3.2023 \(\int \frac{x^4}{\sqrt{a+\frac{b}{x^3}}} \, dx\)

Optimal. Leaf size=270 \[ -\frac{7 b x^2 \sqrt{a+\frac{b}{x^3}}}{20 a^2}-\frac{7 \sqrt{2+\sqrt{3}} b^{5/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{20 \sqrt [4]{3} a^2 \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{x^5 \sqrt{a+\frac{b}{x^3}}}{5 a} \]

[Out]

(-7*b*Sqrt[a + b/x^3]*x^2)/(20*a^2) + (Sqrt[a + b/x^3]*x^5)/(5*a) - (7*Sqrt[2 +
Sqrt[3]]*b^(5/3)*(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]])/(20
*3^(1/4)*a^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])

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Rubi [A]  time = 0.36841, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ -\frac{7 b x^2 \sqrt{a+\frac{b}{x^3}}}{20 a^2}-\frac{7 \sqrt{2+\sqrt{3}} b^{5/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{20 \sqrt [4]{3} a^2 \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}+\frac{x^5 \sqrt{a+\frac{b}{x^3}}}{5 a} \]

Antiderivative was successfully verified.

[In]  Int[x^4/Sqrt[a + b/x^3],x]

[Out]

(-7*b*Sqrt[a + b/x^3]*x^2)/(20*a^2) + (Sqrt[a + b/x^3]*x^5)/(5*a) - (7*Sqrt[2 +
Sqrt[3]]*b^(5/3)*(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]])/(20
*3^(1/4)*a^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])

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Rubi in Sympy [A]  time = 17.904, size = 226, normalized size = 0.84 \[ \frac{x^{5} \sqrt{a + \frac{b}{x^{3}}}}{5 a} - \frac{7 \cdot 3^{\frac{3}{4}} b^{\frac{5}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \frac{\sqrt [3]{a} \sqrt [3]{b}}{x} + \frac{b^{\frac{2}{3}}}{x^{2}}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{60 a^{2} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{a + \frac{b}{x^{3}}}} - \frac{7 b x^{2} \sqrt{a + \frac{b}{x^{3}}}}{20 a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(a+b/x**3)**(1/2),x)

[Out]

x**5*sqrt(a + b/x**3)/(5*a) - 7*3**(3/4)*b**(5/3)*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_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))/(60*a**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)) - 7*b*x**2*sqrt(a + b/x**3)/(20*a**2)

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Mathematica [C]  time = 0.380012, size = 188, normalized size = 0.7 \[ \frac{-3 \sqrt [3]{-b} \left (-4 a^2 x^6+3 a b x^3+7 b^2\right )-7 i 3^{3/4} \sqrt [3]{a} b^2 x \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-b}}{\sqrt [3]{a} x}-1\right )} \sqrt{\frac{\frac{(-b)^{2/3}}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+x^2}{x^2}} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b}}{\sqrt [3]{a} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{60 a^2 \sqrt [3]{-b} x \sqrt{a+\frac{b}{x^3}}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^4/Sqrt[a + b/x^3],x]

[Out]

(-3*(-b)^(1/3)*(7*b^2 + 3*a*b*x^3 - 4*a^2*x^6) - (7*I)*3^(3/4)*a^(1/3)*b^2*Sqrt[
(-1)^(5/6)*(-1 + (-b)^(1/3)/(a^(1/3)*x))]*x*Sqrt[((-b)^(2/3)/a^(2/3) + ((-b)^(1/
3)*x)/a^(1/3) + x^2)/x^2]*EllipticF[ArcSin[Sqrt[-(-1)^(5/6) - (I*(-b)^(1/3))/(a^
(1/3)*x)]/3^(1/4)], (-1)^(1/3)])/(60*a^2*(-b)^(1/3)*Sqrt[a + b/x^3]*x)

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Maple [B]  time = 0.018, size = 2017, normalized size = 7.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(a+b/x^3)^(1/2),x)

[Out]

-1/20/((a*x^3+b)/x^3)^(1/2)/x*(a*x^3+b)/a^3/(-a^2*b)^(1/3)*(14*I*3^(1/2)*(-(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^2*b^2-28*I*(-a^2
*b)^(1/3)*3^(1/2)*(-(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*a*b^2+14*I*(-a^2*b)^(2/3)*3^(1/2)*(-(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))*b^2-4*I*(-a^2*b)^(1/3)*3^(1/2)*(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)*x^3*a^2-14*(-(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^2*b^2+28*(-(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^2*b)^(1/3)*x*a*b^2-14*(-(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)*Elli
pticF((-(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^2*b)^(2/3)*b^2+12*x^3*
(a*x^4+b*x)^(1/2)*a^2*(-a^2*b)^(1/3)*(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)+7*I*(-a^2*b)^(1/3)*3^(1/2)*(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)*a*b-21*b*(a*x^4+b*x)^(1/2)*a*(-a^2*b)^(1/3)*(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*(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. \[ \int \frac{x^{4}}{\sqrt{a + \frac{b}{x^{3}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(a + b/x^3),x, algorithm="maxima")

[Out]

integrate(x^4/sqrt(a + b/x^3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{\sqrt{\frac{a x^{3} + b}{x^{3}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(a + b/x^3),x, algorithm="fricas")

[Out]

integral(x^4/sqrt((a*x^3 + b)/x^3), x)

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Sympy [A]  time = 3.73112, size = 46, normalized size = 0.17 \[ - \frac{x^{5} \Gamma \left (- \frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{3}, \frac{1}{2} \\ - \frac{2}{3} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{3}}} \right )}}{3 \sqrt{a} \Gamma \left (- \frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(a+b/x**3)**(1/2),x)

[Out]

-x**5*gamma(-5/3)*hyper((-5/3, 1/2), (-2/3,), b*exp_polar(I*pi)/(a*x**3))/(3*sqr
t(a)*gamma(-2/3))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\sqrt{a + \frac{b}{x^{3}}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/sqrt(a + b/x^3),x, algorithm="giac")

[Out]

integrate(x^4/sqrt(a + b/x^3), x)