3.1041 \(\int \frac{\left (a+b x^4\right )^{3/4}}{x^2} \, dx\)

Optimal. Leaf size=97 \[ -\frac{\left (a+b x^4\right )^{3/4}}{x}+\frac{3 b x^3}{2 \sqrt [4]{a+b x^4}}+\frac{3 \sqrt{a} \sqrt{b} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt [4]{a+b x^4}} \]

[Out]

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

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Rubi [A]  time = 0.152491, antiderivative size = 97, 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 \[ -\frac{\left (a+b x^4\right )^{3/4}}{x}+\frac{3 b x^3}{2 \sqrt [4]{a+b x^4}}+\frac{3 \sqrt{a} \sqrt{b} x \sqrt [4]{\frac{a}{b x^4}+1} E\left (\left .\frac{1}{2} \cot ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(3/4)/x^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0380396, size = 63, normalized size = 0.65 \[ \frac{b x^4 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )-a-b x^4}{x \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(-a - b*x^4 + b*x^4*(1 + (b*x^4)/a)^(1/4)*Hypergeometric2F1[1/4, 3/4, 7/4, -((b*
x^4)/a)])/(x*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{2}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(3/4)/x^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(3/4)/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a)^(3/4)/x^2, x)

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Sympy [A]  time = 3.20371, size = 41, normalized size = 0.42 \[ \frac{a^{\frac{3}{4}} \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(3/4)/x^2, x)