Integrand size = 24, antiderivative size = 617 \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \, dx=\frac {7776 a^2 x \left (a-b x^2\right )^{2/3}}{1729}-\frac {252}{247} a x \left (a-b x^2\right )^{5/3}-\frac {3}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )-\frac {31104 a^3 x}{1729 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {15552 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{1729 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {10368 \sqrt {2} 3^{3/4} a^{10/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{1729 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \]
7776/1729*a^2*x*(-b*x^2+a)^(2/3)-252/247*a*x*(-b*x^2+a)^(5/3)-3/19*x*(-b*x ^2+a)^(5/3)*(b*x^2+3*a)-31104/1729*a^3*x/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^( 1/2)))+10368/1729*3^(3/4)*a^(10/3)*(a^(1/3)-(-b*x^2+a)^(1/3))*EllipticF((- (-b*x^2+a)^(1/3)+a^(1/3)*(1+3^(1/2)))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2 ))),2*I-I*3^(1/2))*2^(1/2)*((a^(2/3)+a^(1/3)*(-b*x^2+a)^(1/3)+(-b*x^2+a)^( 2/3))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)/b/x/(-a^(1/3)*(a^(1 /3)-(-b*x^2+a)^(1/3))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)-155 52/1729*3^(1/4)*a^(10/3)*(a^(1/3)-(-b*x^2+a)^(1/3))*EllipticE((-(-b*x^2+a) ^(1/3)+a^(1/3)*(1+3^(1/2)))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2))),2*I-I* 3^(1/2))*((a^(2/3)+a^(1/3)*(-b*x^2+a)^(1/3)+(-b*x^2+a)^(2/3))/(-(-b*x^2+a) ^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^(1/2)*(1/2*6^(1/2)+1/2*2^(1/2))/b/x/(-a^(1/ 3)*(a^(1/3)-(-b*x^2+a)^(1/3))/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))^2)^( 1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 8.21 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.29 \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \, dx=\frac {x \left (a-b x^2\right )^{2/3} \left (21 a \left (45 a^2+10 a b x^2+b^2 x^4\right ) \operatorname {Gamma}\left (-\frac {2}{3}\right ) \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},\frac {1}{2},\frac {7}{2},\frac {b x^2}{a}\right )+8 b x^2 \left (18 a^2+9 a b x^2+b^2 x^4\right ) \operatorname {Gamma}\left (\frac {1}{3}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{2},\frac {9}{2},\frac {b x^2}{a}\right )+4 b \left (3 a x+b x^3\right )^2 \operatorname {Gamma}\left (\frac {1}{3}\right ) \, _3F_2\left (\frac {1}{3},\frac {3}{2},2;1,\frac {9}{2};\frac {b x^2}{a}\right )\right )}{105 a \left (1-\frac {b x^2}{a}\right )^{2/3} \operatorname {Gamma}\left (-\frac {2}{3}\right )} \]
(x*(a - b*x^2)^(2/3)*(21*a*(45*a^2 + 10*a*b*x^2 + b^2*x^4)*Gamma[-2/3]*Hyp ergeometric2F1[-2/3, 1/2, 7/2, (b*x^2)/a] + 8*b*x^2*(18*a^2 + 9*a*b*x^2 + b^2*x^4)*Gamma[1/3]*Hypergeometric2F1[1/3, 3/2, 9/2, (b*x^2)/a] + 4*b*(3*a *x + b*x^3)^2*Gamma[1/3]*HypergeometricPFQ[{1/3, 3/2, 2}, {1, 9/2}, (b*x^2 )/a]))/(105*a*(1 - (b*x^2)/a)^(2/3)*Gamma[-2/3])
Time = 0.53 (sec) , antiderivative size = 669, normalized size of antiderivative = 1.08, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {318, 27, 299, 211, 233, 833, 760, 2418}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \, dx\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle -\frac {3 \int -4 a b \left (a-b x^2\right )^{2/3} \left (7 b x^2+15 a\right )dx}{19 b}-\frac {3}{19} x \left (3 a+b x^2\right ) \left (a-b x^2\right )^{5/3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {12}{19} a \int \left (a-b x^2\right )^{2/3} \left (7 b x^2+15 a\right )dx-\frac {3}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {12}{19} a \left (\frac {216}{13} a \int \left (a-b x^2\right )^{2/3}dx-\frac {21}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {3}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {12}{19} a \left (\frac {216}{13} a \left (\frac {4}{7} a \int \frac {1}{\sqrt [3]{a-b x^2}}dx+\frac {3}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {21}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {3}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {12}{19} a \left (\frac {216}{13} a \left (\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {6 a \sqrt {-b x^2} \int \frac {\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}}{7 b x}\right )-\frac {21}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {3}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {12}{19} a \left (\frac {216}{13} a \left (\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {6 a \sqrt {-b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}\right )}{7 b x}\right )-\frac {21}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {3}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {12}{19} a \left (\frac {216}{13} a \left (\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {6 a \sqrt {-b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\right )}{7 b x}\right )-\frac {21}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {3}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {12}{19} a \left (\frac {216}{13} a \left (\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {6 a \sqrt {-b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {2 \sqrt {-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )}{7 b x}\right )-\frac {21}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {3}{19} x \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )\) |
