3.1026 \(\int \frac{\left (a+b x^4\right )^{3/4}}{x^3} \, dx\)

Optimal. Leaf size=98 \[ \frac{3 b x^2}{2 \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{2 x^2}-\frac{3 \sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt [4]{a+b x^4}} \]

[Out]

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

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Rubi [A]  time = 0.134753, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{3 b x^2}{2 \sqrt [4]{a+b x^4}}-\frac{\left (a+b x^4\right )^{3/4}}{2 x^2}-\frac{3 \sqrt{a} \sqrt{b} \sqrt [4]{\frac{b x^4}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )\right |2\right )}{2 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(3/4)/x^3,x]

[Out]

(3*b*x^2)/(2*(a + b*x^4)^(1/4)) - (a + b*x^4)^(3/4)/(2*x^2) - (3*Sqrt[a]*Sqrt[b]
*(1 + (b*x^4)/a)^(1/4)*EllipticE[ArcTan[(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. \[ - \frac{3 a b \int ^{x^{2}} \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{4} + \frac{3 b x^{2}}{2 \sqrt [4]{a + b x^{4}}} - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0410525, size = 67, normalized size = 0.68 \[ \frac{3 b x^4 \sqrt [4]{\frac{b x^4}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^4}{a}\right )-2 \left (a+b x^4\right )}{4 x^2 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*(a + b*x^4) + 3*b*x^4*(1 + (b*x^4)/a)^(1/4)*Hypergeometric2F1[1/4, 1/2, 3/2,
-((b*x^4)/a)])/(4*x^2*(a + b*x^4)^(1/4))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(3/4)/x^3,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(3/4)/x^3, x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a)^(3/4)/x^3, x)

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Sympy [A]  time = 3.15916, size = 32, normalized size = 0.33 \[ - \frac{a^{\frac{3}{4}}{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{1}{2} \\ \frac{1}{2} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(3/4)/x^3, x)