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

Optimal. Leaf size=145 \[ \frac{3 b^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 a^{3/2} \sqrt [4]{a+b x^2}}-\frac{3 b^3 x}{20 a^2 \sqrt [4]{a+b x^2}}+\frac{3 b^2 \left (a+b x^2\right )^{3/4}}{20 a^2 x}-\frac{\left (a+b x^2\right )^{3/4}}{5 x^5}-\frac{b \left (a+b x^2\right )^{3/4}}{10 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.154334, antiderivative size = 145, 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 \[ \frac{3 b^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{20 a^{3/2} \sqrt [4]{a+b x^2}}-\frac{3 b^3 x}{20 a^2 \sqrt [4]{a+b x^2}}+\frac{3 b^2 \left (a+b x^2\right )^{3/4}}{20 a^2 x}-\frac{\left (a+b x^2\right )^{3/4}}{5 x^5}-\frac{b \left (a+b x^2\right )^{3/4}}{10 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(3/4)/x^6,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a + b*x**2)**(3/4)/(5*x**5) - b*(a + b*x**2)**(3/4)/(10*a*x**3) - 3*b**3*Integ
ral((a + b*x**2)**(-1/4), x)/(40*a**2) + 3*b**2*(a + b*x**2)**(3/4)/(20*a**2*x)

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Mathematica [C]  time = 0.0495263, size = 94, normalized size = 0.65 \[ \frac{-8 a^3-12 a^2 b x^2-3 b^3 x^6 \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )+2 a b^2 x^4+6 b^3 x^6}{40 a^2 x^5 \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-8*a^3 - 12*a^2*b*x^2 + 2*a*b^2*x^4 + 6*b^3*x^6 - 3*b^3*x^6*(1 + (b*x^2)/a)^(1/
4)*Hypergeometric2F1[1/4, 1/2, 3/2, -((b*x^2)/a)])/(40*a^2*x^5*(a + b*x^2)^(1/4)
)

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Maple [F]  time = 0.044, 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((b*x^2+a)^(3/4)/x^6,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(3/4)/x^6, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^2 + a)^(3/4)/x^6, x)

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Sympy [A]  time = 5.12632, size = 34, normalized size = 0.23 \[ - \frac{a^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{2}, - \frac{3}{4} \\ - \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError