3.116 \(\int \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^3 \, dx\)

Optimal. Leaf size=668 \[ \frac{3746304 \sqrt{2} 3^{3/4} a^{16/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{267995 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{5619456 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{16/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{267995 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{11238912 a^5 x}{267995 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{2809728 a^4 x \left (a-b x^2\right )^{2/3}}{267995}+\frac{1404864 a^3 x \left (a-b x^2\right )^{5/3}}{191425}-\frac{33264 a^2 x \left (a-b x^2\right )^{8/3}}{14725}-\frac{432}{775} a x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac{3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2 \]

[Out]

(2809728*a^4*x*(a - b*x^2)^(2/3))/267995 + (1404864*a^3*x*(a - b*x^2)^(5/3))/191
425 - (33264*a^2*x*(a - b*x^2)^(8/3))/14725 - (432*a*x*(a - b*x^2)^(8/3)*(3*a +
b*x^2))/775 - (3*x*(a - b*x^2)^(8/3)*(3*a + b*x^2)^2)/31 - (11238912*a^5*x)/(267
995*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (5619456*3^(1/4)*Sqrt[2 + Sqr
t[3]]*a^(16/3)*(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]*Ellip
ticE[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]])/(267995*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)]) + (3746304*Sq
rt[2]*3^(3/4)*a^(16/3)*(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]])/(267995*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 = 1.29296, antiderivative size = 668, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{3746304 \sqrt{2} 3^{3/4} a^{16/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{267995 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{5619456 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{16/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{267995 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{11238912 a^5 x}{267995 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac{2809728 a^4 x \left (a-b x^2\right )^{2/3}}{267995}+\frac{1404864 a^3 x \left (a-b x^2\right )^{5/3}}{191425}-\frac{33264 a^2 x \left (a-b x^2\right )^{8/3}}{14725}-\frac{432}{775} a x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac{3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2 \]

Antiderivative was successfully verified.

[In]  Int[(a - b*x^2)^(5/3)*(3*a + b*x^2)^3,x]

[Out]

(2809728*a^4*x*(a - b*x^2)^(2/3))/267995 + (1404864*a^3*x*(a - b*x^2)^(5/3))/191
425 - (33264*a^2*x*(a - b*x^2)^(8/3))/14725 - (432*a*x*(a - b*x^2)^(8/3)*(3*a +
b*x^2))/775 - (3*x*(a - b*x^2)^(8/3)*(3*a + b*x^2)^2)/31 - (11238912*a^5*x)/(267
995*((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (5619456*3^(1/4)*Sqrt[2 + Sqr
t[3]]*a^(16/3)*(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]*Ellip
ticE[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]])/(267995*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)]) + (3746304*Sq
rt[2]*3^(3/4)*a^(16/3)*(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]])/(267995*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 = 102.624, size = 544, normalized size = 0.81 \[ - \frac{5619456 \sqrt [4]{3} a^{\frac{16}{3}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a - b x^{2}} + \left (a - b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{267995 b x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}}} + \frac{3746304 \sqrt{2} \cdot 3^{\frac{3}{4}} a^{\frac{16}{3}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a - b x^{2}} + \left (a - b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{267995 b x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}}} + \frac{11238912 a^{5} x}{267995 \left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )} + \frac{2809728 a^{4} x \left (a - b x^{2}\right )^{\frac{2}{3}}}{267995} + \frac{1404864 a^{3} x \left (a - b x^{2}\right )^{\frac{5}{3}}}{191425} - \frac{33264 a^{2} x \left (a - b x^{2}\right )^{\frac{8}{3}}}{14725} - \frac{432 a x \left (a - b x^{2}\right )^{\frac{8}{3}} \left (3 a + b x^{2}\right )}{775} - \frac{3 x \left (a - b x^{2}\right )^{\frac{8}{3}} \left (3 a + b x^{2}\right )^{2}}{31} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**2+a)**(5/3)*(b*x**2+3*a)**3,x)

[Out]

