3.1042 \(\int \frac{\left (a+b x^4\right )^{3/4}}{x^6} \, dx\)

Optimal. Leaf size=99 \[ \frac{3 b^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{a} \sqrt [4]{a+b x^4}}-\frac{3 b}{5 x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{5 x^5} \]

[Out]

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

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Rubi [A]  time = 0.159725, antiderivative size = 99, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ \frac{3 b^{3/2} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{a} \sqrt [4]{a+b x^4}}-\frac{3 b}{5 x \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{5 x^5} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(3/4)/x^6,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0480042, size = 83, normalized size = 0.84 \[ \frac{-a^2+2 b^2 x^8 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-4 a b x^4-3 b^2 x^8}{5 a x^5 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(-a^2 - 4*a*b*x^4 - 3*b^2*x^8 + 2*b^2*x^8*(1 + (b*x^4)/a)^(1/4)*Hypergeometric2F
1[1/4, 3/4, 7/4, -((b*x^4)/a)])/(5*a*x^5*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{6}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(3/4)/x^6,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(3/4)/x^6, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a)^(3/4)/x^6, x)

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Sympy [A]  time = 5.28071, size = 31, normalized size = 0.31 \[ - \frac{b^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(3/4)/x^6, x)