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

Optimal. Leaf size=30 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{2 \sqrt{b}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0375919, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[a + b*x^4],x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 4.28193, size = 26, normalized size = 0.87 \[ \frac{\operatorname{atanh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a + b x^{4}}} \right )}}{2 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atanh(sqrt(b)*x**2/sqrt(a + b*x**4))/(2*sqrt(b))

_______________________________________________________________________________________

Mathematica [A]  time = 0.0153819, size = 30, normalized size = 1. \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a+b x^4}}\right )}{2 \sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[a + b*x^4],x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.008, size = 24, normalized size = 0.8 \[{\frac{1}{2}\ln \left ( \sqrt{b}{x}^{2}+\sqrt{b{x}^{4}+a} \right ){\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(b*x^4+a)^(1/2),x)

[Out]

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

_______________________________________________________________________________________

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

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.246733, size = 1, normalized size = 0.03 \[ \left [\frac{\log \left (-2 \, \sqrt{b x^{4} + a} b x^{2} -{\left (2 \, b x^{4} + a\right )} \sqrt{b}\right )}{4 \, \sqrt{b}}, \frac{\arctan \left (\frac{\sqrt{-b} x^{2}}{\sqrt{b x^{4} + a}}\right )}{2 \, \sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 3.55987, size = 20, normalized size = 0.67 \[ \frac{\operatorname{asinh}{\left (\frac{\sqrt{b} x^{2}}{\sqrt{a}} \right )}}{2 \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

asinh(sqrt(b)*x**2/sqrt(a))/(2*sqrt(b))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.223648, size = 34, normalized size = 1.13 \[ -\frac{{\rm ln}\left ({\left | -\sqrt{b} x^{2} + \sqrt{b x^{4} + a} \right |}\right )}{2 \, \sqrt{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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