3.765 \(\int \frac{\sqrt{a+c x^4}}{x} \, dx\)

Optimal. Leaf size=43 \[ \frac{1}{2} \sqrt{a+c x^4}-\frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right ) \]

[Out]

Sqrt[a + c*x^4]/2 - (Sqrt[a]*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]])/2

_______________________________________________________________________________________

Rubi [A]  time = 0.0673404, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{1}{2} \sqrt{a+c x^4}-\frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + c*x^4]/x,x]

[Out]

Sqrt[a + c*x^4]/2 - (Sqrt[a]*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]])/2

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.69605, size = 34, normalized size = 0.79 \[ - \frac{\sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + c x^{4}}}{\sqrt{a}} \right )}}{2} + \frac{\sqrt{a + c x^{4}}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**4+a)**(1/2)/x,x)

[Out]

-sqrt(a)*atanh(sqrt(a + c*x**4)/sqrt(a))/2 + sqrt(a + c*x**4)/2

_______________________________________________________________________________________

Mathematica [A]  time = 0.0699035, size = 43, normalized size = 1. \[ \frac{1}{2} \sqrt{a+c x^4}-\frac{1}{2} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+c x^4}}{\sqrt{a}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + c*x^4]/x,x]

[Out]

Sqrt[a + c*x^4]/2 - (Sqrt[a]*ArcTanh[Sqrt[a + c*x^4]/Sqrt[a]])/2

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 41, normalized size = 1. \[{\frac{1}{2}\sqrt{c{x}^{4}+a}}-{\frac{1}{2}\sqrt{a}\ln \left ({\frac{1}{{x}^{2}} \left ( 2\,a+2\,\sqrt{a}\sqrt{c{x}^{4}+a} \right ) } \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^4+a)^(1/2)/x,x)

[Out]

1/2*(c*x^4+a)^(1/2)-1/2*a^(1/2)*ln((2*a+2*a^(1/2)*(c*x^4+a)^(1/2))/x^2)

_______________________________________________________________________________________

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(sqrt(c*x^4 + a)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.277143, size = 1, normalized size = 0.02 \[ \left [\frac{1}{4} \, \sqrt{a} \log \left (\frac{c x^{4} - 2 \, \sqrt{c x^{4} + a} \sqrt{a} + 2 \, a}{x^{4}}\right ) + \frac{1}{2} \, \sqrt{c x^{4} + a}, -\frac{1}{2} \, \sqrt{-a} \arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right ) + \frac{1}{2} \, \sqrt{c x^{4} + a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/4*sqrt(a)*log((c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(a) + 2*a)/x^4) + 1/2*sqrt(c*x^4
+ a), -1/2*sqrt(-a)*arctan(sqrt(c*x^4 + a)/sqrt(-a)) + 1/2*sqrt(c*x^4 + a)]

_______________________________________________________________________________________

Sympy [A]  time = 4.81135, size = 66, normalized size = 1.53 \[ - \frac{\sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a}}{\sqrt{c} x^{2}} \right )}}{2} + \frac{a}{2 \sqrt{c} x^{2} \sqrt{\frac{a}{c x^{4}} + 1}} + \frac{\sqrt{c} x^{2}}{2 \sqrt{\frac{a}{c x^{4}} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**4+a)**(1/2)/x,x)

[Out]

-sqrt(a)*asinh(sqrt(a)/(sqrt(c)*x**2))/2 + a/(2*sqrt(c)*x**2*sqrt(a/(c*x**4) + 1
)) + sqrt(c)*x**2/(2*sqrt(a/(c*x**4) + 1))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.217072, size = 49, normalized size = 1.14 \[ \frac{a \arctan \left (\frac{\sqrt{c x^{4} + a}}{\sqrt{-a}}\right )}{2 \, \sqrt{-a}} + \frac{1}{2} \, \sqrt{c x^{4} + a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*a*arctan(sqrt(c*x^4 + a)/sqrt(-a))/sqrt(-a) + 1/2*sqrt(c*x^4 + a)