Integrand size = 24, antiderivative size = 815 \[ \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^3} \, dx=\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}-\frac {x \left (a-b x^2\right )^{2/3}}{18 a \left (3 a+b x^2\right )}+\frac {x}{18 a \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 )}{18\ 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 )}{18\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{54\ 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 )}{18\ 2^{2/3} a^{5/6} \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 )}{12\ 3^{3/4} a^{2/3} 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 )}{9 \sqrt {2} \sqrt [4]{3} a^{2/3} 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}}} \] Output:
1/3*x*(-b*x^2+a)^(2/3)/(b*x^2+3*a)^2-1/18*x*(-b*x^2+a)^(2/3)/a/(b*x^2+3*a) +1/18*x/a/((1-3^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3))+1/108*arctan(3^(1/2)*a^(1 /2)/b^(1/2)/x)*2^(1/3)*3^(1/2)/a^(5/6)/b^(1/2)+1/108*arctan(3^(1/2)*a^(1/6 )*(a^(1/3)-2^(1/3)*(-b*x^2+a)^(1/3))/b^(1/2)/x)*2^(1/3)*3^(1/2)/a^(5/6)/b^ (1/2)-1/108*arctanh(b^(1/2)*x/a^(1/2))*2^(1/3)/a^(5/6)/b^(1/2)+1/36*arctan h(b^(1/2)*x/a^(1/6)/(a^(1/3)+2^(1/3)*(-b*x^2+a)^(1/3)))*2^(1/3)/a^(5/6)/b^ (1/2)+1/36*(1/2*6^(1/2)+1/2*2^(1/2))*(a^(1/3)-(-b*x^2+a)^(1/3))*((a^(2/3)+ a^(1/3)*(-b*x^2+a)^(1/3)+(-b*x^2+a)^(2/3))/((1-3^(1/2))*a^(1/3)-(-b*x^2+a) ^(1/3))^2)^(1/2)*EllipticE(((1+3^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3))/((1-3^(1 /2))*a^(1/3)-(-b*x^2+a)^(1/3)),2*I-I*3^(1/2))*3^(1/4)/a^(2/3)/b/x/(-a^(1/3 )*(a^(1/3)-(-b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3))^2)^(1/ 2)-1/54*(a^(1/3)-(-b*x^2+a)^(1/3))*((a^(2/3)+a^(1/3)*(-b*x^2+a)^(1/3)+(-b* x^2+a)^(2/3))/((1-3^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3))^2)^(1/2)*EllipticF((( 1+3^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3))/((1-3^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3) ),2*I-I*3^(1/2))*2^(1/2)*3^(3/4)/a^(2/3)/b/x/(-a^(1/3)*(a^(1/3)-(-b*x^2+a) ^(1/3))/((1-3^(1/2))*a^(1/3)-(-b*x^2+a)^(1/3))^2)^(1/2)
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.15 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.31 \[ \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^3} \, dx=\frac {-\frac {b x^3 \sqrt [3]{1-\frac {b x^2}{a}} \operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},1,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}{a^2}+\frac {27 x \left (3 a-4 b x^2+\frac {b^2 x^4}{a}+\frac {9 a \left (3 a+b x^2\right ) \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )}{9 a \operatorname {AppellF1}\left (\frac {1}{2},\frac {1}{3},1,\frac {3}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+2 b x^2 \left (-\operatorname {AppellF1}\left (\frac {3}{2},\frac {1}{3},2,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )+\operatorname {AppellF1}\left (\frac {3}{2},\frac {4}{3},1,\frac {5}{2},\frac {b x^2}{a},-\frac {b x^2}{3 a}\right )\right )}\right )}{\left (3 a+b x^2\right )^2}}{486 \sqrt [3]{a-b x^2}} \] Input:
Integrate[(a - b*x^2)^(5/3)/(3*a + b*x^2)^3,x]
Output:
(-((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^2) + (27*x*(3*a - 4*b*x^2 + (b^2*x^4)/a + (9*a*(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)/(486*(a - b*x^2)^(1/3))
Time = 0.71 (sec) , antiderivative size = 882, normalized size of antiderivative = 1.08, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {315, 27, 373, 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.