(-3*x*(a - b*x^2)^(5/3)*(3*a + b*x^2))/19 + (12*a*((-21*x*(a - b*x^2)^(5/3 ))/13 + (216*a*((3*x*(a - b*x^2)^(2/3))/7 - (6*a*Sqrt[-(b*x^2)]*((-2*Sqrt[ -(b*x^2)])/((1 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3)) + (3^(1/4)*Sqrt[2 + Sqrt[3]]*a^(1/3)*(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]])/(Sqrt[-(b*x^2) ]*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)]) - (2*Sqrt[2 - Sqrt[3]]*(1 + Sqrt[3])*a^(1/3)*(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[Arc Sin[((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]])/(3^(1/4)*Sqrt[-(b*x^2)]*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)])))/(7*b*x)))/13))/19
3.2.10.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x*((a + b*x^2)^p/(2*p + 1 )), x] + Simp[2*a*(p/(2*p + 1)) Int[(a + b*x^2)^(p - 1), x], x] /; FreeQ[ {a, b}, x] && GtQ[p, 0] && (IntegerQ[4*p] || IntegerQ[6*p])
Int[((a_) + (b_.)*(x_)^2)^(-1/3), x_Symbol] :> Simp[3*(Sqrt[b*x^2]/(2*b*x)) Subst[Int[x/Sqrt[-a + x^3], x], x, (a + b*x^2)^(1/3)], x] /; FreeQ[{a, b }, x]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2), x_Symbol] :> Simp[d*x *((a + b*x^2)^(p + 1)/(b*(2*p + 3))), x] - Simp[(a*d - b*c*(2*p + 3))/(b*(2 *p + 3)) Int[(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && NeQ[2*p + 3, 0]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S imp[1/(b*(2*(p + q) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b *c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G tQ[q, 1] && NeQ[2*(p + q) + 1, 0] && !IGtQ[p, 1] && IntBinomialQ[a, b, c, d, 2, p, q, x]
Int[1/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3]], s = Denom[Rt[b/a, 3]]}, Simp[2*Sqrt[2 - Sqrt[3]]*(s + r*x)*(Sqrt[(s^2 - r*s *x + r^2*x^2)/((1 - Sqrt[3])*s + r*x)^2]/(3^(1/4)*r*Sqrt[a + b*x^3]*Sqrt[(- s)*((s + r*x)/((1 - Sqrt[3])*s + r*x)^2)]))*EllipticF[ArcSin[((1 + Sqrt[3]) *s + r*x)/((1 - Sqrt[3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b}, x ] && NegQ[a]
Int[(x_)/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = Numer[Rt[b/a, 3] ], s = Denom[Rt[b/a, 3]]}, Simp[(-(1 + Sqrt[3]))*(s/r) Int[1/Sqrt[a + b*x ^3], x], x] + Simp[1/r Int[((1 + Sqrt[3])*s + r*x)/Sqrt[a + b*x^3], x], x ]] /; FreeQ[{a, b}, x] && NegQ[a]
Int[((c_) + (d_.)*(x_))/Sqrt[(a_) + (b_.)*(x_)^3], x_Symbol] :> With[{r = N umer[Simplify[(1 + Sqrt[3])*(d/c)]], s = Denom[Simplify[(1 + Sqrt[3])*(d/c) ]]}, Simp[2*d*s^3*(Sqrt[a + b*x^3]/(a*r^2*((1 - Sqrt[3])*s + r*x))), x] + S imp[3^(1/4)*Sqrt[2 + Sqrt[3]]*d*s*(s + r*x)*(Sqrt[(s^2 - r*s*x + r^2*x^2)/( (1 - Sqrt[3])*s + r*x)^2]/(r^2*Sqrt[a + b*x^3]*Sqrt[(-s)*((s + r*x)/((1 - S qrt[3])*s + r*x)^2)]))*EllipticE[ArcSin[((1 + Sqrt[3])*s + r*x)/((1 - Sqrt[ 3])*s + r*x)], -7 + 4*Sqrt[3]], x]] /; FreeQ[{a, b, c, d}, x] && NegQ[a] && EqQ[b*c^3 - 2*(5 + 3*Sqrt[3])*a*d^3, 0]
\[\int \left (-b \,x^{2}+a \right )^{\frac {2}{3}} \left (b \,x^{2}+3 a \right )^{2}d x\]
\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \, dx=\int { {\left (b x^{2} + 3 \, a\right )}^{2} {\left (-b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \]
Time = 1.44 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.16 \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \, dx=9 a^{\frac {8}{3}} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + 2 a^{\frac {5}{3}} b x^{3} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {3}{2} \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} + \frac {a^{\frac {2}{3}} b^{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5} \]
9*a**(8/3)*x*hyper((-2/3, 1/2), (3/2,), b*x**2*exp_polar(2*I*pi)/a) + 2*a* *(5/3)*b*x**3*hyper((-2/3, 3/2), (5/2,), b*x**2*exp_polar(2*I*pi)/a) + a** (2/3)*b**2*x**5*hyper((-2/3, 5/2), (7/2,), b*x**2*exp_polar(2*I*pi)/a)/5
\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \, dx=\int { {\left (b x^{2} + 3 \, a\right )}^{2} {\left (-b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \]
\[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \, dx=\int { {\left (b x^{2} + 3 \, a\right )}^{2} {\left (-b x^{2} + a\right )}^{\frac {2}{3}} \,d x } \]
Timed out. \[ \int \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )^2 \, dx=\int {\left (a-b\,x^2\right )}^{2/3}\,{\left (b\,x^2+3\,a\right )}^2 \,d x \]