3/4)/x^11,x)

[Out]

int((b*x^4+a)^(3/4)/x^11,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^11,x, algorithm="maxima")

[Out]

integrate((b*x^4 + a)^(3/4)/x^11, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{11}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^11,x, algorithm="fricas")

[Out]

integral((b*x^4 + a)^(3/4)/x^11, x)

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Sympy [A]  time = 12.7167, size = 34, normalized size = 0.23 \[ - \frac{a^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, - \frac{3}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{10 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(3/4)/x**11,x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{x^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(3/4)/x^11,x, algorithm="giac")

[Out]

integrate((b*x^4 + a)^(3/4)/x^11, x)