Integrand size = 24, antiderivative size = 96 \[ \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx=-\frac {\arctan \left (\frac {1+\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b}}}{\sqrt {3}}\right )}{\sqrt {3} \sqrt [3]{a+b}}+\frac {\log \left (1-x^3\right )}{6 \sqrt [3]{a+b}}-\frac {\log \left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}} \]
1/6*ln(-x^3+1)/(a+b)^(1/3)-1/2*ln((a+b)^(1/3)-(b*x^3+a)^(1/3))/(a+b)^(1/3) -1/3*arctan(1/3*(1+2*(b*x^3+a)^(1/3)/(a+b)^(1/3))*3^(1/2))/(a+b)^(1/3)*3^( 1/2)
Result contains complex when optimal does not.
Time = 0.37 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.90 \[ \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx=\frac {2 \sqrt {-6+6 i \sqrt {3}} \arctan \left (\frac {1+\frac {\left (-1-i \sqrt {3}\right ) \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b}}}{\sqrt {3}}\right )-i \left (-i+\sqrt {3}\right ) \left (\log \left (\left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right ) \left (2 \sqrt [3]{a+b}+\sqrt [3]{a+b x^3}-i \sqrt {3} \sqrt [3]{a+b x^3}\right )\right )-2 \log \left (2 \sqrt [3]{a+b}+\left (1+i \sqrt {3}\right ) \sqrt [3]{a+b x^3}\right )\right )}{12 \sqrt [3]{a+b}} \]
(2*Sqrt[-6 + (6*I)*Sqrt[3]]*ArcTan[(1 + ((-1 - I*Sqrt[3])*(a + b*x^3)^(1/3 ))/(a + b)^(1/3))/Sqrt[3]] - I*(-I + Sqrt[3])*(Log[((a + b)^(1/3) - (a + b *x^3)^(1/3))*(2*(a + b)^(1/3) + (a + b*x^3)^(1/3) - I*Sqrt[3]*(a + b*x^3)^ (1/3))] - 2*Log[2*(a + b)^(1/3) + (1 + I*Sqrt[3])*(a + b*x^3)^(1/3)]))/(12 *(a + b)^(1/3))
Time = 0.21 (sec) , antiderivative size = 100, normalized size of antiderivative = 1.04, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {946, 67, 16, 1082, 217}
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.
| \(\displaystyle \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx\) |
| \(\Big \downarrow \) 946 |
| \(\displaystyle \frac {1}{3} \int \frac {1}{\left (1-x^3\right ) \sqrt [3]{b x^3+a}}dx^3\) |
| \(\Big \downarrow \) 67 |
| \(\displaystyle \frac {1}{3} \left (\frac {3 \int \frac {1}{\sqrt [3]{a+b}-\sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}}{2 \sqrt [3]{a+b}}-\frac {3}{2} \int \frac {1}{x^6+(a+b)^{2/3}+\sqrt [3]{a+b} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}+\frac {\log \left (1-x^3\right )}{2 \sqrt [3]{a+b}}\right )\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {1}{3} \left (-\frac {3}{2} \int \frac {1}{x^6+(a+b)^{2/3}+\sqrt [3]{a+b} \sqrt [3]{b x^3+a}}d\sqrt [3]{b x^3+a}+\frac {\log \left (1-x^3\right )}{2 \sqrt [3]{a+b}}-\frac {3 \log \left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {1}{3} \left (\frac {3 \int \frac {1}{-x^6-3}d\left (\frac {2 \sqrt [3]{b x^3+a}}{\sqrt [3]{a+b}}+1\right )}{\sqrt [3]{a+b}}+\frac {\log \left (1-x^3\right )}{2 \sqrt [3]{a+b}}-\frac {3 \log \left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {1}{3} \left (-\frac {\sqrt {3} \arctan \left (\frac {\frac {2 \sqrt [3]{a+b x^3}}{\sqrt [3]{a+b}}+1}{\sqrt {3}}\right )}{\sqrt [3]{a+b}}+\frac {\log \left (1-x^3\right )}{2 \sqrt [3]{a+b}}-\frac {3 \log \left (\sqrt [3]{a+b}-\sqrt [3]{a+b x^3}\right )}{2 \sqrt [3]{a+b}}\right )\) |
(-((Sqrt[3]*ArcTan[(1 + (2*(a + b*x^3)^(1/3))/(a + b)^(1/3))/Sqrt[3]])/(a + b)^(1/3)) + Log[1 - x^3]/(2*(a + b)^(1/3)) - (3*Log[(a + b)^(1/3) - (a + b*x^3)^(1/3)])/(2*(a + b)^(1/3)))/3
3.1.96.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m - n + 1, 0]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Time = 0.91 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.96
| method | result | size |
| pseudoelliptic | \(-\frac {\arctan \left (\frac {\left (\left (a +b \right )^{\frac {1}{3}}+2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3 \left (a +b \right )^{\frac {1}{3}}}\right ) \sqrt {3}+\ln \left (\left (b \,x^{3}+a \right )^{\frac {1}{3}}-\left (a +b \right )^{\frac {1}{3}}\right )-\frac {\ln \left (\left (b \,x^{3}+a \right )^{\frac {2}{3}}+\left (a +b \right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}}+\left (a +b \right )^{\frac {2}{3}}\right )}{2}}{3 \left (a +b \right )^{\frac {1}{3}}}\) | \(92\) |
-1/3/(a+b)^(1/3)*(arctan(1/3*((a+b)^(1/3)+2*(b*x^3+a)^(1/3))*3^(1/2)/(a+b) ^(1/3))*3^(1/2)+ln((b*x^3+a)^(1/3)-(a+b)^(1/3))-1/2*ln((b*x^3+a)^(2/3)+(a+ b)^(1/3)*(b*x^3+a)^(1/3)+(a+b)^(2/3)))
Leaf count of result is larger than twice the leaf count of optimal. 162 vs. \(2 (75) = 150\).
