Integrand size = 24, antiderivative size = 254 \[ \int \frac {x}{\left (-3 a^3+x^3\right )^3 \left (2 a^3+x^3\right )^3} \, dx=\frac {-7887 a^9 x^2+22 a^6 x^5+931 a^3 x^8-104 x^{11}}{202500 a^{15} \left (6 a^6+a^3 x^3-x^6\right )^2}+\frac {671 \arctan \left (\frac {\sqrt {3} a+2 \sqrt [6]{3} x}{3 a}\right )}{253125\ 3^{5/6} a^{16}}+\frac {89 \arctan \left (\frac {\sqrt {3} a-2^{2/3} \sqrt {3} x}{3 a}\right )}{28125 \sqrt [3]{2} \sqrt {3} a^{16}}+\frac {89 \log \left (\sqrt [3]{2} a+x\right )}{84375 \sqrt [3]{2} a^{16}}+\frac {671 \log \left (-\sqrt [3]{3} a+x\right )}{759375 \sqrt [3]{3} a^{16}}-\frac {89 \log \left (2^{2/3} a^2-\sqrt [3]{2} a x+x^2\right )}{168750 \sqrt [3]{2} a^{16}}-\frac {671 \log \left (3^{2/3} a^2+\sqrt [3]{3} a x+x^2\right )}{1518750 \sqrt [3]{3} a^{16}} \] Output:
1/202500*(-7887*a^9*x^2+22*a^6*x^5+931*a^3*x^8-104*x^11)/a^15/(6*a^6+a^3*x ^3-x^6)^2+671/759375*arctan(1/3*(3^(1/2)*a+2*3^(1/6)*x)/a)*3^(1/6)/a^16+89 /168750*arctan(1/3*(3^(1/2)*a-2^(2/3)*x*3^(1/2))/a)*2^(2/3)*3^(1/2)/a^16+8 9/168750*ln(2^(1/3)*a+x)*2^(2/3)/a^16+671/2278125*ln(-3^(1/3)*a+x)*3^(2/3) /a^16-89/337500*ln(2^(2/3)*a^2-2^(1/3)*a*x+x^2)*2^(2/3)/a^16-671/4556250*l n(3^(2/3)*a^2+3^(1/3)*a*x+x^2)*3^(2/3)/a^16
Time = 0.18 (sec) , antiderivative size = 241, normalized size of antiderivative = 0.95 \[ \int \frac {x}{\left (-3 a^3+x^3\right )^3 \left (2 a^3+x^3\right )^3} \, dx=\frac {\frac {10125 a^7 \left (-13 a^3 x^2+x^5\right )}{\left (6 a^6+a^3 x^3-x^6\right )^2}+\frac {45 a \left (827 a^3 x^2-104 x^5\right )}{-6 a^6-a^3 x^3+x^6}-4806\ 2^{2/3} \sqrt {3} \arctan \left (\frac {-a+2^{2/3} x}{\sqrt {3} a}\right )+8052 \sqrt [6]{3} \arctan \left (\frac {1}{\sqrt {3}}+\frac {2 x}{3^{5/6} a}\right )+4806\ 2^{2/3} \log \left (2 a+2^{2/3} x\right )+2684\ 3^{2/3} \log \left (3 a-3^{2/3} x\right )-2403\ 2^{2/3} \log \left (2 a^2-2^{2/3} a x+\sqrt [3]{2} x^2\right )-1342\ 3^{2/3} \log \left (3 a^2+3^{2/3} a x+\sqrt [3]{3} x^2\right )}{9112500 a^{16}} \] Input:
Integrate[x/((-3*a^3 + x^3)^3*(2*a^3 + x^3)^3),x]
Output:
((10125*a^7*(-13*a^3*x^2 + x^5))/(6*a^6 + a^3*x^3 - x^6)^2 + (45*a*(827*a^ 3*x^2 - 104*x^5))/(-6*a^6 - a^3*x^3 + x^6) - 4806*2^(2/3)*Sqrt[3]*ArcTan[( -a + 2^(2/3)*x)/(Sqrt[3]*a)] + 8052*3^(1/6)*ArcTan[1/Sqrt[3] + (2*x)/(3^(5 /6)*a)] + 4806*2^(2/3)*Log[2*a + 2^(2/3)*x] + 2684*3^(2/3)*Log[3*a - 3^(2/ 3)*x] - 2403*2^(2/3)*Log[2*a^2 - 2^(2/3)*a*x + 2^(1/3)*x^2] - 1342*3^(2/3) *Log[3*a^2 + 3^(2/3)*a*x + 3^(1/3)*x^2])/(9112500*a^16)
Time = 1.12 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.28, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {972, 27, 1049, 27, 1049, 27, 1049, 27, 1054, 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.
