3.2024 \(\int \frac{x}{\sqrt{a+\frac{b}{x^3}}} \, dx\)

Optimal. Leaf size=248 \[ \frac{\sqrt{2+\sqrt{3}} b^{2/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt [4]{3} a \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{x^2 \sqrt{a+\frac{b}{x^3}}}{2 a} \]

[Out]

(Sqrt[a + b/x^3]*x^2)/(2*a) + (Sqrt[2 + Sqrt[3]]*b^(2/3)*(a^(1/3) + b^(1/3)/x)*S
qrt[(a^(2/3) + b^(2/3)/x^2 - (a^(1/3)*b^(1/3))/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/
3)/x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(
1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]])/(2*3^(1/4)*a*Sqrt[a + b/x^3]*Sqrt[(a^(1/3)*
(a^(1/3) + b^(1/3)/x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^2])

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Rubi [A]  time = 0.264369, antiderivative size = 248, 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 \[ \frac{\sqrt{2+\sqrt{3}} b^{2/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt [4]{3} a \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{x^2 \sqrt{a+\frac{b}{x^3}}}{2 a} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[a + b/x^3],x]

[Out]

(Sqrt[a + b/x^3]*x^2)/(2*a) + (Sqrt[2 + Sqrt[3]]*b^(2/3)*(a^(1/3) + b^(1/3)/x)*S
qrt[(a^(2/3) + b^(2/3)/x^2 - (a^(1/3)*b^(1/3))/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/
3)/x)^2]*EllipticF[ArcSin[((1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(
1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]])/(2*3^(1/4)*a*Sqrt[a + b/x^3]*Sqrt[(a^(1/3)*
(a^(1/3) + b^(1/3)/x))/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)^2])

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Rubi in Sympy [A]  time = 12.0511, size = 201, normalized size = 0.81 \[ \frac{3^{\frac{3}{4}} b^{\frac{2}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \frac{\sqrt [3]{a} \sqrt [3]{b}}{x} + \frac{b^{\frac{2}{3}}}{x^{2}}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{6 a \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{a + \frac{b}{x^{3}}}} + \frac{x^{2} \sqrt{a + \frac{b}{x^{3}}}}{2 a} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3**(3/4)*b**(2/3)*sqrt((a**(2/3) - a**(1/3)*b**(1/3)/x + b**(2/3)/x**2)/(a**(1/3
)*(1 + sqrt(3)) + b**(1/3)/x)**2)*sqrt(sqrt(3) + 2)*(a**(1/3) + b**(1/3)/x)*elli
ptic_f(asin((-a**(1/3)*(-1 + sqrt(3)) + b**(1/3)/x)/(a**(1/3)*(1 + sqrt(3)) + b*
*(1/3)/x)), -7 - 4*sqrt(3))/(6*a*sqrt(a**(1/3)*(a**(1/3) + b**(1/3)/x)/(a**(1/3)
*(1 + sqrt(3)) + b**(1/3)/x)**2)*sqrt(a + b/x**3)) + x**2*sqrt(a + b/x**3)/(2*a)

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Mathematica [C]  time = 0.39224, size = 174, normalized size = 0.7 \[ \frac{a x^3+b}{2 a x \sqrt{a+\frac{b}{x^3}}}+\frac{i b \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-b}}{\sqrt [3]{a} x}-1\right )} \sqrt{\frac{(-b)^{2/3}}{a^{2/3} x^2}+\frac{\sqrt [3]{-b}}{\sqrt [3]{a} x}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b}}{\sqrt [3]{a} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{2 \sqrt [4]{3} a^{2/3} \sqrt [3]{-b} \sqrt{a+\frac{b}{x^3}}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x/Sqrt[a + b/x^3],x]

[Out]

(b + a*x^3)/(2*a*Sqrt[a + b/x^3]*x) + ((I/2)*b*Sqrt[(-1)^(5/6)*(-1 + (-b)^(1/3)/
(a^(1/3)*x))]*Sqrt[1 + (-b)^(2/3)/(a^(2/3)*x^2) + (-b)^(1/3)/(a^(1/3)*x)]*Ellipt
icF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-b)^(1/3))/(a^(1/3)*x)]/3^(1/4)], (-1)^(1/3)])
/(3^(1/4)*a^(2/3)*(-b)^(1/3)*Sqrt[a + b/x^3])

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Maple [B]  time = 0.015, size = 1793, normalized size = 7.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(a+b/x^3)^(1/2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{a + \frac{b}{x^{3}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/sqrt(a + b/x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x}{\sqrt{\frac{a x^{3} + b}{x^{3}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x/sqrt((a*x^3 + b)/x^3), x)

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Sympy [A]  time = 2.81385, size = 42, normalized size = 0.17 \[ - \frac{x^{2} \Gamma \left (- \frac{2}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{2} \\ \frac{1}{3} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt{a} \Gamma \left (\frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**2*gamma(-2/3)*hyper((-2/3, 1/2), (1/3,), b*exp_polar(I*pi)/(a*x**3))/(3*sqrt
(a)*gamma(1/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{\sqrt{a + \frac{b}{x^{3}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x/sqrt(a + b/x^3), x)