Integrand size = 24, antiderivative size = 653 \[ \int \frac {\left (3 a+b x^2\right )^4}{\left (a-b x^2\right )^{7/3}} \, dx=-\frac {3240}{91} a x \left (a-b x^2\right )^{2/3}-\frac {81}{13} x \left (a-b x^2\right )^{2/3} \left (3 a+b x^2\right )-\frac {9 x \left (3 a+b x^2\right )^2}{2 \sqrt [3]{a-b x^2}}+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}-\frac {36936 a^2 x}{91 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}-\frac {18468 \sqrt [4]{3} \sqrt {2+\sqrt {3}} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{91 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {12312 \sqrt {2} 3^{3/4} a^{7/3} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{91 b x \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}} \]
-3240/91*a*x*(-b*x^2+a)^(2/3)-81/13*x*(-b*x^2+a)^(2/3)*(b*x^2+3*a)-9/2*x*( b*x^2+3*a)^2/(-b*x^2+a)^(1/3)+3/2*x*(b*x^2+3*a)^3/(-b*x^2+a)^(4/3)-36936/9 1*a^2*x/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))+12312/91*3^(3/4)*a^(7/3)*( 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)-18468/91*3^(1/4)*a^(7/3)*(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 = 15.06 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.15 \[ \int \frac {\left (3 a+b x^2\right )^4}{\left (a-b x^2\right )^{7/3}} \, dx=-\frac {3 \left (1647 a^3 x-4743 a^2 b x^3+177 a b^2 x^5+7 b^3 x^7-4104 a^2 x \left (a-b x^2\right ) \sqrt [3]{1-\frac {b x^2}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {1}{2},\frac {3}{2},\frac {b x^2}{a}\right )\right )}{91 \left (a-b x^2\right )^{4/3}} \]
(-3*(1647*a^3*x - 4743*a^2*b*x^3 + 177*a*b^2*x^5 + 7*b^3*x^7 - 4104*a^2*x* (a - b*x^2)*(1 - (b*x^2)/a)^(1/3)*Hypergeometric2F1[1/3, 1/2, 3/2, (b*x^2) /a]))/(91*(a - b*x^2)^(4/3))
Time = 0.55 (sec) , antiderivative size = 706, normalized size of antiderivative = 1.08, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {315, 27, 401, 27, 403, 27, 299, 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.
| \(\displaystyle \int \frac {\left (3 a+b x^2\right )^4}{\left (a-b x^2\right )^{7/3}} \, dx\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}-\frac {3 \int -\frac {4 a b \left (3 a-5 b x^2\right ) \left (b x^2+3 a\right )^2}{\left (a-b x^2\right )^{4/3}}dx}{8 a b}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} \int \frac {\left (3 a-5 b x^2\right ) \left (b x^2+3 a\right )^2}{\left (a-b x^2\right )^{4/3}}dx+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}\) |
| \(\Big \downarrow \) 401 |
| \(\displaystyle \frac {3}{2} \left (\frac {3 \int \frac {12 a b \left (b x^2+a\right ) \left (b x^2+3 a\right )}{\sqrt [3]{a-b x^2}}dx}{2 a b}-\frac {3 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}\right )+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} \left (18 \int \frac {\left (b x^2+a\right ) \left (b x^2+3 a\right )}{\sqrt [3]{a-b x^2}}dx-\frac {3 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}\right )+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}\) |
| \(\Big \downarrow \) 403 |
| \(\displaystyle \frac {3}{2} \left (18 \left (-\frac {3 \int -\frac {2 a b \left (9 b x^2+7 a\right )}{\sqrt [3]{a-b x^2}}dx}{13 b}-\frac {3}{13} x \left (a-b x^2\right )^{2/3} \left (a+b x^2\right )\right )-\frac {3 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}\right )+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{2} \left (18 \left (\frac {6}{13} a \int \frac {9 b x^2+7 a}{\sqrt [3]{a-b x^2}}dx-\frac {3}{13} x \left (a-b x^2\right )^{2/3} \left (a+b x^2\right )\right )-\frac {3 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}\right )+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}\) |
| \(\Big \downarrow \) 299 |
