3.2029 \(\int \frac{x^3}{\sqrt{a+\frac{b}{x^3}}} \, dx\)

Optimal. Leaf size=542 \[ \frac{5 b^{4/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} a^{5/3} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}-\frac{5 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{16 a^{5/3} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{5 b^{4/3} \sqrt{a+\frac{b}{x^3}}}{8 a^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}-\frac{5 b x \sqrt{a+\frac{b}{x^3}}}{8 a^2}+\frac{x^4 \sqrt{a+\frac{b}{x^3}}}{4 a} \]

[Out]

(5*b^(4/3)*Sqrt[a + b/x^3])/(8*a^2*((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)) - (5*b*S
qrt[a + b/x^3]*x)/(8*a^2) + (Sqrt[a + b/x^3]*x^4)/(4*a) - (5*3^(1/4)*Sqrt[2 - Sq
rt[3]]*b^(4/3)*(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]])/(16*a
^(5/3)*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]) + (5*b^(4/3)*(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[ArcSi
n[((1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 -
4*Sqrt[3]])/(4*Sqrt[2]*3^(1/4)*a^(5/3)*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 = 0.80934, antiderivative size = 542, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{5 b^{4/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{4 \sqrt{2} \sqrt [4]{3} a^{5/3} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}-\frac{5 \sqrt [4]{3} \sqrt{2-\sqrt{3}} b^{4/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{16 a^{5/3} \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{5 b^{4/3} \sqrt{a+\frac{b}{x^3}}}{8 a^2 \left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}-\frac{5 b x \sqrt{a+\frac{b}{x^3}}}{8 a^2}+\frac{x^4 \sqrt{a+\frac{b}{x^3}}}{4 a} \]

Antiderivative was successfully verified.

[In]  Int[x^3/Sqrt[a + b/x^3],x]

[Out]

(5*b^(4/3)*Sqrt[a + b/x^3])/(8*a^2*((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)) - (5*b*S
qrt[a + b/x^3]*x)/(8*a^2) + (Sqrt[a + b/x^3]*x^4)/(4*a) - (5*3^(1/4)*Sqrt[2 - Sq
rt[3]]*b^(4/3)*(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]])/(16*a
^(5/3)*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]) + (5*b^(4/3)*(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[ArcSi
n[((1 - Sqrt[3])*a^(1/3) + b^(1/3)/x)/((1 + Sqrt[3])*a^(1/3) + b^(1/3)/x)], -7 -
4*Sqrt[3]])/(4*Sqrt[2]*3^(1/4)*a^(5/3)*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 = 46.3919, size = 452, normalized size = 0.83 \[ \frac{x^{4} \sqrt{a + \frac{b}{x^{3}}}}{4 a} + \frac{5 b^{\frac{4}{3}} \sqrt{a + \frac{b}{x^{3}}}}{8 a^{2} \left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )} - \frac{5 b x \sqrt{a + \frac{b}{x^{3}}}}{8 a^{2}} - \frac{5 \sqrt [4]{3} b^{\frac{4}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \frac{\sqrt [3]{a} \sqrt [3]{b}}{x} + \frac{b^{\frac{2}{3}}}{x^{2}}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right ) E\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{16 a^{\frac{5}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{a + \frac{b}{x^{3}}}} + \frac{5 \sqrt{2} \cdot 3^{\frac{3}{4}} b^{\frac{4}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \frac{\sqrt [3]{a} \sqrt [3]{b}}{x} + \frac{b^{\frac{2}{3}}}{x^{2}}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{24 a^{\frac{5}{3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{a + \frac{b}{x^{3}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**4*sqrt(a + b/x**3)/(4*a) + 5*b**(4/3)*sqrt(a + b/x**3)/(8*a**2*(a**(1/3)*(1 +
sqrt(3)) + b**(1/3)/x)) - 5*b*x*sqrt(a + b/x**3)/(8*a**2) - 5*3**(1/4)*b**(4/3)
*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 + sqrt(3)) + b**(1/3)/x)), -7 -
4*sqrt(3))/(16*a**(5/3)*sqrt(a**(1/3)*(a**(1/3) + b**(1/3)/x)/(a**(1/3)*(1 + sq
rt(3)) + b**(1/3)/x)**2)*sqrt(a + b/x**3)) + 5*sqrt(2)*3**(3/4)*b**(4/3)*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))/(24*a**(5/3)*sq
rt(a**(1/3)*(a**(1/3) + b**(1/3)/x)/(a**(1/3)*(1 + sqrt(3)) + b**(1/3)/x)**2)*sq
rt(a + b/x**3))