-5619456*3**(1/4)*a**(16/3)*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))/(267995*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)) + 3746304*sqrt(2)*3**(3/4)*a**(16/3)*
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))*ellipt
ic_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))/(267995*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)) + 1
1238912*a**5*x/(267995*(a**(1/3)*(-1 + sqrt(3)) + (a - b*x**2)**(1/3))) + 280972
8*a**4*x*(a - b*x**2)**(2/3)/267995 + 1404864*a**3*x*(a - b*x**2)**(5/3)/191425
- 33264*a**2*x*(a - b*x**2)**(8/3)/14725 - 432*a*x*(a - b*x**2)**(8/3)*(3*a + b*
x**2)/775 - 3*x*(a - b*x**2)**(8/3)*(3*a + b*x**2)**2/31

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Mathematica [C]  time = 0.0849968, size = 110, normalized size = 0.16 \[ \frac{3 \left (6243840 a^5 x \sqrt [3]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )+5815935 a^5 x-5312355 a^4 b x^3-1675114 a^3 b^2 x^5+749658 a^2 b^3 x^7+378651 a b^4 x^9+43225 b^5 x^{11}\right )}{1339975 \sqrt [3]{a-b x^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a - b*x^2)^(5/3)*(3*a + b*x^2)^3,x]

[Out]

(3*(5815935*a^5*x - 5312355*a^4*b*x^3 - 1675114*a^3*b^2*x^5 + 749658*a^2*b^3*x^7
+ 378651*a*b^4*x^9 + 43225*b^5*x^11 + 6243840*a^5*x*(1 - (b*x^2)/a)^(1/3)*Hyper
geometric2F1[1/3, 1/2, 3/2, (b*x^2)/a]))/(1339975*(a - b*x^2)^(1/3))

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Maple [F]  time = 0.045, size = 0, normalized size = 0. \[ \int \left ( -b{x}^{2}+a \right ) ^{{\frac{5}{3}}} \left ( b{x}^{2}+3\,a \right ) ^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^2+a)^(5/3)*(b*x^2+3*a)^3,x)

[Out]

int((-b*x^2+a)^(5/3)*(b*x^2+3*a)^3,x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + 3 \, a\right )}^{3}{\left (-b x^{2} + a\right )}^{\frac{5}{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + 3*a)^3*(-b*x^2 + a)^(5/3),x, algorithm="maxima")

[Out]

integrate((b*x^2 + 3*a)^3*(-b*x^2 + a)^(5/3), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (b^{4} x^{8} + 8 \, a b^{3} x^{6} + 18 \, a^{2} b^{2} x^{4} - 27 \, a^{4}\right )}{\left (-b x^{2} + a\right )}^{\frac{2}{3}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + 3*a)^3*(-b*x^2 + a)^(5/3),x, algorithm="fricas")

[Out]

integral(-(b^4*x^8 + 8*a*b^3*x^6 + 18*a^2*b^2*x^4 - 27*a^4)*(-b*x^2 + a)^(2/3),
x)

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Sympy [A]  time = 21.0275, size = 139, normalized size = 0.21 \[ 27 a^{\frac{14}{3}} x{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} - \frac{18 a^{\frac{8}{3}} b^{2} x^{5}{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{5} - \frac{8 a^{\frac{5}{3}} b^{3} x^{7}{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{7} - \frac{a^{\frac{2}{3}} b^{4} x^{9}{{}_{2}F_{1}\left (\begin{matrix} - \frac{2}{3}, \frac{9}{2} \\ \frac{11}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x**2+a)**(5/3)*(b*x**2+3*a)**3,x)

[Out]

27*a**(14/3)*x*hyper((-2/3, 1/2), (3/2,), b*x**2*exp_polar(2*I*pi)/a) - 18*a**(8
/3)*b**2*x**5*hyper((-2/3, 5/2), (7/2,), b*x**2*exp_polar(2*I*pi)/a)/5 - 8*a**(5
/3)*b**3*x**7*hyper((-2/3, 7/2), (9/2,), b*x**2*exp_polar(2*I*pi)/a)/7 - a**(2/3
)*b**4*x**9*hyper((-2/3, 9/2), (11/2,), b*x**2*exp_polar(2*I*pi)/a)/9

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (b x^{2} + 3 \, a\right )}^{3}{\left (-b x^{2} + a\right )}^{\frac{5}{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + 3*a)^3*(-b*x^2 + a)^(5/3),x, algorithm="giac")

[Out]

integrate((b*x^2 + 3*a)^3*(-b*x^2 + a)^(5/3), x)