| \(\displaystyle \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^3} \, dx\) |
| \(\Big \downarrow \) 315 |
| \(\displaystyle \frac {\int \frac {16 a b^2 x^2}{3 \sqrt [3]{a-b x^2} \left (b x^2+3 a\right )^2}dx}{12 a b}+\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{9} b \int \frac {x^2}{\sqrt [3]{a-b x^2} \left (b x^2+3 a\right )^2}dx+\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 373 |
| \(\displaystyle \frac {4}{9} b \left (\frac {\int \frac {3 a-b x^2}{3 \sqrt [3]{a-b x^2} \left (b x^2+3 a\right )}dx}{8 a b}-\frac {x \left (a-b x^2\right )^{2/3}}{8 a b \left (3 a+b x^2\right )}\right )+\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {4}{9} b \left (\frac {\int \frac {3 a-b x^2}{\sqrt [3]{a-b x^2} \left (b x^2+3 a\right )}dx}{24 a b}-\frac {x \left (a-b x^2\right )^{2/3}}{8 a b \left (3 a+b x^2\right )}\right )+\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 405 |
| \(\displaystyle \frac {4}{9} b \left (\frac {6 a \int \frac {1}{\sqrt [3]{a-b x^2} \left (b x^2+3 a\right )}dx-\int \frac {1}{\sqrt [3]{a-b x^2}}dx}{24 a b}-\frac {x \left (a-b x^2\right )^{2/3}}{8 a b \left (3 a+b x^2\right )}\right )+\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 233 |
| \(\displaystyle \frac {4}{9} b \left (\frac {\frac {3 \sqrt {-b x^2} \int \frac {\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}}{2 b x}+6 a \int \frac {1}{\sqrt [3]{a-b x^2} \left (b x^2+3 a\right )}dx}{24 a b}-\frac {x \left (a-b x^2\right )^{2/3}}{8 a b \left (3 a+b x^2\right )}\right )+\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 305 |
| \(\displaystyle \frac {4}{9} b \left (\frac {\frac {3 \sqrt {-b x^2} \int \frac {\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}}{2 b x}+6 a \left (\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}\right )}{24 a b}-\frac {x \left (a-b x^2\right )^{2/3}}{8 a b \left (3 a+b x^2\right )}\right )+\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 833 |
| \(\displaystyle \frac {4}{9} b \left (\frac {\frac {3 \sqrt {-b x^2} \left (\left (1+\sqrt {3}\right ) \sqrt [3]{a} \int \frac {1}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}\right )}{2 b x}+6 a \left (\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}\right )}{24 a b}-\frac {x \left (a-b x^2\right )^{2/3}}{8 a b \left (3 a+b x^2\right )}\right )+\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 760 |
| \(\displaystyle \frac {4}{9} b \left (\frac {\frac {3 \sqrt {-b x^2} \left (-\int \frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\sqrt {-b x^2}}d\sqrt [3]{a-b x^2}-\frac {2 \sqrt {2-\sqrt {3}} \left (1+\sqrt {3}\right ) \sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right ) \sqrt {\frac {a^{2/3}+\sqrt [3]{a} \sqrt [3]{a-b x^2}+\left (a-b x^2\right )^{2/3}}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}{\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}}\right ),-7+4 \sqrt {3}\right )}{\sqrt [4]{3} \sqrt {-b x^2} \sqrt {-\frac {\sqrt [3]{a} \left (\sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )}{\left (\left (1-\sqrt {3}\right ) \sqrt [3]{a}-\sqrt [3]{a-b x^2}\right )^2}}}\right )}{2 b x}+6 a \left (\frac {\arctan \left (\frac {\sqrt {3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a-b x^2}\right )}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\arctan \left (\frac {\sqrt {3} \sqrt {a}}{\sqrt {b} x}\right )}{2\ 2^{2/3} \sqrt {3} a^{5/6} \sqrt {b}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a-b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt {b}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt {b}}\right )}{24 a b}-\frac {x \left (a-b x^2\right )^{2/3}}{8 a b \left (3 a+b x^2\right )}\right )+\frac {x \left (a-b x^2\right )^{2/3}}{3 \left (3 a+b x^2\right )^2}\) |
| \(\Big \downarrow \) 2418 |
| \(\displaystyle \frac {\left (a-b x^2\right )^{2/3} x}{3 \left (b x^2+3 a\right )^2}+\frac {4}{9} b \left (\frac {6 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}}{24 a b}-\frac {x \left (a-b x^2\right )^{2/3}}{8 a b \left (b x^2+3 a\right )}\right )\) |
Input:
Int[(a - b*x^2)^(5/3)/(3*a + b*x^2)^3,x]
Output:
(x*(a - b*x^2)^(2/3))/(3*(3*a + b*x^2)^2) + (4*b*(-1/8*(x*(a - b*x^2)^(2/3 ))/(a*b*(3*a + b*x^2)) + (6*a*(ArcTan[(Sqrt[3]*Sqrt[a])/(Sqrt[b]*x)]/(2*2^ (2/3)*Sqrt[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]) - A rcTanh[(Sqrt[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)*Sq rt[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 - Sqrt[3])*a^(1/3) - (a - b*x^2)^(1/3))^2]*EllipticE[ArcSin[((1 + Sqr t[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))/(24*a*b)))/9
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[(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[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_ ), x_Symbol] :> Simp[e*(e*x)^(m - 1)*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q + 1)/(2*(b*c - a*d)*(p + 1))), x] - Simp[e^2/(2*(b*c - a*d)*(p + 1)) Int[(e *x)^(m - 2)*(a + b*x^2)^(p + 1)*(c + d*x^2)^q*Simp[c*(m - 1) + d*(m + 2*p + 2*q + 3)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[m, 1] && LeQ[m, 3] && IntBinomialQ[a, b, c, d, e, m, 2, p, q, x]
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]
\[\int \frac {\left (-b \,x^{2}+a \right )^{\frac {5}{3}}}{\left (b \,x^{2}+3 a \right )^{3}}d x\]
Input:
int((-b*x^2+a)^(5/3)/(b*x^2+3*a)^3,x)
Output:
int((-b*x^2+a)^(5/3)/(b*x^2+3*a)^3,x)
Timed out. \[ \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^3} \, dx=\text {Timed out} \] Input:
integrate((-b*x^2+a)^(5/3)/(b*x^2+3*a)^3,x, algorithm="fricas")
Output:
Timed out
\[ \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^3} \, dx=\int \frac {\left (a - b x^{2}\right )^{\frac {5}{3}}}{\left (3 a + b x^{2}\right )^{3}}\, dx \] Input:
