3.2004 \(\int \sqrt{a+\frac{b}{x^3}} \, dx\)

Optimal. Leaf size=507 \[ -\frac{\sqrt{2} 3^{3/4} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{\sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+x \sqrt{a+\frac{b}{x^3}}-\frac{3 \sqrt [3]{b} \sqrt{a+\frac{b}{x^3}}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}} \]

[Out]

(-3*b^(1/3)*Sqrt[a + b/x^3])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x) + Sqrt[a + b/x^
3]*x + (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*b^(1/3)*(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]*EllipticE[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*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]) - (Sqrt[2]*3^(3/4)*a^(1/3)*b^(1/
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 - Sqrt[3])*a^(1/3) + b^(
1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]])/(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 = 0.648055, antiderivative size = 507, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.454 \[ -\frac{\sqrt{2} 3^{3/4} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{\sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{3 \sqrt [4]{3} \sqrt{2-\sqrt{3}} \sqrt [3]{a} \sqrt [3]{b} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{2 \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+x \sqrt{a+\frac{b}{x^3}}-\frac{3 \sqrt [3]{b} \sqrt{a+\frac{b}{x^3}}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b/x^3],x]

[Out]

(-3*b^(1/3)*Sqrt[a + b/x^3])/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x) + Sqrt[a + b/x^
3]*x + (3*3^(1/4)*Sqrt[2 - Sqrt[3]]*a^(1/3)*b^(1/3)*(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]*EllipticE[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*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]) - (Sqrt[2]*3^(3/4)*a^(1/3)*b^(1/
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 - Sqrt[3])*a^(1/3) + b^(
1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 - 4*Sqrt[3]])/(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 = 34.3546, size = 418, normalized size = 0.82 \[ \frac{3 \sqrt [4]{3} \sqrt [3]{a} \sqrt [3]{b} \sqrt{\frac{a^{\frac{2}{3}} - \frac{\sqrt [3]{a} \sqrt [3]{b}}{x} + \frac{b^{\frac{2}{3}}}{x^{2}}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{2 \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{a + \frac{b}{x^{3}}}} - \frac{\sqrt{2} \cdot 3^{\frac{3}{4}} \sqrt [3]{a} \sqrt [3]{b} \sqrt{\frac{a^{\frac{2}{3}} - \frac{\sqrt [3]{a} \sqrt [3]{b}}{x} + \frac{b^{\frac{2}{3}}}{x^{2}}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{\sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{a + \frac{b}{x^{3}}}} - \frac{3 \sqrt [3]{b} \sqrt{a + \frac{b}{x^{3}}}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} + x \sqrt{a + \frac{b}{x^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3*3**(1/4)*a**(1/3)*b**(1/3)*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)*elliptic_e(asin((-a**(1/3)*(-1 + sqrt(3)) + b**(1/3)/x)/(a**(1/3)*(1 + s
qrt(3)) + b**(1/3)/x)), -7 - 4*sqrt(3))/(2*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)) - sqrt(2)*3**(3/4)*
a**(1/3)*b**(1/3)*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)*(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))/(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)) - 3*b**(1/3)*sqrt(a + b/x**3)/(a**(1/3)*(1 + sqrt(3
)) + b**(1/3)/x) + x*sqrt(a + b/x**3)

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Mathematica [C]  time = 1.3442, size = 351, normalized size = 0.69 \[ x \sqrt{a+\frac{b}{x^3}} \left (\frac{3 (-1)^{2/3} \sqrt [3]{b} \left (\sqrt [3]{a} x+\sqrt [3]{b}\right ) \sqrt{\frac{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} x \left (\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} x\right )}{\left (\sqrt [3]{a} x+\sqrt [3]{b}\right )^2}} \sqrt{\frac{(-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}}{\sqrt [3]{a} x+\sqrt [3]{b}}} \left (\left (1+i \sqrt{3}\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\left (3+i \sqrt{3}\right ) \sqrt [3]{a} x}{\sqrt [3]{a} x+\sqrt [3]{b}}}}{\sqrt{2}}\right )|\frac{-i+\sqrt{3}}{i+\sqrt{3}}\right )+\left (-3-i \sqrt{3}\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\left (3+i \sqrt{3}\right ) \sqrt [3]{a} x}{\sqrt [3]{a} x+\sqrt [3]{b}}}}{\sqrt{2}}\right )|\frac{-i+\sqrt{3}}{i+\sqrt{3}}\right )\right )}{2 \left ((-1)^{2/3}-1\right ) \left (a^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3}\right )}+\frac{3 \sqrt [3]{a} x}{\sqrt [3]{a} x+\sqrt [3]{b}}-2\right ) \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[a + b/x^3],x]

[Out]

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

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Maple [B]  time = 0.037, size = 2864, normalized size = 5.7 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+b/x^3)^(1/2),x)

[Out]

-2*((a*x^3+b)/x^3)^(1/2)*x/a*(3*I*((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)*EllipticE((-(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)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^
2*b)^(1/3)))^(1/2)*3^(1/2)*(x*(a*x^3+b))^(1/2)*a*b-3*I*(-a^2*b)^(1/3)*3^(1/2)*(x
*(a*x^3+b))^(1/2)*x^2*a-6*(-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)+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)-3)*x*a/(I*3^(1/2)-1)/(-a
*x+(-a^2*b)^(1/3)))^(1/2)*(x*(a*x^3+b))^(1/2)*x^2*a+9*(-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)+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)*EllipticE((-(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)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(x*(a*x^3+b))^(1/2)*x^2*a-
3*I*(-a^2*b)^(2/3)*3^(1/2)*(x*(a*x^3+b))^(1/2)*x+12*(-a^2*b)^(2/3)*((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
)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(x*(a*x^3+b))^(1/2)*x-18*(-a
^2*b)^(2/3)*((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)*EllipticE((-(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)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(x*
(a*x^3+b))^(1/2)*x+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*b-3*I*3^(1/2)*(x*(a*x^3+b))^(1/2)*x^3*a^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)*x^3*a^2+6*((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)-3)*x*a/(I*3^(1/2)-1)/(-a
*x+(-a^2*b)^(1/3)))^(1/2)*(x*(a*x^3+b))^(1/2)*a*b-9*((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)*Elli
pticE((-(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)-3)*x*a/(I*3^(
1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*(x*(a*x^3+b))^(1/2)*a*b-3*I*(-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)+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)*EllipticE((-(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)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*3^(1/2)*(x*(a*x
^3+b))^(1/2)*x^2*a-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^3*a
^2+9*(x*(a*x^3+b))^(1/2)*x^3*a^2+9*(-a^2*b)^(1/3)*(x*(a*x^3+b))^(1/2)*x^2*a+6*I*
(-a^2*b)^(2/3)*((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)*EllipticE((-(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)-3)*x*a/(I*3^(1/2)-1)/(-a*x+(-a^2*b)^(1/3)))^(1/2)*
3^(1/2)*(x*(a*x^3+b))^(1/2)*x+9*(-a^2*b)^(2/3)*(x*(a*x^3+b))^(1/2)*x-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)*a*b)/(a*x^3+b)/(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. \[ \int \sqrt{a + \frac{b}{x^{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a + b/x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt((a*x^3 + b)/x^3), x)

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Sympy [A]  time = 2.7907, size = 42, normalized size = 0.08 \[ - \frac{\sqrt{a} x \Gamma \left (- \frac{1}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{2}, - \frac{1}{3} \\ \frac{2}{3} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{3}}} \right )}}{3 \Gamma \left (\frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+b/x**3)**(1/2),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a + b/x^3), x)