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

Optimal. Leaf size=312 \[ -2 \sqrt{x^3+1} x \sqrt{\frac{a}{x^3}}+\frac{2 \left (1+\sqrt{3}\right ) \sqrt{x^3+1} x^2 \sqrt{\frac{a}{x^3}}}{\left (1+\sqrt{3}\right ) x+1}-\frac{\left (1-\sqrt{3}\right ) (x+1) \sqrt{\frac{x^2-x+1}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} x^2 \sqrt{\frac{a}{x^3}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x+1}{\left (1+\sqrt{3}\right ) x+1}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt [4]{3} \sqrt{\frac{x (x+1)}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} \sqrt{x^3+1}}-\frac{2 \sqrt [4]{3} (x+1) \sqrt{\frac{x^2-x+1}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} x^2 \sqrt{\frac{a}{x^3}} E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x+1}{\left (1+\sqrt{3}\right ) x+1}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt{\frac{x (x+1)}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} \sqrt{x^3+1}} \]

[Out]

-2*Sqrt[a/x^3]*x*Sqrt[1 + x^3] + (2*(1 + Sqrt[3])*Sqrt[a/x^3]*x^2*Sqrt[1 + x^3])
/(1 + (1 + Sqrt[3])*x) - (2*3^(1/4)*Sqrt[a/x^3]*x^2*(1 + x)*Sqrt[(1 - x + x^2)/(
1 + (1 + Sqrt[3])*x)^2]*EllipticE[ArcCos[(1 + (1 - Sqrt[3])*x)/(1 + (1 + Sqrt[3]
)*x)], (2 + Sqrt[3])/4])/(Sqrt[(x*(1 + x))/(1 + (1 + Sqrt[3])*x)^2]*Sqrt[1 + x^3
]) - ((1 - Sqrt[3])*Sqrt[a/x^3]*x^2*(1 + x)*Sqrt[(1 - x + x^2)/(1 + (1 + Sqrt[3]
)*x)^2]*EllipticF[ArcCos[(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 = 0.44452, antiderivative size = 312, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316 \[ -2 \sqrt{x^3+1} x \sqrt{\frac{a}{x^3}}+\frac{2 \left (1+\sqrt{3}\right ) \sqrt{x^3+1} x^2 \sqrt{\frac{a}{x^3}}}{\left (1+\sqrt{3}\right ) x+1}-\frac{\left (1-\sqrt{3}\right ) (x+1) \sqrt{\frac{x^2-x+1}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} x^2 \sqrt{\frac{a}{x^3}} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x+1}{\left (1+\sqrt{3}\right ) x+1}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt [4]{3} \sqrt{\frac{x (x+1)}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} \sqrt{x^3+1}}-\frac{2 \sqrt [4]{3} (x+1) \sqrt{\frac{x^2-x+1}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} x^2 \sqrt{\frac{a}{x^3}} E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x+1}{\left (1+\sqrt{3}\right ) x+1}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{\sqrt{\frac{x (x+1)}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} \sqrt{x^3+1}} \]

Antiderivative was successfully verified.

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

[Out]

