Integrand size = 24, antiderivative size = 668 \[ \int \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^3 \, dx=\frac {2809728 a^4 x \left (a-b x^2\right )^{2/3}}{267995}+\frac {1404864 a^3 x \left (a-b x^2\right )^{5/3}}{191425}-\frac {33264 a^2 x \left (a-b x^2\right )^{8/3}}{14725}-\frac {432}{775} a x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )-\frac {3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2-\frac {11238912 a^5 x}{267995 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {5619456 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{16/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{267995 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {3746304 \sqrt {2} 3^{3/4} a^{16/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{267995 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \]
2809728/267995*a^4*x*(-b*x^2+a)^(2/3)+1404864/191425*a^3*x*(-b*x^2+a)^(5/3 )-33264/14725*a^2*x*(-b*x^2+a)^(8/3)-432/775*a*x*(-b*x^2+a)^(8/3)*(b*x^2+3 *a)-3/31*x*(-b*x^2+a)^(8/3)*(b*x^2+3*a)^2-11238912/267995*a^5*x/(-(-b*x^2+ a)^(1/3)+a^(1/3)*(1-3^(1/2)))+3746304/267995*3^(3/4)*a^(16/3)*(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)-5619456/267995*3^(1/4)*a^(16/3)*(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 = 12.77 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.16 \[ \int \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^3 \, dx=\frac {3 \left (5815935 a^5 x-5312355 a^4 b x^3-1675114 a^3 b^2 x^5+749658 a^2 b^3 x^7+378651 a b^4 x^9+43225 b^5 x^{11}+6243840 a^5 x \sqrt [3]{1-\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {b x^2}{a}\right )\right )}{1339975 \sqrt [3]{a-b x^2}} \]
(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)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2)/a]))/(1339975*(a - b*x^2)^ (1/3))
Time = 0.56 (sec) , antiderivative size = 726, normalized size of antiderivative = 1.09, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {318, 27, 403, 27, 299, 211, 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.
| \(\displaystyle \int \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^3 \, dx\) |
| \(\Big \downarrow \) 318 |
| \(\displaystyle -\frac {3 \int -48 a b \left (a-b x^2\right )^{5/3} \left (b x^2+2 a\right ) \left (b x^2+3 a\right )dx}{31 b}-\frac {3}{31} x \left (3 a+b x^2\right )^2 \left (a-b x^2\right )^{8/3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {144}{31} a \int \left (a-b x^2\right )^{5/3} \left (b x^2+2 a\right ) \left (b x^2+3 a\right )dx-\frac {3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {144}{31} a \left (-\frac {3 \int -4 a b \left (a-b x^2\right )^{5/3} \left (8 b x^2+13 a\right )dx}{25 b}-\frac {3}{25} x \left (2 a+b x^2\right ) \left (a-b x^2\right )^{8/3}\right )-\frac {3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {144}{31} a \left (\frac {12}{25} a \int \left (a-b x^2\right )^{5/3} \left (8 b x^2+13 a\right )dx-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (2 a+b x^2\right )\right )-\frac {3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {144}{31} a \left (\frac {12}{25} a \left (\frac {271}{19} a \int \left (a-b x^2\right )^{5/3}dx-\frac {24}{19} x \left (a-b x^2\right )^{8/3}\right )-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (2 a+b x^2\right )\right )-\frac {3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {144}{31} a \left (\frac {12}{25} a \left (\frac {271}{19} a \left (\frac {10}{13} a \int \left (a-b x^2\right )^{2/3}dx+\frac {3}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {24}{19} x \left (a-b x^2\right )^{8/3}\right )-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (2 a+b x^2\right )\right )-\frac {3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2\) |
| \(\Big \downarrow \) 211 |
