∖int∘ ____ a+ b_ x3 ___________

3.1991 \(\int \frac{\sqrt{a+\frac{b}{x^3}}}{x} \, dx\)

Optimal. Leaf size=43 \[ \frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )-\frac{2}{3} \sqrt{a+\frac{b}{x^3}} \]

[Out]

(-2*Sqrt[a + b/x^3])/3 + (2*Sqrt[a]*ArcTanh[Sqrt[a + b/x^3]/Sqrt[a]])/3

_______________________________________________________________________________________

Rubi [A] time = 0.0765857, 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 \[ \frac{2}{3} \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+\frac{b}{x^3}}}{\sqrt{a}}\right )-\frac{2}{3} \sqrt{a+\frac{b}{x^3}} \]

Antiderivative was successfully veriﬁed.

[In] Int[Sqrt[a + b/x^3]/x,x]

[Out]

(-2*Sqrt[a + b/x^3])/3 + (2*Sqrt[a]*ArcTanh[Sqrt[a + b/x^3]/Sqrt[a]])/3

_______________________________________________________________________________________

Rubi in Sympy [A] time = 6.64285, size = 37, normalized size = 0.86 \[ \frac{2 \sqrt{a} \operatorname{atanh}{\left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{a}} \right )}}{3} - \frac{2 \sqrt{a + \frac{b}{x^{3}}}}{3} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(a)*atanh(sqrt(a + b/x**3)/sqrt(a))/3 - 2*sqrt(a + b/x**3)/3

_______________________________________________________________________________________

Mathematica [A] time = 0.047376, size = 75, normalized size = 1.74 \[ \frac{2 \sqrt{a} x^{3/2} \sqrt{a+\frac{b}{x^3}} \tanh ^{-1}\left (\frac{\sqrt{a} x^{3/2}}{\sqrt{a x^3+b}}\right )}{3 \sqrt{a x^3+b}}-\frac{2}{3} \sqrt{a+\frac{b}{x^3}} \]

Antiderivative was successfully veriﬁed.

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

[Out]

(-2*Sqrt[a + b/x^3])/3 + (2*Sqrt[a]*Sqrt[a + b/x^3]*x^(3/2)*ArcTanh[(Sqrt[a]*x^(

3/2))/Sqrt[b + a*x^3]])/(3*Sqrt[b + a*x^3])

_______________________________________________________________________________________

Maple [C] time = 0.042, size = 3339, normalized size = 77.7 \[ \text{output too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In] int((a+b/x^3)^(1/2)/x,x)

[Out]

-2/3*((a*x^3+b)/x^3)^(1/2)/a*(6*I*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*

b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/

(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I

*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2

)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*

3^(1/2)-3))^(1/2))*3^(1/2)*x^4*a^2-6*I*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(

-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2

)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3

))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(I*

3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+

3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*3^(1/2)*x^4*a^2-12*I*(-a^2*

b)^(1/3)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1

/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1

/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b

)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)

))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*3^(1/2

)*x^3*a+12*I*(-a^2*b)^(1/3)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/

3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+

(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/

2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/

(-a*x+(-a^2*b)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/

2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*3^(1/2)*x^3*a+6*I*(-a^2*b)^(2/3)*(-(I*

3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(

1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2

)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2

)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3

^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*3^(1/2)*x^2-6*I*(-a^

2*b)^(2/3)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^

(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^

(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2

*b)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1

/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+

1)/(I*3^(1/2)-3))^(1/2))*3^(1/2)*x^2-6*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(

-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2

)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3

))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3

^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1

)/(I*3^(1/2)-3))^(1/2))*x^4*a^2+6*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*

b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/

(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I

*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(I*3^(1/

2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I

*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*x^4*a^2+12*(-a^2*b)^(1/3)*(-(I*3

^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1

/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)

*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)

