3.2043 \(\int \frac{x^4}{\left (a+\frac{b}{x^3}\right )^{3/2}} \, dx\)

Optimal. Leaf size=291 \[ -\frac{91 b x^2 \sqrt{a+\frac{b}{x^3}}}{60 a^3}+\frac{13 x^5 \sqrt{a+\frac{b}{x^3}}}{15 a^2}-\frac{91 \sqrt{2+\sqrt{3}} b^{5/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{60 \sqrt [4]{3} a^3 \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}-\frac{2 x^5}{3 a \sqrt{a+\frac{b}{x^3}}} \]

[Out]

(-91*b*Sqrt[a + b/x^3]*x^2)/(60*a^3) - (2*x^5)/(3*a*Sqrt[a + b/x^3]) + (13*Sqrt[
a + b/x^3]*x^5)/(15*a^2) - (91*Sqrt[2 + Sqrt[3]]*b^(5/3)*(a^(1/3) + b^(1/3)/x)*S
qrt[(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]])/(60*3^(1/4)*a^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 = 0.456815, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ -\frac{91 b x^2 \sqrt{a+\frac{b}{x^3}}}{60 a^3}+\frac{13 x^5 \sqrt{a+\frac{b}{x^3}}}{15 a^2}-\frac{91 \sqrt{2+\sqrt{3}} b^{5/3} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right ) \sqrt{\frac{a^{2/3}-\frac{\sqrt [3]{a} \sqrt [3]{b}}{x}+\frac{b^{2/3}}{x^2}}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1-\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}{\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}}\right )|-7-4 \sqrt{3}\right )}{60 \sqrt [4]{3} a^3 \sqrt{a+\frac{b}{x^3}} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )}{\left (\left (1+\sqrt{3}\right ) \sqrt [3]{a}+\frac{\sqrt [3]{b}}{x}\right )^2}}}-\frac{2 x^5}{3 a \sqrt{a+\frac{b}{x^3}}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b/x^3)^(3/2),x]

[Out]

(-91*b*Sqrt[a + b/x^3]*x^2)/(60*a^3) - (2*x^5)/(3*a*Sqrt[a + b/x^3]) + (13*Sqrt[
a + b/x^3]*x^5)/(15*a^2) - (91*Sqrt[2 + Sqrt[3]]*b^(5/3)*(a^(1/3) + b^(1/3)/x)*S
qrt[(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]])/(60*3^(1/4)*a^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 = 23.9799, size = 248, normalized size = 0.85 \[ - \frac{2 x^{5}}{3 a \sqrt{a + \frac{b}{x^{3}}}} + \frac{13 x^{5} \sqrt{a + \frac{b}{x^{3}}}}{15 a^{2}} - \frac{91 \cdot 3^{\frac{3}{4}} b^{\frac{5}{3}} \sqrt{\frac{a^{\frac{2}{3}} - \frac{\sqrt [3]{a} \sqrt [3]{b}}{x} + \frac{b^{\frac{2}{3}}}{x^{2}}}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right ) F\left (\operatorname{asin}{\left (\frac{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}}{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}} \right )}\middle | -7 - 4 \sqrt{3}\right )}{180 a^{3} \sqrt{\frac{\sqrt [3]{a} \left (\sqrt [3]{a} + \frac{\sqrt [3]{b}}{x}\right )}{\left (\sqrt [3]{a} \left (1 + \sqrt{3}\right ) + \frac{\sqrt [3]{b}}{x}\right )^{2}}} \sqrt{a + \frac{b}{x^{3}}}} - \frac{91 b x^{2} \sqrt{a + \frac{b}{x^{3}}}}{60 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*x**5/(3*a*sqrt(a + b/x**3)) + 13*x**5*sqrt(a + b/x**3)/(15*a**2) - 91*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))/(180*a**3*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)) - 91*b*x**2*sqrt(a + b/x**3)/(6
0*a**3)

