3.673 \(\int x^2 \sqrt [3]{a+b x^2} \, dx\)

Optimal. Leaf size=290 \[ \frac{6\ 3^{3/4} \sqrt{2-\sqrt{3}} a^2 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{55 b^2 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac{6 a x \sqrt [3]{a+b x^2}}{55 b}+\frac{3}{11} x^3 \sqrt [3]{a+b x^2} \]

[Out]

(6*a*x*(a + b*x^2)^(1/3))/(55*b) + (3*x^3*(a + b*x^2)^(1/3))/11 + (6*3^(3/4)*Sqr
t[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]*E
llipticF[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^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 = 0.41542, antiderivative size = 290, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{6\ 3^{3/4} \sqrt{2-\sqrt{3}} a^2 \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{b x^2+a}}\right )|-7+4 \sqrt{3}\right )}{55 b^2 x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a+b x^2}\right )^2}}}+\frac{6 a x \sqrt [3]{a+b x^2}}{55 b}+\frac{3}{11} x^3 \sqrt [3]{a+b x^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + b*x^2)^(1/3),x]

[Out]

(6*a*x*(a + b*x^2)^(1/3))/(55*b) + (3*x^3*(a + b*x^2)^(1/3))/11 + (6*3^(3/4)*Sqr
t[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]*E
llipticF[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^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 = 15.7747, size = 240, normalized size = 0.83 \[ \frac{6 \cdot 3^{\frac{3}{4}} a^{2} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a + b x^{2}} + \left (a + b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}} \sqrt{- \sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a + b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{55 b^{2} x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a + b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a + b x^{2}}\right )^{2}}}} + \frac{6 a x \sqrt [3]{a + b x^{2}}}{55 b} + \frac{3 x^{3} \sqrt [3]{a + b x^{2}}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

6*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**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)) + 6*a*x*(a + b*x**2)**(1/3)/(55*b) + 3*x**3*(a +
b*x**2)**(1/3)/11

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Mathematica [C]  time = 0.0553379, size = 78, normalized size = 0.27 \[ \frac{3 x \left (-2 a^2 \left (\frac{b x^2}{a}+1\right )^{2/3} \, _2F_1\left (\frac{1}{2},\frac{2}{3};\frac{3}{2};-\frac{b x^2}{a}\right )+2 a^2+7 a b x^2+5 b^2 x^4\right )}{55 b \left (a+b x^2\right )^{2/3}} \]

Antiderivative was successfully verified.

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

[Out]

(3*x*(2*a^2 + 7*a*b*x^2 + 5*b^2*x^4 - 2*a^2*(1 + (b*x^2)/a)^(2/3)*Hypergeometric
2F1[1/2, 2/3, 3/2, -((b*x^2)/a)]))/(55*b*(a + b*x^2)^(2/3))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{x}^{2}\sqrt [3]{b{x}^{2}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x^2+a)^(1/3),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + a\right )}^{\frac{1}{3}} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)*x^2,x, algorithm="maxima")

[Out]

integrate((b*x^2 + a)^(1/3)*x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)*x^2,x, algorithm="fricas")

[Out]

integral((b*x^2 + a)^(1/3)*x^2, x)

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Sympy [A]  time = 2.37917, size = 29, normalized size = 0.1 \[ \frac{\sqrt [3]{a} x^{3}{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**(1/3)*x**3*hyper((-1/3, 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. \[ \int{\left (b x^{2} + a\right )}^{\frac{1}{3}} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(1/3)*x^2,x, algorithm="giac")

[Out]

integrate((b*x^2 + a)^(1/3)*x^2, x)