3.1058 \(\int \frac{\left (a+b x^4\right )^{5/4}}{x^{11}} \, dx\)

Optimal. Leaf size=122 \[ -\frac{b^{5/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{24 \sqrt{a} \left (a+b x^4\right )^{3/4}}-\frac{b^2 \sqrt [4]{a+b x^4}}{24 a x^2}-\frac{\left (a+b x^4\right )^{5/4}}{10 x^{10}}-\frac{b \sqrt [4]{a+b x^4}}{12 x^6} \]

[Out]

-(b*(a + b*x^4)^(1/4))/(12*x^6) - (b^2*(a + b*x^4)^(1/4))/(24*a*x^2) - (a + b*x^
4)^(5/4)/(10*x^10) - (b^(5/2)*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^
2)/Sqrt[a]]/2, 2])/(24*Sqrt[a]*(a + b*x^4)^(3/4))

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Rubi [A]  time = 0.182314, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{b^{5/2} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{24 \sqrt{a} \left (a+b x^4\right )^{3/4}}-\frac{b^2 \sqrt [4]{a+b x^4}}{24 a x^2}-\frac{\left (a+b x^4\right )^{5/4}}{10 x^{10}}-\frac{b \sqrt [4]{a+b x^4}}{12 x^6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(5/4)/x^11,x]

[Out]

-(b*(a + b*x^4)^(1/4))/(12*x^6) - (b^2*(a + b*x^4)^(1/4))/(24*a*x^2) - (a + b*x^
4)^(5/4)/(10*x^10) - (b^(5/2)*(1 + (b*x^4)/a)^(3/4)*EllipticF[ArcTan[(Sqrt[b]*x^
2)/Sqrt[a]]/2, 2])/(24*Sqrt[a]*(a + b*x^4)^(3/4))

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Rubi in Sympy [A]  time = 18.562, size = 105, normalized size = 0.86 \[ - \frac{b \sqrt [4]{a + b x^{4}}}{12 x^{6}} - \frac{\left (a + b x^{4}\right )^{\frac{5}{4}}}{10 x^{10}} - \frac{b^{2} \sqrt [4]{a + b x^{4}}}{24 a x^{2}} - \frac{b^{\frac{5}{2}} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{24 \sqrt{a} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-b*(a + b*x**4)**(1/4)/(12*x**6) - (a + b*x**4)**(5/4)/(10*x**10) - b**2*(a + b*
x**4)**(1/4)/(24*a*x**2) - b**(5/2)*(1 + b*x**4/a)**(3/4)*elliptic_f(atan(sqrt(b
)*x**2/sqrt(a))/2, 2)/(24*sqrt(a)*(a + b*x**4)**(3/4))

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Mathematica [C]  time = 0.0616857, size = 97, normalized size = 0.8 \[ \frac{-2 \left (12 a^3+34 a^2 b x^4+27 a b^2 x^8+5 b^3 x^{12}\right )-5 b^3 x^{12} \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )}{240 a x^{10} \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(12*a^3 + 34*a^2*b*x^4 + 27*a*b^2*x^8 + 5*b^3*x^12) - 5*b^3*x^12*(1 + (b*x^4
)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -((b*x^4)/a)])/(240*a*x^10*(a + b*x^
4)^(3/4))

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Maple [F]  time = 0.052, 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((b*x^4+a)^(5/4)/x^11,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(5/4)/x^11, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{11}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a)^(5/4)/x^11, x)

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Sympy [A]  time = 18.7596, size = 34, normalized size = 0.28 \[ - \frac{a^{\frac{5}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, - \frac{5}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{10 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(5/4)*hyper((-5/2, -5/4), (-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. \[ \int \frac{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}{x^{11}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(5/4)/x^11, x)