3.1120 \(\int \frac{1}{x^3 \left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=82 \[ -\frac{\sqrt [4]{a+b x^4}}{2 a x^2}-\frac{\sqrt{b} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \left (a+b x^4\right )^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.107731, antiderivative size = 82, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{\sqrt [4]{a+b x^4}}{2 a x^2}-\frac{\sqrt{b} \left (\frac{b x^4}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt{a} \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + b*x^4)^(3/4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 10.7267, size = 70, normalized size = 0.85 \[ - \frac{\sqrt [4]{a + b x^{4}}}{2 a x^{2}} - \frac{\sqrt{b} \left (1 + \frac{b x^{4}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{2 \sqrt{a} \left (a + b x^{4}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(b*x**4+a)**(3/4),x)

[Out]

-(a + b*x**4)**(1/4)/(2*a*x**2) - sqrt(b)*(1 + b*x**4/a)**(3/4)*elliptic_f(atan(
sqrt(b)*x**2/sqrt(a))/2, 2)/(2*sqrt(a)*(a + b*x**4)**(3/4))

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Mathematica [C]  time = 0.0453499, size = 70, normalized size = 0.85 \[ \frac{-b x^4 \left (\frac{b x^4}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^4}{a}\right )-2 \left (a+b x^4\right )}{4 a x^2 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + b*x^4)^(3/4)),x]

[Out]

(-2*(a + b*x^4) - b*x^4*(1 + (b*x^4)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -
((b*x^4)/a)])/(4*a*x^2*(a + b*x^4)^(3/4))

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{3}} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(b*x^4+a)^(3/4),x)

[Out]

int(1/x^3/(b*x^4+a)^(3/4),x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/4)*x^3),x, algorithm="maxima")

[Out]

integrate(1/((b*x^4 + a)^(3/4)*x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/4)*x^3),x, algorithm="fricas")

[Out]

integral(1/((b*x^4 + a)^(3/4)*x^3), x)

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Sympy [A]  time = 3.06132, size = 31, normalized size = 0.38 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, \frac{3}{4} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2 a^{\frac{3}{4}} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(b*x**4+a)**(3/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{4} + a\right )}^{\frac{3}{4}} x^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^4 + a)^(3/4)*x^3),x, algorithm="giac")

[Out]

integrate(1/((b*x^4 + a)^(3/4)*x^3), x)