| \(\displaystyle \frac {3}{2} \left (18 \left (\frac {6}{13} a \left (\frac {76}{7} a \int \frac {1}{\sqrt [3]{a-b x^2}}dx-\frac {27}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {3}{13} x \left (a-b x^2\right )^{2/3} \left (a+b x^2\right )\right )-\frac {3 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}\right )+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {3}{2} \left (18 \left (\frac {6}{13} a \left (-\frac {114 a \sqrt {-b x^2} \int \frac {\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}}{7 b x}-\frac {27}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {3}{13} x \left (a-b x^2\right )^{2/3} \left (a+b x^2\right )\right )-\frac {3 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}\right )+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {3}{2} \left (18 \left (\frac {6}{13} a \left (-\frac {114 a \sqrt {-b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}\right )}{7 b x}-\frac {27}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {3}{13} x \left (a-b x^2\right )^{2/3} \left (a+b x^2\right )\right )-\frac {3 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}\right )+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {3}{2} \left (18 \left (\frac {6}{13} a \left (-\frac {114 a \sqrt {-b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\right )}{7 b x}-\frac {27}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {3}{13} x \left (a-b x^2\right )^{2/3} \left (a+b x^2\right )\right )-\frac {3 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}\right )+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {3}{2} \left (18 \left (\frac {6}{13} a \left (-\frac {114 a \sqrt {-b x^2} \left (-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}+\frac {\sqrt [4]{3} \sqrt {2+\sqrt {3}} \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} E\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )|-7+4 \sqrt {3}\right )}{\sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}-\frac {2 \sqrt {-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )}{7 b x}-\frac {27}{7} x \left (a-b x^2\right )^{2/3}\right )-\frac {3}{13} x \left (a-b x^2\right )^{2/3} \left (a+b x^2\right )\right )-\frac {3 x \left (3 a+b x^2\right )^2}{\sqrt [3]{a-b x^2}}\right )+\frac {3 x \left (3 a+b x^2\right )^3}{2 \left (a-b x^2\right )^{4/3}}\) |
(3*x*(3*a + b*x^2)^3)/(2*(a - b*x^2)^(4/3)) + (3*((-3*x*(3*a + b*x^2)^2)/( a - b*x^2)^(1/3) + 18*((-3*x*(a - b*x^2)^(2/3)*(a + b*x^2))/13 + (6*a*((-2 7*x*(a - b*x^2)^(2/3))/7 - (114*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]*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]])/(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[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)))/2
3.2.35.3.1 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)^(-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[(a*d - c*b)*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(2*a*b*(p + 1))), x] - Simp[1/(2*a*b*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 2)*S imp[c*(a*d - c*b*(2*p + 3)) + d*(a*d*(2*(q - 1) + 1) - b*c*(2*(p + q) + 1)) *x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, - 1] && GtQ[q, 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[(-(b*e - a*f))*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^ q/(a*b*2*(p + 1))), x] + Simp[1/(a*b*2*(p + 1)) Int[(a + b*x^2)^(p + 1)*( c + d*x^2)^(q - 1)*Simp[c*(b*e*2*(p + 1) + b*e - a*f) + d*(b*e*2*(p + 1) + (b*e - a*f)*(2*q + 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && L tQ[p, -1] && GtQ[q, 0]
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]
\[\int \frac {\left (b \,x^{2}+3 a \right )^{4}}{\left (-b \,x^{2}+a \right )^{\frac {7}{3}}}d x\]
\[ \int \frac {\left (3 a+b x^2\right )^4}{\left (a-b x^2\right )^{7/3}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{4}}{{\left (-b x^{2} + a\right )}^{\frac {7}{3}}} \,d x } \]
integral(-(b^4*x^8 + 12*a*b^3*x^6 + 54*a^2*b^2*x^4 + 108*a^3*b*x^2 + 81*a^ 4)*(-b*x^2 + a)^(2/3)/(b^3*x^6 - 3*a*b^2*x^4 + 3*a^2*b*x^2 - a^3), x)
\[ \int \frac {\left (3 a+b x^2\right )^4}{\left (a-b x^2\right )^{7/3}} \, dx=\int \frac {\left (3 a + b x^{2}\right )^{4}}{\left (a - b x^{2}\right )^{\frac {7}{3}}}\, dx \]
\[ \int \frac {\left (3 a+b x^2\right )^4}{\left (a-b x^2\right )^{7/3}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{4}}{{\left (-b x^{2} + a\right )}^{\frac {7}{3}}} \,d x } \]
\[ \int \frac {\left (3 a+b x^2\right )^4}{\left (a-b x^2\right )^{7/3}} \, dx=\int { \frac {{\left (b x^{2} + 3 \, a\right )}^{4}}{{\left (-b x^{2} + a\right )}^{\frac {7}{3}}} \,d x } \]
Timed out. \[ \int \frac {\left (3 a+b x^2\right )^4}{\left (a-b x^2\right )^{7/3}} \, dx=\int \frac {{\left (b\,x^2+3\,a\right )}^4}{{\left (a-b\,x^2\right )}^{7/3}} \,d x \]