*EllipticF((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^

(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*x^3*a-12*(-a^2*b)^(1/

3)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-

a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((

I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3

)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1

/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^

(1/2)-3))^(1/2))*x^3*a-6*(-a^2*b)^(2/3)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/(-a*x+

(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I*3^(1/

2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/

3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticF((-(I*3^(1/2)-3)*x*a/(I*

3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),((I*3^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+

1)/(I*3^(1/2)-3))^(1/2))*x^2+6*(-a^2*b)^(2/3)*(-(I*3^(1/2)-3)*x*a/(I*3^(1/2)-1)/

(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))/(I

*3^(1/2)+1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*((I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*

b)^(1/3))/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*EllipticPi((-(I*3^(1/2)-3)*

x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2),(I*3^(1/2)-1)/(I*3^(1/2)-3),((I*3

^(1/2)+3)*(I*3^(1/2)-1)/(I*3^(1/2)+1)/(I*3^(1/2)-3))^(1/2))*x^2+I*(1/a^2*x*(-a*x

+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^

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

)^(1/2)*a*(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*x+(-a^2*b

)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(1/2))/(x*(a*x^3+b))^(

1/2)/(I*3^(1/2)-3)/(1/a^2*x*(-a*x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)+2*a*

x+(-a^2*b)^(1/3))*(I*3^(1/2)*(-a^2*b)^(1/3)-2*a*x-(-a^2*b)^(1/3)))^(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^3)/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A] time = 0.363743, size = 1, normalized size = 0.02 \[ \left [\frac{1}{6} \, \sqrt{a} \log \left (-8 \, a^{2} x^{6} - 8 \, a b x^{3} - b^{2} - 4 \,{\left (2 \, a x^{6} + b x^{3}\right )} \sqrt{a} \sqrt{\frac{a x^{3} + b}{x^{3}}}\right ) - \frac{2}{3} \, \sqrt{\frac{a x^{3} + b}{x^{3}}}, \frac{1}{3} \, \sqrt{-a} \arctan \left (\frac{2 \, a x^{3} \sqrt{\frac{a x^{3} + b}{x^{3}}}}{{\left (2 \, a x^{3} + b\right )} \sqrt{-a}}\right ) - \frac{2}{3} \, \sqrt{\frac{a x^{3} + b}{x^{3}}}\right ] \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*sqrt(a)*log(-8*a^2*x^6 - 8*a*b*x^3 - b^2 - 4*(2*a*x^6 + b*x^3)*sqrt(a)*sqrt

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

((a*x^3 + b)/x^3)/((2*a*x^3 + b)*sqrt(-a))) - 2/3*sqrt((a*x^3 + b)/x^3)]

_______________________________________________________________________________________

Sympy [A] time = 5.53078, size = 76, normalized size = 1.77 \[ \frac{2 \sqrt{a} \operatorname{asinh}{\left (\frac{\sqrt{a} x^{\frac{3}{2}}}{\sqrt{b}} \right )}}{3} - \frac{2 a x^{\frac{3}{2}}}{3 \sqrt{b} \sqrt{\frac{a x^{3}}{b} + 1}} - \frac{2 \sqrt{b}}{3 x^{\frac{3}{2}} \sqrt{\frac{a x^{3}}{b} + 1}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

2*sqrt(a)*asinh(sqrt(a)*x**(3/2)/sqrt(b))/3 - 2*a*x**(3/2)/(3*sqrt(b)*sqrt(a*x**

3/b + 1)) - 2*sqrt(b)/(3*x**(3/2)*sqrt(a*x**3/b + 1))

_______________________________________________________________________________________

GIAC/XCAS [A] time = 0.321774, size = 49, normalized size = 1.14 \[ -\frac{2 \, a \arctan \left (\frac{\sqrt{a + \frac{b}{x^{3}}}}{\sqrt{-a}}\right )}{3 \, \sqrt{-a}} - \frac{2}{3} \, \sqrt{a + \frac{b}{x^{3}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[next] [prev] [prev-tail] [front] [up]