-2*Sqrt[a/x^3]*x*Sqrt[1 + x^3] + (2*(1 + Sqrt[3])*Sqrt[a/x^3]*x^2*Sqrt[1 + x^3])
/(1 + (1 + Sqrt[3])*x) - (2*3^(1/4)*Sqrt[a/x^3]*x^2*(1 + x)*Sqrt[(1 - x + x^2)/(
1 + (1 + Sqrt[3])*x)^2]*EllipticE[ArcCos[(1 + (1 - Sqrt[3])*x)/(1 + (1 + Sqrt[3]
)*x)], (2 + Sqrt[3])/4])/(Sqrt[(x*(1 + x))/(1 + (1 + Sqrt[3])*x)^2]*Sqrt[1 + x^3
]) - ((1 - Sqrt[3])*Sqrt[a/x^3]*x^2*(1 + x)*Sqrt[(1 - x + x^2)/(1 + (1 + Sqrt[3]
)*x)^2]*EllipticF[ArcCos[(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 = 21.1696, size = 282, normalized size = 0.9 \[ - \frac{2 \sqrt [4]{3} x^{2} \sqrt{\frac{a}{x^{3}}} \sqrt{\frac{x^{2} - x + 1}{\left (x \left (1 + \sqrt{3}\right ) + 1\right )^{2}}} \left (x + 1\right ) E\left (\operatorname{acos}{\left (\frac{x \left (- \sqrt{3} + 1\right ) + 1}{x \left (1 + \sqrt{3}\right ) + 1} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{\sqrt{\frac{x \left (x + 1\right )}{\left (x \left (1 + \sqrt{3}\right ) + 1\right )^{2}}} \sqrt{x^{3} + 1}} - \frac{2 \cdot 3^{\frac{3}{4}} x^{2} \sqrt{\frac{a}{x^{3}}} \sqrt{\frac{x^{2} - x + 1}{\left (x \left (1 + \sqrt{3}\right ) + 1\right )^{2}}} \left (- \frac{\sqrt{3}}{2} + \frac{1}{2}\right ) \left (x + 1\right ) F\left (\operatorname{acos}{\left (\frac{x \left (- \sqrt{3} + 1\right ) + 1}{x \left (1 + \sqrt{3}\right ) + 1} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{3 \sqrt{\frac{x \left (x + 1\right )}{\left (x \left (1 + \sqrt{3}\right ) + 1\right )^{2}}} \sqrt{x^{3} + 1}} + \frac{x^{2} \sqrt{\frac{a}{x^{3}}} \left (4 + 4 \sqrt{3}\right ) \sqrt{x^{3} + 1}}{x \left (2 + 2 \sqrt{3}\right ) + 2} - 2 x \sqrt{\frac{a}{x^{3}}} \sqrt{x^{3} + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*3**(1/4)*x**2*sqrt(a/x**3)*sqrt((x**2 - x + 1)/(x*(1 + sqrt(3)) + 1)**2)*(x +
1)*elliptic_e(acos((x*(-sqrt(3) + 1) + 1)/(x*(1 + sqrt(3)) + 1)), sqrt(3)/4 + 1
/2)/(sqrt(x*(x + 1)/(x*(1 + sqrt(3)) + 1)**2)*sqrt(x**3 + 1)) - 2*3**(3/4)*x**2*
sqrt(a/x**3)*sqrt((x**2 - x + 1)/(x*(1 + sqrt(3)) + 1)**2)*(-sqrt(3)/2 + 1/2)*(x
+ 1)*elliptic_f(acos((x*(-sqrt(3) + 1) + 1)/(x*(1 + sqrt(3)) + 1)), sqrt(3)/4 +
1/2)/(3*sqrt(x*(x + 1)/(x*(1 + sqrt(3)) + 1)**2)*sqrt(x**3 + 1)) + x**2*sqrt(a/
x**3)*(4 + 4*sqrt(3))*sqrt(x**3 + 1)/(x*(2 + 2*sqrt(3)) + 2) - 2*x*sqrt(a/x**3)*
sqrt(x**3 + 1)

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Mathematica [C]  time = 0.461757, size = 165, normalized size = 0.53 \[ -\frac{\sqrt{\frac{2}{3}} \left ((-1)^{2/3}-1\right ) a \sqrt{\frac{x-\sqrt [3]{-1}}{\left (1+\sqrt [3]{-1}\right ) x}} \sqrt{\frac{(x+1) \left (2 x+i \sqrt{3}-1\right )}{x^2}} \left (\left (1+\sqrt [3]{-1}\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} (x+1)}{\left (-1+(-1)^{2/3}\right ) x}}\right )|1+(-1)^{2/3}\right )-F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} (x+1)}{\left (-1+(-1)^{2/3}\right ) x}}\right )|1+(-1)^{2/3}\right )\right )}{\sqrt{x^3+1} \sqrt{\frac{a}{x^3}}} \]

Antiderivative was successfully verified.

