3.991 \(\int \frac{\sqrt [4]{a+b x^4}}{x} \, dx\)

Optimal. Leaf size=66 \[ \sqrt [4]{a+b x^4}-\frac{1}{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac{1}{2} \sqrt [4]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.100755, antiderivative size = 66, 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 \[ \sqrt [4]{a+b x^4}-\frac{1}{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right )-\frac{1}{2} \sqrt [4]{a} \tanh ^{-1}\left (\frac{\sqrt [4]{a+b x^4}}{\sqrt [4]{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^4)^(1/4)/x,x]

[Out]

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

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Rubi in Sympy [A]  time = 10.5008, size = 56, normalized size = 0.85 \[ - \frac{\sqrt [4]{a} \operatorname{atan}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{2} - \frac{\sqrt [4]{a} \operatorname{atanh}{\left (\frac{\sqrt [4]{a + b x^{4}}}{\sqrt [4]{a}} \right )}}{2} + \sqrt [4]{a + b x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**4+a)**(1/4)/x,x)

[Out]

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

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Mathematica [C]  time = 0.0426291, size = 61, normalized size = 0.92 \[ \frac{3 \left (a+b x^4\right )-a \left (\frac{a}{b x^4}+1\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{a}{b x^4}\right )}{3 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^4)^(1/4)/x,x]

[Out]

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

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int{\frac{1}{x}\sqrt [4]{b{x}^{4}+a}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^4+a)^(1/4)/x,x)

[Out]

int((b*x^4+a)^(1/4)/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)^(1/4)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29539, size = 116, normalized size = 1.76 \[ a^{\frac{1}{4}} \arctan \left (\frac{a^{\frac{1}{4}}}{\sqrt{\sqrt{b x^{4} + a} + \sqrt{a}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\right ) - \frac{1}{4} \, a^{\frac{1}{4}} \log \left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} + a^{\frac{1}{4}}\right ) + \frac{1}{4} \, a^{\frac{1}{4}} \log \left ({\left (b x^{4} + a\right )}^{\frac{1}{4}} - a^{\frac{1}{4}}\right ) +{\left (b x^{4} + a\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.7672, size = 42, normalized size = 0.64 \[ - \frac{\sqrt [4]{b} x \Gamma \left (- \frac{1}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, - \frac{1}{4} \\ \frac{3}{4} \end{matrix}\middle |{\frac{a e^{i \pi }}{b x^{4}}} \right )}}{4 \Gamma \left (\frac{3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**4+a)**(1/4)/x,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.22896, size = 247, normalized size = 3.74 \[ -\frac{1}{4} \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} + 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right ) - \frac{1}{4} \, \sqrt{2} \left (-a\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (-a\right )^{\frac{1}{4}} - 2 \,{\left (b x^{4} + a\right )}^{\frac{1}{4}}\right )}}{2 \, \left (-a\right )^{\frac{1}{4}}}\right ) - \frac{1}{8} \, \sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right ) + \frac{1}{8} \, \sqrt{2} \left (-a\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2}{\left (b x^{4} + a\right )}^{\frac{1}{4}} \left (-a\right )^{\frac{1}{4}} + \sqrt{b x^{4} + a} + \sqrt{-a}\right ) +{\left (b x^{4} + a\right )}^{\frac{1}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^4 + a)^(1/4)/x,x, algorithm="giac")

[Out]

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