3.1020 \(\int \frac{\left (a+b x^4\right )^{3/4}}{x} \, dx\)

Optimal. Leaf size=70 \[ \frac{1}{2} a^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac{1}{2} a^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )+\frac{1}{3} \left (a+b x^4\right )^{3/4} \]

[Out]

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

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Rubi [A]  time = 0.105679, antiderivative size = 70, 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 \[ \frac{1}{2} a^{3/4} \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac{1}{2} a^{3/4} \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )+\frac{1}{3} \left (a+b x^4\right )^{3/4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(3/4)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.4827, size = 58, normalized size = 0.83 \[ \frac{a^{\frac{3}{4}} \operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{2} - \frac{a^{\frac{3}{4}} \operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{2} + \frac{\left (a + b x^{4}\right )^{\frac{3}{4}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [C]  time = 0.0459377, size = 58, normalized size = 0.83 \[ \frac{-3 a \sqrt [4]{\frac{a}{b x^4}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{4};\frac{5}{4};-\frac{a}{b x^4}\right )+a+b x^4}{3 \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

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

[Out]

(a + b*x^4 - 3*a*(1 + a/(b*x^4))^(1/4)*Hypergeometric2F1[1/4, 1/4, 5/4, -(a/(b*x
^4))])/(3*(a + b*x^4)^(1/4))

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \[ \int{\frac{1}{x} \left ( b{x}^{4}+a \right ) ^{{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(3/4)/x,x)

[Out]

int((b*x^4+a)^(3/4)/x,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,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.270956, size = 166, normalized size = 2.37 \[ -{\left (a^{3}\right )}^{\frac{1}{4}} \arctan \left (\frac{{\left (a^{3}\right )}^{\frac{3}{4}}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} + \sqrt{\sqrt{b x^{4} + a} a^{4} + \sqrt{a^{3}} a^{3}}}\right ) - \frac{1}{4} \,{\left (a^{3}\right )}^{\frac{1}{4}} \log \left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} +{\left (a^{3}\right )}^{\frac{3}{4}}\right ) + \frac{1}{4} \,{\left (a^{3}\right )}^{\frac{1}{4}} \log \left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{2} -{\left (a^{3}\right )}^{\frac{3}{4}}\right ) + \frac{1}{3} \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(a^3)^(1/4)*arctan((a^3)^(3/4)/((b*x^4 + a)^(1/4)*a^2 + sqrt(sqrt(b*x^4 + a)*a^
4 + sqrt(a^3)*a^3))) - 1/4*(a^3)^(1/4)*log((b*x^4 + a)^(1/4)*a^2 + (a^3)^(3/4))
+ 1/4*(a^3)^(1/4)*log((b*x^4 + a)^(1/4)*a^2 - (a^3)^(3/4)) + 1/3*(b*x^4 + a)^(3/
4)

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Sympy [A]  time = 4.39877, size = 44, normalized size = 0.63 \[ - \frac{b^{\frac{3}{4}} x^{3} \Gamma \left (- \frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, - \frac{3}{4} \\ \frac{1}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 \Gamma \left (\frac{1}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.224656, size = 250, normalized size = 3.57 \[ -\frac{1}{4} \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right ) - \frac{1}{4} \, \sqrt{2} \left (-a\right )^{\frac{3}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right ) + \frac{1}{8} \, \sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right ) - \frac{1}{8} \, \sqrt{2} \left (-a\right )^{\frac{3}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right ) + \frac{1}{3} \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*sqrt(2)*(-a)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(b*x^4 + a)^(
1/4))/(-a)^(1/4)) - 1/4*sqrt(2)*(-a)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/
4) - 2*(b*x^4 + a)^(1/4))/(-a)^(1/4)) + 1/8*sqrt(2)*(-a)^(3/4)*ln(sqrt(2)*(b*x^4
+ a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a)) - 1/8*sqrt(2)*(-a)^(3/4)*ln
(-sqrt(2)*(b*x^4 + a)^(1/4)*(-a)^(1/4) + sqrt(b*x^4 + a) + sqrt(-a)) + 1/3*(b*x^
4 + a)^(3/4)