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

[Out]

-((Sqrt[2/3]*(-1 + (-1)^(2/3))*a*Sqrt[(-(-1)^(1/3) + x)/((1 + (-1)^(1/3))*x)]*Sq
rt[((1 + x)*(-1 + I*Sqrt[3] + 2*x))/x^2]*((1 + (-1)^(1/3))*EllipticE[ArcSin[Sqrt
[((-1)^(2/3)*(1 + x))/((-1 + (-1)^(2/3))*x)]], 1 + (-1)^(2/3)] - EllipticF[ArcSi
n[Sqrt[((-1)^(2/3)*(1 + x))/((-1 + (-1)^(2/3))*x)]], 1 + (-1)^(2/3)]))/(Sqrt[a/x
^3]*Sqrt[1 + x^3]))

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Maple [C]  time = 0.096, size = 1784, normalized size = 5.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*(a/x^3)^(1/2)*x/(x^3+1)^(1/2)*(4*I*3^(1/2)*(x*(x^3+1))^(1/2)*((I*3^(1/2)+3)*x
/(I*3^(1/2)+1)/(1+x))^(1/2)*((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)*EllipticE(((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+I*3^(1/2)*(-x*(1+x)*(I*3^(1/2)+2*x-1)*(I*3^(1/2)-2*x+1))^(1/2)*x^3+2*I*3^(1/
2)*(x*(x^3+1))^(1/2)*x^2-4*(x*(x^3+1))^(1/2)*((I*3^(1/2)+3)*x/(I*3^(1/2)+1)/(1+x
))^(1/2)*((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^2+6*(x*(x^3+1))
^(1/2)*((I*3^(1/2)+3)*x/(I*3^(1/2)+1)/(1+x))^(1/2)*((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)*EllipticE(((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^2+2*I*3^(1/2)*(x*(x^3+1))^(1/2)*((I*3^(1/2)+3)*x/(I*3
^(1/2)+1)/(1+x))^(1/2)*((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)*EllipticE(((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^2
-2*I*3^(1/2)*(x*(x^3+1))^(1/2)*x^3-8*(x*(x^3+1))^(1/2)*((I*3^(1/2)+3)*x/(I*3^(1/
2)+1)/(1+x))^(1/2)*((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+12*(x
*(x^3+1))^(1/2)*((I*3^(1/2)+3)*x/(I*3^(1/2)+1)/(1+x))^(1/2)*((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)*Ellipti
cE(((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+I*3^(1/2)*(-x*(1+x)*(I*3^(1/2)+2*x-1)*(I*3^(
1/2)-2*x+1))^(1/2)-4*(x*(x^3+1))^(1/2)*((I*3^(1/2)+3)*x/(I*3^(1/2)+1)/(1+x))^(1/
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))+6*(x*(x^3+1))^(1/2)*((I
*3^(1/2)+3)*x/(I*3^(1/2)+1)/(1+x))^(1/2)*((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)*EllipticE(((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))+2*I*3^(1/2)*(x*(x^3+1))^(1/2)*((I*3^(1/2)+3)*x/(I*3^(1/2)+1)/(1+x
))^(1/2)*((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)*EllipticE(((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))-6*(x*(x^3+1))^(1/
2)*x^3+3*(-x*(1+x)*(I*3^(1/2)+2*x-1)*(I*3^(1/2)-2*x+1))^(1/2)*x^3-2*I*3^(1/2)*(x
*(x^3+1))^(1/2)*x+6*(x*(x^3+1))^(1/2)*x^2-6*(x*(x^3+1))^(1/2)*x+3*(-x*(1+x)*(I*3
^(1/2)+2*x-1)*(I*3^(1/2)-2*x+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. \[ \int \frac{\sqrt{\frac{a}{x^{3}}}}{\sqrt{x^{3} + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a/x^3)/sqrt(x^3 + 1), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\frac{a}{x^{3}}}}{\sqrt{\left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(sqrt(a/x^3)/sqrt(x^3 + 1), x)