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

Optimal. Leaf size=116 \[ \frac{x (x+1) \sqrt{\frac{x^2-x+1}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} \sqrt{\frac{a}{x}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x+1}{\left (1+\sqrt{3}\right ) x+1}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt [4]{3} \sqrt{\frac{x (x+1)}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} \sqrt{x^3+1}} \]

[Out]

(Sqrt[a/x]*x*(1 + x)*Sqrt[(1 - x + x^2)/(1 + (1 + Sqrt[3])*x)^2]*EllipticF[ArcCo
s[(1 + (1 - Sqrt[3])*x)/(1 + (1 + Sqrt[3])*x)], (2 + Sqrt[3])/4])/(3^(1/4)*Sqrt[
(x*(1 + x))/(1 + (1 + Sqrt[3])*x)^2]*Sqrt[1 + x^3])

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Rubi [A]  time = 0.1479, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ \frac{x (x+1) \sqrt{\frac{x^2-x+1}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} \sqrt{\frac{a}{x}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x+1}{\left (1+\sqrt{3}\right ) x+1}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt [4]{3} \sqrt{\frac{x (x+1)}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} \sqrt{x^3+1}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a/x]/Sqrt[1 + x^3],x]

[Out]

(Sqrt[a/x]*x*(1 + x)*Sqrt[(1 - x + x^2)/(1 + (1 + Sqrt[3])*x)^2]*EllipticF[ArcCo
s[(1 + (1 - Sqrt[3])*x)/(1 + (1 + Sqrt[3])*x)], (2 + Sqrt[3])/4])/(3^(1/4)*Sqrt[
(x*(1 + x))/(1 + (1 + Sqrt[3])*x)^2]*Sqrt[1 + x^3])

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Rubi in Sympy [A]  time = 9.03999, size = 100, normalized size = 0.86 \[ \frac{3^{\frac{3}{4}} x \sqrt{\frac{a}{x}} \sqrt{\frac{x^{2} - x + 1}{\left (x \left (1 + \sqrt{3}\right ) + 1\right )^{2}}} \left (x + 1\right ) F\left (\operatorname{acos}{\left (\frac{x \left (- \sqrt{3} + 1\right ) + 1}{x \left (1 + \sqrt{3}\right ) + 1} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{3 \sqrt{\frac{x \left (x + 1\right )}{\left (x \left (1 + \sqrt{3}\right ) + 1\right )^{2}}} \sqrt{x^{3} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3**(3/4)*x*sqrt(a/x)*sqrt((x**2 - x + 1)/(x*(1 + sqrt(3)) + 1)**2)*(x + 1)*ellip
tic_f(acos((x*(-sqrt(3) + 1) + 1)/(x*(1 + sqrt(3)) + 1)), sqrt(3)/4 + 1/2)/(3*sq
rt(x*(x + 1)/(x*(1 + sqrt(3)) + 1)**2)*sqrt(x**3 + 1))

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Mathematica [A]  time = 0.16617, size = 106, normalized size = 0.91 \[ -\frac{2 \sqrt [6]{-1} \sqrt{-\sqrt [6]{-1} \left (\frac{1}{x}+(-1)^{2/3}\right )} \sqrt{\frac{(-1)^{2/3}}{x^2}+\frac{\sqrt [3]{-1}}{x}+1} x^2 \sqrt{\frac{a}{x}} F\left (\sin ^{-1}\left (\frac{\sqrt{-(-1)^{5/6} \left (1+\frac{1}{x}\right )}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{\sqrt [4]{3} \sqrt{x^3+1}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[Sqrt[a/x]/Sqrt[1 + x^3],x]

[Out]

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

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Maple [C]  time = 0.146, size = 232, normalized size = 2. \[ 4\,{\frac{x\sqrt{{x}^{3}+1} \left ( i\sqrt{3}+1 \right ) \left ( 1+x \right ) ^{2}}{\sqrt{x \left ({x}^{3}+1 \right ) } \left ( i\sqrt{3}+3 \right ) \sqrt{-x \left ( 1+x \right ) \left ( i\sqrt{3}+2\,x-1 \right ) \left ( i\sqrt{3}-2\,x+1 \right ) }}\sqrt{{\frac{a}{x}}}\sqrt{{\frac{ \left ( i\sqrt{3}+3 \right ) x}{ \left ( i\sqrt{3}+1 \right ) \left ( 1+x \right ) }}}\sqrt{{\frac{i\sqrt{3}+2\,x-1}{ \left ( -1+i\sqrt{3} \right ) \left ( 1+x \right ) }}}\sqrt{{\frac{i\sqrt{3}-2\,x+1}{ \left ( i\sqrt{3}+1 \right ) \left ( 1+x \right ) }}}{\it EllipticF} \left ( \sqrt{{\frac{ \left ( i\sqrt{3}+3 \right ) x}{ \left ( i\sqrt{3}+1 \right ) \left ( 1+x \right ) }}},\sqrt{{\frac{ \left ( -3+i\sqrt{3} \right ) \left ( i\sqrt{3}+1 \right ) }{ \left ( -1+i\sqrt{3} \right ) \left ( i\sqrt{3}+3 \right ) }}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{a}{x}}}{\sqrt{x^{3} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a/x)/sqrt(x^3 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(sqrt(a/x)/sqrt(x^3 + 1), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{a}{x}}}{\sqrt{\left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a/x)/sqrt((x + 1)*(x**2 - x + 1)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a/x)/sqrt(x^3 + 1), x)