| \(\displaystyle \int \frac {x}{\left (x^3-3 a^3\right )^3 \left (2 a^3+x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 972 |
| \(\displaystyle \frac {\int -\frac {2 x \left (13 a^3+5 x^3\right )}{\left (3 a^3-x^3\right )^2 \left (2 a^3+x^3\right )^3}dx}{90 a^6}-\frac {x^2}{90 a^6 \left (3 a^3-x^3\right )^2 \left (2 a^3+x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {x \left (13 a^3+5 x^3\right )}{\left (3 a^3-x^3\right )^2 \left (2 a^3+x^3\right )^3}dx}{45 a^6}-\frac {x^2}{90 a^6 \left (3 a^3-x^3\right )^2 \left (2 a^3+x^3\right )^2}\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle -\frac {\frac {\int \frac {a^3 x \left (83 a^3+196 x^3\right )}{\left (3 a^3-x^3\right ) \left (2 a^3+x^3\right )^3}dx}{45 a^6}+\frac {28 x^2}{45 a^3 \left (3 a^3-x^3\right ) \left (2 a^3+x^3\right )^2}}{45 a^6}-\frac {x^2}{90 a^6 \left (3 a^3-x^3\right )^2 \left (2 a^3+x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\int \frac {x \left (83 a^3+196 x^3\right )}{\left (3 a^3-x^3\right ) \left (2 a^3+x^3\right )^3}dx}{45 a^3}+\frac {28 x^2}{45 a^3 \left (3 a^3-x^3\right ) \left (2 a^3+x^3\right )^2}}{45 a^6}-\frac {x^2}{90 a^6 \left (3 a^3-x^3\right )^2 \left (2 a^3+x^3\right )^2}\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle -\frac {\frac {-\frac {\int -\frac {12 a^3 x \left (362 a^3+103 x^3\right )}{\left (3 a^3-x^3\right ) \left (2 a^3+x^3\right )^2}dx}{60 a^6}-\frac {103 x^2}{20 a^3 \left (2 a^3+x^3\right )^2}}{45 a^3}+\frac {28 x^2}{45 a^3 \left (3 a^3-x^3\right ) \left (2 a^3+x^3\right )^2}}{45 a^6}-\frac {x^2}{90 a^6 \left (3 a^3-x^3\right )^2 \left (2 a^3+x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\int \frac {x \left (362 a^3+103 x^3\right )}{\left (3 a^3-x^3\right ) \left (2 a^3+x^3\right )^2}dx}{5 a^3}-\frac {103 x^2}{20 a^3 \left (2 a^3+x^3\right )^2}}{45 a^3}+\frac {28 x^2}{45 a^3 \left (3 a^3-x^3\right ) \left (2 a^3+x^3\right )^2}}{45 a^6}-\frac {x^2}{90 a^6 \left (3 a^3-x^3\right )^2 \left (2 a^3+x^3\right )^2}\) |
| \(\Big \downarrow \) 1049 |
| \(\displaystyle -\frac {\frac {\frac {\frac {26 x^2}{5 a^3 \left (2 a^3+x^3\right )}-\frac {\int -\frac {6 a^3 x \left (749 a^3-26 x^3\right )}{\left (3 a^3-x^3\right ) \left (2 a^3+x^3\right )}dx}{30 a^6}}{5 a^3}-\frac {103 x^2}{20 a^3 \left (2 a^3+x^3\right )^2}}{45 a^3}+\frac {28 x^2}{45 a^3 \left (3 a^3-x^3\right ) \left (2 a^3+x^3\right )^2}}{45 a^6}-\frac {x^2}{90 a^6 \left (3 a^3-x^3\right )^2 \left (2 a^3+x^3\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int \frac {x \left (749 a^3-26 x^3\right )}{\left (3 a^3-x^3\right ) \left (2 a^3+x^3\right )}dx}{5 a^3}+\frac {26 x^2}{5 a^3 \left (2 a^3+x^3\right )}}{5 a^3}-\frac {103 x^2}{20 a^3 \left (2 a^3+x^3\right )^2}}{45 a^3}+\frac {28 x^2}{45 a^3 \left (3 a^3-x^3\right ) \left (2 a^3+x^3\right )^2}}{45 a^6}-\frac {x^2}{90 a^6 \left (3 a^3-x^3\right )^2 \left (2 a^3+x^3\right )^2}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle -\frac {\frac {\frac {\frac {\int \left (\frac {801 x}{5 \left (2 a^3+x^3\right )}-\frac {671 x}{5 \left (x^3-3 a^3\right )}\right )dx}{5 a^3}+\frac {26 x^2}{5 a^3 \left (2 a^3+x^3\right )}}{5 a^3}-\frac {103 x^2}{20 a^3 \left (2 a^3+x^3\right )^2}}{45 a^3}+\frac {28 x^2}{45 a^3 \left (3 a^3-x^3\right ) \left (2 a^3+x^3\right )^2}}{45 a^6}-\frac {x^2}{90 a^6 \left (3 a^3-x^3\right )^2 \left (2 a^3+x^3\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {x^2}{90 a^6 \left (3 a^3-x^3\right )^2 \left (2 a^3+x^3\right )^2}-\frac {\frac {28 x^2}{45 a^3 \left (3 a^3-x^3\right ) \left (2 a^3+x^3\right )^2}+\frac {\frac {\frac {26 x^2}{5 a^3 \left (2 a^3+x^3\right )}+\frac {\frac {267 \log \left (2^{2/3} a^2-\sqrt [3]{2} a x+x^2\right )}{10 \sqrt [3]{2} a}+\frac {671 \log \left (3^{2/3} a^2+\sqrt [3]{3} a x+x^2\right )}{30 \sqrt [3]{3} a}-\frac {267 \sqrt {3} \arctan \left (\frac {a-2^{2/3} x}{\sqrt {3} a}\right )}{5 \sqrt [3]{2} a}-\frac {671 \arctan \left (\frac {2 x}{3^{5/6} a}+\frac {1}{\sqrt {3}}\right )}{5\ 3^{5/6} a}-\frac {671 \log \left (\sqrt [3]{3} a-x\right )}{15 \sqrt [3]{3} a}-\frac {267 \log \left (\sqrt [3]{2} a+x\right )}{5 \sqrt [3]{2} a}}{5 a^3}}{5 a^3}-\frac {103 x^2}{20 a^3 \left (2 a^3+x^3\right )^2}}{45 a^3}}{45 a^6}\) |
Input:
Int[x/((-3*a^3 + x^3)^3*(2*a^3 + x^3)^3),x]
Output:
-1/90*x^2/(a^6*(3*a^3 - x^3)^2*(2*a^3 + x^3)^2) - ((28*x^2)/(45*a^3*(3*a^3 - x^3)*(2*a^3 + x^3)^2) + ((-103*x^2)/(20*a^3*(2*a^3 + x^3)^2) + ((26*x^2 )/(5*a^3*(2*a^3 + x^3)) + ((-267*Sqrt[3]*ArcTan[(a - 2^(2/3)*x)/(Sqrt[3]*a )])/(5*2^(1/3)*a) - (671*ArcTan[1/Sqrt[3] + (2*x)/(3^(5/6)*a)])/(5*3^(5/6) *a) - (671*Log[3^(1/3)*a - x])/(15*3^(1/3)*a) - (267*Log[2^(1/3)*a + x])/( 5*2^(1/3)*a) + (267*Log[2^(2/3)*a^2 - 2^(1/3)*a*x + x^2])/(10*2^(1/3)*a) + (671*Log[3^(2/3)*a^2 + 3^(1/3)*a*x + x^2])/(30*3^(1/3)*a))/(5*a^3))/(5*a^ 3))/(45*a^3))/(45*a^6)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_), x_Symbol] :> Simp[(-b)*(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x ^n)^(q + 1)/(a*e*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*b*(m + 1) + n*( b*c - a*d)*(p + 1) + d*b*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{ a, b, c, d, e, m, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[p, -1] & & IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ ))^(q_)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*(g*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*g*n*(b*c - a*d)*(p + 1))) , x] + Simp[1/(a*n*(b*c - a*d)*(p + 1)) Int[(g*x)^m*(a + b*x^n)^(p + 1)*( c + d*x^n)^q*Simp[c*(b*e - a*f)*(m + 1) + e*n*(b*c - a*d)*(p + 1) + d*(b*e - a*f)*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m, q}, x] && IGtQ[n, 0] && LtQ[p, -1]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.62
| method | result | size |
