Integrand size = 26, antiderivative size = 193 \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=\frac {\left (a x^2+x^3\right )^{2/3} \left (-7280 a^5+5460 a^4 x-4680 a^3 x^2+4212 a^2 x^3-3888 a x^4+3645 x^5\right )}{21870 x}+\frac {\left (728 \sqrt {3} a^6-6561 \sqrt {3} b^2\right ) \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{a x^2+x^3}}\right )}{6561}+\frac {\left (-728 a^6+6561 b^2\right ) \log \left (-x+\sqrt [3]{a x^2+x^3}\right )}{6561}+\frac {\left (728 a^6-6561 b^2\right ) \log \left (x^2+x \sqrt [3]{a x^2+x^3}+\left (a x^2+x^3\right )^{2/3}\right )}{13122} \]
1/21870*(a*x^2+x^3)^(2/3)*(-7280*a^5+5460*a^4*x-4680*a^3*x^2+4212*a^2*x^3- 3888*a*x^4+3645*x^5)/x+1/6561*(728*3^(1/2)*a^6-6561*3^(1/2)*b^2)*arctan(3^ (1/2)*x/(x+2*(a*x^2+x^3)^(1/3)))+1/6561*(-728*a^6+6561*b^2)*ln(-x+(a*x^2+x ^3)^(1/3))+1/13122*(728*a^6-6561*b^2)*ln(x^2+x*(a*x^2+x^3)^(1/3)+(a*x^2+x^ 3)^(2/3))
Time = 0.70 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.31 \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=\frac {-21840 a^6 x-5460 a^5 x^2+2340 a^4 x^3-1404 a^3 x^4+972 a^2 x^5-729 a x^6+10935 x^7+10 \sqrt {3} \left (728 a^6-6561 b^2\right ) x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {\sqrt {3} \sqrt [3]{x}}{\sqrt [3]{x}+2 \sqrt [3]{a+x}}\right )+10 \left (-728 a^6+6561 b^2\right ) x^{2/3} \sqrt [3]{a+x} \log \left (-\sqrt [3]{x}+\sqrt [3]{a+x}\right )+3640 a^6 x^{2/3} \sqrt [3]{a+x} \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{a+x}+(a+x)^{2/3}\right )-32805 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (x^{2/3}+\sqrt [3]{x} \sqrt [3]{a+x}+(a+x)^{2/3}\right )}{65610 \sqrt [3]{x^2 (a+x)}} \]
(-21840*a^6*x - 5460*a^5*x^2 + 2340*a^4*x^3 - 1404*a^3*x^4 + 972*a^2*x^5 - 729*a*x^6 + 10935*x^7 + 10*Sqrt[3]*(728*a^6 - 6561*b^2)*x^(2/3)*(a + x)^( 1/3)*ArcTan[(Sqrt[3]*x^(1/3))/(x^(1/3) + 2*(a + x)^(1/3))] + 10*(-728*a^6 + 6561*b^2)*x^(2/3)*(a + x)^(1/3)*Log[-x^(1/3) + (a + x)^(1/3)] + 3640*a^6 *x^(2/3)*(a + x)^(1/3)*Log[x^(2/3) + x^(1/3)*(a + x)^(1/3) + (a + x)^(2/3) ] - 32805*b^2*x^(2/3)*(a + x)^(1/3)*Log[x^(2/3) + x^(1/3)*(a + x)^(1/3) + (a + x)^(2/3)])/(65610*(x^2*(a + x))^(1/3))
Leaf count is larger than twice the leaf count of optimal. \(416\) vs. \(2(193)=386\).
Time = 0.56 (sec) , antiderivative size = 416, normalized size of antiderivative = 2.16, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {2450, 2009}
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.
