3.835 \(\int \frac{1}{x^6 \left (a+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=126 \[ -\frac{3 b^{5/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{4 a^{5/2} \left (a+b x^2\right )^{3/4}}-\frac{3 b^2 \sqrt [4]{a+b x^2}}{4 a^3 x}+\frac{3 b \sqrt [4]{a+b x^2}}{10 a^2 x^3}-\frac{\sqrt [4]{a+b x^2}}{5 a x^5} \]

[Out]

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

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Rubi [A]  time = 0.133766, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{3 b^{5/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{4 a^{5/2} \left (a+b x^2\right )^{3/4}}-\frac{3 b^2 \sqrt [4]{a+b x^2}}{4 a^3 x}+\frac{3 b \sqrt [4]{a+b x^2}}{10 a^2 x^3}-\frac{\sqrt [4]{a+b x^2}}{5 a x^5} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^6*(a + b*x^2)^(3/4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 15.9738, size = 112, normalized size = 0.89 \[ - \frac{\sqrt [4]{a + b x^{2}}}{5 a x^{5}} + \frac{3 b \sqrt [4]{a + b x^{2}}}{10 a^{2} x^{3}} - \frac{3 b^{2} \sqrt [4]{a + b x^{2}}}{4 a^{3} x} - \frac{3 b^{\frac{5}{2}} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{4 a^{\frac{5}{2}} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**6/(b*x**2+a)**(3/4),x)

[Out]

-(a + b*x**2)**(1/4)/(5*a*x**5) + 3*b*(a + b*x**2)**(1/4)/(10*a**2*x**3) - 3*b**
2*(a + b*x**2)**(1/4)/(4*a**3*x) - 3*b**(5/2)*(1 + b*x**2/a)**(3/4)*elliptic_f(a
tan(sqrt(b)*x/sqrt(a))/2, 2)/(4*a**(5/2)*(a + b*x**2)**(3/4))

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Mathematica [C]  time = 0.0631611, size = 94, normalized size = 0.75 \[ \frac{-8 a^3+4 a^2 b x^2-15 b^3 x^6 \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right )-18 a b^2 x^4-30 b^3 x^6}{40 a^3 x^5 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^6*(a + b*x^2)^(3/4)),x]

[Out]

(-8*a^3 + 4*a^2*b*x^2 - 18*a*b^2*x^4 - 30*b^3*x^6 - 15*b^3*x^6*(1 + (b*x^2)/a)^(
3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -((b*x^2)/a)])/(40*a^3*x^5*(a + b*x^2)^(3/
4))

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{6}} \left ( b{x}^{2}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^6/(b*x^2+a)^(3/4),x)

[Out]

int(1/x^6/(b*x^2+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/4)*x^6),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(3/4)*x^6), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{6}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/4)*x^6),x, algorithm="fricas")

[Out]

integral(1/((b*x^2 + a)^(3/4)*x^6), x)

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Sympy [A]  time = 5.07331, size = 32, normalized size = 0.25 \[ - \frac{{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, \frac{3}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac{3}{4}} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**6/(b*x**2+a)**(3/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{3}{4}} x^{6}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)^(3/4)*x^6),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(3/4)*x^6), x)