| risch | \(\frac {-\frac {26 x^{11}}{50625 a^{15}}+\frac {931 x^{8}}{202500 a^{12}}+\frac {11 x^{5}}{101250 a^{9}}-\frac {2629 x^{2}}{67500 a^{6}}}{\left (3 a^{3}-x^{3}\right )^{2} \left (2 a^{3}+x^{3}\right )^{2}}+\frac {89 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 a^{48} \textit {\_Z}^{3}-1\right )}{\sum }\textit {\_R} \ln \left (\left (2022624591 \textit {\_R}^{3} a^{48}-542540405\right ) x +1027844802 a^{81} \textit {\_R}^{5}+423621380 a^{33} \textit {\_R}^{2}\right )\right )}{84375}+\frac {671 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 a^{48} \textit {\_Z}^{3}-1\right )}{\sum }\textit {\_R} \ln \left (\left (5083075283 \textit {\_R}^{3} a^{48}-2319387615\right ) x +1812670266 a^{81} \textit {\_R}^{5}+1270864140 a^{33} \textit {\_R}^{2}\right )\right )}{759375}\) | \(157\) |
| default | \(-\frac {\frac {\frac {445}{54} a^{3} x^{2}-\frac {185}{81} x^{5}}{\left (3 a^{3}-x^{3}\right )^{2}}-\frac {671 \,3^{\frac {2}{3}} \ln \left (x -3^{\frac {1}{3}} \left (a^{3}\right )^{\frac {1}{3}}\right )}{729 \left (a^{3}\right )^{\frac {1}{3}}}+\frac {671 \,3^{\frac {2}{3}} \ln \left (x^{2}+3^{\frac {1}{3}} \left (a^{3}\right )^{\frac {1}{3}} x +3^{\frac {2}{3}} \left (a^{3}\right )^{\frac {2}{3}}\right )}{1458 \left (a^{3}\right )^{\frac {1}{3}}}-\frac {671 \,3^{\frac {1}{6}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 \,3^{\frac {2}{3}} x}{3 \left (a^{3}\right )^{\frac {1}{3}}}+1\right )}{3}\right )}{243 \left (a^{3}\right )^{\frac {1}{3}}}}{3125 a^{15}}-\frac {\frac {\frac {355}{36} a^{3} x^{2}+\frac {35}{9} x^{5}}{\left (2 a^{3}+x^{3}\right )^{2}}-\frac {89 \,2^{\frac {2}{3}} \ln \left (x +2^{\frac {1}{3}} \left (a^{3}\right )^{\frac {1}{3}}\right )}{54 \left (a^{3}\right )^{\frac {1}{3}}}+\frac {89 \,2^{\frac {2}{3}} \ln \left (x^{2}-2^{\frac {1}{3}} \left (a^{3}\right )^{\frac {1}{3}} x +2^{\frac {2}{3}} \left (a^{3}\right )^{\frac {2}{3}}\right )}{108 \left (a^{3}\right )^{\frac {1}{3}}}+\frac {89 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (\frac {2^{\frac {2}{3}} x}{\left (a^{3}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{54 \left (a^{3}\right )^{\frac {1}{3}}}}{3125 a^{15}}\) | \(244\) |
Input:
int(x/(-3*a^3+x^3)^3/(2*a^3+x^3)^3,x,method=_RETURNVERBOSE)
Output:
36*(-13/911250/a^15*x^11+931/7290000/a^12*x^8+11/3645000/a^9*x^5-2629/2430 000/a^6*x^2)/(3*a^3-x^3)^2/(2*a^3+x^3)^2+89/84375*sum(_R*ln((2022624591*_R ^3*a^48-542540405)*x+1027844802*a^81*_R^5+423621380*a^33*_R^2),_R=RootOf(2 *_Z^3*a^48-1))+671/759375*sum(_R*ln((5083075283*_R^3*a^48-2319387615)*x+18 12670266*a^81*_R^5+1270864140*a^33*_R^2),_R=RootOf(3*_Z^3*a^48-1))
