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

Optimal. Leaf size=292 \[ \frac{\left (1+\sqrt{3}\right ) \sqrt{x^3+1} \sqrt{a x^3}}{x \left (\left (1+\sqrt{3}\right ) x+1\right )}-\frac{\left (1-\sqrt{3}\right ) (x+1) \sqrt{\frac{x^2-x+1}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} \sqrt{a x^3} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x+1}{\left (1+\sqrt{3}\right ) x+1}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2 \sqrt [4]{3} x \sqrt{\frac{x (x+1)}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} \sqrt{x^3+1}}-\frac{\sqrt [4]{3} (x+1) \sqrt{\frac{x^2-x+1}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} \sqrt{a x^3} E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x+1}{\left (1+\sqrt{3}\right ) x+1}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{x \sqrt{\frac{x (x+1)}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} \sqrt{x^3+1}} \]

[Out]

((1 + Sqrt[3])*Sqrt[a*x^3]*Sqrt[1 + x^3])/(x*(1 + (1 + Sqrt[3])*x)) - (3^(1/4)*S
qrt[a*x^3]*(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])/(x*Sqrt[(x*(1 +
x))/(1 + (1 + Sqrt[3])*x)^2]*Sqrt[1 + x^3]) - ((1 - Sqrt[3])*Sqrt[a*x^3]*(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])/(2*3^(1/4)*x*Sqrt[(x*(1 + x))/(1 +
(1 + Sqrt[3])*x)^2]*Sqrt[1 + x^3])

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Rubi [A]  time = 0.424752, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263 \[ \frac{\left (1+\sqrt{3}\right ) \sqrt{x^3+1} \sqrt{a x^3}}{x \left (\left (1+\sqrt{3}\right ) x+1\right )}-\frac{\left (1-\sqrt{3}\right ) (x+1) \sqrt{\frac{x^2-x+1}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} \sqrt{a x^3} F\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x+1}{\left (1+\sqrt{3}\right ) x+1}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{2 \sqrt [4]{3} x \sqrt{\frac{x (x+1)}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} \sqrt{x^3+1}}-\frac{\sqrt [4]{3} (x+1) \sqrt{\frac{x^2-x+1}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} \sqrt{a x^3} E\left (\cos ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) x+1}{\left (1+\sqrt{3}\right ) x+1}\right )|\frac{1}{4} \left (2+\sqrt{3}\right )\right )}{x \sqrt{\frac{x (x+1)}{\left (\left (1+\sqrt{3}\right ) x+1\right )^2}} \sqrt{x^3+1}} \]

Antiderivative was successfully verified.

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

[Out]

((1 + Sqrt[3])*Sqrt[a*x^3]*Sqrt[1 + x^3])/(x*(1 + (1 + Sqrt[3])*x)) - (3^(1/4)*S
qrt[a*x^3]*(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])/(x*Sqrt[(x*(1 +
x))/(1 + (1 + Sqrt[3])*x)^2]*Sqrt[1 + x^3]) - ((1 - Sqrt[3])*Sqrt[a*x^3]*(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])/(2*3^(1/4)*x*Sqrt[(x*(1 + x))/(1 +
(1 + Sqrt[3])*x)^2]*Sqrt[1 + x^3])

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Rubi in Sympy [A]  time = 19.2598, size = 253, normalized size = 0.87 \[ - \frac{\sqrt [4]{3} \sqrt{a x^{3}} \sqrt{\frac{x^{2} - x + 1}{\left (x \left (1 + \sqrt{3}\right ) + 1\right )^{2}}} \left (x + 1\right ) E\left (\operatorname{acos}{\left (\frac{x \left (- \sqrt{3} + 1\right ) + 1}{x \left (1 + \sqrt{3}\right ) + 1} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{x \sqrt{\frac{x \left (x + 1\right )}{\left (x \left (1 + \sqrt{3}\right ) + 1\right )^{2}}} \sqrt{x^{3} + 1}} - \frac{3^{\frac{3}{4}} \sqrt{a x^{3}} \sqrt{\frac{x^{2} - x + 1}{\left (x \left (1 + \sqrt{3}\right ) + 1\right )^{2}}} \left (- \frac{\sqrt{3}}{2} + \frac{1}{2}\right ) \left (x + 1\right ) F\left (\operatorname{acos}{\left (\frac{x \left (- \sqrt{3} + 1\right ) + 1}{x \left (1 + \sqrt{3}\right ) + 1} \right )}\middle | \frac{\sqrt{3}}{4} + \frac{1}{2}\right )}{3 x \sqrt{\frac{x \left (x + 1\right )}{\left (x \left (1 + \sqrt{3}\right ) + 1\right )^{2}}} \sqrt{x^{3} + 1}} + \frac{\sqrt{a x^{3}} \left (2 + 2 \sqrt{3}\right ) \sqrt{x^{3} + 1}}{x \left (x \left (2 + 2 \sqrt{3}\right ) + 2\right )} \]

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]

-3**(1/4)*sqrt(a*x**3)*sqrt((x**2 - x + 1)/(x*(1 + sqrt(3)) + 1)**2)*(x + 1)*ell
iptic_e(acos((x*(-sqrt(3) + 1) + 1)/(x*(1 + sqrt(3)) + 1)), sqrt(3)/4 + 1/2)/(x*
sqrt(x*(x + 1)/(x*(1 + sqrt(3)) + 1)**2)*sqrt(x**3 + 1)) - 3**(3/4)*sqrt(a*x**3)
*sqrt((x**2 - x + 1)/(x*(1 + sqrt(3)) + 1)**2)*(-sqrt(3)/2 + 1/2)*(x + 1)*ellipt
ic_f(acos((x*(-sqrt(3) + 1) + 1)/(x*(1 + sqrt(3)) + 1)), sqrt(3)/4 + 1/2)/(3*x*s
qrt(x*(x + 1)/(x*(1 + sqrt(3)) + 1)**2)*sqrt(x**3 + 1)) + sqrt(a*x**3)*(2 + 2*sq
rt(3))*sqrt(x**3 + 1)/(x*(x*(2 + 2*sqrt(3)) + 2))

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Mathematica [C]  time = 0.570049, size = 174, normalized size = 0.6 \[ \frac{a x \left (x^3+\frac{\left (1-(-1)^{2/3}\right ) \sqrt{\frac{x-\sqrt [3]{-1}}{\left (1+\sqrt [3]{-1}\right ) x}} \sqrt{\frac{(x+1) \left (2 x+i \sqrt{3}-1\right )}{x^2}} x^2 \left (\left (1+\sqrt [3]{-1}\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} (x+1)}{\left (-1+(-1)^{2/3}\right ) x}}\right )|1+(-1)^{2/3}\right )-F\left (\sin ^{-1}\left (\sqrt{\frac{(-1)^{2/3} (x+1)}{\left (-1+(-1)^{2/3}\right ) x}}\right )|1+(-1)^{2/3}\right )\right )}{\sqrt{6}}+1\right )}{\sqrt{x^3+1} \sqrt{a x^3}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [C]  time = 0.376, size = 1521, normalized size = 5.2 \[ \text{result too large to display} \]

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

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. \[{\rm integral}\left (\frac{\sqrt{a x^{3}}}{\sqrt{x^{3} + 1}}, x\right ) \]

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. \[ \int \frac{\sqrt{a x^{3}}}{\sqrt{\left (x + 1\right ) \left (x^{2} - x + 1\right )}}\, dx \]

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. \[ \int \frac{\sqrt{a x^{3}}}{\sqrt{x^{3} + 1}}\,{d x} \]

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)