| \(\displaystyle \int \frac {\left (x^3-b\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx\) |
| \(\Big \downarrow \) 2450 |
| \(\displaystyle \int \left (\frac {x^6}{\sqrt [3]{a x^2+x^3}}-\frac {b^2}{\sqrt [3]{a x^2+x^3}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {728 a^6 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{2187 \sqrt {3} \sqrt [3]{a x^2+x^3}}-\frac {364 a^6 x^{2/3} \sqrt [3]{a+x} \log (x)}{6561 \sqrt [3]{a x^2+x^3}}-\frac {364 a^6 x^{2/3} \sqrt [3]{a+x} \log \left (\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}-1\right )}{2187 \sqrt [3]{a x^2+x^3}}-\frac {728 a^5 \left (a x^2+x^3\right )^{2/3}}{2187 x}+\frac {182}{729} a^4 \left (a x^2+x^3\right )^{2/3}-\frac {52}{243} a^3 x \left (a x^2+x^3\right )^{2/3}+\frac {26}{135} a^2 x^2 \left (a x^2+x^3\right )^{2/3}+\frac {\sqrt {3} b^2 x^{2/3} \sqrt [3]{a+x} \arctan \left (\frac {2 \sqrt [3]{a+x}}{\sqrt {3} \sqrt [3]{x}}+\frac {1}{\sqrt {3}}\right )}{\sqrt [3]{a x^2+x^3}}+\frac {b^2 x^{2/3} \sqrt [3]{a+x} \log (x)}{2 \sqrt [3]{a x^2+x^3}}+\frac {3 b^2 x^{2/3} \sqrt [3]{a+x} \log \left (\frac {\sqrt [3]{a+x}}{\sqrt [3]{x}}-1\right )}{2 \sqrt [3]{a x^2+x^3}}-\frac {8}{45} a x^3 \left (a x^2+x^3\right )^{2/3}+\frac {1}{6} x^4 \left (a x^2+x^3\right )^{2/3}\) |
(182*a^4*(a*x^2 + x^3)^(2/3))/729 - (728*a^5*(a*x^2 + x^3)^(2/3))/(2187*x) - (52*a^3*x*(a*x^2 + x^3)^(2/3))/243 + (26*a^2*x^2*(a*x^2 + x^3)^(2/3))/1 35 - (8*a*x^3*(a*x^2 + x^3)^(2/3))/45 + (x^4*(a*x^2 + x^3)^(2/3))/6 - (728 *a^6*x^(2/3)*(a + x)^(1/3)*ArcTan[1/Sqrt[3] + (2*(a + x)^(1/3))/(Sqrt[3]*x ^(1/3))])/(2187*Sqrt[3]*(a*x^2 + x^3)^(1/3)) + (Sqrt[3]*b^2*x^(2/3)*(a + x )^(1/3)*ArcTan[1/Sqrt[3] + (2*(a + x)^(1/3))/(Sqrt[3]*x^(1/3))])/(a*x^2 + x^3)^(1/3) - (364*a^6*x^(2/3)*(a + x)^(1/3)*Log[x])/(6561*(a*x^2 + x^3)^(1 /3)) + (b^2*x^(2/3)*(a + x)^(1/3)*Log[x])/(2*(a*x^2 + x^3)^(1/3)) - (364*a ^6*x^(2/3)*(a + x)^(1/3)*Log[-1 + (a + x)^(1/3)/x^(1/3)])/(2187*(a*x^2 + x ^3)^(1/3)) + (3*b^2*x^(2/3)*(a + x)^(1/3)*Log[-1 + (a + x)^(1/3)/x^(1/3)]) /(2*(a*x^2 + x^3)^(1/3))
3.24.99.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[Expan dIntegrand[Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j, n, p}, x] && (Po lyQ[Pq, x] || PolyQ[Pq, x^n]) && !IntegerQ[p] && NeQ[n, j]
Time = 0.64 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.06
| method | result | size |
| pseudoelliptic | \(-\frac {728 \left (-\frac {\left (a^{6}-\frac {6561 b^{2}}{728}\right ) x \ln \left (\frac {\left (x^{2} \left (a +x \right )\right )^{\frac {2}{3}}+\left (x^{2} \left (a +x \right )\right )^{\frac {1}{3}} x +x^{2}}{x^{2}}\right )}{2}+\sqrt {3}\, x \left (a^{6}-\frac {6561 b^{2}}{728}\right ) \arctan \left (\frac {\left (2 \left (x^{2} \left (a +x \right )\right )^{\frac {1}{3}}+x \right ) \sqrt {3}}{3 x}\right )+\left (a^{6}-\frac {6561 b^{2}}{728}\right ) x \ln \left (\frac {\left (x^{2} \left (a +x \right )\right )^{\frac {1}{3}}-x}{x}\right )+3 \left (a^{5}-\frac {3}{4} a^{4} x +\frac {9}{14} a^{3} x^{2}-\frac {81}{140} a^{2} x^{3}+\frac {243}{455} a \,x^{4}-\frac {729}{1456} x^{5}\right ) \left (x^{2} \left (a +x \right )\right )^{\frac {2}{3}}\right ) x^{11} a^{6}}{6561 {\left (-\left (x^{2} \left (a +x \right )\right )^{\frac {1}{3}}+x \right )}^{6} {\left (\left (x^{2} \left (a +x \right )\right )^{\frac {2}{3}}+x \left (x +\left (x^{2} \left (a +x \right )\right )^{\frac {1}{3}}\right )\right )}^{6}}\) | \(204\) |