integrate((-b*x**2+a)**(5/3)/(b*x**2+3*a)**3,x)
Output:
Integral((a - b*x**2)**(5/3)/(3*a + b*x**2)**3, x)
\[ \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^3} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {5}{3}}}{{\left (b x^{2} + 3 \, a\right )}^{3}} \,d x } \] Input:
integrate((-b*x^2+a)^(5/3)/(b*x^2+3*a)^3,x, algorithm="maxima")
Output:
integrate((-b*x^2 + a)^(5/3)/(b*x^2 + 3*a)^3, x)
\[ \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^3} \, dx=\int { \frac {{\left (-b x^{2} + a\right )}^{\frac {5}{3}}}{{\left (b x^{2} + 3 \, a\right )}^{3}} \,d x } \] Input:
integrate((-b*x^2+a)^(5/3)/(b*x^2+3*a)^3,x, algorithm="giac")
Output:
integrate((-b*x^2 + a)^(5/3)/(b*x^2 + 3*a)^3, x)
Timed out. \[ \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^3} \, dx=\int \frac {{\left (a-b\,x^2\right )}^{5/3}}{{\left (b\,x^2+3\,a\right )}^3} \,d x \] Input:
int((a - b*x^2)^(5/3)/(3*a + b*x^2)^3,x)
Output:
int((a - b*x^2)^(5/3)/(3*a + b*x^2)^3, x)
\[ \int \frac {\left (a-b x^2\right )^{5/3}}{\left (3 a+b x^2\right )^3} \, dx=\frac {3 \left (-b \,x^{2}+a \right )^{\frac {2}{3}} x +54 \left (\int \frac {\left (-b \,x^{2}+a \right )^{\frac {2}{3}}}{-b^{4} x^{8}-8 a \,b^{3} x^{6}-18 a^{2} b^{2} x^{4}+27 a^{4}}d x \right ) a^{4}+36 \left (\int \frac {\left (-b \,x^{2}+a \right )^{\frac {2}{3}}}{-b^{4} x^{8}-8 a \,b^{3} x^{6}-18 a^{2} b^{2} x^{4}+27 a^{4}}d x \right ) a^{3} b \,x^{2}+6 \left (\int \frac {\left (-b \,x^{2}+a \right )^{\frac {2}{3}}}{-b^{4} x^{8}-8 a \,b^{3} x^{6}-18 a^{2} b^{2} x^{4}+27 a^{4}}d x \right ) a^{2} b^{2} x^{4}+90 \left (\int \frac {\left (-b \,x^{2}+a \right )^{\frac {2}{3}} x^{4}}{-b^{4} x^{8}-8 a \,b^{3} x^{6}-18 a^{2} b^{2} x^{4}+27 a^{4}}d x \right ) a^{2} b^{2}+60 \left (\int \frac {\left (-b \,x^{2}+a \right )^{\frac {2}{3}} x^{4}}{-b^{4} x^{8}-8 a \,b^{3} x^{6}-18 a^{2} b^{2} x^{4}+27 a^{4}}d x \right ) a \,b^{3} x^{2}+10 \left (\int \frac {\left (-b \,x^{2}+a \right )^{\frac {2}{3}} x^{4}}{-b^{4} x^{8}-8 a \,b^{3} x^{6}-18 a^{2} b^{2} x^{4}+27 a^{4}}d x \right ) b^{4} x^{4}}{15 b^{2} x^{4}+90 a b \,x^{2}+135 a^{2}} \] Input:
int((-b*x^2+a)^(5/3)/(b*x^2+3*a)^3,x)
Output:
(3*(a - b*x**2)**(2/3)*x + 54*int((a - b*x**2)**(2/3)/(27*a**4 - 18*a**2*b **2*x**4 - 8*a*b**3*x**6 - b**4*x**8),x)*a**4 + 36*int((a - b*x**2)**(2/3) /(27*a**4 - 18*a**2*b**2*x**4 - 8*a*b**3*x**6 - b**4*x**8),x)*a**3*b*x**2 + 6*int((a - b*x**2)**(2/3)/(27*a**4 - 18*a**2*b**2*x**4 - 8*a*b**3*x**6 - b**4*x**8),x)*a**2*b**2*x**4 + 90*int(((a - b*x**2)**(2/3)*x**4)/(27*a**4 - 18*a**2*b**2*x**4 - 8*a*b**3*x**6 - b**4*x**8),x)*a**2*b**2 + 60*int((( a - b*x**2)**(2/3)*x**4)/(27*a**4 - 18*a**2*b**2*x**4 - 8*a*b**3*x**6 - b* *4*x**8),x)*a*b**3*x**2 + 10*int(((a - b*x**2)**(2/3)*x**4)/(27*a**4 - 18* a**2*b**2*x**4 - 8*a*b**3*x**6 - b**4*x**8),x)*b**4*x**4)/(15*(9*a**2 + 6* a*b*x**2 + b**2*x**4))