| \(\displaystyle \frac {144}{31} a \left (\frac {12}{25} a \left (\frac {271}{19} a \left (\frac {10}{13} a \left (\frac {4}{7} a \int \frac {1}{\sqrt [3]{a-b x^2}}dx+\frac {3}{7} x \left (a-b x^2\right )^{2/3}\right )+\frac {3}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {24}{19} x \left (a-b x^2\right )^{8/3}\right )-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (2 a+b x^2\right )\right )-\frac {3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {144}{31} a \left (\frac {12}{25} a \left (\frac {271}{19} a \left (\frac {10}{13} a \left (\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {6 a \sqrt {-b x^2} \int \frac {\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}}{7 b x}\right )+\frac {3}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {24}{19} x \left (a-b x^2\right )^{8/3}\right )-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (2 a+b x^2\right )\right )-\frac {3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {144}{31} a \left (\frac {12}{25} a \left (\frac {271}{19} a \left (\frac {10}{13} a \left (\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {6 a \sqrt {-b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}\right )}{7 b x}\right )+\frac {3}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {24}{19} x \left (a-b x^2\right )^{8/3}\right )-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (2 a+b x^2\right )\right )-\frac {3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {144}{31} a \left (\frac {12}{25} a \left (\frac {271}{19} a \left (\frac {10}{13} a \left (\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {6 a \sqrt {-b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\right )}{7 b x}\right )+\frac {3}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {24}{19} x \left (a-b x^2\right )^{8/3}\right )-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (2 a+b x^2\right )\right )-\frac {3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {144}{31} a \left (\frac {12}{25} a \left (\frac {271}{19} a \left (\frac {10}{13} a \left (\frac {3}{7} x \left (a-b x^2\right )^{2/3}-\frac {6 a \sqrt {-b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {2 \sqrt {-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )}{7 b x}\right )+\frac {3}{13} x \left (a-b x^2\right )^{5/3}\right )-\frac {24}{19} x \left (a-b x^2\right )^{8/3}\right )-\frac {3}{25} x \left (a-b x^2\right )^{8/3} \left (2 a+b x^2\right )\right )-\frac {3}{31} x \left (a-b x^2\right )^{8/3} \left (3 a+b x^2\right )^2\) |
(-3*x*(a - b*x^2)^(8/3)*(3*a + b*x^2)^2)/31 + (144*a*((-3*x*(a - b*x^2)^(8 /3)*(2*a + b*x^2))/25 + (12*a*((-24*x*(a - b*x^2)^(8/3))/19 + (271*a*((3*x *(a - b*x^2)^(5/3))/13 + (10*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 - Sqr t[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[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]])/(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))/25))/31
3.2.16.3.1 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[((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.)*((e_) + (f_.)*( x_)^2), x_Symbol] :> Simp[f*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(b*(2*(p + q + 1) + 1))), x] + Simp[1/(b*(2*(p + q + 1) + 1)) Int[(a + b*x^2)^p*(c + d*x^2)^(q - 1)*Simp[c*(b*e - a*f + b*e*2*(p + q + 1)) + (d*(b*e - a*f) + f*2*q*(b*c - a*d) + b*d*e*2*(p + q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[q, 0] && NeQ[2*(p + q + 1) + 1, 0]
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]
\[\int \left (-b \,x^{2}+a \right )^{\frac {5}{3}} \left (b \,x^{2}+3 a \right )^{3}d x\]
\[ \int \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^3 \, dx=\int { {\left (b x^{2} + 3 \, a\right )}^{3} {\left (-b x^{2} + a\right )}^{\frac {5}{3}} \,d x } \]
Time = 2.32 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.21 \[ \int \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^3 \, dx=27 a^{\frac {14}{3}} x {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )} - \frac {18 a^{\frac {8}{3}} b^{2} x^{5} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{5} - \frac {8 a^{\frac {5}{3}} b^{3} x^{7} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {7}{2} \\ \frac {9}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{7} - \frac {a^{\frac {2}{3}} b^{4} x^{9} {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, \frac {9}{2} \\ \frac {11}{2} \end {matrix}\middle | {\frac {b x^{2} e^{2 i \pi }}{a}} \right )}}{9} \]
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
\[ \int \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^3 \, dx=\int { {\left (b x^{2} + 3 \, a\right )}^{3} {\left (-b x^{2} + a\right )}^{\frac {5}{3}} \,d x } \]
\[ \int \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^3 \, dx=\int { {\left (b x^{2} + 3 \, a\right )}^{3} {\left (-b x^{2} + a\right )}^{\frac {5}{3}} \,d x } \]
Timed out. \[ \int \left (a-b x^2\right )^{5/3} \left (3 a+b x^2\right )^3 \, dx=\int {\left (a-b\,x^2\right )}^{5/3}\,{\left (b\,x^2+3\,a\right )}^3 \,d x \]