Time = 0.26 (sec) , antiderivative size = 387, normalized size of antiderivative = 4.03 \[ \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx=\left [\frac {3 \, \sqrt {\frac {1}{3}} {\left (a + b\right )} \sqrt {\frac {{\left (-a - b\right )}^{\frac {1}{3}}}{a + b}} \log \left (\frac {2 \, b x^{3} + 3 \, \sqrt {\frac {1}{3}} {\left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a + b\right )} - {\left (a + b\right )} {\left (-a - b\right )}^{\frac {1}{3}} - 2 \, {\left (b x^{3} + a\right )}^{\frac {2}{3}} {\left (-a - b\right )}^{\frac {2}{3}}\right )} \sqrt {\frac {{\left (-a - b\right )}^{\frac {1}{3}}}{a + b}} + 3 \, a - 3 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (-a - b\right )}^{\frac {2}{3}} + b}{x^{3} - 1}\right ) + {\left (-a - b\right )}^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (-a - b\right )}^{\frac {1}{3}} + {\left (-a - b\right )}^{\frac {2}{3}}\right ) - 2 \, {\left (-a - b\right )}^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} + {\left (-a - b\right )}^{\frac {1}{3}}\right )}{6 \, {\left (a + b\right )}}, -\frac {6 \, \sqrt {\frac {1}{3}} {\left (a + b\right )} \sqrt {-\frac {{\left (-a - b\right )}^{\frac {1}{3}}}{a + b}} \arctan \left (\sqrt {\frac {1}{3}} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} - {\left (-a - b\right )}^{\frac {1}{3}}\right )} \sqrt {-\frac {{\left (-a - b\right )}^{\frac {1}{3}}}{a + b}}\right ) - {\left (-a - b\right )}^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} - {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (-a - b\right )}^{\frac {1}{3}} + {\left (-a - b\right )}^{\frac {2}{3}}\right ) + 2 \, {\left (-a - b\right )}^{\frac {2}{3}} \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} + {\left (-a - b\right )}^{\frac {1}{3}}\right )}{6 \, {\left (a + b\right )}}\right ] \]
[1/6*(3*sqrt(1/3)*(a + b)*sqrt((-a - b)^(1/3)/(a + b))*log((2*b*x^3 + 3*sq rt(1/3)*((b*x^3 + a)^(1/3)*(a + b) - (a + b)*(-a - b)^(1/3) - 2*(b*x^3 + a )^(2/3)*(-a - b)^(2/3))*sqrt((-a - b)^(1/3)/(a + b)) + 3*a - 3*(b*x^3 + a) ^(1/3)*(-a - b)^(2/3) + b)/(x^3 - 1)) + (-a - b)^(2/3)*log((b*x^3 + a)^(2/ 3) - (b*x^3 + a)^(1/3)*(-a - b)^(1/3) + (-a - b)^(2/3)) - 2*(-a - b)^(2/3) *log((b*x^3 + a)^(1/3) + (-a - b)^(1/3)))/(a + b), -1/6*(6*sqrt(1/3)*(a + b)*sqrt(-(-a - b)^(1/3)/(a + b))*arctan(sqrt(1/3)*(2*(b*x^3 + a)^(1/3) - ( -a - b)^(1/3))*sqrt(-(-a - b)^(1/3)/(a + b))) - (-a - b)^(2/3)*log((b*x^3 + a)^(2/3) - (b*x^3 + a)^(1/3)*(-a - b)^(1/3) + (-a - b)^(2/3)) + 2*(-a - b)^(2/3)*log((b*x^3 + a)^(1/3) + (-a - b)^(1/3)))/(a + b)]
\[ \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx=- \int \frac {x^{2}}{x^{3} \sqrt [3]{a + b x^{3}} - \sqrt [3]{a + b x^{3}}}\, dx \]
