3.123 \(\int \frac{\left (3 a+b x^2\right )^3}{\sqrt [3]{a-b x^2}} \, dx\)

Optimal. Leaf size=628 \[ \frac{71712 \sqrt{2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{1729 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{107568 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{1729 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{215136 a^3 x}{1729 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{15768 a^2 x \left (a-b x^2\right )^{2/3}}{1729}-\frac{324}{247} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{3}{19} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \]

[Out]

(-15768*a^2*x*(a - b*x^2)^(2/3))/1729 - (324*a*x*(a - b*x^2)^(2/3)*(3*a + b*x^2)
)/247 - (3*x*(a - b*x^2)^(2/3)*(3*a + b*x^2)^2)/19 - (215136*a^3*x)/(1729*((1 -
Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (107568*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(10
/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]])/(1729*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)]) + (71712*Sqrt[2]*3^(3/4)*a
^(10/3)*(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[Ar
cSin[((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]])/(1729*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 = 1.00488, antiderivative size = 628, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ \frac{71712 \sqrt{2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} F\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{1729 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{107568 \sqrt [4]{3} \sqrt{2+\sqrt{3}} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt{\frac{a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\sin ^{-1}\left (\frac{\left (1+\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt{3}\right )}{1729 b x \sqrt{-\frac{\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac{215136 a^3 x}{1729 \left (\left (1-\sqrt{3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac{15768 a^2 x \left (a-b x^2\right )^{2/3}}{1729}-\frac{324}{247} a x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac{3}{19} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \]

Antiderivative was successfully verified.

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

[Out]

(-15768*a^2*x*(a - b*x^2)^(2/3))/1729 - (324*a*x*(a - b*x^2)^(2/3)*(3*a + b*x^2)
)/247 - (3*x*(a - b*x^2)^(2/3)*(3*a + b*x^2)^2)/19 - (215136*a^3*x)/(1729*((1 -
Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))) - (107568*3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(10
/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]])/(1729*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)]) + (71712*Sqrt[2]*3^(3/4)*a
^(10/3)*(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[Ar
cSin[((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]])/(1729*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 = 81.7608, size = 507, normalized size = 0.81 \[ - \frac{107568 \sqrt [4]{3} a^{\frac{10}{3}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a - b x^{2}} + \left (a - b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}} \sqrt{\sqrt{3} + 2} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right ) E\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{1729 b x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}}} + \frac{71712 \sqrt{2} \cdot 3^{\frac{3}{4}} a^{\frac{10}{3}} \sqrt{\frac{a^{\frac{2}{3}} + \sqrt [3]{a} \sqrt [3]{a - b x^{2}} + \left (a - b x^{2}\right )^{\frac{2}{3}}}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right ) F\left (\operatorname{asin}{\left (\frac{\sqrt [3]{a} \left (1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}}{- \sqrt [3]{a} \left (-1 + \sqrt{3}\right ) - \sqrt [3]{a - b x^{2}}} \right )}\middle | -7 + 4 \sqrt{3}\right )}{1729 b x \sqrt{- \frac{\sqrt [3]{a} \left (\sqrt [3]{a} - \sqrt [3]{a - b x^{2}}\right )}{\left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )^{2}}}} + \frac{215136 a^{3} x}{1729 \left (\sqrt [3]{a} \left (-1 + \sqrt{3}\right ) + \sqrt [3]{a - b x^{2}}\right )} - \frac{15768 a^{2} x \left (a - b x^{2}\right )^{\frac{2}{3}}}{1729} - \frac{324 a x \left (a - b x^{2}\right )^{\frac{2}{3}} \left (3 a + b x^{2}\right )}{247} - \frac{3 x \left (a - b x^{2}\right )^{\frac{2}{3}} \left (3 a + b x^{2}\right )^{2}}{19} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-107568*3**(1/4)*a**(10/3)*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))/(1729*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)) + 71712*sqrt(2)*3**(3/4)*a**(10/3)*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))/(1729*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)) + 215136*a
**3*x/(1729*(a**(1/3)*(-1 + sqrt(3)) + (a - b*x**2)**(1/3))) - 15768*a**2*x*(a -
b*x**2)**(2/3)/1729 - 324*a*x*(a - b*x**2)**(2/3)*(3*a + b*x**2)/247 - 3*x*(a -
b*x**2)**(2/3)*(3*a + b*x**2)**2/19

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Mathematica [C]  time = 0.0757102, size = 88, normalized size = 0.14 \[ \frac{3 \left (23904 a^3 x \sqrt [3]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{3},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )-8343 a^3 x+7041 a^2 b x^3+1211 a b^2 x^5+91 b^3 x^7\right )}{1729 \sqrt [3]{a-b x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(-8343*a^3*x + 7041*a^2*b*x^3 + 1211*a*b^2*x^5 + 91*b^3*x^7 + 23904*a^3*x*(1
- (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a]))/(1729*(a - b*x^
2)^(1/3))

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Maple [F]  time = 0.04, size = 0, normalized size = 0. \[ \int{ \left ( b{x}^{2}+3\,a \right ) ^{3}{\frac{1}{\sqrt [3]{-b{x}^{2}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{b^{3} x^{6} + 9 \, a b^{2} x^{4} + 27 \, a^{2} b x^{2} + 27 \, a^{3}}{{\left (-b x^{2} + a\right )}^{\frac{1}{3}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b^3*x^6 + 9*a*b^2*x^4 + 27*a^2*b*x^2 + 27*a^3)/(-b*x^2 + a)^(1/3), x)

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Sympy [A]  time = 8.90166, size = 129, normalized size = 0.21 \[ 27 a^{\frac{8}{3}} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} + 9 a^{\frac{5}{3}} b x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} + \frac{9 a^{\frac{2}{3}} b^{2} x^{5}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{5} + \frac{b^{3} x^{7}{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{3}, \frac{7}{2} \\ \frac{9}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )}}{7 \sqrt [3]{a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

27*a**(8/3)*x*hyper((1/3, 1/2), (3/2,), b*x**2*exp_polar(2*I*pi)/a) + 9*a**(5/3)
*b*x**3*hyper((1/3, 3/2), (5/2,), b*x**2*exp_polar(2*I*pi)/a) + 9*a**(2/3)*b**2*
x**5*hyper((1/3, 5/2), (7/2,), b*x**2*exp_polar(2*I*pi)/a)/5 + b**3*x**7*hyper((
1/3, 7/2), (9/2,), b*x**2*exp_polar(2*I*pi)/a)/(7*a**(1/3))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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