Integrand size = 24, antiderivative size = 818 \[ \int \frac {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^3} \, dx=\frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}+\frac {x \left (a-b x^2\right )^{2/3}}{36 a^2 \left (3 a+b x^2\right )}-\frac {x}{36 a^2 \left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{72\ 2^{2/3} \sqrt {3} a^{11/6} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{72\ 2^{2/3} \sqrt {3} a^{11/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{216\ 2^{2/3} a^{11/6} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{72\ 2^{2/3} a^{11/6} \sqrt {b}}-\frac {\sqrt {2+\sqrt {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 )}{24\ 3^{3/4} a^{5/3} 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 {\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 )}{18 \sqrt {2} \sqrt [4]{3} a^{5/3} 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}}} \]
1/12*x*(-b*x^2+a)^(2/3)/a/(b*x^2+3*a)^2+1/36*x*(-b*x^2+a)^(2/3)/a^2/(b*x^2 +3*a)-1/36*x/a^2/(-(-b*x^2+a)^(1/3)+a^(1/3)*(1-3^(1/2)))+1/144*arctanh(x*b ^(1/2)/a^(1/6)/(a^(1/3)+2^(1/3)*(-b*x^2+a)^(1/3)))*2^(1/3)/a^(11/6)/b^(1/2 )-1/432*arctanh(x*b^(1/2)/a^(1/2))*2^(1/3)/a^(11/6)/b^(1/2)+1/432*arctan(a ^(1/6)*(a^(1/3)-2^(1/3)*(-b*x^2+a)^(1/3))*3^(1/2)/x/b^(1/2))*2^(1/3)/a^(11 /6)*3^(1/2)/b^(1/2)+1/432*arctan(3^(1/2)*a^(1/2)/x/b^(1/2))*2^(1/3)/a^(11/ 6)*3^(1/2)/b^(1/2)+1/108*(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))*((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)*3^(3/4)/a^(5/3)/b/x*2^(1/2)/(-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)-1/72*(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))*3^(1/4)/a^(5/3)/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 6 vs. order 4 in optimal.
Time = 10.24 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.31 \[ \int \frac {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^3} \, dx=\frac {\frac {b x^3 \sqrt [3]{1-\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},1,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}{a^3}+\frac {27 x \left (6-\frac {5 b x^2}{a}-\frac {b^2 x^4}{a^2}+\frac {18 \left (3 a+b x^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}{9 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+2 b x^2 \left (-\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )\right )}\right )}{\left (3 a+b x^2\right )^2}}{972 \sqrt [3]{a-b x^2}} \]
((b*x^3*(1 - (b*x^2)/a)^(1/3)*AppellF1[3/2, 1/3, 1, 5/2, (b*x^2)/a, -1/3*( b*x^2)/a])/a^3 + (27*x*(6 - (5*b*x^2)/a - (b^2*x^4)/a^2 + (18*(3*a + b*x^2 )*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a])/(9*a*AppellF1[1/2 , 1/3, 1, 3/2, (b*x^2)/a, -1/3*(b*x^2)/a] + 2*b*x^2*(-AppellF1[3/2, 1/3, 2 , 5/2, (b*x^2)/a, -1/3*(b*x^2)/a] + AppellF1[3/2, 4/3, 1, 5/2, (b*x^2)/a, -1/3*(b*x^2)/a]))))/(3*a + b*x^2)^2)/(972*(a - b*x^2)^(1/3))
Time = 0.71 (sec) , antiderivative size = 878, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {314, 27, 402, 27, 405, 233, 305, 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.
| \(\displaystyle \int \frac {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 314 |
| \(\displaystyle \frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}-\frac {\int -\frac {9 a-5 b x^2}{3 \sqrt [3]{a-b x^2} \left (b x^2+3 a\right )^2}dx}{12 a}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {9 a-5 b x^2}{\sqrt [3]{a-b x^2} \left (b x^2+3 a\right )^2}dx}{36 a}+\frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 402 |
| \(\displaystyle \frac {\frac {x \left (a-b x^2\right )^{2/3}}{a \left (3 a+b x^2\right )}-\frac {\int -\frac {8 a b \left (b x^2+6 a\right )}{\sqrt [3]{a-b x^2} \left (b x^2+3 a\right )}dx}{24 a^2 b}}{36 a}+\frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\int \frac {b x^2+6 a}{\sqrt [3]{a-b x^2} \left (b x^2+3 a\right )}dx}{3 a}+\frac {x \left (a-b x^2\right )^{2/3}}{a \left (3 a+b x^2\right )}}{36 a}+\frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 405 |
| \(\displaystyle \frac {\frac {\int \frac {1}{\sqrt [3]{a-b x^2}}dx+3 a \int \frac {1}{\sqrt [3]{a-b x^2} \left (b x^2+3 a\right )}dx}{3 a}+\frac {x \left (a-b x^2\right )^{2/3}}{a \left (3 a+b x^2\right )}}{36 a}+\frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {\frac {3 a \int \frac {1}{\sqrt [3]{a-b x^2} \left (b x^2+3 a\right )}dx-\frac {3 \sqrt {-b x^2} \int \frac {\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}}{2 b x}}{3 a}+\frac {x \left (a-b x^2\right )^{2/3}}{a \left (3 a+b x^2\right )}}{36 a}+\frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 305 |