Time = 0.09 (sec) , antiderivative size = 395, normalized size of antiderivative = 1.56 \[ \int \frac {x}{\left (-3 a^3+x^3\right )^3 \left (2 a^3+x^3\right )^3} \, dx=-\frac {354915 \, a^{10} x^{2} - 990 \, a^{7} x^{5} - 41895 \, a^{4} x^{8} + 4680 \, a x^{11} - 28836 \cdot 2^{\frac {1}{6}} \sqrt {\frac {1}{6}} {\left (36 \, a^{12} + 12 \, a^{9} x^{3} - 11 \, a^{6} x^{6} - 2 \, a^{3} x^{9} + x^{12}\right )} \arctan \left (\frac {2^{\frac {1}{6}} \sqrt {\frac {1}{6}} {\left (2^{\frac {1}{3}} a - 2 \, x\right )}}{a}\right ) + 1342 \cdot 3^{\frac {2}{3}} {\left (36 \, a^{12} + 12 \, a^{9} x^{3} - 11 \, a^{6} x^{6} - 2 \, a^{3} x^{9} + x^{12}\right )} \log \left (3^{\frac {2}{3}} a^{2} + 3^{\frac {1}{3}} a x + x^{2}\right ) + 2403 \cdot 2^{\frac {2}{3}} {\left (36 \, a^{12} + 12 \, a^{9} x^{3} - 11 \, a^{6} x^{6} - 2 \, a^{3} x^{9} + x^{12}\right )} \log \left (2^{\frac {2}{3}} a^{2} - 2^{\frac {1}{3}} a x + x^{2}\right ) - 2684 \cdot 3^{\frac {2}{3}} {\left (36 \, a^{12} + 12 \, a^{9} x^{3} - 11 \, a^{6} x^{6} - 2 \, a^{3} x^{9} + x^{12}\right )} \log \left (-3^{\frac {1}{3}} a + x\right ) - 4806 \cdot 2^{\frac {2}{3}} {\left (36 \, a^{12} + 12 \, a^{9} x^{3} - 11 \, a^{6} x^{6} - 2 \, a^{3} x^{9} + x^{12}\right )} \log \left (2^{\frac {1}{3}} a + x\right ) - 8052 \cdot 3^{\frac {1}{6}} {\left (36 \, a^{12} + 12 \, a^{9} x^{3} - 11 \, a^{6} x^{6} - 2 \, a^{3} x^{9} + x^{12}\right )} \arctan \left (\frac {3^{\frac {1}{6}} {\left (3^{\frac {1}{3}} a + 2 \, x\right )}}{3 \, a}\right )}{9112500 \, {\left (36 \, a^{28} + 12 \, a^{25} x^{3} - 11 \, a^{22} x^{6} - 2 \, a^{19} x^{9} + a^{16} x^{12}\right )}} \] Input:
integrate(x/(-3*a^3+x^3)^3/(2*a^3+x^3)^3,x, algorithm="fricas")
Output:
-1/9112500*(354915*a^10*x^2 - 990*a^7*x^5 - 41895*a^4*x^8 + 4680*a*x^11 - 28836*2^(1/6)*sqrt(1/6)*(36*a^12 + 12*a^9*x^3 - 11*a^6*x^6 - 2*a^3*x^9 + x ^12)*arctan(2^(1/6)*sqrt(1/6)*(2^(1/3)*a - 2*x)/a) + 1342*3^(2/3)*(36*a^12 + 12*a^9*x^3 - 11*a^6*x^6 - 2*a^3*x^9 + x^12)*log(3^(2/3)*a^2 + 3^(1/3)*a *x + x^2) + 2403*2^(2/3)*(36*a^12 + 12*a^9*x^3 - 11*a^6*x^6 - 2*a^3*x^9 + x^12)*log(2^(2/3)*a^2 - 2^(1/3)*a*x + x^2) - 2684*3^(2/3)*(36*a^12 + 12*a^ 9*x^3 - 11*a^6*x^6 - 2*a^3*x^9 + x^12)*log(-3^(1/3)*a + x) - 4806*2^(2/3)* (36*a^12 + 12*a^9*x^3 - 11*a^6*x^6 - 2*a^3*x^9 + x^12)*log(2^(1/3)*a + x) - 8052*3^(1/6)*(36*a^12 + 12*a^9*x^3 - 11*a^6*x^6 - 2*a^3*x^9 + x^12)*arct an(1/3*3^(1/6)*(3^(1/3)*a + 2*x)/a))/(36*a^28 + 12*a^25*x^3 - 11*a^22*x^6 - 2*a^19*x^9 + a^16*x^12)
Time = 0.41 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.51 \[ \int \frac {x}{\left (-3 a^3+x^3\right )^3 \left (2 a^3+x^3\right )^3} \, dx=\frac {- 7887 a^{9} x^{2} + 22 a^{6} x^{5} + 931 a^{3} x^{8} - 104 x^{11}}{7290000 a^{27} + 2430000 a^{24} x^{3} - 2227500 a^{21} x^{6} - 405000 a^{18} x^{9} + 202500 a^{15} x^{12}} + \frac {\operatorname {RootSum} {\left (1201354980468750 t^{3} - 704969, \left ( t \mapsto t \log {\left (\frac {52845738528617361187934875488281250 t^{5} a}{3343617664699171541} - \frac {25000218167269785908203125 t^{2} a}{3343617664699171541} + x \right )} \right )\right )} + \operatorname {RootSum} {\left (1313681671142578125 t^{3} - 302111711, \left ( t \mapsto t \log {\left (\frac {52845738528617361187934875488281250 t^{5} a}{3343617664699171541} - \frac {25000218167269785908203125 t^{2} a}{3343617664699171541} + x \right )} \right )\right )}}{a^{16}} \] Input:
integrate(x/(-3*a**3+x**3)**3/(2*a**3+x**3)**3,x)
Output:
(-7887*a**9*x**2 + 22*a**6*x**5 + 931*a**3*x**8 - 104*x**11)/(7290000*a**2 7 + 2430000*a**24*x**3 - 2227500*a**21*x**6 - 405000*a**18*x**9 + 202500*a **15*x**12) + (RootSum(1201354980468750*_t**3 - 704969, Lambda(_t, _t*log( 52845738528617361187934875488281250*_t**5*a/3343617664699171541 - 25000218 167269785908203125*_t**2*a/3343617664699171541 + x))) + RootSum(1313681671 142578125*_t**3 - 302111711, Lambda(_t, _t*log(528457385286173611879348754 88281250*_t**5*a/3343617664699171541 - 25000218167269785908203125*_t**2*a/ 3343617664699171541 + x))))/a**16
Time = 0.11 (sec) , antiderivative size = 216, normalized size of antiderivative = 0.85 \[ \int \frac {x}{\left (-3 a^3+x^3\right )^3 \left (2 a^3+x^3\right )^3} \, dx=-\frac {7887 \, a^{9} x^{2} - 22 \, a^{6} x^{5} - 931 \, a^{3} x^{8} + 104 \, x^{11}}{202500 \, {\left (36 \, a^{27} + 12 \, a^{24} x^{3} - 11 \, a^{21} x^{6} - 2 \, a^{18} x^{9} + a^{15} x^{12}\right )}} - \frac {89 \, \sqrt {3} 2^{\frac {2}{3}} \arctan \left (-\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} a - 2 \, x\right )}}{6 \, a}\right )}{168750 \, a^{16}} - \frac {671 \cdot 3^{\frac {2}{3}} \log \left (3^{\frac {2}{3}} a^{2} + 3^{\frac {1}{3}} a x + x^{2}\right )}{4556250 \, a^{16}} - \frac {89 \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {2}{3}} a^{2} - 2^{\frac {1}{3}} a x + x^{2}\right )}{337500 \, a^{16}} + \frac {671 \cdot 3^{\frac {2}{3}} \log \left (-3^{\frac {1}{3}} a + x\right )}{2278125 \, a^{16}} + \frac {89 \cdot 2^{\frac {2}{3}} \log \left (2^{\frac {1}{3}} a + x\right )}{168750 \, a^{16}} + \frac {671 \cdot 3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {1}{6}} {\left (3^{\frac {1}{3}} a + 2 \, x\right )}}{3 \, a}\right )}{759375 \, a^{16}} \] Input:
integrate(x/(-3*a^3+x^3)^3/(2*a^3+x^3)^3,x, algorithm="maxima")
Output:
-1/202500*(7887*a^9*x^2 - 22*a^6*x^5 - 931*a^3*x^8 + 104*x^11)/(36*a^27 + 12*a^24*x^3 - 11*a^21*x^6 - 2*a^18*x^9 + a^15*x^12) - 89/168750*sqrt(3)*2^ (2/3)*arctan(-1/6*sqrt(3)*2^(2/3)*(2^(1/3)*a - 2*x)/a)/a^16 - 671/4556250* 3^(2/3)*log(3^(2/3)*a^2 + 3^(1/3)*a*x + x^2)/a^16 - 89/337500*2^(2/3)*log( 2^(2/3)*a^2 - 2^(1/3)*a*x + x^2)/a^16 + 671/2278125*3^(2/3)*log(-3^(1/3)*a + x)/a^16 + 89/168750*2^(2/3)*log(2^(1/3)*a + x)/a^16 + 671/759375*3^(1/6 )*arctan(1/3*3^(1/6)*(3^(1/3)*a + 2*x)/a)/a^16