Time = 0.29 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.15 \[ \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx=-\frac {\frac {2 \, \sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + {\left (a + b\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a + b\right )}^{\frac {1}{3}}}\right )}{{\left (a + b\right )}^{\frac {1}{3}}} - \frac {b \log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a + b\right )}^{\frac {1}{3}} + {\left (a + b\right )}^{\frac {2}{3}}\right )}{{\left (a + b\right )}^{\frac {1}{3}}} + \frac {2 \, b \log \left ({\left (b x^{3} + a\right )}^{\frac {1}{3}} - {\left (a + b\right )}^{\frac {1}{3}}\right )}{{\left (a + b\right )}^{\frac {1}{3}}}}{6 \, b} \]
-1/6*(2*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + (a + b)^(1/3)) /(a + b)^(1/3))/(a + b)^(1/3) - b*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3 )*(a + b)^(1/3) + (a + b)^(2/3))/(a + b)^(1/3) + 2*b*log((b*x^3 + a)^(1/3) - (a + b)^(1/3))/(a + b)^(1/3))/b
Time = 3.12 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.18 \[ \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx=-\frac {{\left (a + b\right )}^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, {\left (b x^{3} + a\right )}^{\frac {1}{3}} + {\left (a + b\right )}^{\frac {1}{3}}\right )}}{3 \, {\left (a + b\right )}^{\frac {1}{3}}}\right )}{\sqrt {3} a + \sqrt {3} b} + \frac {\log \left ({\left (b x^{3} + a\right )}^{\frac {2}{3}} + {\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (a + b\right )}^{\frac {1}{3}} + {\left (a + b\right )}^{\frac {2}{3}}\right )}{6 \, {\left (a + b\right )}^{\frac {1}{3}}} - \frac {\log \left ({\left | {\left (b x^{3} + a\right )}^{\frac {1}{3}} - {\left (a + b\right )}^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a + b\right )}^{\frac {1}{3}}} \]
-(a + b)^(2/3)*arctan(1/3*sqrt(3)*(2*(b*x^3 + a)^(1/3) + (a + b)^(1/3))/(a + b)^(1/3))/(sqrt(3)*a + sqrt(3)*b) + 1/6*log((b*x^3 + a)^(2/3) + (b*x^3 + a)^(1/3)*(a + b)^(1/3) + (a + b)^(2/3))/(a + b)^(1/3) - 1/3*log(abs((b*x ^3 + a)^(1/3) - (a + b)^(1/3)))/(a + b)^(1/3)
Time = 0.61 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.64 \[ \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx=\frac {\ln \left ({\left (b\,x^3+a\right )}^{1/3}-\frac {9\,a+9\,b}{9\,{\left (-a-b\right )}^{2/3}}\right )}{3\,{\left (-a-b\right )}^{1/3}}+\frac {\ln \left ({\left (b\,x^3+a\right )}^{1/3}-\frac {{\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (9\,a+9\,b\right )}{36\,{\left (-a-b\right )}^{2/3}}\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-a-b\right )}^{1/3}}-\frac {\ln \left ({\left (b\,x^3+a\right )}^{1/3}-\frac {{\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}^2\,\left (9\,a+9\,b\right )}{36\,{\left (-a-b\right )}^{2/3}}\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{6\,{\left (-a-b\right )}^{1/3}} \]
log((a + b*x^3)^(1/3) - (9*a + 9*b)/(9*(- a - b)^(2/3)))/(3*(- a - b)^(1/3 )) + (log((a + b*x^3)^(1/3) - ((3^(1/2)*1i - 1)^2*(9*a + 9*b))/(36*(- a - b)^(2/3)))*(3^(1/2)*1i - 1))/(6*(- a - b)^(1/3)) - (log((a + b*x^3)^(1/3) - ((3^(1/2)*1i + 1)^2*(9*a + 9*b))/(36*(- a - b)^(2/3)))*(3^(1/2)*1i + 1)) /(6*(- a - b)^(1/3))
\[ \int \frac {x^2}{\left (1-x^3\right ) \sqrt [3]{a+b x^3}} \, dx=-\left (\int \frac {x^{2}}{\left (b \,x^{3}+a \right )^{\frac {1}{3}} x^{3}-\left (b \,x^{3}+a \right )^{\frac {1}{3}}}d x \right ) \]