| \(\displaystyle \frac {\frac {3 a \left (\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}\right )-\frac {3 \sqrt {-b x^2} \int \frac {\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}}{2 b x}}{3 a}+\frac {x \left (a-b x^2\right )^{2/3}}{a \left (3 a+b x^2\right )}}{36 a}+\frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {\frac {3 a \left (\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}\right )-\frac {3 \sqrt {-b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}\right )}{2 b x}}{3 a}+\frac {x \left (a-b x^2\right )^{2/3}}{a \left (3 a+b x^2\right )}}{36 a}+\frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {\frac {3 a \left (\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}\right )-\frac {3 \sqrt {-b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\right )}{2 b x}}{3 a}+\frac {x \left (a-b x^2\right )^{2/3}}{a \left (3 a+b x^2\right )}}{36 a}+\frac {x \left (a-b x^2\right )^{2/3}}{12 a \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {\left (a-b x^2\right )^{2/3} x}{12 a \left (b x^2+3 a\right )^2}+\frac {\frac {\left (a-b x^2\right )^{2/3} x}{a \left (b x^2+3 a\right )}+\frac {3 a \left (\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{a}+\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}\right )-\frac {3 \sqrt {-b x^2} \left (\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-b x^2} \sqrt [3]{a}+\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 {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-b x^2} \sqrt [3]{a}+\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 {2 \sqrt {-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right )}{2 b x}}{3 a}}{36 a}\) |
(x*(a - b*x^2)^(2/3))/(12*a*(3*a + b*x^2)^2) + ((x*(a - b*x^2)^(2/3))/(a*( 3*a + b*x^2)) + (3*a*(ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(2*2^(2/3)*Sqr t[3]*a^(5/6)*Sqrt[b]) + ArcTan[(Sqrt[3]*a^(1/6)*(a^(1/3) - 2^(1/3)*(a - b* x^2)^(1/3)))/(Sqrt[b]*x)]/(2*2^(2/3)*Sqrt[3]*a^(5/6)*Sqrt[b]) - ArcTanh[(S qrt[b]*x)/Sqrt[a]]/(6*2^(2/3)*a^(5/6)*Sqrt[b]) + ArcTanh[(Sqrt[b]*x)/(a^(1 /6)*(a^(1/3) + 2^(1/3)*(a - b*x^2)^(1/3)))]/(2*2^(2/3)*a^(5/6)*Sqrt[b])) - (3*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 - Sq rt[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[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)])))/(2*b*x))/(3*a))/(36*a)
3.2.14.3.1 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[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Wit h[{q = Rt[-b/a, 2]}, Simp[q*(ArcTan[Sqrt[3]/(q*x)]/(2*2^(2/3)*Sqrt[3]*a^(1/ 3)*d)), x] + (Simp[q*(ArcTanh[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a + b*x^2)^ (1/3))]/(2*2^(2/3)*a^(1/3)*d)), x] - Simp[q*(ArcTanh[q*x]/(6*2^(2/3)*a^(1/3 )*d)), x] + Simp[q*(ArcTan[Sqrt[3]*((a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3))/( a^(1/3)*q*x))]/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d)), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && NegQ[b/a]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim p[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] + Simp[1/(2*a* (p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*p + 3) + d *(2*(p + q + 1) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && LtQ[0, 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 + 1)/(a*2*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*2*(b*c - a*d)*(p + 1)) Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(b*e - a*f) + e*2*(b*c - a*d) *(p + 1) + d*(b*e - a*f)*(2*(p + q + 2) + 1)*x^2, x], x], x] /; FreeQ[{a, b , c, d, e, f, q}, x] && LtQ[p, -1]
Int[(((a_) + (b_.)*(x_)^2)^(p_)*((e_) + (f_.)*(x_)^2))/((c_) + (d_.)*(x_)^2 ), x_Symbol] :> Simp[f/d Int[(a + b*x^2)^p, x], x] + Simp[(d*e - c*f)/d Int[(a + b*x^2)^p/(c + d*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, p}, 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]
\[\int \frac {\left (-b \,x^{2}+a \right )^{\frac {2}{3}}}{\left (b \,x^{2}+3 a \right )^{3}}d x\]
Timed out. \[ \int \frac {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^3} \, dx=\int \frac {\left (a - b x^{2}\right )^{\frac {2}{3}}}{\left (3 a + b x^{2}\right )^{3}}\, dx \]
\[ \int \frac {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^3} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {2}{3}}}{{\left (b x^{2} + 3 \, a\right )}^{3}} \,d x } \]
\[ \int \frac {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^3} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {2}{3}}}{{\left (b x^{2} + 3 \, a\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a-b x^2\right )^{2/3}}{\left (3 a+b x^2\right )^3} \, dx=\int \frac {{\left (a-b\,x^2\right )}^{2/3}}{{\left (b\,x^2+3\,a\right )}^3} \,d x \]