-728/6561*(-1/2*(a^6-6561/728*b^2)*x*ln(((x^2*(a+x))^(2/3)+(x^2*(a+x))^(1/ 3)*x+x^2)/x^2)+3^(1/2)*x*(a^6-6561/728*b^2)*arctan(1/3*(2*(x^2*(a+x))^(1/3 )+x)*3^(1/2)/x)+(a^6-6561/728*b^2)*x*ln(((x^2*(a+x))^(1/3)-x)/x)+3*(a^5-3/ 4*a^4*x+9/14*a^3*x^2-81/140*a^2*x^3+243/455*a*x^4-729/1456*x^5)*(x^2*(a+x) )^(2/3))*x^11*a^6/(-(x^2*(a+x))^(1/3)+x)^6/((x^2*(a+x))^(2/3)+x*(x+(x^2*(a +x))^(1/3)))^6
Time = 0.28 (sec) , antiderivative size = 185, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=-\frac {10 \, \sqrt {3} {\left (728 \, a^{6} - 6561 \, b^{2}\right )} x \arctan \left (\frac {\sqrt {3} x + 2 \, \sqrt {3} {\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{3 \, x}\right ) + 10 \, {\left (728 \, a^{6} - 6561 \, b^{2}\right )} x \log \left (-\frac {x - {\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}}}{x}\right ) - 5 \, {\left (728 \, a^{6} - 6561 \, b^{2}\right )} x \log \left (\frac {x^{2} + {\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}} x + {\left (a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{x^{2}}\right ) + 3 \, {\left (7280 \, a^{5} - 5460 \, a^{4} x + 4680 \, a^{3} x^{2} - 4212 \, a^{2} x^{3} + 3888 \, a x^{4} - 3645 \, x^{5}\right )} {\left (a x^{2} + x^{3}\right )}^{\frac {2}{3}}}{65610 \, x} \]
-1/65610*(10*sqrt(3)*(728*a^6 - 6561*b^2)*x*arctan(1/3*(sqrt(3)*x + 2*sqrt (3)*(a*x^2 + x^3)^(1/3))/x) + 10*(728*a^6 - 6561*b^2)*x*log(-(x - (a*x^2 + x^3)^(1/3))/x) - 5*(728*a^6 - 6561*b^2)*x*log((x^2 + (a*x^2 + x^3)^(1/3)* x + (a*x^2 + x^3)^(2/3))/x^2) + 3*(7280*a^5 - 5460*a^4*x + 4680*a^3*x^2 - 4212*a^2*x^3 + 3888*a*x^4 - 3645*x^5)*(a*x^2 + x^3)^(2/3))/x
\[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=\int \frac {\left (- b + x^{3}\right ) \left (b + x^{3}\right )}{\sqrt [3]{x^{2} \left (a + x\right )}}\, dx \]
\[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=\int { \frac {{\left (x^{3} + b\right )} {\left (x^{3} - b\right )}}{{\left (a x^{2} + x^{3}\right )}^{\frac {1}{3}}} \,d x } \]
Time = 0.31 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.02 \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=-\frac {10 \, \sqrt {3} {\left (728 \, a^{7} - 6561 \, a b^{2}\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (\frac {a}{x} + 1\right )}^{\frac {1}{3}} + 1\right )}\right ) - 5 \, {\left (728 \, a^{7} - 6561 \, a b^{2}\right )} \log \left ({\left (\frac {a}{x} + 1\right )}^{\frac {2}{3}} + {\left (\frac {a}{x} + 1\right )}^{\frac {1}{3}} + 1\right ) + 10 \, {\left (728 \, a^{7} - 6561 \, a b^{2}\right )} \log \left ({\left | {\left (\frac {a}{x} + 1\right )}^{\frac {1}{3}} - 1 \right |}\right ) + \frac {3 \, {\left (7280 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {17}{3}} - 41860 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {14}{3}} + 99320 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {11}{3}} - 123812 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {8}{3}} + 84592 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {5}{3}} - 29165 \, a^{7} {\left (\frac {a}{x} + 1\right )}^{\frac {2}{3}}\right )} x^{6}}{a^{6}}}{65610 \, a} \]
-1/65610*(10*sqrt(3)*(728*a^7 - 6561*a*b^2)*arctan(1/3*sqrt(3)*(2*(a/x + 1 )^(1/3) + 1)) - 5*(728*a^7 - 6561*a*b^2)*log((a/x + 1)^(2/3) + (a/x + 1)^( 1/3) + 1) + 10*(728*a^7 - 6561*a*b^2)*log(abs((a/x + 1)^(1/3) - 1)) + 3*(7 280*a^7*(a/x + 1)^(17/3) - 41860*a^7*(a/x + 1)^(14/3) + 99320*a^7*(a/x + 1 )^(11/3) - 123812*a^7*(a/x + 1)^(8/3) + 84592*a^7*(a/x + 1)^(5/3) - 29165* a^7*(a/x + 1)^(2/3))*x^6/a^6)/a
Timed out. \[ \int \frac {\left (-b+x^3\right ) \left (b+x^3\right )}{\sqrt [3]{a x^2+x^3}} \, dx=-\int \frac {\left (x^3+b\right )\,\left (b-x^3\right )}{{\left (x^3+a\,x^2\right )}^{1/3}} \,d x \]