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

Optimal. Leaf size=588 \[ \frac{24 \sqrt{2} 3^{3/4} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{13 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{36 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{13 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{72 a^2 x}{13 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{18}{13} a x \left (a-b x^2\right )^{2/3}-\frac{3}{13} x \left (a-b x^2\right )^{5/3} \]

[Out]

(18*a*x*(a - b*x^2)^(2/3))/13 - (3*x*(a - b*x^2)^(5/3))/13 - (72*a^2*x)/(13*((1
- Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (36*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(7/3)
*(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 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[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]])/(13*b*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)]) + (24*Sqrt[2]*3^(3/4)*a^(7/3)*(
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 - 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]])/(13*b*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 = 0.806444, antiderivative size = 588, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{24 \sqrt{2} 3^{3/4} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{13 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{36 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{13 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{72 a^2 x}{13 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{18}{13} a x \left (a-b x^2\right )^{2/3}-\frac{3}{13} x \left (a-b x^2\right )^{5/3} \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^2)^(2/3)*(3*a + b*x^2),x]

[Out]

(18*a*x*(a - b*x^2)^(2/3))/13 - (3*x*(a - b*x^2)^(5/3))/13 - (72*a^2*x)/(13*((1
- Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (36*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(7/3)
*(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 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[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]])/(13*b*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)]) + (24*Sqrt[2]*3^(3/4)*a^(7/3)*(
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 - 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]])/(13*b*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 = 39.1913, size = 469, normalized size = 0.8 \[ - \frac{36 \sqrt [4]{3} a^{\frac{7}{3}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a - b x^{2}} + \left (a - b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{13 b x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}}} + \frac{24 \sqrt{2} \cdot 3^{\frac{3}{4}} a^{\frac{7}{3}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a - b x^{2}} + \left (a - b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{13 b x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}}} + \frac{72 a^{2} x}{13 \left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )} + \frac{18 a x \left (a - b x^{2}\right )^{\frac{2}{3}}}{13} - \frac{3 x \left (a - b x^{2}\right )^{\frac{5}{3}}}{13} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-36*3**(1/4)*a**(7/3)*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_e(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))/(13*b*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)) + 24*sqrt(2)*3**(3/4)*a**(7/3)*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)*(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))/(13*b*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)) + 72*a**2*x/(13*(a**(1/
3)*(-1 + sqrt(3)) + (a - b*x**2)**(1/3))) + 18*a*x*(a - b*x**2)**(2/3)/13 - 3*x*
(a - b*x**2)**(5/3)/13

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Mathematica [C]  time = 0.0495926, size = 76, normalized size = 0.13 \[ -\frac{3 \left (-8 a^2 x \sqrt [3]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )-5 a^2 x+4 a b x^3+b^2 x^5\right )}{13 \sqrt [3]{a-b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*(-5*a^2*x + 4*a*b*x^3 + b^2*x^5 - 8*a^2*x*(1 - (b*x^2)/a)^(1/3)*Hypergeometr
ic2F1[1/3, 1/2, 3/2, (b*x^2)/a]))/(13*(a - b*x^2)^(1/3))

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Maple [F]  time = 0.038, size = 0, normalized size = 0. \[ \int \left ( -b{x}^{2}+a \right ) ^{{\frac{2}{3}}} \left ( b{x}^{2}+3\,a \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^2+a)^(2/3)*(b*x^2+3*a),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + 3*a)*(-b*x^2 + a)^(2/3), x)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 5.37092, size = 63, normalized size = 0.11 \[ 3 a^{\frac{5}{3}} x{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} + \frac{a^{\frac{2}{3}} b x^{3}{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + 3*a)*(-b*x^2 + a)^(2/3), x)