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

Optimal. Leaf size=311 \[ \frac{48 a^2 x \sqrt [3]{a+b x^2}}{935 b}+\frac{48\ 3^{3/4} \sqrt{2-\sqrt{3}} a^3 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{935 b^2 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac{24}{187} a x^3 \sqrt [3]{a+b x^2}+\frac{3}{17} x^3 \left (a+b x^2\right )^{4/3} \]

[Out]

(48*a^2*x*(a + b*x^2)^(1/3))/(935*b) + (24*a*x^3*(a + b*x^2)^(1/3))/187 + (3*x^3
*(a + b*x^2)^(4/3))/17 + (48*3^(3/4)*Sqrt[2 - Sqrt[3]]*a^3*(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 - Sq
rt[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]
])/(935*b^2*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 = 0.493214, antiderivative size = 311, 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 \[ \frac{48 a^2 x \sqrt [3]{a+b x^2}}{935 b}+\frac{48\ 3^{3/4} \sqrt{2-\sqrt{3}} a^3 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{935 b^2 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac{24}{187} a x^3 \sqrt [3]{a+b x^2}+\frac{3}{17} x^3 \left (a+b x^2\right )^{4/3} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x^2)^(4/3),x]

[Out]

(48*a^2*x*(a + b*x^2)^(1/3))/(935*b) + (24*a*x^3*(a + b*x^2)^(1/3))/187 + (3*x^3
*(a + b*x^2)^(4/3))/17 + (48*3^(3/4)*Sqrt[2 - Sqrt[3]]*a^3*(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 - Sq
rt[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]
])/(935*b^2*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 = 21.0105, size = 260, normalized size = 0.84 \[ \frac{48 \cdot 3^{\frac{3}{4}} a^{3} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a + b x^{2}} + \left (a + b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{935 b^{2} x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}}} + \frac{48 a^{2} x \sqrt [3]{a + b x^{2}}}{935 b} + \frac{24 a x^{3} \sqrt [3]{a + b x^{2}}}{187} + \frac{3 x^{3} \left (a + b x^{2}\right )^{\frac{4}{3}}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

48*3**(3/4)*a**3*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))/
(935*b**2*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)) + 48*a**2*x*(a + b*x**2)**(1/3)/(935*b) + 24*a*
x**3*(a + b*x**2)**(1/3)/187 + 3*x**3*(a + b*x**2)**(4/3)/17

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Mathematica [C]  time = 0.0705652, size = 90, normalized size = 0.29 \[ \frac{3 \left (-16 a^3 x \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{3}{2};-\frac{b x^2}{a}\right )+16 a^3 x+111 a^2 b x^3+150 a b^2 x^5+55 b^3 x^7\right )}{935 b \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(16*a^3*x + 111*a^2*b*x^3 + 150*a*b^2*x^5 + 55*b^3*x^7 - 16*a^3*x*(1 + (b*x^2
)/a)^(2/3)*Hypergeometric2F1[1/2, 2/3, 3/2, -((b*x^2)/a)]))/(935*b*(a + b*x^2)^(
2/3))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^2+a)^(4/3),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(4/3)*x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{4} + a x^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^4 + a*x^2)*(b*x^2 + a)^(1/3), x)

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Sympy [A]  time = 4.4478, size = 29, normalized size = 0.09 \[ \frac{a^{\frac{4}{3}} x^{3}{{}_{2}F_{1}\left (\begin{matrix} - \frac{4}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(4/3)*x**3*hyper((-4/3, 3/2), (5/2,), b*x**2*exp_polar(I*pi)/a)/3

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(4/3)*x^2, x)