3.1033 \(\int \frac{\left (a+b x^4\right )^{3/4}}{x^4} \, dx\)

Optimal. Leaf size=75 \[ \frac{1}{2} b^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac{1}{2} b^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )-\frac{\left (a+b x^4\right )^{3/4}}{3 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0571237, antiderivative size = 75, 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 \[ \frac{1}{2} b^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\frac{1}{2} b^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )-\frac{\left (a+b x^4\right )^{3/4}}{3 x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(3/4)/x^4,x]

[Out]

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

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Rubi in Sympy [A]  time = 6.81954, size = 65, normalized size = 0.87 \[ \frac{b^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2} + \frac{b^{\frac{3}{4}} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2} - \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{3 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

b**(3/4)*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/2 + b**(3/4)*atanh(b**(1/4)*x/(a +
b*x**4)**(1/4))/2 - (a + b*x**4)**(3/4)/(3*x**3)

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Mathematica [A]  time = 0.0855721, size = 95, normalized size = 1.27 \[ \frac{1}{4} b^{3/4} \left (-\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\log \left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right )-\frac{\left (a+b x^4\right )^{3/4}}{3 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{\frac{1}{{x}^{4}} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(3/4)/x^4,x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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Sympy [A]  time = 5.03848, size = 42, normalized size = 0.56 \[ \frac{a^{\frac{3}{4}} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{3}{4} \\ \frac{1}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 x^{3} \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^4 + a)^(3/4)/x^4, x)