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 274, normalized size of antiderivative = 1.08 \[ \int \frac {x}{\left (-3 a^3+x^3\right )^3 \left (2 a^3+x^3\right )^3} \, dx=\frac {89 \, \sqrt {3} 2^{\frac {2}{3}} {\left (\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}\right )}^{2} \arctan \left (\frac {\sqrt {3} 2^{\frac {2}{3}} {\left (2 \, x + 2^{\frac {1}{3}} \left (-a^{3}\right )^{\frac {1}{3}}\right )}}{6 \, \left (-a^{3}\right )^{\frac {1}{3}}}\right )}{168750 \, a^{16}} - \frac {89 \cdot 2^{\frac {2}{3}} {\left (\frac {1}{2} i \, \sqrt {3} + \frac {1}{2}\right )}^{2} \log \left (x^{2} + 2^{\frac {1}{3}} \left (-a^{3}\right )^{\frac {1}{3}} x + 2^{\frac {2}{3}} \left (-a^{3}\right )^{\frac {2}{3}}\right )}{337500 \, a^{16}} - \frac {671 \cdot 3^{\frac {2}{3}} \log \left (x^{2} + 3^{\frac {1}{3}} {\left (a^{3}\right )}^{\frac {1}{3}} x + 3^{\frac {2}{3}} {\left (a^{3}\right )}^{\frac {2}{3}}\right )}{4556250 \, a^{16}} + \frac {671 \cdot 3^{\frac {1}{6}} \arctan \left (\frac {3^{\frac {1}{6}} {\left (2 \, x + 3^{\frac {1}{3}} {\left (a^{3}\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a^{3}\right )}^{\frac {1}{3}}}\right )}{759375 \, a^{16}} + \frac {89 \cdot 2^{\frac {2}{3}} \left (-a^{3}\right )^{\frac {2}{3}} \log \left ({\left | x - 2^{\frac {1}{3}} \left (-a^{3}\right )^{\frac {1}{3}} \right |}\right )}{168750 \, a^{18}} + \frac {671 \cdot 3^{\frac {2}{3}} {\left (a^{3}\right )}^{\frac {2}{3}} \log \left ({\left | x - 3^{\frac {1}{3}} {\left (a^{3}\right )}^{\frac {1}{3}} \right |}\right )}{2278125 \, a^{18}} - \frac {7887 \, a^{9} x^{2} - 22 \, a^{6} x^{5} - 931 \, a^{3} x^{8} + 104 \, x^{11}}{202500 \, {\left (6 \, a^{6} + a^{3} x^{3} - x^{6}\right )}^{2} a^{15}} \] Input:
integrate(x/(-3*a^3+x^3)^3/(2*a^3+x^3)^3,x, algorithm="giac")
Output:
89/168750*sqrt(3)*2^(2/3)*(1/2*I*sqrt(3) + 1/2)^2*arctan(1/6*sqrt(3)*2^(2/ 3)*(2*x + 2^(1/3)*(-a^3)^(1/3))/(-a^3)^(1/3))/a^16 - 89/337500*2^(2/3)*(1/ 2*I*sqrt(3) + 1/2)^2*log(x^2 + 2^(1/3)*(-a^3)^(1/3)*x + 2^(2/3)*(-a^3)^(2/ 3))/a^16 - 671/4556250*3^(2/3)*log(x^2 + 3^(1/3)*(a^3)^(1/3)*x + 3^(2/3)*( a^3)^(2/3))/a^16 + 671/759375*3^(1/6)*arctan(1/3*3^(1/6)*(2*x + 3^(1/3)*(a ^3)^(1/3))/(a^3)^(1/3))/a^16 + 89/168750*2^(2/3)*(-a^3)^(2/3)*log(abs(x - 2^(1/3)*(-a^3)^(1/3)))/a^18 + 671/2278125*3^(2/3)*(a^3)^(2/3)*log(abs(x - 3^(1/3)*(a^3)^(1/3)))/a^18 - 1/202500*(7887*a^9*x^2 - 22*a^6*x^5 - 931*a^3 *x^8 + 104*x^11)/((6*a^6 + a^3*x^3 - x^6)^2*a^15)
Time = 10.15 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.26 \[ \int \frac {x}{\left (-3 a^3+x^3\right )^3 \left (2 a^3+x^3\right )^3} \, dx=\frac {89\,{1201354980468750}^{2/3}\,\ln \left (x+2^{1/3}\,a^{81}\,{\left (\frac {1}{a^{48}}\right )}^{5/3}\right )\,{\left (\frac {1}{a^{48}}\right )}^{1/3}}{1201354980468750}-\frac {\frac {2629\,x^2}{67500\,a^6}-\frac {11\,x^5}{101250\,a^9}-\frac {931\,x^8}{202500\,a^{12}}+\frac {26\,x^{11}}{50625\,a^{15}}}{36\,a^{12}+12\,a^9\,x^3-11\,a^6\,x^6-2\,a^3\,x^9+x^{12}}+\frac {671\,{1313681671142578125}^{2/3}\,\ln \left (x-3^{1/3}\,a^{81}\,{\left (\frac {1}{a^{48}}\right )}^{5/3}\right )\,{\left (\frac {1}{a^{48}}\right )}^{1/3}}{1313681671142578125}+\frac {89\,{1201354980468750}^{2/3}\,\ln \left (x-\frac {2^{1/3}\,a^{81}\,{\left (\frac {1}{a^{48}}\right )}^{5/3}}{2}-\frac {2^{1/3}\,\sqrt {3}\,a^{81}\,{\left (\frac {1}{a^{48}}\right )}^{5/3}\,1{}\mathrm {i}}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1}{a^{48}}\right )}^{1/3}}{2402709960937500}-\frac {89\,{1201354980468750}^{2/3}\,\ln \left (x-\frac {2^{1/3}\,a^{81}\,{\left (\frac {1}{a^{48}}\right )}^{5/3}}{2}+\frac {2^{1/3}\,\sqrt {3}\,a^{81}\,{\left (\frac {1}{a^{48}}\right )}^{5/3}\,1{}\mathrm {i}}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1}{a^{48}}\right )}^{1/3}}{2402709960937500}-\frac {671\,{1313681671142578125}^{2/3}\,\ln \left (x+\frac {3^{1/3}\,a^{81}\,{\left (\frac {1}{a^{48}}\right )}^{5/3}}{2}-\frac {3^{5/6}\,a^{81}\,{\left (\frac {1}{a^{48}}\right )}^{5/3}\,1{}\mathrm {i}}{2}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1}{a^{48}}\right )}^{1/3}}{2627363342285156250}+\frac {671\,{1313681671142578125}^{2/3}\,\ln \left (x+\frac {3^{1/3}\,a^{81}\,{\left (\frac {1}{a^{48}}\right )}^{5/3}}{2}+\frac {3^{5/6}\,a^{81}\,{\left (\frac {1}{a^{48}}\right )}^{5/3}\,1{}\mathrm {i}}{2}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\,{\left (\frac {1}{a^{48}}\right )}^{1/3}}{2627363342285156250} \] Input:
int(-x/((2*a^3 + x^3)^3*(3*a^3 - x^3)^3),x)
Output:
(89*1201354980468750^(2/3)*log(x + 2^(1/3)*a^81*(1/a^48)^(5/3))*(1/a^48)^( 1/3))/1201354980468750 - ((2629*x^2)/(67500*a^6) - (11*x^5)/(101250*a^9) - (931*x^8)/(202500*a^12) + (26*x^11)/(50625*a^15))/(36*a^12 + x^12 - 2*a^3 *x^9 - 11*a^6*x^6 + 12*a^9*x^3) + (671*1313681671142578125^(2/3)*log(x - 3 ^(1/3)*a^81*(1/a^48)^(5/3))*(1/a^48)^(1/3))/1313681671142578125 + (89*1201 354980468750^(2/3)*log(x - (2^(1/3)*a^81*(1/a^48)^(5/3))/2 - (2^(1/3)*3^(1 /2)*a^81*(1/a^48)^(5/3)*1i)/2)*(3^(1/2)*1i - 1)*(1/a^48)^(1/3))/2402709960 937500 - (89*1201354980468750^(2/3)*log(x - (2^(1/3)*a^81*(1/a^48)^(5/3))/ 2 + (2^(1/3)*3^(1/2)*a^81*(1/a^48)^(5/3)*1i)/2)*(3^(1/2)*1i + 1)*(1/a^48)^ (1/3))/2402709960937500 - (671*1313681671142578125^(2/3)*log(x + (3^(1/3)* a^81*(1/a^48)^(5/3))/2 - (3^(5/6)*a^81*(1/a^48)^(5/3)*1i)/2)*(3^(1/2)*1i + 1)*(1/a^48)^(1/3))/2627363342285156250 + (671*1313681671142578125^(2/3)*l og(x + (3^(1/3)*a^81*(1/a^48)^(5/3))/2 + (3^(5/6)*a^81*(1/a^48)^(5/3)*1i)/ 2)*(3^(1/2)*1i - 1)*(1/a^48)^(1/3))/2627363342285156250
\[ \int \frac {x}{\left (-3 a^3+x^3\right )^3 \left (2 a^3+x^3\right )^3} \, dx=-\left (\int \frac {x}{216 a^{18}+108 a^{15} x^{3}-90 a^{12} x^{6}-35 a^{9} x^{9}+15 a^{6} x^{12}+3 a^{3} x^{15}-x^{18}}d x \right ) \] Input:
int(x/(-3*a^3+x^3)^3/(2*a^3+x^3)^3,x)
Output:
- int(x/(216*a**18 + 108*a**15*x**3 - 90*a**12*x**6 - 35*a**9*x**9 + 15*a **6*x**12 + 3*a**3*x**15 - x**18),x)