3.2025 \(\int \frac{1}{\sqrt{a+\frac{b}{x^3}} x^2} \, dx\)

Optimal. Leaf size=221 \[ -\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}} \]

[Out]

(-2*Sqrt[2 + Sqrt[3]]*(a^(1/3) + b^(1/3)/x)*Sqrt[(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 - Sqr
t[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]]
)/(3^(1/4)*b^(1/3)*Sqrt[a + b/x^3]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)/x))/((1 + Sq
rt[3])*a^(1/3) + b^(1/3)/x)^2])

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Rubi [A]  time = 0.192498, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ -\frac{2 \sqrt{2+\sqrt{3}} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{\sqrt [4]{3} \sqrt [3]{b} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b/x^3]*x^2),x]

[Out]

(-2*Sqrt[2 + Sqrt[3]]*(a^(1/3) + b^(1/3)/x)*Sqrt[(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 - Sqr
t[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]]
)/(3^(1/4)*b^(1/3)*Sqrt[a + b/x^3]*Sqrt[(a^(1/3)*(a^(1/3) + b^(1/3)/x))/((1 + Sq
rt[3])*a^(1/3) + b^(1/3)/x)^2])

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Rubi in Sympy [A]  time = 6.86967, size = 185, normalized size = 0.84 \[ - \frac{2 \cdot 3^{\frac{3}{4}} \sqrt{\frac{a^{\frac{2}{3}} - \frac{\sqrt [3]{a} \sqrt [3]{b}}{x} + \frac{b^{\frac{2}{3}}}{x^{2}}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{3 \sqrt [3]{b} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{a + \frac{b}{x^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*3**(3/4)*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)*elliptic_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))/(3*b**(1/3)*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))

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Mathematica [C]  time = 0.209595, size = 142, normalized size = 0.64 \[ -\frac{2 i \sqrt [3]{a} \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-b}}{\sqrt [3]{a} x}-1\right )} \sqrt{\frac{(-b)^{2/3}}{a^{2/3} x^2}+\frac{\sqrt [3]{-b}}{\sqrt [3]{a} x}+1} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b}}{\sqrt [3]{a} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{\sqrt [4]{3} \sqrt [3]{-b} \sqrt{a+\frac{b}{x^3}}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/(Sqrt[a + b/x^3]*x^2),x]

[Out]

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

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Maple [B]  time = 0.021, size = 437, normalized size = 2. \[ -4\,{\frac{ \left ( a{x}^{3}+b \right ) \left ( i\sqrt{3}{x}^{2}{a}^{2}-2\,i\sqrt [3]{-{a}^{2}b}\sqrt{3}xa+i \left ( -{a}^{2}b \right ) ^{2/3}\sqrt{3}-{a}^{2}{x}^{2}+2\,\sqrt [3]{-{a}^{2}b}xa- \left ( -{a}^{2}b \right ) ^{2/3} \right ) }{\sqrt [3]{-{a}^{2}b}xa\sqrt{x \left ( a{x}^{3}+b \right ) } \left ( i\sqrt{3}-3 \right ) }\sqrt{-{\frac{ \left ( i\sqrt{3}-3 \right ) xa}{ \left ( i\sqrt{3}-1 \right ) \left ( -ax+\sqrt [3]{-{a}^{2}b} \right ) }}}\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-{a}^{2}b}+2\,ax+\sqrt [3]{-{a}^{2}b}}{ \left ( i\sqrt{3}+1 \right ) \left ( -ax+\sqrt [3]{-{a}^{2}b} \right ) }}}\sqrt{{\frac{i\sqrt{3}\sqrt [3]{-{a}^{2}b}-2\,ax-\sqrt [3]{-{a}^{2}b}}{ \left ( i\sqrt{3}-1 \right ) \left ( -ax+\sqrt [3]{-{a}^{2}b} \right ) }}}{\it EllipticF} \left ( \sqrt{-{\frac{ \left ( i\sqrt{3}-3 \right ) xa}{ \left ( i\sqrt{3}-1 \right ) \left ( -ax+\sqrt [3]{-{a}^{2}b} \right ) }}},\sqrt{{\frac{ \left ( i\sqrt{3}+3 \right ) \left ( i\sqrt{3}-1 \right ) }{ \left ( i\sqrt{3}+1 \right ) \left ( i\sqrt{3}-3 \right ) }}} \right ){\frac{1}{\sqrt{{\frac{a{x}^{3}+b}{{x}^{3}}}}}}{\frac{1}{\sqrt{{\frac{x \left ( -ax+\sqrt [3]{-{a}^{2}b} \right ) \left ( i\sqrt{3}\sqrt [3]{-{a}^{2}b}+2\,ax+\sqrt [3]{-{a}^{2}b} \right ) \left ( i\sqrt{3}\sqrt [3]{-{a}^{2}b}-2\,ax-\sqrt [3]{-{a}^{2}b} \right ) }{{a}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^2/(a+b/x^3)^(1/2),x)

[Out]

-4/((a*x^3+b)/x^3)^(1/2)/x*(a*x^3+b)/(-a^2*b)^(1/3)/a*(-(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))*(I*3^(1/2)*x^2*a^2-2*I*(-a^2*b)^(1/3)*3^(
1/2)*x*a+I*(-a^2*b)^(2/3)*3^(1/2)-a^2*x^2+2*(-a^2*b)^(1/3)*x*a-(-a^2*b)^(2/3))/(
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. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{3}}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^3)*x^2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(a + b/x^3)*x^2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{x^{2} \sqrt{\frac{a x^{3} + b}{x^{3}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^3)*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.32182, size = 37, normalized size = 0.17 \[ - \frac{\Gamma \left (\frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{4}{3} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt{a} x \Gamma \left (\frac{4}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-gamma(1/3)*hyper((1/3, 1/2), (4/3,), b*exp_polar(I*pi)/(a*x**3))/(3*sqrt(a)*x*g
amma(4/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{a + \frac{b}{x^{3}}} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(a + b/x^3)*x^2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(a + b/x^3)*x^2), x)