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Mathematica [C]  time = 0.447876, size = 188, normalized size = 0.65 \[ \frac{-3 \sqrt [3]{-b} \left (-12 a^2 x^6+39 a b x^3+91 b^2\right )-91 i 3^{3/4} \sqrt [3]{a} b^2 x \sqrt{(-1)^{5/6} \left (\frac{\sqrt [3]{-b}}{\sqrt [3]{a} x}-1\right )} \sqrt{\frac{\frac{(-b)^{2/3}}{a^{2/3}}+\frac{\sqrt [3]{-b} x}{\sqrt [3]{a}}+x^2}{x^2}} F\left (\sin ^{-1}\left (\frac{\sqrt{-\frac{i \sqrt [3]{-b}}{\sqrt [3]{a} x}-(-1)^{5/6}}}{\sqrt [4]{3}}\right )|\sqrt [3]{-1}\right )}{180 a^3 \sqrt [3]{-b} x \sqrt{a+\frac{b}{x^3}}} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^4/(a + b/x^3)^(3/2),x]

[Out]

(-3*(-b)^(1/3)*(91*b^2 + 39*a*b*x^3 - 12*a^2*x^6) - (91*I)*3^(3/4)*a^(1/3)*b^2*S
qrt[(-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)])/(180*a^3*(-b)^(1/3)*Sqrt[a + b/x^3]*x)

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Maple [B]  time = 0.023, size = 2302, normalized size = 7.9 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(a+b/x^3)^(3/2),x)

[Out]

-1/60/((a*x^3+b)/x^3)^(3/2)/x^5*(a*x^3+b)/a^4/(-a^2*b)^(1/3)*(182*I*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))*(-(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)*3^(1/2)*(x*(a*x^3+b))^(1/2)
*x^2*a^2*b^2-364*I*(-a^2*b)^(1/3)*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))*(-(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)*3^(1/2)*(x*(a*x^3+b))^(1/2)*x*a*b^2+182*I*(-a^2*b)^(2/3)*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))*(-(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)*3^(1/2)*(x*(a*x^3+b
))^(1/2)*b^2-12*I*(-a^2*b)^(1/3)*(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*(a*x^3+b))^(1/2)*x^3*a^2-182*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))*(-(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)*(x*(a*x^3+b))^(1/2)*x^2*a^2*b^
2+364*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))*(-(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)*(-a^2*b)^
(1/3)*(x*(a*x^3+b))^(1/2)*x*a*b^2-182*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))*(-(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)*(-a^2*b)^(2/3)*(x*(a*x^3+b))^(1/2)*b^2+36*(-a^2*b)^(1/3)*
(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*(a*x^
3+b))^(1/2)*x^3*a^2+51*I*(-a^2*b)^(1/3)*(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*(a*x^3+b))^(1/2)*a*b+40*I*(-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)*3^(1/2)*x*a*b^2-153*(-
a^2*b)^(1/3)*(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*(a*x^3+b))^(1/2)*a*b-120*(-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*b^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. \[ \int \frac{x^{4}}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(a + b/x^3)^(3/2),x, algorithm="maxima")

[Out]

integrate(x^4/(a + b/x^3)^(3/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{7}}{{\left (a x^{3} + b\right )} \sqrt{\frac{a x^{3} + b}{x^{3}}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(a + b/x^3)^(3/2),x, algorithm="fricas")

[Out]

integral(x^7/((a*x^3 + b)*sqrt((a*x^3 + b)/x^3)), x)

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Sympy [A]  time = 4.08182, size = 46, normalized size = 0.16 \[ - \frac{x^{5} \Gamma \left (- \frac{5}{3}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{5}{3}, \frac{3}{2} \\ - \frac{2}{3} \end{matrix}\middle |{\frac{b e^{i \pi }}{a x^{3}}} \right )}}{3 a^{\frac{3}{2}} \Gamma \left (- \frac{2}{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (a + \frac{b}{x^{3}}\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(a + b/x^3)^(3/2),x, algorithm="giac")

[Out]

integrate(x^4/(a + b/x^3)^(3/2), x)