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Mathematica [C]  time = 1.43155, size = 356, normalized size = 0.66 \[ \frac{5 b x \left (-\frac{b^{2/3}}{a^{2/3}}+\frac{\sqrt [3]{b} x}{\sqrt [3]{a}}-x^2\right )-\frac{5 (-1)^{2/3} b^{4/3} \left (\sqrt [3]{a} x+\sqrt [3]{b}\right )^2 \sqrt{\frac{\left (1+\sqrt [3]{-1}\right ) \sqrt [3]{a} x \left (\sqrt [3]{b}-\sqrt [3]{-1} \sqrt [3]{a} x\right )}{\left (\sqrt [3]{a} x+\sqrt [3]{b}\right )^2}} \sqrt{\frac{(-1)^{2/3} \sqrt [3]{a} x+\sqrt [3]{b}}{\sqrt [3]{a} x+\sqrt [3]{b}}} \left (\left (1+i \sqrt{3}\right ) F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\left (3+i \sqrt{3}\right ) \sqrt [3]{a} x}{\sqrt [3]{a} x+\sqrt [3]{b}}}}{\sqrt{2}}\right )|\frac{-i+\sqrt{3}}{i+\sqrt{3}}\right )+\left (-3-i \sqrt{3}\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{\left (3+i \sqrt{3}\right ) \sqrt [3]{a} x}{\sqrt [3]{a} x+\sqrt [3]{b}}}}{\sqrt{2}}\right )|\frac{-i+\sqrt{3}}{i+\sqrt{3}}\right )\right )}{2 \left ((-1)^{2/3}-1\right ) a}+2 x^3 \left (a x^3+b\right )}{8 a x^2 \sqrt{a+\frac{b}{x^3}}} \]

Warning: Unable to verify antiderivative.

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

[Out]

(5*b*x*(-(b^(2/3)/a^(2/3)) + (b^(1/3)*x)/a^(1/3) - x^2) + 2*x^3*(b + a*x^3) - (5
*(-1)^(2/3)*b^(4/3)*(b^(1/3) + a^(1/3)*x)^2*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 + Sq
rt[3])] + (1 + I*Sqrt[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)
)*a))/(8*a*Sqrt[a + b/x^3]*x^2)

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Maple [B]  time = 0.019, size = 2586, normalized size = 4.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3/(a+b/x^3)^(1/2),x)

[Out]

1/4/((a*x^3+b)/x^3)^(1/2)/x*(a*x^3+b)/a^3*(5*I*(-(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)*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))*3^(1/2)*a*b^2-5*I*(-a^2*b)^(1/3)*(-(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)*Elliptic
E((-(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))*3^(1/2)*x^2*a*b-10*(-a^2*b)^(1
/3)*(-(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))*x^2*a*b+15*
(-a^2*b)^(1/3)*(-(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)*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))*
x^2*a*b-5*I*(-a^2*b)^(1/3)*3^(1/2)*x^2*a*b+20*(-a^2*b)^(2/3)*(-(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))*x*b-30*(-a^2*b)^(2/3)*(-(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)*Ellip
ticE((-(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))*x*b+I*(a*x^4+b*x)^(1/2)*(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^2*a^2-5*I*(-a^2*b)^
(2/3)*3^(1/2)*x*b-5*I*3^(1/2)*x^3*a^2*b+10*(-(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))*a*b^2-15*(-(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)*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))*a*b^2+10*I*(-a^2*b)^(2/3)*(-(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)*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))*3^(1/2)*x*b-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*x^4+b*x)^(1/2)*x^2*a^2+15*a^2*b*x^3+15*(-a^2*b)^(1/3)*x^
2*a*b+15*(-a^2*b)^(2/3)*x*b)/(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. \[ \int \frac{x^{3}}{\sqrt{a + \frac{b}{x^{3}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^3/sqrt(a + b/x^3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.36918, size = 46, normalized size = 0.08 \[ - \frac{x^{4} \Gamma \left (- \frac{4}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{1}{2} \\ - \frac{1}{3} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt{a} \Gamma \left (- \frac{1}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x**4*gamma(-4/3)*hyper((-4/3, 1/2), (-1/3,), b*exp_polar(I*pi)/(a*x**3))/(3*sqr
t(a)*gamma(-1/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^3/sqrt(a + b/x^3), x)