3.834 \(\int \frac{1}{x^4 \left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=102 \[ \frac{5 b^{3/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{6 a^{3/2} \left (a+b x^2\right )^{3/4}}+\frac{5 b \sqrt [4]{a+b x^2}}{6 a^2 x}-\frac{\sqrt [4]{a+b x^2}}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.0965869, antiderivative size = 102, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{5 b^{3/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{6 a^{3/2} \left (a+b x^2\right )^{3/4}}+\frac{5 b \sqrt [4]{a+b x^2}}{6 a^2 x}-\frac{\sqrt [4]{a+b x^2}}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 11.5788, size = 88, normalized size = 0.86 \[ - \frac{\sqrt [4]{a + b x^{2}}}{3 a x^{3}} + \frac{5 b \sqrt [4]{a + b x^{2}}}{6 a^{2} x} + \frac{5 b^{\frac{3}{2}} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{6 a^{\frac{3}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(b*x**2+a)**(3/4),x)

[Out]

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

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Mathematica [C]  time = 0.053337, size = 83, normalized size = 0.81 \[ \frac{-4 a^2+5 b^2 x^4 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right )+6 a b x^2+10 b^2 x^4}{12 a^2 x^3 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

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

[Out]

(-4*a^2 + 6*a*b*x^2 + 10*b^2*x^4 + 5*b^2*x^4*(1 + (b*x^2)/a)^(3/4)*Hypergeometri
c2F1[1/2, 3/4, 3/2, -((b*x^2)/a)])/(12*a^2*x^3*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(b*x^2+a)^(3/4),x)

[Out]

int(1/x^4/(b*x^2+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/4)*x^4),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(3/4)*x^4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/4)*x^4),x, algorithm="fricas")

[Out]

integral(1/((b*x^2 + a)^(3/4)*x^4), x)

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Sympy [A]  time = 3.8991, size = 32, normalized size = 0.31 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{2}, \frac{3}{4} \\ - \frac{1}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3 a^{\frac{3}{4}} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**4/(b*x**2+a)**(3/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{4}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/4)*x^4),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(3/4)*x^4), x)