3.2056 \(\int \sqrt{a+\frac{b}{x^4}} x^3 \, dx\)

Optimal. Leaf size=47 \[ \frac{1}{4} x^4 \sqrt{a+\frac{b}{x^4}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 \sqrt{a}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0869045, antiderivative size = 47, 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 \[ \frac{1}{4} x^4 \sqrt{a+\frac{b}{x^4}}+\frac{b \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^4}}}{\sqrt{a}}\right )}{4 \sqrt{a}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^4]*x^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 6.9488, size = 39, normalized size = 0.83 \[ \frac{x^{4} \sqrt{a + \frac{b}{x^{4}}}}{4} + \frac{b \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{4}}}}{\sqrt{a}} \right )}}{4 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(a+b/x**4)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0676393, size = 64, normalized size = 1.36 \[ \frac{1}{4} x^2 \sqrt{a+\frac{b}{x^4}} \left (\frac{b \log \left (\sqrt{a} \sqrt{a x^4+b}+a x^2\right )}{\sqrt{a} \sqrt{a x^4+b}}+x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + b/x^4]*x^2*(x^2 + (b*Log[a*x^2 + Sqrt[a]*Sqrt[b + a*x^4]])/(Sqrt[a]*Sq
rt[b + a*x^4])))/4

_______________________________________________________________________________________

Maple [A]  time = 0.024, size = 68, normalized size = 1.5 \[{\frac{{x}^{2}}{4}\sqrt{{\frac{a{x}^{4}+b}{{x}^{4}}}} \left ({x}^{2}\sqrt{a{x}^{4}+b}\sqrt{a}+b\ln \left ({x}^{2}\sqrt{a}+\sqrt{a{x}^{4}+b} \right ) \right ){\frac{1}{\sqrt{a{x}^{4}+b}}}{\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(a+b/x^4)^(1/2),x)

[Out]

1/4*((a*x^4+b)/x^4)^(1/2)*x^2*(x^2*(a*x^4+b)^(1/2)*a^(1/2)+b*ln(x^2*a^(1/2)+(a*x
^4+b)^(1/2)))/(a*x^4+b)^(1/2)/a^(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(sqrt(a + b/x^4)*x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.254519, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} + \sqrt{a} b \log \left (-2 \, a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} -{\left (2 \, a x^{4} + b\right )} \sqrt{a}\right )}{8 \, a}, \frac{a x^{4} \sqrt{\frac{a x^{4} + b}{x^{4}}} - \sqrt{-a} b \arctan \left (\frac{\sqrt{-a}}{\sqrt{\frac{a x^{4} + b}{x^{4}}}}\right )}{4 \, a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/8*(2*a*x^4*sqrt((a*x^4 + b)/x^4) + sqrt(a)*b*log(-2*a*x^4*sqrt((a*x^4 + b)/x^
4) - (2*a*x^4 + b)*sqrt(a)))/a, 1/4*(a*x^4*sqrt((a*x^4 + b)/x^4) - sqrt(-a)*b*ar
ctan(sqrt(-a)/sqrt((a*x^4 + b)/x^4)))/a]

_______________________________________________________________________________________

Sympy [A]  time = 8.70102, size = 44, normalized size = 0.94 \[ \frac{\sqrt{b} x^{2} \sqrt{\frac{a x^{4}}{b} + 1}}{4} + \frac{b \operatorname{asinh}{\left (\frac{\sqrt{a} x^{2}}{\sqrt{b}} \right )}}{4 \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(a+b/x**4)**(1/2),x)

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.232942, size = 55, normalized size = 1.17 \[ \frac{1}{4} \, \sqrt{a x^{4} + b} x^{2} - \frac{b{\rm ln}\left ({\left | -\sqrt{a} x^{2} + \sqrt{a x^{4} + b} \right |}\right )}{4 \, \sqrt{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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