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

Optimal. Leaf size=293 \[ -\frac{27\ 3^{3/4} \sqrt{2-\sqrt{3}} a^2 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{55 b^3 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac{27 a x \sqrt [3]{a+b x^2}}{55 b^2}+\frac{3 x^3 \sqrt [3]{a+b x^2}}{11 b} \]

[Out]

(-27*a*x*(a + b*x^2)^(1/3))/(55*b^2) + (3*x^3*(a + b*x^2)^(1/3))/(11*b) - (27*3^
(3/4)*Sqrt[2 - Sqrt[3]]*a^2*(a^(1/3) - (a + b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3
)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1
/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))/((1 - Sqrt[
3])*a^(1/3) - (a + b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(55*b^3*x*Sqrt[-((a^(1/3)*(a
^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2)])

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Rubi [A]  time = 0.419613, antiderivative size = 293, 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 \[ -\frac{27\ 3^{3/4} \sqrt{2-\sqrt{3}} a^2 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{55 b^3 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}-\frac{27 a x \sqrt [3]{a+b x^2}}{55 b^2}+\frac{3 x^3 \sqrt [3]{a+b x^2}}{11 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^2)^(2/3),x]

[Out]

(-27*a*x*(a + b*x^2)^(1/3))/(55*b^2) + (3*x^3*(a + b*x^2)^(1/3))/(11*b) - (27*3^
(3/4)*Sqrt[2 - Sqrt[3]]*a^2*(a^(1/3) - (a + b*x^2)^(1/3))*Sqrt[(a^(2/3) + a^(1/3
)*(a + b*x^2)^(1/3) + (a + b*x^2)^(2/3))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1
/3))^2]*EllipticF[ArcSin[((1 + Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))/((1 - Sqrt[
3])*a^(1/3) - (a + b*x^2)^(1/3))], -7 + 4*Sqrt[3]])/(55*b^3*x*Sqrt[-((a^(1/3)*(a
^(1/3) - (a + b*x^2)^(1/3)))/((1 - Sqrt[3])*a^(1/3) - (a + b*x^2)^(1/3))^2)])

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Rubi in Sympy [A]  time = 15.5465, size = 243, normalized size = 0.83 \[ - \frac{27 \cdot 3^{\frac{3}{4}} a^{2} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a + b x^{2}} + \left (a + b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{55 b^{3} x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}}} - \frac{27 a x \sqrt [3]{a + b x^{2}}}{55 b^{2}} + \frac{3 x^{3} \sqrt [3]{a + b x^{2}}}{11 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-27*3**(3/4)*a**2*sqrt((a**(2/3) + a**(1/3)*(a + b*x**2)**(1/3) + (a + b*x**2)**
(2/3))/(a**(1/3)*(-1 + sqrt(3)) + (a + b*x**2)**(1/3))**2)*sqrt(-sqrt(3) + 2)*(a
**(1/3) - (a + b*x**2)**(1/3))*elliptic_f(asin((a**(1/3)*(1 + sqrt(3)) - (a + b*
x**2)**(1/3))/(-a**(1/3)*(-1 + sqrt(3)) - (a + b*x**2)**(1/3))), -7 + 4*sqrt(3))
/(55*b**3*x*sqrt(-a**(1/3)*(a**(1/3) - (a + b*x**2)**(1/3))/(a**(1/3)*(-1 + sqrt
(3)) + (a + b*x**2)**(1/3))**2)) - 27*a*x*(a + b*x**2)**(1/3)/(55*b**2) + 3*x**3
*(a + b*x**2)**(1/3)/(11*b)

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Mathematica [C]  time = 0.0574078, size = 79, normalized size = 0.27 \[ \frac{3 \left (9 a^2 x \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{3}{2};-\frac{b x^2}{a}\right )-9 a^2 x-4 a b x^3+5 b^2 x^5\right )}{55 b^2 \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(-9*a^2*x - 4*a*b*x^3 + 5*b^2*x^5 + 9*a^2*x*(1 + (b*x^2)/a)^(2/3)*Hypergeomet
ric2F1[1/2, 2/3, 3/2, -((b*x^2)/a)]))/(55*b^2*(a + b*x^2)^(2/3))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{{x}^{4} \left ( b{x}^{2}+a \right ) ^{-{\frac{2}{3}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^2+a)^(2/3),x)

[Out]

int(x^4/(b*x^2+a)^(2/3),x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(b*x^2 + a)^(2/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{4}}{{\left (b x^{2} + a\right )}^{\frac{2}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral(x^4/(b*x^2 + a)^(2/3), x)

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Sympy [A]  time = 2.53838, size = 27, normalized size = 0.09 \[ \frac{x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{2}{3}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{5 a^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*hyper((2/3, 5/2), (7/2,), b*x**2*exp_polar(I*pi)/a)/(5*a**(2/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(b*x^2 + a)^(2/3), x)